Dynamic Trip Modelling: From Shopping Centres to the Internet (GeoJournal Library, 84) 1402043457, 9781402043451

Citation preview

The GeoJournal Library

Dynamic Trip Modelling From Shopping Centres to the Internet by

Robert G.V. Baker

Dynamic Trip Modelling

The GeoJournal Library Volume 84 Managing Editor:

Max Barlow, Toronto, Canada

Founding Series Editor: Wolf Tietze, Helmstedt, Germany

Editorial Board:

Paul Claval, France Yehuda Gradus, Israel Sam Ock Park, South Korea Herman van der Wusten, The Netherlands

The titles published in this series are listed at the end of this volume.

Dynamic Trip Modelling From Shopping Centres to the Internet

by

ROBERT G.V. BAKER School of Human and Environmental Studies, University of New England, Australia

A C.I.P. Catalogue record for this book is available from the Library of Congress

ISBN-10 ISBN-13 ISBN-10 ISBN-13

1-4020-4345-7 (HB) 978-1-4020-4345-1 (HB) 1-4020-4346-5 (e-book) 978-1-4020-4346-8 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

www.springer.com

Printed on acid-free paper

All Rights Reserved © 2006 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands.

To Sue, Kristen and Cameron

Contents Preface

ix

Illustrations

xi

Chapter 1: Introduction

1

1.1 Shopping Change 1.2 Definitions of Retail Forms Underpinning the Model 1.3 The Time-space Convergence 1.4 A Way Forward

1 6 16 18

Chapter 2: An Introduction to Retail and Consumer Modelling

21

2.1 Definition 2.2 A Justification for Modelling 2.3 The Art of Modelling 2.4 Model-building and its Weaknesses 2.5 Examples of Retail and Consumer Modelling 2.6 A Vision for Dynamic Trip Modelling

21 22 23 29 31 75

Chapter 3: Dynamic Trip Modelling

77

3.1 Background to the RASTT Model 3.2 Space and Time-discounting Shopping Trips 3.3 Characteristics of Space-discounting Behaviour 3.4 The Time-discounting Model 3.5 The Fourier Transform and Aggregate Periodic Trips 3.6 Estimating Shopping Centre Hours 3.7 Two Dimensional Space-time Modelling 3.8 Estimating Market Penetration with an Extension of Shopping Hours 3.9 Stochastic Space-time Trips 3.10 Space-time Modelling Shopping Trips: A Summary 3.11 Dynamic Shopping Trip Modelling

77 79 84 97 116 126 130 134 138 151 156

viii

Contents

Chapter 4: Empirical Testing of the RASTT Model in Time and Space 4.1 Introduction 4.2 Background to the Research Methodology 4.3 The Empirical Method 4.4 The Sydney Project: Long Term Time Change of Shopping Trips (1980-1998) 4.5 Changes in Time-space Trip Behaviour in the Sydney Project 4.6 Application of the RASTT Model to Unplanned Shopping Centres: Armidale in Regional New South Wales, 1995 4.7 Application of the RASTT Model to Planned Shopping Centres: Auckland, New Zealand, 2000 4.8 Is there a ‘Global’ RASTT Model?

157

157 158 161 167 200 237 247 256

Chapter 5: Dynamic Modelling of the Internet

265

5.1 Introduction 5.2 The RASTT Model and Internet Transactions 5.3 Deriving the RASTT Model for Internet Transactions 5.4 Empirical Evidence 5.5 Applications to Shopping Transactions 5.6 Summary

265 268 273 277 289 290

Chapter 6: The Socio-Economic and Planning Consequences of Changes to Shopping Trips

293

6.1 The Problem of Shopping Times and Shopping Places 6.2 The Role of Parking and Walking 6.3 The Vacant Shop Problem in Australia 6.4 The Role of the Large Supermarket or Superstore 6.5 The Role of Planned Regional Shopping Centres 6.6 Policy Implications for Modelling Shopping Trip Change 6.7 Retail Planning as a ‘Wicked Problem’

293 294 298 304 317 321 326

Chapter 7: Conclusions

327

B ib liography

339

Preface The thesis of this book is that there are one set of equations that can define any trip between an origin and destination. The idea originally came from work that I did when applying the hydrodynamic analogy to study congested traffic flows in 1981. However, I was disappointed to find out that much of the mathematical work had already been done decades earlier. When I looked for a new application, I realised that shopping centre demand could be like a longitudinal wave, governed by centre opening and closing times. Further, a solution to the differential equation was the gravity model and this suggested that time was somehow part of distance decay. This was published in 1985 and represented a different approach to spatial interaction modelling. The next step was to translate the abstract theory into something that could be tested empirically. To this end, I am grateful to my Ph.D supervisor, Professor Barry Garner who taught me that it is not sufficient just to have a theoretical model. This book is an outcome of this on-going quest to look at how the evolution of the model performs against real world data. This is a far more difficult process than numerical simulations, but the results have been more valuable to policy formulation, and closer to what I think is spatial science. The testing and application of the model required the compilation of shopping centre surveys and an Internet data set. I would like to thank the management of the shopping centres for allowing me to survey consumers. I have had to observe, in some cases, a lag period to use some of the data as part of commercial confidentiality agreements. I am indebted to David Marshall who supervised the collection and collation of the shopping centre data. The Internet data was kindly made available by Dr Les Cottrell and Dr Gerod Williams from their network at the Stanford Linear Accelerator Centre. I also appreciate the work on this project by Troy MacKay, Brett Carson and Raj Rajaratnam and their discussions over many cups of coffee. My thanks also extends to Mike Roach who undertook a huge cartographic task and Sue Baker for a difficult proofing job. I am also indebted to the Australian Research Council for grants to complete the shopping centre surveys and Internet analysis. Finally, thanks to my family, Sue, Kristen and Cameron, my brother Ross, my parents, Ellen and Douglas, and Alpha O without whose support during the good times and bad, this book would not have happened.

Illustrations List of Figures Figure 1.1 Meadowhall Regional Planned Shopping Centre, Sheffield, 1999 Figure 1.2 The Macellum on the Dupondius coin (AD 65) in the reign of the Emperor Nero Figure 1.3 Market Share of Pharmaceutical Products in Australia, 1997 Figure 1.4 The Generation of an Internet Tree showing the Aspatial Connectivity from 100,000 Internet Routers and the Hierarchical Structures that develop from a few Highly Connected Nodes Figure 1.5 The Time-space Convergence Showing the Cone of Time and Space Interaction Relative to Changes in Technology Figure 2.1 A Flow Diagram Showing the Evolution of the Gravity Model in the Context of Consumer Behaviour Figure 2.2 A Process of Building Relevant, Testable and Reproducible Models Figure 2.3 The Regression of Mean Trip Frequency and the β Coefficient of the Gravity Model for Shopping Trips to the Sydney Shopping Centres 1980/82 and 1988/89 Figure 2.4 An Extension of Shopping Hours reduces the Slope of the Gravity Model ( E ) where there is an Increased Propensity for Households to Travel to Planned Shopping Centre O rather than Shop Locally. Inset Photographs: (right), Vacant Shops (signed) in the Abbotsford Shopping Centre from Competition from MarketPlace Leichhardt (left), Sydney, New South Wales, May, 1999 Figure 2.5 Loschian Modifications in Christaller’s Hexagonal Trade Areas and the Northwest Retail Hierarchy for Canberra, Australian Capital Territory Figure 2.6 The Location of 24-hour Coles Supermarkets in Sydney 1996 Figure 2.7 Consumer Equilibrium Analysis for Shopping Time and Visits to a Shopping Centre

xii

Illustrations

Figure 2.8 Impact of Price Shifts (top) and Demand Curve Formation (bottom) Figure 2.9 The Trading Hour Consumption Curve for Shopping Centre TimeVisit Allocations Figure 2.10 Percentage Composition of Socio-economic Status of Late Night Shoppers (Armidale, NSW, November, 1995) Figure 2.11 Shopping Preferences for Extended Hours by Socio-economic Groups Figure 2.12 Analogous Engel Curves Relative to the Size of the Centre (a) Percentage of HDI Respondents in the Sample Plotted with HDI Trip Frequency and (b) a Model showing Normal and Inferior Engel Lines with Centre Scale Figure 2.13 The Utility of Shopping at a Hierarchy of Malls in Sydney with Trip Distance, 1988/89 Figure 2.14 The Relative Utility Distribution with (kD)/N for Sydney Project 1988/89 and 1996/98 Figure 2.15 The Regression of the Gravity Coefficient β and U2N2/MD2 for the Aggregate Sydney Project in 1988/89 and 1996/98 (excluding the regional Bankstown Square samples) Figure 2.16 Quadratic Distributions of the Gravity Coefficient (top) and Mean Trip Frequency (bottom) with Centre Size (Sydney Project 1980/82 and 1988/89). The Point of Inflections are at N = 147 and 145 Centre Destinations, respectively, for Small (negative slope) and Large (positive slope) Centre Behaviour Figure 2.17 The Relationship between Trip Frequency and the Percentage of Multi-purpose Shopping (Sydney Project 1988/89) Figure 2.18 The Relationship between the Gravity Coefficient and the Percentage of Multi-purpose Shopping Squared Divided by the Transfer Coefficient (Sydney Project 1988/89) Figure 2.19 The Distribution of MPS changing with Centre Size (Sydney Project 1988/89). The Point of Inflection is at N = 167 Centre Destinations for Small (negative slope) and Large (positive slope) Centre Behaviour Figure 2.20 The Gravity Model as a Negative Exponential Distribution away from a Shopping Centre for E 1 > E 2

Illustrations

xiii

Figure 2.21 The Trade-off between Constant Logarithmic Supply of Destinations and Constant Negative Exponential Demand for a Shopping Centre C , if Consumers Minimise Trip Distance Figure 3.1 The Fourier Transform of exp-g2t for Space-discounting Consumers Figure 3.2 Changing Market Areas for Space-discounting Behaviour for Successive Time Periods Figure 3.3 Changing Trip Distributions over a Day for Space-discounting Shopping Figure 3.4 Changing Morning and Afternoon Distributions for Westfield Burwood and Ashfield Mall for pre-Christmas Space- and Time-discounting Trip Behaviour, respectively: Sydney Project 1988/89 Figure 3.5 The Negative Exponential Self-reciprocity between Trip Frequency ( f, 2f, 3f to 6f ) and Post Office Distance for Westfield Burwood (Sample: 15/12/88A), Sydney Data Set, 1988/89 Figure 3.6 The Fourier Frequency Assignment Ψ(f) with Shopping Time (t) Figure 3.7 Testing the Shopping Time Hypothesis in Figure 3.6 with Distance Zones 1 and 2 in the Westfield Burwood pre-Christmas Rush Sample (15/12/88A) with Box and Whisker Plots Figure 3.8 The Relationship between Mean Shopping Duration (p) and Destinations Visited m (1988/89) showing the Shift Right towards ‘Large Centre’ Behaviour Figure 3.9 The Regression of Intra-centre Shopping Frequency (f = p/2T) and the Mean Shopping Duration p per Trading Week T ( f = m× k /2T) showing the Shift Right towards ‘Large Centre’ Behaviour Figure 3.10 Comparison for the Theoretical m/2T and m/4T Values with the p/T Empirical Estimates from the Sydney 1988/89 Data Set Figure 3.11 Regression showing the Relationship between Two Forms of the Intra-centre Shopping Frequency (f = mk/2T)) and (f = Mk ) for the Sydney Project 1988/89

Figure 3.12 Regression showing the Relationship between Two Forms of the Intra-centre Shopping Frequency (f =mk/2T)) and (f = Mk ) for the Sydney Project 1996/98

Figure 3.13 Simulation of the Space-time Shopping Distributions for 49.5, 70 and 100 Trading Hours (per week) for ‘Small Centre’ Behaviour

xiv

Illustrations

Figure 3.14 Simulation of the Time-space Shopping Distributions for 49.5, 70 and 100 Trading Hours (per week) for ‘Large Centre’ Behaviour Figure 3.15 The Aggregation of Population Demand Waves (Sf) of Different Frequencies Figure 3.16 The Demand Wave φ3 showing the Equal Likelihood of Visiting Three Equally-sized Centres (n =3) over the Trading Week Figure 3.17 (a) The Graph of a sinc Function and the Gaussian Wave Packet (dotted line) and; (b) the Gaussian Wave Packet for T = 0.5, 1.0 and the Probability Density Function for T = 1.0 (dotted line) Figure 3.18 The Probability Density of the Weekly Grocery Trip as a Gaussian Wave Packet with the Shift Towards More Frequent Trips with Extended Shopping Hours Figure 3.19 The Higher Frequency Shift with the Extension of Shopping Hours in the Sydney Project from Regulated Hours in 1988/89 to Deregulated Hours in 1996/98 Figure 3.20 The Theoretical and Empirical Frequency Distributions for MarketPlace Leichhardt (8/12/88A) and Bankstown Square (23/3/89M) Figure 3.21 Bessel Functions of Order Zero and One Figure 3.22 The Relationship between the Gravity Coefficient and Mean Trip Post Office Distance in the Sydney 1980/82, 1988/89 and 1996/98 Data Sets Figure 3.23 The Relationship between Retail Floorspace and the Number of Retail Destinations: Y = -0.811+ 0.313X , R 2 = 0.92 Figure 3.24 The Aggregate Household Time ‘Doughnut’ for MarketPlace Leichhardt 8/12/88A through an Extension of Trading Hours from 49.5 to 60 hours per week Figure 3.25 A Venn Diagram showing Set Relationships among Selected Random Processes Figure 3.26 The Shopping Trip Pattern for Space-discounting Behaviour (α = 2) compared to Non-Space-discounting Behaviour (α = 1.3) Figure 3.27 State Transition Flow Chart for Trip-chaining from a Residential State to n Shopping Centres Figure 3.28 The Poisson Distribution for n Shopping States

Illustrations

xv

Figure 3.29 The Erlangian Distribution for N Stops over a Distance D Figure 4.1 The Location of the Planned Shopping Centres used in the Sydney Project, Australia with the Period or Year of Sampling Figure 4.2 Sample Questionnaire used in the Sydney, Armidale and Auckland Projects Figure 4.3 An Example of the Segmentation of One Kilometre Concentric Aggregation Bands from MarketPlace Leichhardt, Sydney Project 1996/98, where the Respondents Pointed to which Band their Residence is Located Figure 4.4 Regression between Postcode Centroid Distance and Segment Nominated Distance (Aggregate Sydney Project 1996/98) Figure 4.5 Box and Whisker Plots for Trip Distance at Bankstown Square for Equivalent Morning (top) and Afternoon Samples (bottom): 1989, 1997 and 1998 Figure 4.6 Box and Whisker Plots for Trip Frequency at the Regional PSC Bankstown Square for Equivalent Morning (top) and Afternoon (bottom) Samples: 1989, 1997 and 1998 Figure 4.7 Box and Whisker Plots for Trip Frequency at the Community PSC Ashfield Mall for Equivalent Morning 1989 and 1998 (top) and Afternoon Samples: 1989, 1997 and 1998 (bottom) , Figure 4.8 Box and Whisker Plots for Total Populations Perception of the Level of Shopping Satisfaction for 1988/89 and 1996/98 Figure 4.9 Box and Whisker Plots for Equivalent Samples of the Time Spent Shopping (Duration/min/trip) at Pre-Christmas Westfield Burwood Afternoon Samples: 1988, 1996 and 1997 Figure 4.10 Box and Whisker Plots for Total Samples of Shops Visited per trip over the Decade from 1988/89 to 1996/98 Figure 4.11 Box and Whisker Plots for Total Samples of the Changes in the Socio-economic Index over the Decade from 1988/89 to 1996/98 Figure 4.12 Regressions for the Time-space Convergence (β = k 2/M) for the 1980/82, 1988/89 and 1996/98 Data Sets for the Sydney Project Figure 4.13 The Stages in the Evolution of a New Definition of the Gravity Coefficient (using 1988/89 and 1996/98 data): First Approximation Regresses the Deterministic and Probabilistic Forms of the Transport Coefficient M; Second Approximation Regresses a Revised and Standardised Form of M Eliminating the

xvi

Illustrations

Autocorrelation; Third Approximation Corrects the Double Counting in the Deterministic Form of M Figure 4.14 Regression between Raw (top, R^2 = 0.666) and Standardised (bottom, R^2 = 0.84 ) Probabilistic Forms of the Gravity Coefficient. The Dotted Lines are the 95% Confidence Lines from the True Mean of the Regression Figure 4.15 Aggregate Transfer Mobility (M) with Centre Size (Number of Shopping Destinations) for the 1988/89 Sydney Data Set Figure 4.16 Aggregate Transfer Mobility (M) with Centre Size (Number of Shopping Destinations) for the 1996/98 Sydney Data Set Figure 4.17 The Aggregate Curve for Trading Hours Regulated (1980/82 and 1988/89; 15 samples) and Deregulated (1996/98; 17 samples) Data from the Sydney Project Figure 4.18 (top left) Quadratic Regression of Population Index and Distance Decay from Bankstown Square 3/11/80 (1km Bands); (top right) Log-Linear Regression of Population Index and Distance Decay from Bankstown Square 3/11/80 (1km bands); (bottom left) Quadratic Regression of Population Distance and Distance Decay from Bankstown Square 3/11/80 (1km bands); (bottom right) Quadratic Regression of Population Density and Distance Decay from Bankstown Square 3/11/80 (1.5km bands) Figure 4.19 The Gravity Trip Distribution for Ashfield Mall 23/3/98 (Afternoon) for (left) 10 Zones with DW-statistic of 0.758 and (right) 9 Zones with an Improved DW-statistic of 1.380 Figure 4.20 The Park Test Regressing the Logarithm of Frequency Variance and Trip Distance and Testing for Significance Figure 4.21 The Bankstown Square 1997 Afternoon Gravity Regression (left) Including and (right) Excluding the 13.5km Point Figure 4.22 Quadratic Regression of the Gravity Coefficient-Trading Hour Hypothesis for the Sydney Project 1988/89 Figure 4.23 Quadratic Regression of the Gravity Coefficient-Trading Hour Hypothesis for the Sydney Project 1996/98 Figure 4.24 Three Dimensional Contour Model of Changing Time-space Behaviour at the Community Centre MarketPlace Leichhardt, from the Regulated 49.5 hours in 1988 to a Supply Average of 64.7 hours in 1998 Figure 4.25 Three Dimensional Contour Model of Changing Time-space Behaviour at the Regional Centre Bankstown Square from the Regulated 49.5 hours in 1989 to a Supply Average of 61.6 hours in 1998

Illustrations xvii

Figure 4.26 Linear Regression between (p/2T) and (mw/2T) for (left) the 1996/98 Sydney Data Set and for (right) the Aggregate Sydney Data Set (1988/89 and 1996/98) Figure 4.27 Linear Regression between Inter-location Trip Frequency ( f ) and Intra-centre Frequency k, ( f = Mk ), for (left) 1988/89 and (right) 1996/98 Sydney Data Sets

Figure 4.28 Linear Regression for MPS and Trip Frequency, Sydney Project 1988/89, 1996/98 and Aggregate Regression Figure 4.29 Quadratic Regression for Percentage of MPS and Centre Scale (Number of Destinations), Sydney Project 1988/89, 1996/98 and Aggregate Regression Figure 4.30 The Relationship between the Percentage of MPS and HDI Respondents in the Sydney Project (1988/89 and 1996/98) Figure 4.31 The Relationship between the Percentage of MPS and HDI × Trip Frequency Squared (per week) in the Sydney Project (1988/89 and 1996/98) Figure 4.32 The Regression of the Gravity Coefficient β and U2N2/MD2 for the Sydney Project Excluding Bankstown Square Data (left) 1988/89 and (right) 1996/98 Figure 4.33 The Location of Armidale, New South Wales Figure 4.34 The Scatter Plot of the Five Armidale 1995 Samples Compared to Sydney 1980/82, 1988/89 and 1996/98 Regressions of  = k2 /M Figure 4.35 The Scatter Plot of the Five Armidale 1995 Samples Compared to Sydney 1988/89 and 1996/98 Regressions of f =± Mk

Figure 4.36 The Scatter Plot of the Five Armidale 1995 Samples Compared to the Aggregated 1988/89 and 1996/98 Regression of p/2T = m × k/2T for Sydney

Figure 4.37 The Scatter Plot of the Five Armidale 1995 Samples Compared to Sydney 1988/89 and 1996/98 Aggregate Regression of MPS = h k Figure 4.38 Quadratic Regression of MPS Percentage and Centre Destinations showing the Armidale Samples as Positive Type 2 MPS

Figure 4.39 The Scatter Plot of the Five Armidale 1995 Samples Compared to the Sydney 1988/89 and 1996/98 Regression of MPS = HDI × k2

xviii Illustrations

Figure 4.40 The Scatter Plot of the Five Armidale 1995 Samples Compared to Sydney 1988/89 and 1996/98 Regressions of U= D× k / N Figure 4.41 Location Map of the Three Planned Shopping Centres Sampled in the Auckland Survey, Thursday April 6, 2000 Figure 4.42 The Scatter Plot of the Six Auckland 2000 Samples Compared to Sydney 1980/82, 1988/89 and 1996/98 Regressions of  = k 2 /M Figure 4.43 The Scatter Plot of the Six Auckland 2000 Samples Compared to Sydney 1988/89 and 1996/98 Regressions of f = ± Mk

Figure 4.44 The Scatter Plot of the Six Auckland 2000 Samples Compared to Sydney 1988/89 and 1996/98 Regressions of p/2T = m × k/2T

Figure 4.45 The Scatter Plot of the Six Auckland 2000 Samples Compared to the Aggregate Sydney 1988/89 and 1996/98 Regressions of MPS = h k Figure 4.46 The Scatter Plot of the Six Auckland 2000 Samples Compared to Sydney 1988/89 and 1996/98 Regressions of MPS = HDI × k 2 Figure 4.47 The Scatter Plot of the Six Auckland 2000 Samples Compared to Sydney 1988/89 and 1996/98 Regressions of U= D × k/N Figure 4.48 The Regression of Sydney 1980/82, 1988/89 and 1996/98, Armidale 1995 and Auckland 2000 for  = k2/M

Figure 4.49 (top) The Aggregate Regression of Sydney 1988/89, Armidale 1995, Sydney 1996/98 and Auckland 2000 of f = ± Mk (bottom) Post-1993 Extended Hours Data (Excluding Sydney 1988/89 Points) (28 Samples) showing an Improved R-squared Value of 0.60 Figure 4.50 The Regression of Sydney 1988/89, Armidale 1995, Sydney 1996/98 and Auckland 2000 of p/2T = m × k/2T Figure 4.51 The Regression of Sydney 1988/89, Armidale 1995, Sydney 1996/98 and Auckland 2000 for MPS = h k Figure 4.52 The Regression of Sydney 1988/89, Armidale 1995, Sydney 1996/98 and Auckland 2000 for MPS = HDI × k2 Figure 4.53 The Regression of Sydney 1988/89, Armidale 1995, Sydney 1996/98 and Auckland 2000 for U= D × k/N Figure 5.1 The Location of the hepnrc.hep.net.gif Monitoring Site (top) and the Remote Hosts (bottom)

Illustrations

xix

Figure 5.2 (top) A Range of Possible Time-space Distributions that could apply to Internet Demand are Simulated for β = 0.0001, T = 24 hours, x0 = 0 to x0 = 10,000 km and a scaled φo max = 10 for a Sequence of k Values where k =0.1, 0.2,... 1.0 (bottom). A Three-dimensional Plot Visualising a likely form of the Demand Wave for k= 0.1 (Baker, 2001) Figure 5.3 The Equal Likelihood of J umping Forwards in Time to Sites in Auckland or Backwards to Perth from the ith Sydney Site defines the Underpinnings of the Type of Differential Equations in Equations (5.16) to (5.18) (Baker, 2001) Figure 5.4 Contour Density Plot for a Simulation of the RASTT Model with k = 0.385 (left) and a 3-Dimensional Plot (right)

Figure 5.5 The Ping Time Distribution for Traffic in the hepnrc.hep.net.gif Network for 2000

Figure 5.6 The Internet Traffic Wave for hepnrc.hep.net.gif using Packet Loss Averages for 2000. The Range 0 to 168 hours represents Monday to Sunday. The US Origin-destination Pairs (-130 0 W to -60 0W Longitudes) show the Capacity to handle Peak Traffic Times with Small Amplitudes. This is not the Case with Connections to Europe and Asia

Figure 5.7 The Cumulative Frequency of Traffic Contributions from Successive 75km Zones from the hepnrc.hep.net.gif Monitoring Site Relative to Latency Periods 5-15ms to 75-85ms.

Figure 5.8 (top) The Gravity Model for Traffic Densities for the hepnrc.hep.net.gif Monitoring Site in 2000 for Average Ping Times for Latencies less than 25ms; (bottom left) the Three-dimensional of φ =A exp-0.005D sin 0.12t showing Time Gaussian Behaviour over 24-hours and Two-Dimensional Density Plot (bottom right) Figure 5.9 (top left) The Gravity Model for Traffic Densities for the hepnrc.hep.net.gif Monitoring Site in 2000 for Average Ping Times 5-15ms; (top right) the Three-dimensional Plot for 24 hours for φ =58 exp-0.015 D sin 0.208 t; (bottom left) the Three-dimensional Plot for 168 hours for φ =58 exp-0.015 D sin 0.208 t; and (bottom right) a Density Plot Figure 5.10 (top left) The Gravity Model for Traffic Densities for the hepnrc.hep.net.gif Monitoring Site in 2000 for Average Ping Times 15-25ms; (top right) the Three-dimensional Plot for 24 hours for φ =6.9 exp-0.004 D sin 0.108 t; (bottom left) the Three-dimensional plot for 168 hours for φ =6.9 exp0.004 D sin 0.108 t; and (bottom right) a Contour Density Plot

xx

Illustrations

Figure 5.11 The Linear Regression between Latency Δt and Distance Range Δx showing Time Gaussian Behaviour for the hepnrc.hep.net.gif Site

Figure 5.12 The Plot of Periodic Pairs for the hepnrc.hep.net.gif Site Relative to the Phase Line Standardised to the Earth’s Rotation

Figure 6.1 (top left) Frequency Distributions for Number of Shops Visited; (top right) Total Walking Distance (compared to a normal distribution); (bottom left) Maximum Walking Distance (compared to a normal distribution); and (bottom right) Time Spent Shopping (compared to a normal distribution) Figure 6.2 Log-linear Regression of Total Walking Distance from Carparks in the Tracking Armidale 1995 Data Set Figure 6.3 The Location of Sample Centres in the New South Wales Retail Hierarchy Figure 6.4 The Regression of UCL 1996 Population and Population Change (1996-2001) for the Selected Case Studies Figure 6.5 Main Street Mayfield showing a Vacant Shop, Financial Planner and Pawnbroker forming a Sequence of Shops in what was Prime Retail Space a Decade earlier. The former Cake Shop, now Vacant, offers the First Three Months Rent Free (notice on the door) Figure 6.6 The Redeveloped Supermarket Site in the Centre of Main Street Mayfield showing the Current Tenants as the Salvation Army Second Hand Shop and a $2 Shop Figure 6.7 Extra Employees per One Hundred Thousand Dollars of Turnover, NSW Figure 6.8 The Collapse of Rental Income from Prime Retail Properties surrounding Regional Shopping Centres (1990 to 1995) Figure 6.9 Vacant Shops in Sheffield in 1999, nine years after the Opening of the Meadowhall Regional Shopping Centre Figure 6.10 Vacant Shops in Morley, Leeds in 1999, 12 months after the Opening of the White Rose Centre, 3km away

List of Tab les Table 1.1 Classification of Planned Shopping Centres Table 1.2 Proposed Floorspace Assignments for the Proposed Woolworths Supermarket, Inverell, New South Wales 2000 Table 1.3 Change in Independent Supermarkets and Food Specialty Stores in Australia 1992-1999 Table 2.1 Survey of Tenancy Changes in Local Centres, Canberra, 1998 Table 2.2 Shopping Centres used in the Canberra Household Shopping Preference Survey 1996, 1997 Table 2.3 A Review of the Important Attributes Considered in Studies of Consumer Behaviour Table 2.4 An Individual’s Assessment of their Level of Shopping Satisfaction (Utility) Table 2.5 Characteristics of ‘Small’ and ‘Large’ Centre Behaviour Table 2.6 Source Matrix of Literature Associations for Multi-purpose Shopping Table 3.1 A Classification of Relevant Equations using Space-Time Operators Table 3.2 Time-space Characteristics of Three Malls in the Sydney 1988/89 Data Set Table 3.3 Various Estimates of the Intra-centre Shopping Frequency (per week) in the Sydney 1988/89 Data Set Table 3.4 SE and FA Estimates of Mean Trading Hours for the 1988/89 Sydney Data Set Table 4.1 Occupation Weighting for the Index of Disposable Income (IDI) Table 4.2 Number of Retail Outlets for Sampled PSCs Sydney Project 1980/82, 1988/89 and 1996/98 Table 4.3 Sample Sizes, Sydney Project, Equivalent Time Samples (shaded) for 1988/89 and 1996/98 Table 4.4 Trip Distance Comparison, Sydney Project, Equivalent Samples (shaded) for 1988/89 and 1996/98

xxii Illustrations

Table 4.5 Mean Trip Frequency Comparison, Sydney Project, Equivalent Samples (shaded) and Pre-Christmas Samples (bold) for 1988/89 and 1996/98 Table 4.6 Comparison in the Mean Level of Shopping Satisfaction, Sydney Project, Equivalent Samples (shaded) and Pre-Christmas Samples (bold) for 1988/89 and 1996/98 Table 4.7 The Percentage of Samples in the 1988/89 and 1996/98 Data Sets (The Westfield Chatswood Sample in 1989 of 17.61 is included in the Mean Shopping Satisfaction, but not in 1996/98 because of refurbishment.) Table 4.8 Comparison in the Mean Shopping Time, Sydney Project for Equivalent Samples (shaded) and Pre-Christmas Samples (bold) for 1988/89 and 1996/98. The Total Time Spent Shopping (Frequency × Time Spent Shopping per trip) per week is in brackets (the asterisk describes the Mann-Whitney Significance at the 0.05 Level) Table 4.9 Comparison in the Mean Shops Visited for 1988/89 and 1996/98 in the Sydney Project, Equivalent Samples (shaded) and Pre-Christmas Samples (bold) Table 4.10 Comparison in the Socio-economic Index for 1988/89 and 1996/98 in the Sydney Project, Equivalent Samples (shaded) and Pre-Christmas Samples (bold) Table 4.11 A Summary of Occupational Types from Respondents from Sydney Project (Ordinary Font, 1988/89; Bold Font 1996 or 1997; Bold Italic Font 1997 and 1998 Samples) Table 4.12 Summary of Nature of Trip Purpose from Respondents from Sydney Project (Ordinary Font, 1988/89; Bold Font 1996 or 1997; Bold Italic Font 1997 and 1998 Samples) Table 4.13 Changing Behaviour from Particular Trip Purpose and Socioeconomic Groups from 1988/89 to 1996/98 (1988/89 Samples, Normal Font; 1996/98 Samples, Bold Font) (HI High Income; LI Low Income, according to the Index of Disposable Income) Table 4.14 Population Index and Density Assignments for 1km Bands: Bankstown Square Morning Sample, 3/11/1980 Table 4.15 The Estimation of the Gravity Coefficient for Bankstown Square using Two Assignment Procedures (DW = Durban-Watson Statistic); BSMBankstown Square Morning Sample, BSA- Bankstown Square Afternoon Sample

Illustrations xxiii

Table 4.16 Mean Trip Frequency and Variance per Concentric Band: Sydney 1996/98 Data Set Table 4.17 Comparison in Variance in Trip Frequency for 1988/89 and 1996/98 Samples in the Sydney Project for Equivalent Samples (shaded) and PreChristmas Samples (bold) Table 4.18 Comparison in the Spatial and Time-based (in brackets) Gravity Coefficients for 1988/89 and 1996/98 in the Sydney Project, Equivalent Samples (shaded) and Pre-Christmas Samples (bold) Table 4.19 The Sample Sizes in the Armidale Survey, November 1995 Table 4.20 Comparison between Armidale LGA, Primary Trade Area and NSW LGA Averages for Selected Socio-economic Groups, 1991 Table 4.21 Population Change in Armidale and Region 1991–1996 Table 4.22 Numbers of Students Enrolled at Armidale Campus of UNE Table 4.23 Full-time Equivalent Staff Numbers at UNE, Armidale Table 4.24 Summary of Statistics of Armidale Surveys (bold), November 1995, Compared to the Sydney Project 1996/98 Table 4.25 Sample Sizes of Centre Surveys, Auckland, April 6, 2000 Table 4.26 Summary Statistics of Auckland Surveys, November 1995, Compared to the Sydney Project 1996/98 Table 4.27 Comparison between Auckland Samples with Selected Sydney Samples in 1997/98 taken on the Thursday before Easter Table 6.1 Walking Distance Statistics from Armidale Carparks Table 6.2 Selected Case Studies of Retail Vacant Shops in Nonmetropolitan New South Wales. All Elements not stated as CBD Values are Main Streets only Table 6.3 Multi-purpose Retail Functions within Oberon Retail Establishments Table 6.4 Changes in Landuse of Retail Establishments in Oberon Table 6.5 List of Shop Closures and New Businesses within 18 Months of the Major Supermarket Re-locating to an Edge-of-centre Site, Mayfield NSW 1995-1997

xxiv Illustrations

Table 6.6 Comparison in Retail Employment Statistics between Australian States 1995-2000 Table 6.7 Ratio of Full-time to Part-time Employment for NSW and WA, 1980 and 1992 Table 6.8 NSW Employment and Retail Structural Ratios Table 6.9 Changes in Employment Structure at Coles Supermarkets, ACT 1992-95 Table 6.10 Changes in Employment Structures at Coles, Armidale Table 6.11 Floorspace Equivalent per Person Employed for NSW and Queensland Table 6.12 Westfield Marion, Adelaide Floorspace and Turnover Statistics compared to Westfield Carindale, Brisbane (both regulated hours and no Sunday trading) and Westfield Miranda and Parramatta, Sydney (deregulated hours) for 1999 and 2001

CHAPTER 1 Introduction 1.1

Shopping Change

Shopping is an essential part of our day-to-day existence and the organization of retailing has implications for every household. The Concise Oxford Dictionary defines ‘shopping’ as ‘goods purchased in shops’ and ‘retail’ as ‘the sale of goods in small quantities direct to consumers’ (Brown, 1992a). These definitions are suggestive of the interdependence between ‘shopping’ and ‘retailing’, where the former is ‘consumer-based’ and the latter ‘producer-based’. Shopping is underpinned by connecting consumer demand to producer supply and the mechanism for this exchange is the shopping trip. This implies the importance of flows of consumers through a time-space fabric of shopping opportunities. Conversely, the nature of retailing is producer-focused, encompassing the ways in which firms interact with each other and sell goods to consumers. Whether the dynamics of this demand and supply interaction is driven by consumer or producer sovereignty is a hotly contested issue. Nevertheless, it is the thesis here that consumer trips to and within shopping centres are what maintain their viability, much like the analogy of blood-flows to different organs of the human body. If flows are stopped or reduced, precincts die. It is for this reason that it is important to study the dynamics of these consumer flows at different places, times and scales so that retail planning can be pro-active rather than reactive to sustain healthy shopping precincts. The nature of shopping is not only an integral part of our society, but also an indicator of the rapid societal changes in the socio-economics of households and technological innovations. Consumers can access the Internet at home to shop globally, penetrating national borders or different time zones. The idea of the traditional neighbourhood shopping centre has been replaced, in part, by the onestop shop at supermarkets, hypermarkets or planned shopping centres (Figure 1.1). The result has been substantial structural change in landuse patterns within retail hierarchies over a short time period. Town centres that were once vibrant have been replaced by chains of vacant shops and urban obsolescence. The decline of town centres and neighbourhood stores have therefore been one of the major ramifications of this new global retail order where consumers have fundamentally changed ‘when’ and ‘where’ they undertake their shopping trips. The reasons for this retail ‘shock wave’ enveloping many centres are complex, but could involve an interplay of socio-economic change with household mobility and expenditure, technological change in the nature of flows and regulatory change in the ideology of public policy. Any one or combinations of these factors can fundamentally affect the nature of consumer flows and the health of precincts in retail hierarchies.

1

2

C hapter 1

The main function of retailing is to act as an intermediary between the consumer on one hand and the producer or wholesaler on the other, in the physical distribution of particular goods or services. Such a perspective requires some qualifications. Retailers are now not only fundamental to the buying, selling, storage and delivery of goods, but are increasingly becoming multi-range and multi-purpose in what they offer, and when and where they offer them. Consequently, there has been a fundamental change, from retailers originally selling specialised merchandise, to a multiplicity of goods and services. The supermarket is very much part of this transformation. It now offers a diversity of food and non-food merchandise within its floorspace. In other words, the supermarket has become a shopping centre within a shopping centre. Consumers can now conduct their banking, buy a book or flowers on the floorspace of a supermarket, twenty-four hours-a-day, seven-days-a-week. This structural change should have implications for the nature of the multi-purpose trip. A particular dynamic model will be used extensively in this study (the so-called retail aggregate space-time trip or RASTT model) because it can be applied to a range of trip configurations, from local trips to a neighbourhood centre to global flows of Internet traffic. The model will provide a unifying structure to the analysis because it is process-based rather than subject-based and therefore it has the potential to look at retail change over different scales of space and time. The RASTT model was developed back in the mid-1980s and was a key component of the ‘Storewars’ debates on retail trading hours and location of floorspace in the 1990s within Australia. ‘Storewars’ is a term coined to describe the process of powerful retail corporations taking market share of smaller traders using various market strategies and public policy positions (Wrigley, 1994). Particular attention in this study will focus on changes in retail locations and shopping times. The RASTT model has therefore been intimately linked in public policy, by showing the predicted affects and accrued advantages to large retailers of changing the length and structure of the shopping week (Baker, 1994b) and by locating retail developments away from town centres (Baker, 1995). This process will be examined in this study, showing the link between a model and its policy implications. Many of the issues that became globally significant in the 1990s (such as out-oftown retail developments) are not new. Indeed, such issues have been recorded within antiquity. For example, in ancient Rome, Claudius had asked consuls to license him to hold a market on his estates (Sherwin-White, 1966). Other cases sometimes led to protests. In Book V, Letters of the Younger Pliny (Radice, 1963, 137-138), in a letter written in 105AD to Julius Valarianus, Pliny records: A praetorian senator named Sollers asked the Senate for permission to hold a weekly market on his property. This was opposed by representatives of the town of Vicetia, with Tuscilius Nominatus acting on their behalf, and the case was adjourned until a subsequent meeting of the Senate.

The issue was whether markets held weekly on private land could compete with the public daily markets in forums (Figure 1.2) and thus change the nature of consumer flows. Ancient Rome gave the Senate the ability to adjudicate on this issue because

Introduction

3

of questions of public order (Sherwin-White, 1966, 319). Such a basis of order still underpins Planning Acts of Parliament today. Further, this example shows timespace retail developments have been a source of debate not just peculiar to 21st century policy. The negative idea of retail change and loss of market share was the same for the representatives of Vicetia as for small business in places such as the UK, Australia and the US. The location of shopping destinations and their time of trading was crucial then as it is now.

Figure 1.1 Meadowhall Regional Planned Shopping Centre, Sheffield, 1999

Figure 1.2 The Macellum (Rome’s shopping mall) on the Dupondius coin (AD 65) in the reign of the Emperor Nero

4

C hapter 1

For space, classical economics often avoids the ‘where’ question for the location of economic activity, yet this has a major bearing on the cost and efficiency of distribution. The space variable is also central to any analysis of a retail market and to avoid it is to present an irrelevant view of retailing. This is why a geographic approach is just as important as an econometric approach to retail analysis. Space has many meanings, ranging from the perception of space to the mathematical conception of space. We will view space largely in a narrow physical context and our variables will be measurable through various survey methods. This is not to say that the perception of space is not important to retailing. Stilwell (1994) argued that a key aspect for political economists is the interaction of space with society. Space is seen as a social product and does not occur randomly. Since the 1970s, human geographers have dwelt on what constructs determine ‘place’ and what the relationship is between ‘space’ and ‘place’. Is place just personalised space and is it diametrically opposed to the freedom of space? Such a sense of place is important, particularly to the design of shopping centres. People have a simultaneous need for security of place and freedom of space and this seems to be part of the contradictory dualism in the human condition. People’s shopping activity is not solely price-driven (as neo-classical economics tells us) so an understanding of these issues is important to retail analysis but not necessarily to this study. Time paths very much underpin the evolution of the movement theory of this study and follows on from the conception of time geography developed by Hagerstrand (1970) and the Lund School in the 1970s. Time geography assigns an individual’s activities in a 24-hour day and ‘travel’ as a continuous temporal sequence of activities within geographical space (Kwan, 1999). The location and trading hours of a shop are time-space constraints on these activities and their regulation limits the person’s freedom of choice. There is a time supply and demand for shopping opportunities and the trade-off, at an aggregate level, depends on societal priorities. This allocation of shopping time opportunities affects accessibility and the viability of location. Consequently, there is a strong link between shopping times and retail landuse. This connection has often been missed in retail policy, but it is an important corollary of dynamic trip modelling. A further important question to arise in large scale spatial interaction is whether technological advances in production and transportation have lead to a situation whereby the constraints imposed by distance on economic life tend to evaporate over time. Karl Marx made the prediction of ‘the annihilation of space by time’ (Stilwell, 1994). Is global access of the Internet the ultimate mechanism for the annihilation of the ‘tyranny of distance’? This is a common view of the Internet and its growth highlights the rapid changes in technology and communication, restructuring the nature of shopping on a world scale. This is leading to the acceleration, in a ‘timespace compression’, of retail form and process (Janelle, 1968; Baker, 2002). Shopping and consumption are just a click away on a computer screen. Yet there is still stability in traditional time-space shopping relationships. For example, despite technological change and deregulation, a majority of people still shop once a week at

Introduction

5

a supermarket and a substantial number at their nearest shopping centre. What provides the ‘glue’ for this stability? The reasons for this are difficult to disentangle from the tapestry of human behaviour. One ingredient that we could measure is the frequency of spatial interaction. The very concept of place and the human identification with specific spaces, depends on a certain repetitiveness. The frequency of shopping trips can be an indicator of store or centre loyalty. A person who shops at a particular centre regularly can be counted in a quantitative assessment. Further, the definition of a place depends on certain predictability in behaviour. Frequency, therefore, not only underpins place but also can be assigned a number and can be used for prediction. It is argued here to be an essential ingredient in the realisation and reproduction of space and time (and place). This connectivity is a fundamental part of the time-space model and we use frequency to solve the assignment problem of ‘where’ and ‘when’, on average, people shop. Changes in space and time in the differential equation (the engine driving the RASTT model) are equal to the trip frequency. ‘How often’ is therefore a proxy for understanding the nature of trip behaviour. Another ingredient is the mobility underpinnings of socio-economic groups. Elderly consumers will access shopping opportunities by adapting distance minimisation trips to reduce the total effort in shopping. Likewise, women with young children may prefer smaller shopping centres rather the regional planned shopping centres with parking and access problems. Their aim would also be to minimise walking distance. Such groups are more likely to shop in the mornings than the afternoons. Conversely, ‘time poor’ consumers, such as ‘professionals’ in two income households, prefer to shop in the afternoons and evenings, particularly on the way home from work. These households provide instability in the time-space relationship because of the fluidity in their time budgets. They are more likely to shop on the Internet and are the targeted market for retailers in the evolution of the new retail order of shopping. Therefore, if such groups are substantially present in samples, then they would have a strong influence on the time-space characteristics of the shopping distributions. Likewise, the timing of the sample and its location would also affect these distributions, but this could also be autocorrelated with socioeconomics. Therefore, the socio-economic mix of the sample could be a major factor in determining the ‘when’, where’ and ‘how often’ shopping occurs in aggregate distributions. The blood supply exchanging oxygen, carbon dioxide, energy and waste with the organs of the body provides an analogy to the flow of people from an origin to a destination. When shopping, people exchange money for goods and services and help maintain viable retailers and healthy shopping precincts. This exchange occurs (like the different organs of the body) at a range of retail forms, such as the supermarket or discount department store. Therefore, to understand the flows of consumption we need to understand the structure of these retail units at the points of exchange and how these are changing over time.

6

C hapter 1

1.2

Definitions of Retail Forms Underpinning the Model

1.2.1

PLANNED SHOPPING CENTRES

Planned shopping centres (PSCs) are a consequence of the rise in household mobility through increasing car ownership and the decreasing attractiveness and accessibility of the Central Business District (CBD) of large cities (due to such factors as, increasing congestion, changing population characteristics and suburban sprawl). They are synonymous with carparks and one-stop shopping (Figure 1.1). These centres cater not only for daily needs, but also for higher order specialised goods and services for a large segment of the urban population. They represent a quantum jump in retail capitalism, replacing traditional shopping centres that evolved from central places in previous centuries. Shopping functions have devolved from the central business district (CBD) of large cities to the suburbs, and this suburbanisation of retailing has placed significant stress on Local Government in their location planning for new retail capacity. PSCs are designed to operate as a unit at a single time to meet the trading requirements of a specific trade area under single management. They are a major generator of trips to satisfy consumer demand and are the basis for much of the dynamic trip modelling in this study. Reynolds (1993) points out that there is no uniform definition of a planned shopping centre. For example, PSCs in the UK, typically, have over 4 650 sq m of gross leasable floor space (GLFS) with carparking provision and three or more retail units. However, in Germany, this area is set with a GLFS of 15,000 sq m and is defined by several factors, namely, a spatial concentration of specialist non-food, food or service outlets of various sizes; a number of smaller specialist outlets in combination, as a general rule, with one or more dominant operators; a large shared parking area; a central management; and a set of common functions (such as marketing and publicity). Within Australia, planned centres have a lower floorspace boundary, similar to the British definition, and they can be classified broadly into a retail hierarchy based on floorspace and the number of anchors (Table 1.1). Traditionally, within such larger PSCs in the US, there is, typically, a large department store, which is the major attractor of trade, and smaller specialty and convenience outlets that share the trade attracted by the larger stores. In Australia (and some centres in the UK and Canada), supermarkets are also major anchors and this difference in the mix of anchors underpins what type of model is selected to study trips to the PSC. Planned shopping centres (PSCs) were a retail innovation arriving from the US to Australia in the 1960s (with Roselands opening in Sydney in 1966). Stimson (1985) states that, in general, the retail industry in Australia has been adaptive of innovations that have occurred overseas, especially from North America. Furthermore, he argues that it is likely that the pressures within the industry to adopt increasingly frequent innovations in the industry will become greater as market share competition becomes more aggressively pursued by ownership entities in the industry. However, PSCs were more slow to spread to the UK, because of what Reynolds (1993) described as,

Introduction

7

the most rigorous and restrictive land-use planning systems. This changed dramatically after 1985 when 6.6 million sq m of floorspace was added to UK stock during 198692 (Reynolds, 1993). Likewise, shopping hour liberalisation occurred at around the same time allowing out-of-town locations to be time accessible to consumers and therefore profitable. Deregulation therefore accelerated structural change and allowed for the proliferation of PSCs as a result of a parallel shift from regulatory to deregulatory policy for both space and time. The RASTT model fundamentally looks at trips to and from a planned shopping centre (or mall). The PSC is considered to be, theoretically, a point density of demand where these trips are focused in space and time. They converge at the mall because of the benefits of shopping there in terms of time, effort and choice in agglomerating shopping opportunities under the one roof. PSCs offer one-stop shopping and this is increasingly attractive to mobile and ‘time-poor’ households. Why visit ten shops in a chain of visitations when they can be grouped together in one trip and accessible from a centralised carpark? The assumptions of the RASTT model (such as, nearest neighbour shopping and continuity of demand) are quite defensible at planned shopping centres because of the agglomeration of supply points and the convergence of demand. Further, each centre can be defined by common trading hours, although in Australia it has been found that the major supermarket chains have traded substantially outside the hours set for the other retailers. Planned shopping centres are therefore very amenable for modelling because they allow for partitioning demand in a defined space and time with realistic assumptions. 1.2.2

DEPARTMENT STORES

The department store, first developed in Europe in the mid-nineteenth century (see Adburgham, 1981), was popularised in Australia by consumers of the 1920s requiring the choice of an extended range of quality stock with the aura of European and North American fashion and culture. It was the market for overseas and upmarket fashion, where the latest trends from the continent could be found on the clothing racks of the department store. By combining a whole range of retail functions under the one roof, it was the forerunner of the planned shopping centre. Convenience, quality and choice were the trademarks of these stores. Each had their own cafeteria for shoppers to sit and discuss what they had observed shopping and were self-contained in terms of credit and service. Yet one of the major casualties of attitudinal and socio-economic change in household shopping behaviour has been the decline of this form of retailing. Consumers had a preference in the 1990s for shopping in specialty shops for quality, discount department stores for price and ‘category killers’ for choice. Category killers are an interesting creation from the US where the product choice, relative to an increase in floorspace is maximised. For example, a department store may allocate 1,000 sq m and offer 3,000 lines, but a category killer would have 3,000 sq m with 15,000 items saturating the consumer with choice. This decline of the department store was a consequence of the time constraint of households, and their budgets were

8

C hapter 1

more inclined to rank competitive prices ahead of the latest fashions. There were more women in the workforce and more single parent families and less time for shopping. Consumers wanted quality but at lower prices. They were more cynical, discerning and less loyal in their shopping choices. Department stores have had to restructure quickly with such rapid change in household socio-economics and taste. Low demand categories such as dress fabrics, carpets, hardware and food have been discontinued and stores have targeted fashion apparel, homewares and leisure. Discount department stores and supermarkets expanding their product range have made considerable in-roads into department store turnover. Their strategy of quality merchandise at lower prices is what the market is currently demanding. Brand names, once only found at a department store, are now an integral part of merchandise at a discount department store. Their strategy is lower margins for higher turnover. There was once a social stigma of buying at a discount department store, but today this has evaporated. The department store has had to reposition itself. Centre Category Super-regional

Regional

Sub-regional

Community

Neighbourhood

Minimum Requirement Department Store, Two Discount Department Stores, Two Supermarkets, Mass Entertainment Department Store and Two Supermarkets Two Discount Department Stores and at least one Supermarket Supermarket, Discount Department Stores or Two Supermarkets Supermarket

Example

David Jones, Grace Bros, Target, Big W, Franklins, Woolworths, 16 Cinema Complex, 300 Specialty Shops David Jones, K-mart, Woolworths, Franklins, 200 Specialty Shops Target, Big W, Coles, Chandlers, 100 Specialty Shops

Floorspace (sq m) 100,000 +

45,000100,000 20,00045,000

K-mart, Coles, 50 Specialty Shops

12,00020,000

Jewels and 20 Specialty Shops

5000-12,000

Table 1.1 Classification of Planned Shopping Centres in Australia The department store is still a major anchor of PSCs and this is one way of distinguishing US malls, in particular, because of the lack of supermarkets as coanchors. This means that trips to malls in the US should be less periodic than in Australia (where both co-exist in PSCs). Indeed, the RASTT model allows for the definition of a distinctive type of behaviour (termed ‘space-discounting’ behaviour) where shoppers travel there to save time in satisfying demand. This is because choice is maximised by the agglomeration of shopping opportunities at the department store and the competing specialty shops. Department stores were the anchor of choice-

Introduction

9

based trips. Such PSCs have shopping trip generations as normal or gaussian distributions around the centres. Therefore, department stores, as major mall anchors, generate a particular type of trip behaviour to and from a centre and this can be captured within the modelling. 1.2.3

THE SUPERMARKET

The concept of the supermarket came out of the Depression in the US, where consumers, who had limited cash, were offered a large store carrying a wide variety of goods at reduced prices. Up until then, grocery shopping was done at small shops specialising in individual products. Israel Cohen founded the first ‘supermarket’ in 1936 at Washington and remained president of the Giant Food Store until 1977, overseeing its diversification into a wide range of merchandise. The supermarket is an icon of latter twentieth century retailing. The strategies employed by supermarkets show how supermarket design and merchandise presentation is constructed in order to maximise the chance of impulse buying. They locate items, such as, magazines and sweets, near the checkout for queuing patrons to gaze upon or milk and bread at the back of the supermarket to force consumers to walk past the entire aisle. Supermarkets carry a whole range of fresh food traditionally sold through small independent business (Table 1.2). It is not surprising that there has been a decline in the number of small food retailers, particularly after the deregulation of shopping hours in 1992. Data presented from Retail World Annual Reports (1992 and 1999; Table 1.3) show Australia lost 14,272 food and grocery businesses (excluding takeaway and liquor) between 1992 and 1999. Likewise, in England, trading hour deregulation occurred in 1985 for six daysa-week, Sunday trading in 1995 and a trial of 24-hour trading in 1996. In Great Britain, the loss of traditional small business has also been substantial where, in the ‘food, drink, confectionary and tobacco’ category, there has been a 17% decline in single outlets from 115,111 in 1984 to 100,174 in 1992. Food superstores have increased their numbers from 457 in 1986 to 1,102 in 1997 and market share of packaged groceries from 29.9% in 1987 to 53.7% in 1997 (Department of Environment, Transport and Regions or DETR, 1998). This is very similar to the decline in specialty food retailing and independent supermarkets already noted in Australia (Table 1.3). Therefore, a major characteristic of the supermarket has been their growth in size, functions and locations in conjunction with a policy shift to shopping hour liberalisation and a behavioural switch to one-stop shopping. Supermarkets have also been allowed to duplicate, on their floorspace, the high turnover merchandise traditionally traded by other retail business. Sainsbury’s, UK, not only offers food, grocery and variety discount items, but has a bookshop, drycleaner, florist and banking facilities. They have also made significant inroads into the stock traditionally traded by newsagents, pharmacies and health food shops. This is no coincidence. For example, Coles, Australia, stated this strategy publicly when in

10

C hapter 1

Departments Fresh Food and Produce Bakery Delicatessen Meal Solutions Chicken Fish Fruit and Vegetables Meat Dairy Frozen Food Grocery General Merchandise Checkouts & Administration Miscellaneous Total Area

Approx Areas (sq m) 190 215 15 40 70 310 210 135 145 1150 130 500 90 3200

Table 1.2 Proposed Floorspace Assignments for the Proposed Woolworths Supermarket, Inverell, New South Wales 2000 Category

June 1992

June 1999

Gain /Loss

Independent Supermarkets Takeaway Delicatessens Butcher,Fish & Poultry Fruit and Vegetable Bread & Cake Liquor Total

8270

4197

-4073

% Change -49%

20,324 5719 7349

15,357 2136 3924

-4967 -3583 -3425

-24% -62% -47%

3670

1611

-2059

-56%

4711 1847 51,950

3579 1388 32,569

-1132 -459 -19,381

-24% -25% -37%

Table 1.3 Change in Independent Supermarkets and Food Specialty Stores in Australia 1992-1999 (Source: Retail World Annual Reports 1992 and 1999) 1998 they planned to introduce a comprehensive range of health and beauty products, key cutting, photo lab and photocopying on their supermarket floorspace (Coles-Myer Annual Report, 1998, 10). The strategy is now to stock all the high turnover merchandise of all the surrounding specialist small retailers. This expansion of variety effectively means that the range and absolute numbers of small business

Introduction

11

operators under threat from the retail giants has rapidly grown. In February 1998, the CEO of Woolworths, Australia, Reg Clairs, noted that they would be furthering their range of ready prepared meals which would effectively encroach upon all takeaway operators and, in many cases, restaurants and cafes (Weekend Australian, 28.2.98, p 51). Long trading hours allow the chains to capture market share from more and more sectors of the retail economy. The move to deregulate newsagencies and pharmaceutical merchandise is following this strategy and will enable the supermarkets to strengthen their already strong position, as the supermarkets have a labour cost advantage over small business competitors at present (in terms of penalty rates). For example, Figure 1.3 indicates that supermarkets already control approximately 50% of all pharmaceutical sales in Australia, including over 90% of some items previously the domain of chemists. Deregulation is encouraging the major supermarkets to dominate every retail category. The problem is already well advanced. There is a potent mix facing small business competition: long trading hours and a diversified product range undermining their profitability.

Eye care Foot care Family medicine Cough and cold

Commodity

Family planning Analgesics Wound care Sun care Skin care Nutrition & Diet Toiletries/Baby care Hair care Soaps/Talcum/Oil Oral hygiene Feminine hygiene Deodorants

0

10

20

30

40

50

60

70

80

90

100

Percentage Supermarket share (%)

Pharmacy share (%)

Figure 1.3 Market Share of Pharmaceutical Products in Australia, 1997 (Source: Feros Riley and Associates, 1997) The strategy of supermarkets is evidently clear. They have the ability to discount high turnover items (i.e, petrol, milk, bread) over longer hours than small business, lessening the viability of the small business competitors. High turnover items in small business create the revenue to support a variety of less popular stock. The viability of both sectors provides a wider range of goods for consumers and thus a better retail

12

C hapter 1

amenity. The loss of small businesses in a region not only diminishes the range of goods on offer but also the variety of stores to choose from. It is not in the public interest to reduce choice within retail categories. The supermarket strategy is therefore to select any high volume merchandise that can fit on a shelf and trade it long hours. This is one of the major reasons why such a range of small businesses are affected by long trading hours of supermarkets. It culminates in significant and permanent losses of small retailers. This is what Baker (2002) termed, ‘the global vacant shop problem’ since it is affecting retailers in the UK, Australia and other developed economies. When supermarkets are the major anchors of malls (such as, in Australia) they also create a particular type of regular weekly trip behaviour not found in exclusive department store-anchored PSCs. Much of the shopping strategy to Australian PSCs is built around the weekly shopping trip for food and groceries. Further, supermarkets are a major anchor for convenience-based trips. Within the RASTT model, such behaviour aims, in general, to minimise trip distance where consumers time-discount shopping opportunities by consuming at the local supermarket. In other words, they minimise the total effort in food and grocery shopping by accessing the closest supermarket. The idea of distance decay and the gravity model becomes central in the understanding of such regular trips to a PSC. Further, when a supermarket diversifies its merchandise and trades longer hours, it increases the propensity for multi-purpose shopping on the floor of the supermarket, rather than consumers making separate trips to individual retailers in the adjoining precincts. This would also be part of a strategy of shoppers minimising the total effort in shopping. The result, as the evidence suggests, is not only a change in trip strategies in the use of a supermarket for shopping, but also a loss of competing small business in surrounding localities. There is therefore a suggested link between shopping hour liberalisation, diversification of supermarket merchandise, multi-purpose consumer shopping trips and the loss of competing food and non-food retailers from the market. This connection will be explored in this study. 1.2.4

RETAIL PARKS AND POWER CENTRES

Originally in the 1970s, when the retail ‘warehouse’ first appeared in the UK, they were thought unlikely to have an adverse impact on existing trading patterns (Brown, 1989). Retail warehouses were originally associated with the selling of ‘bulky’ or ‘household’ goods which were argued to require large areas for storage and display. The argument followed that such developments were qualitatively different to CBD retailing and had little impact on town centres. Guy (1998, 292) states that this argument is flawed and increasingly at odds with the real situation. The DIY warehouse and garden centres were then viewed as new retail functions unsuited to trading within the composition of traditional shopping centres because of their space and parking requirements. Bromley and Thomas (1989, 134) argue that such developments in off-centre locations to the CBD were considered benign and this was the basis to the idea of a complementary relationship between the retail warehouse phenomenon and the pre-existing retail system. In the UK, such an attitude to this

Introduction

13

relationship in the 1980s could be argued to be a manifestation of the free market principles under Margaret Thatcher. Shopping time was deregulated and landuse constraints relaxed to allow off-centre developments, since the market could better assign resources efficiently. The result has been the emergence of retail warehouses in this environment and the UK government has taken some ten years to seriously scrutinise the assumptions of complementarity between such developments and the CBD (UK Department of Environment, 1992). The idea of complementary retailing does not stand up to close scrutiny. Bromley and Thomas (1993) quote the following examples in the UK that suggested there were problems in this complementary assumption. In Wembley town centre, four furniture stores closed in association with the development of retail warehousing in northwest London. The relocation of four stores to the Swansea Enterprise Park peripheral to the Swansea CBD resulted in the relocation or contraction of five others from other parts of the city. Indeed, they quote that the Swansea relocations and contractions totalled nearly 40% of the retail floorspace of the retail park. The problem is that this type of retailing and its location have been adapted by a new wave of category-specific retailers or category killers in comparison goods such as toys, clothing and footwear. This is a direct duplication of retail function found in town or suburb centres. They require deregulated shopping hours seven-days-week to locate away from town centres. If this is the case, they can form agglomerations of these free-standing large stores (termed, a ‘power centre’) which sell retail categories of almost everything found in a planned shopping centre or town centre. Power centres are now being regarded in the US as a serious threat to regional shopping centres (Rogers, 1996). As Guy (1998, 294) notes, the original type of retail park, restricted to bulky goods and complementing, rather than competing with the town centre, were replaced with comparison goods centres marketing toys, children’s wear, home furnishings, kitchen goods, and office stationary and supplies. This agglomeration of shopping opportunities therefore presents serious competition for town centres. This effect was magnified by the invasion of category killers into the retail parks of the 1990s. Since retail parks are underpinned by the agglomeration of retailers with shared parking, it is reasonable to expect that consumers would also space-discount shopping trips there in order to save time (especially at ‘category killers’). A time minimisation strategy in the RASTT model would create a normal distribution of trips generation surrounding the retail park. Since both regional malls without supermarket anchors or retail parks would probably influence similar space-discounting consumers adopting time minimisation strategies, it would be expected that out-oftown retail parks would more likely affect regional malls and town centres without major supermarket anchors. The corollary is that, in such places as Australia, retail parks would have less impact on competing centres with supermarket anchors because such centres are underpinned by supermarket-based multi-purpose shopping.

14 1.2.5

C hapter 1 THE INTERNET

The freedom of access into the Internet for consumers means that the issues of physical location, travel time or market area may be less relevant and the research frontier has to deal with such things as ‘virtual trips’ and unrestricted shopping opportunities between countries (Baker, 2001). There even appears to be some sort of time substitution for spatial interaction (particularly from time-poor affluent households). The Internet changes the scale of spatial interaction. However, the RASTT model is composed of mathematical operators (or formal process rules) and these allow for an application to global transactions relative to a time boundary (namely, the rotation of the Earth). The equation describing supermarket trips, therefore, can describe Internet traffic with the same operators as a shopping trip, with the only difference being that spatial interaction increases by orders of magnitude in the distance exponent. The Internet forms the physical network of connectivity (such as, optical cables and phone wires), where there are nodes or ‘routers’ that navigate packets of data from one computer to another (Barabasi, 2001). The Internet is therefore spatially specific in that flows occur through physical space. Conversely, in the World Wide Web (WWW) or the Web, links can be easily established arbitrarily as virtual connections between any two computers independent of spatial co-ordinates. The idea of the WWW originated from a hypertext technique where a considerable amount of multimedia is interconnected (Jiang and Ormeling, 2000). It is more content-specific and its properties are analysed by ‘maps’ that tell how the pages are linked together. Both the Internet and the Web can be regarded as a network of nodes and links forming a complex graph defining what is known as ‘cyberspace’. Cyberspace is a computer generated landscape which integrates these networks into a virtual space. Much of the current Internet research involves the application of graph theory to the study of the Internet and the Web (for example, Barabasi and Albert, 1999; Albert et al., 2000; Cohen et al., 2000; Figure 1.4). It is interesting that this connectivity and its theoretical descriptions are expressed in terms of time and that there is little recognition of the spatial domain. For example, The Internet Traffic Report (2001) uses a time-based index describing the round trip travel time of major paths on the Internet (also termed ‘latency’). The distance factor is replaced by how much time it takes to transfer data. Further, the so-called maps plot connectivity and are essentially aspatial. Within a geographical context this is not satisfactory, because the flows of time-dependent Internet traffic around the world are passing through countries and time zones relative to a 24-hour boundary. The transaction times are in milliseconds and the distances cover many thousands of kilometres. These ‘virtual trips’ are a manifestation of the time-space compression, and dynamic trip modelling will be applied to study the movement of such traffic.

Introduction

15

The Internet will be shown to exhibit time-discounting behaviour where the routers minimise the distance within the network. The time boundary, in this case, will be the rotation of the Earth, but flows will also be observed in weekly periods. The scale of spatial interaction has increased, but the space and time operators are independent of the scale of interaction. These operators can apply equally to walking visits to the neighbourhood general store, a car trip to a shopping mall or the global access to dot com retail companies on the Internet.

Figure 1.4 The Generation of an Internet Tree showing the Aspatial Connectivity from 100,000 Internet Routers and the Hierarchical Structures that develop from a few Highly Connected Nodes (Source: Cheswick, 1999)

16

C hapter 1

1.3

The Time-space Convergence

This study tackles the problem of modelling shopping change using a dynamic trip model in space and time. A number of specific trip strategies can be developed and applied to particular consumer trip behaviours. A walk to a local shop and the Internet can be constructed using the same mathematical process, yet they are different because of scale, connectivity, purpose and socio-economics within the interaction. There is a time-space convergence, a concept not new to geography, but very useful in understanding the relationship. There is a dynamic convergence of locations and the evolution of spatial reorganisation from changes in time-space connectivity, particularly from improvements in transportation (Janelle, 1968, 1969; Forer, 1978; Gatrell, 1983). Blaut (1961) argued that every empirical concept of space must be reducible by a chain of definitions to a process and Janelle (1969) states that inherent in Blaut’s view is the implicit existence of a temporal pattern in each and every spatial pattern. The time-space convergence can therefore be presented by an inverted cone of time lines superimposed upon spatial interaction (Figure 1.5). A spectrum of time-space trip possibilities can be presented within the cone. For example, what is the timedistance necessary for a person to walk and buy a book in a shop? The time cone shows how much time is necessary for this trip. A change in technology allows the person to drive a car to a shopping centre to buy the book and in doing so, converges the time-distance between the origin and destination and simultaneously diverges accessibility to other spatial shopping opportunities. A further change in technology, with the computer and the Internet, allows the person to approach the global accessibility of world space near the singularity of the time cone. The cyclic journey between an origin and destination creates the cone, and its time boundary impacts on the timing of the trip, whether by walking, car or the Internet. These are the local effects from the trip and are affected by individual time and economic budgets. Yet, there are also global physical effects from the revolution of the Earth which impact on trips, particularly for the Internet. This global influence can be distinguished in dynamic trip modelling by a further time-based operator. The local cone is defined by spatial and temporal operators (that is, the processes defining the rates of change and the time rate of change) and its shape is determined by the local substitution between space and time. In the 17th century, technology restricted access by the use of horse-drawn transport to establishments, but at the beginning of the 21st century, there are consumers now engaging in the nearly instantaneous global time-space interaction of the Internet. The cone converges towards a singularity, in other words, a discontinuity in the time-space fabric produces an inequality in interaction. This is not hard to see in the two-dimensional representation in Figure 1.5 because there are two quadratic time curves (for the cyclic trip) for every straight distance line between an origin and a destination. For space, the origin-destination

Introduction

17

exchange is linear, but for time, the interaction is cyclic. Consequently, space and time are not equivalent in spatial interaction modelling.

TIME-SPACE CONVERGENCE Ti

m

T h e R e tu r n T r ip el

10 0

in

Δt

e

10

T

el

SCALE P lan e (

10

e

)

C ar; S h op pin g M a ll(

4

in

1

W alk (

10 2

im

10

)

)

-3

Time

Space

Interne t (

)

M a th em atica l S ing ula rity

Figure 1.5 The Time-space Convergence Showing the Cone of Time and Space Interaction Relative to Changes in Technology (Baker, 2005) This study is therefore looking at people engaging in trip behaviour in different time contexts. The way they partition space and time is determined by the operators in the model (although the rates of change can be modified by other factors, such as, technology or socio-economics) and these do not change with the scale of interaction. What is meant by this operator approach to spatial interaction? As verbs in sentences, such as, ‘run’ or ‘walk’, define movement, mathematical operators are the ‘doing’ functions defining process rules for movement. They are the ‘verbs’ in the spatial sentences describing trips between an origin and destination. The same operators apply throughout the time cone, independent of scale, and apply equally to trips to a local store, shopping centre or the Internet. The cone is defined as curvilinear, rather than straight lines, introducing an implicit discontinuity near the apex of the cone. This means that there could be peculiar effects from the ‘rippling’ of time, such as, imaginary time reversals from simultaneously accessing different time zones and the

18

C hapter 1

stationary ‘virtual trip’ from a fixed computer location. The discontinuity also implies that there is also a fundamental uncertainty in interaction within the timespace fabric. This could also have implications. For example, this work was presented at the European Regional Science Association Conference at Dortmund, Germany in September, 2002. There, debate centred on a lively discussion between an economist and a physicist on whether one could make money via the Internet on the daily closures of the world’s stock markets from the differences in time zones. This is not an area of study in this book, but it highlights the importance of this discontinuity in the time cone on flows of transactions, globally, between different locations and time zones. Therefore, dynamic trip modelling has important applications and introduces the conceptual basis for new ideas into spatial interaction modelling. Underpinning this interaction is the mathematical ‘engine’ of the differential equation operating within the framework of the time-space convergence. The result is a time-space spectrum of trip possibilities, from walking to a shop to surfing the Web. 1.4

A Way Forward

The shopping trip underpins the vitality and viability of retail precincts. The movement from an origin to a destination has economic consequences, so the location of destinations is an important corollary to dynamic trip modelling. Time is not benign in spatial interaction modelling because it defines the accessibility of the destinations for shopping trips. These time boundaries partition accessibility to shopping opportunities at different locations. Further, the time operator for the trip transcends the scale of interaction, but does not necessarily annihilate the spatial operator. This study will show that distance still matters in the dynamics of the timespace fabric of shopping opportunities, even for the Internet. This book is therefore a return to the theme of the ‘spacing time and timing space’ project of Carlstein et al. (1978), albeit with new mathematical and empirical underpinnings. It studies changes over time, space and scale. It measures changes over time at the same planned shopping centres in Sydney, Australia, between 1980/82, 1988/89 and 1996/98. Changes over space are studied by sampling shoppers in a pedestrian mall, in regional New South Wales, Australia, in 1994 and cross-culturally to Auckland, New Zealand in 2000. Changes in the scale of interaction are observed by studying the dynamics of an Internet site in the US in 2000. These case studies should provide a robust evaluation of this type of modelling. The ramifications of the RASTT model are widespread, in terms of retail policy and practice. Shopping hours can inflate the market penetration of retail floorspace. There are not only central place hierarchies, but also central time hierarchies where the highest order centres trade 24-hours, seven days-a-week. Shopping hour liberalisation suggests there will be a fundamental restructuring of ‘where’, ‘when’ and ‘how often’ consumers shop. These changes appear to be coincident with the appearance of more vacant shops in town centres. The increase in the volume of trips to destinations away from these town centres should be a major underlying reason, since time deregulation

Introduction

19

allows accessibility that otherwise would not be available. Using the medical analogy, town centres are suffering a loss of ‘economic’ blood from the loss of consumers. This interdependence between shopping places and times therefore has policy ramifications. Deregulating shopping times means that there will be a new order in trip generation with major consequences for the retail landscape. The RASTT model has therefore played a part, with some success, in the ‘Storewars’ debates in Australia over the last fifteen years (for example, Baker 1994b; 1997; 2002). Yet, it is still in the commercial interests of large retailers and property developers not to recognise this interdependence. This is why the epidemic of vacant shops continues unabated. The theme of this study is ‘dynamic trip modelling’ and it offers a way of studying the flows of traffic, locally, regionally and globally. It uses a differential equation not used in the physical sciences where, in the calculus, time is not taken to infinity. In this model, time is a boundary. This allows for the study of how time impacts upon spatial distributions. The corollary to this time partitioning is the gravity model and if this assumption of limiting time is valid, then we should observe gravity-type interactions for walking to the shop, driving to a shopping centre or ‘surfing’ the Web. This spatial interaction between an origin and a destination is summarised conceptually by the time-space convergence of time lines. This time cone, then, is the conceptual framework for the modelling. What is presented in the subsequent chapters is one way of looking at a complex problem of spatial interaction at different scales and times, and does not claim theoretical exclusivity. However, this study does try to show that this type of modelling is relevant, testable and reproducible and has important applications to retail policy and practice. This is what is hoped underpins this book.

CHAPTER 2 An Introduction To Retail and Consumer Modelling 2.1

Definition

A model may be defined as a simplification or abstraction of reality using selected analytical techniques. It is a formal elaboration of an informed guess or hypothesis. Models can contribute in a special way to the refinement of good questions. They may take the form of a map, classification or a functional assignment of mathematical symbols, yet they have all one thing in common, namely, each are approximations of varying degrees of the retail phenomena. Chorley and Haggett in their seminal book, Models in Geography, published in 1967, state that a model is a simplified structuring of reality which presents supposedly significant features or relationships in a generalised form. They describe the many functions that a model may perform in scientific investigation, including: 1. a psychological device which enables complex interactions to be more easily visualised; 2. a normative device which allows broad comparisons to be made; 3. an organisational framework encouraging the collection and manipulation of data; 4. a direct explanatory device; or 5. a constructional framework accompanying the search for theory or for the extension of existing theory. What underpins any model is deductive thinking and its application has had the greatest success in the physical sciences because variables in the complexity can be controlled in laboratory situations. Deductive reasoning develops preliminary hypotheses before data is collected to empirically verify them. The alternative within scientific method is by induction, where data is explored in a search for hypotheses. Inductive reasoning may produce hypotheses that may in turn be false, but nevertheless it can be an important starting point, if meaningful hypotheses cannot be deduced. Deduction and induction often involve the assignment of names or numbers to phenomema. This depends on the nature of the phenomena, how well analysts understand them and the context in which they occur. When a type or kind can be discerned, the assignment is nominal, and if it can be ranked, or form part of a sequence, it is ordinal. Such a process depends on such concepts as ‘category’ or a ‘boundary’ in the assignment of meaning and both can be applied to the same

21

22

C hapter 2

phenomenon. For example, a regional shopping centre is classified by either the number of major anchors or floorspace or both and has a trade area boundary, which can vary according to the time of day or day of the week. The concept is transferable for comparison to other regional shopping centres in different contexts. Table 1.1 on page 8 could then be argued to be a model of shopping centres, even though it is simply a classification. Models, therefore, are used to organise information and to generate meaningful generalisations. Further, they are a lynch-pin for good hypotheses and a focus for empirical testing. They can also provide a ‘crystal ball’ for predictions into the future, if the assumptions and outcomes are scientifically verifiable and reproducible. 2.2

A Justification for Modelling

Public policy is made for communities, rather than individuals, and it is the understanding of generalisations and their objectivity that best serves good policy for communities. For example, the contest between academics in Australia over the 1998 Southeast Queensland Sunday Trading Case, before the Full Bench of the Industrial Relations Commission, is a good example of retail modelling successfully influencing public policy. Numerous small business people presented cases before the Commission of their individual circumstances and hardship, but it was the predictions, rather than these individual accounts (which nevertheless were important to those people), which won the day in court. The Commission was able to observe predictions of vacant shops and the decline of traditional retail precincts in seven days-a-week trading communities. This was one of the major pieces of evidence that convinced the Commission that the costs to small business and local government outweighed the economic arguments of individual choice, from the shopping hour deregulation of major supermarket chains. Generalisations from model predictions can have policy implications, but this puts increased focus on the underlying assumptions. The retail aggregate space-time trip (RASTT) model, used in the above shopping hours case, is underpinned by the assumption of distance minimisation and regular shopping behaviour. Such assumptions may only occur as the determinate in fifty per cent of shopping trips, but for trips to a supermarket, this model is far more relevant than for clothing or gift shopping. Trips to large regional shopping centres increasingly do not fit this assumption, but this conclusion was only reached after looking at the empirical evidence of trip behaviour in a sample of the same shopping centres in Sydney over twenty years (the so-called ‘Sydney Project’; Baker, 1994a, 2002). The modelling of aggregate patterns can provide some explanation, but it is recognised that distance minimisation is relevant only to select situations. For example, trips by less mobile populations (such as, the elderly) or purposes (such as grocery shopping) or the centre size (smaller community to sub-regional shopping malls) all appear to conform to this strategy. The use of the model is then conditional, but it provides a benchmark to define variations in trip processes and change.

Introduction to R etail and C onsum er Modelling

23

Operators provide a way to deal with the understanding of trip processes and change. The problem with the RASTT model (but not for the entropy-maximising model of trip distributions; Baker and Boots, 2005) is that the solution is for only one particular shopping centre and not for all shopping centres (if other centres are visited in the activity set, the model becomes non-linear). The point is that if we understand the mathematics and what its limitations are, in the context in which it is applied, the model can be a powerful tool in disentangling conditional generalisations. For non-linear shopping distributions, distance minimisation is no longer an appropriate behavioural assumption. Does this mean that the model has failed and we abandon modelling altogether? The RASTT model never states that all consumers rationally use space. Such a model then only provides a benchmark to begin classifying consumer trips and shopping centre types. There are therefore many other spatial trip strategies that may not be quantifiable, such as, shopping whilst visiting a family member, but with fifty per cent of trips underpinned by distance minimising strategies, this model is then a starting point to classify spatial behaviour. The differential equation defining change and rates of change in the RASTT shopping model can apply to an individual consumer as well as a population of consumers, because it is mathematically linear (each solution can be added to and, in turn, become a new solution). This is an important property. Further, changing the scale of interaction does not affect the operators (written for space in this model as, G /Gx, and time, G 2/G t 2). A mathematical operator is an instruction to act on a function and it is analogous to an active verb in English where the function is the object of the sentence (Morrison, 1990). Operators, therefore, can transcend the scale of interaction and are equally applicable to individuals or populations.

2.3

The Art of Modelling

How should we then model shopping phenomena and is there a handbook that will tell us the correct way to go about things? There is no ideal representation and no handbook. Rather, it involves intangibles, such as, how do we get ideas and make reasonable guesses as to what are good representations of the observed phenomena? Baker (2002) states that reasoning by analogy reached its zenith in the quantitative revolution and the cross-fertilisation of different ideas between disciplines today is still an important way that geography is evolving. It is a corollary of General Systems Theory. This is where one can use information on the properties of one real system to present the properties of another little known system (Von Bertalanffy, 1962). Chisholm (1967) and others regarded such an approach as an irrelevant distraction, but on balance, it has its limits and increasingly, models must be selfsufficient in terms of their assumptions and empirical relevance (Baker, 2005).

24

C hapter 2

The best example of the construction of a model comes from the development of the gravity model. Spatial interaction modelling defines a theory of movement rather than a theory of location. It has no rigid underlying assumptions, such as in central place theory, and was originally formulated around a series of mathematical models, based on the gravity concept analogous to Newton’s law in physics (Davies, 1976). Interaction between two populations was simply proportional to their size and an inverse function of distance separating them. This gravity model has since evolved and has been applied to many different problems in spatial interaction modelling (for example, traffic movements, population flows and shopping trips). The gravity model occupies a central cog in the evolution of spatial science and is at the edifice in the generation of ideas borrowed from physics and their application to consumer behaviour (Figure 2.1). There have been many attempts to try to generate the form and function of gravity behaviour in spatial interaction, such as, the entropymaximising approach, its role in potential theory, intervening opportunities or spacediscounting behaviour. Yet in the 1980s and 1990s, there was strong doubt as to its value, since the complexity of the real world has proved the calibration process problematical and the specification of gravity models fraught with difficulty. The coefficient E within the gravity model [exp (-Ex)] has been central to gravity model research. Is it more than just a friction-of-distance measure? Research in the 1990s suggests that the gravity coefficient is not solely a distance-constructed entity devoid of context. Consequently, the effect of distance on shopping is not constant everywhere and can vary according to the time of sampling, socio-economic behaviour of composite sub-populations and statistical constructs, such as the size of the aggregation unit and the period used in sampling. Once these limitations are understood, conditional generalisation can be made from gravity modelling and predictions made of future interactions. It is this ability to make such predictions that warrants perseverance with this type of modelling.

How do we construct good models? Learning to apply mathematical skills is different to learning the mathematics itself. There are no precise rules in mathematical modelling, but rather it is an art form, where there is no correct way. There may be a number of steps to produce a relevant, testable and reproducible (RTR) model and they essentially involve how to evaluate good ideas. Yet, what are the origins of these ideas and inspired guesses that become the cornerstones of social science? There is no answer to this question, but once the idea is there, we can proceed to follow these steps or just ‘do it’ and let our peers judge the results. A possible sequence to this process is outlined below in Figure 2.2.

Complex Human Behaviour Clark (1968)

Place Utility Wolpert (1965)

Learning; Search Markov Chains Golledge (1967)

PSYCHOLOGY

Mental Maps Gould and White (1974)

Distant Dependent Utility Function Smith (1975)

Behavioural Measurement Norman and Louviere (1974)

Panel Data Wrigley (1985)

Statistical Purchasing Dunn and Wrigley (1985)

Inventory Purchasing Bacon (1984)

STATISTICS

SOCIOLOGY

Socio-Economic Variables Hanson and Hanson (1981) Place Utility Fields Baker (1982)

Time Discounting Opportunity Baker (1985)

Discrete Choice Random Utility McFadden (1974)

Multi-Purpose Trips Hanson (1980) O'Kelly (1981) Agglomerative and Competitive Effects Fotheringham (1982)

Space Discounting Opportunity Isard (1975)

BEHAVIOUR

Spatial Diffusion Morrill (1968)

Revealed Space Preference Rushton (1969)

CONSUMER

Entropy Trip Distributions Wilson (1967)

Potential Theory Curry (1978) Sheppard (1979)

PHYSICS

GRAVITY MODEL Reilly (1929)

Random Settlement Evolution Curry (1964)

Probability, Potential Utility, Market Areas Huff (1963)

HUMAN BEHAVIOUR

ECONOMICS

Rational Economic Behaviour Berry and Garrison (1958)

Migration Hotelling (1921)

Intervening Opportunity Schneider (1959)

Shopping Centres Ellwood (1954)

Central Place Theory Christaller (1966) Losch (1954)

Introduction to R etail and C onsumer Modelling 25

Figure 2.1 A Flow Diagram Showing the Evolution of the Gravity Model in the Context of Consumer Behaviour (Source: Baker, 2000b)

26

C hapter 2

1. Collect background information to pose a focus question relevant to the community ↓ 2. Select possible variables that can be measured ↓ 3. Exploratory data analysis to discover what the possible significant relationships are between any of these variables and the quality of the data ↓AND/OR 4. Model-building using either statistical or theoretical guesses as to the appropriate distributions ↓ 5. Numerical simulation and visualisation of appropriate distribution and understanding the assumptions behind each model ↓ AND/OR 6. Significance testing and evaluation of different modelling possibilities from ‘real world’ data ↓ 7. Re-application of the selected model to different contexts or re-surveying or compiling different data sets to test the same model for robustness ↓ 8. Application of the results back to the community and public policy. If robust, then further theoretical development to provide new focus questions and the cycle is repeated Figure 2.2 A Process of Building Relevant, Testable and Reproducible Models However, there is no right way of doing things, but often modellers start at (4) and end at (5). They do not go to (6) or (7) because they have not used (2). They forget that much of research is funded from the community and omit (8). An example of applying this type of scheme is in modelling shopping trips to and from shopping centres by the RASTT model. The model is generated from the solution of a time-based ‘diffusion’ model using a differential equation: ∂φ 1 ∂ 2φ = 2 2 ∂x a ∂t

(2.1)

Introduction to R etail and C onsumer Modelling

27

where φ is population density per 100 residents in geographic space, t is the time available for shopping and M = 1/a2, the diffusion constant of transport mobility for trips to the shopping mall. The squiggle ∂ defines what is termed an operator or the ‘verb’ of a mathematical ‘sentence’. There is therefore a trade-off between the time t and the distance d(x-x0) from a spatial demand wave travelling to and from a shopping centre. The δ 2/δ t 2 means that we are particularly interested in the rate this population φ discounts the timing of the trip. Equation (2.1) is part of Step 4 and is just a theoretical guess. The good thing about this guess of time-discounting (that is, consumers accelerating their shopping time opportunities) is that we can impose a trading hour boundary on the trip distributions. Therefore, we can answer a further question: how can the trading hours of a particular destination affect consumer trips?

A solution of Equation 2.1 provides a ‘doorway’ to answer the question, namely: ­sin( kt ) ½ ¾ ¯cos( kt )¿

φ = Aexp ( − βD ) ®

(2.2)

where D is the trip distance, β, the gravity coefficient and k the trip frequency and sin and cos represent the trigonometric functions (of periodicity). A translation could be that the propensity to undertake regular trips (through the periodic functions, sine or cosine) to a shopping centre is discounted by how far away we are from the centre by the gravity model (exp -β D), since there is a logarithmic growth in opportunities (the inverse of the exponential function) to shop elsewhere, further from this centre. Therefore, a corollary to the distance decay (or gravity function) is regular behaviour. Consumers who minimise their trip distance will do so regularly when a trading hour boundary is imposed on their spatial interaction. This is what the solution (Equation 2.2) to our differential equation (Equation 2.1) is stating. How do we know that this solution is relevant to actual shopping trips? We now proceed to Step (6) above. Baker (1994) showed that there was a simple relationship between the gravity coefficient and mean trip frequency in Equation (2.2), namely:

β=

k2 M

(2.3)

The rate of spatial discounting (the slope β of the gravity model) is equivalent to the number of trips made to the shopping centre, relative to the transfer coefficient M. Baker (1994a) tested this relationship empirically in a study of five planned shopping centres in Sydney, in fifteen samples taken from 1980 to 1982 and 1988 to 1989. This relationship is plotted in Figure 2.3 and there is a significant R-squared of 0.51

28

C hapter 2

in the relationship. In other words, approximately 50% of trips to these centres at these times were undertaken by consumers minimising distance and shopping regularly. Further, this type of trip is underpinned by the weekly grocery visit to a supermarket (which is a major anchor in each of these centres). Therefore, our model is more than a theoretical guess. It is relevant to over half of the shopping trips to these malls. 1.2

Gravity Coefficient

1

.8

.6

.4

.2 Y = .057 + .324 * X; R^2 = .507

0 0

.5

1

1.5 Frequency^2

2

2.5

3

Figure 2.3 The Regression of Mean Trip Frequency and the β Coefficient of the Gravity Model for Shopping Trips to the Sydney Shopping Centres 1980/82 and 1988/89

How then is this result relevant to public policy (Step 8 above)? For opening and closing times, defined by (0,T), the inter-locational trip frequency, k, can be defined by k = nπ / T. Equation (2.3) can be manipulated to suggest a hypothesis relevant to the deregulation of trading hours, since it predicts that the period the shops are open (T) will affect the trip frequency and, by Equation (2.3), the gravity co-efficient β. Therefore, one hypothesis could be: does the extension of trading hours (T) increase the attractiveness of the mall for shoppers (inferred through falling β values and increasing market areas) where (n) is the number of shops visited? This hypothesis can be represented mathematically in Equation (2.4):

β=

n 2π 2 MT 2

(2.4)

Introduction to R etail and C onsumer Modelling

29

where π is a constant and part of the period trip assignment. Equation (2.4) is one way of stating a model of time-discounting trips to malls relative to trading hours. This hypothesis is tested in Section 4.4 for a decade of the Sydney Project (1988-1998). There are other model representations of this hypothesis. For example, this relationship can be expressed graphically (Figure 2.4). There is a greater propensity for households to travel to Shopping Centre O beyond C d (because extended hours makes the planned shopping centre more time-accessible to car-owning households). There would also be a contraction of shopping in localities adjacent to shopping centre O. The increased number of trips to centre O beyond C d results in a loss of household expenditure from local shopping and its reallocation to the planned shopping centre. This suggests there would be a substantial loss in profitability in retailers beyond the centre and a rise in vacant shops in traditional neighbouring shopping precincts (particularly beyond C d). An example of the impact (vacant shops) of the extension of shopping hours in 1995 of MarketPlace Leichhardt (in the Sydney Project) on the nearby suburban centre of Haberfield is shown by the vacant shops in 1999 (Figure 2.4). Modelling can therefore take many forms and there is no correct version of reality. Models are just a method to simplify the dynamic retail landscape and help disentangle its complexities. The above example shows one method of generating a retail model relevant to public policy and underpinned by empirical data. We use mathematics to describe the process of change. Equation (2.3) can be equally represented by the graph in Figure 2.4. A theoretical guess is tested empirically in the first cycle (Steps 4 to 6; Figure 2.2) and a new hypothesis generated in the second cycle of model-building and then applied to Step 8. This type of modelling can be termed ‘RTR-chain modelling’, where RTR (relevant, testable and reproducible) models are underpinned by some links between formulation, testing and policy implementation.

2.4

Model-b uilding and its Weaknesses

Openshaw (1989) argued that mathematical modelling from the quantitative revolution was massively theoretical rather than applicable. Many models had virtually no prospect of application or testing, where they were developed largely data -free and thus data-independent and this was no longer justified in the era of computer databases and geographical information systems. Models were often lacking in relevancy and it seemed that the trivial idealised problem was preferred to to the worthwhile complex applied problem. The idea of relevant, testable and reproducible model-building in the previous section is an attempt to respond to Openshaw’s criticism.

30

C hapter 2

L2 L1 Φ Decrease in propensity for

Population Density

β1

C

V

shopping at adjoining sites

Increase in propensity to β2

β1 O Shoppi ng

β2

trip to shopping centre O

L 2 (extended hours) L 1 (restricted hours)

Cd Distance

d

Cent r e

Figure 2.4 An Extension of Shopping Hours reduces the Slope of the Gravity Model (β ) where there is an Increased Propensity for Households to Travel to Planned Shopping Centre O rather than Shop Locally. Inset Photographs: (right), Vacant Shops (signed) in the Haberfield Shopping Centre from Competition from MarketPlace Leichhardt (left), Sydney, New South Wales, May, 1999

Introduction to R etail and C onsumer Modelling

31

Openshaw further identifies problems of model construction. He argues that it is easier to revive old models than build new ones. We are just producing sophisticated versions of older models. These first generation models were simple, yet capable of operational use, but later editions made empirical testing difficult at best but usually impossible, given the complexity or idealising of assumptions for the next generation. This does not have to be the case if care is made in framing the variables used in the model. For example, the retail aggregate space-time trip (RASTT) model, using one set of operators, derives the gravity model of spatial interaction from a differential equation of time-discounting shopping opportunities to malls. Yet, it uses physical variables and constants that can be measured, such as, ‘trip distance’, ‘trip frequency’, ‘number of centre destinations’ and ‘length of trading hours per week’. The advantage of formulating a model in this way is that these physical descriptors can be measured for any context through the same survey instrument. It is therefore devised to compare and contrast indexes of space-time consumer behaviour and cross-cultural studies. The RASTT model would at least provide a framework for any comparative study. This will be shown in Chapter 4. Openshaw identifies another problem of model construction where aggregate patterns are more easily modelled but they are theory-poor. Entropy-maximising models of aggregate trip distributions, defined as gravity models, offer consistency and applicability to a wide variety of contexts but, as a basis for explanation, has led to many problems. The maximising of all possible assignments between origins and destinations to gain equilibrium distributions does not make any sense apart from an exploratory tool. For example, nobody drives down every possible route to a shopping centre before deciding which is the best assignment of transport preference, but population preferences will at least allow probabilities to be assigned to each route. Entropy-maximising can look usefully at assignments over a number of competing shopping centres, but why is the gravity model the final solution in the assignment problem? Conversely, the problem with the RASTT model is that the solution (Equation 2.2) only refers to one particular centre. The transport environment M is only constant for a snapshot of that centre. If more than one centre is being studied, then M varies and the differential equation is non-linear and more difficult to solve. Nevertheless, both models have their strengths and weaknesses. The trick is to understand these and only make conditional generalisations. 2.5 2.5.1

Examples of Retail and Consumer Modelling INTRODUCTION

This section will introduce some different models that have been used to model retailing and consumer behaviour. The list is not intended to be exhaustive, but to show the process of model-building and its value in discovering the nature of retailing and consumer trip behaviour. We will look particularly at the success and

32

C hapter 2

failure of central place theory, a micro-economic theory of trading hours, the nature of shopping trips to a hierarchy of malls and the multi-purpose shopping trip. These are central when exploring the relationship between a particular retail model and the socio-economic political context. Other models and approaches will only be briefly reviewed to provide a wider context of other methods used in analysing consumer behaviour. 2.5.2

CENTRAL PLACE THEORY AND TIME MODELLING

What then determines the location of retail activity? There are distinct relationships between retail centres and their surrounding trade areas and much geographical research has been undertaken in developing theories to account for the location of shopping functions and trade area analysis. One of the most widely known is Central Place Theory which was developed by a German (Walter Christaller) to explain the apparent order in the spatial distribution of urban settlements. Davies (1976) states that this order is most conspicuous in the sizes and spacing of those settlements and, as central places, efficiently provide goods and services to the surrounding populations. It defines the location of services as activities that manifest themselves in retail centres. The description of this ordered arrangement involves geometric arrangements of trade areas. Central Place Theory was originally concerned with the number, size, location and the spacing of towns in an idealised landscape. There are two economic assumptions that underpin the ordered arrangement of the trade areas of towns (Davies, 1976): 1. There must always be some minimum level of consumer demand or minimum threshold in terms of long term profitability that is necessary for a firm to become established; and 2. The effective size of this demand correlates with a maximum level defined by the greatest distance consumers would be prepared to journey to that firm or elsewhere. Those firms involved in daily trade (termed convenience goods or services) will be densely distributed over the landscape and are sustainable from the frequent visits of nearby small populations. Businesses involved in more specialised goods or services are much more widely dispersed and have to exist on less frequent visits from a much larger population. A further constraint is then added that no firms be allowed to make excess profits and so, in the long term, all trade areas for the same good or service approach an equivalent size. The result is that firms offering the same retail functions become spaced at equal distance apart (so the theory goes). Those locations that are more central would attract the more specialised functions as well as the basic convenience goods for the neighbouring populations. These central places then have the greatest number, choice and range of goods and services, and thus

Introduction to R etail and C onsumer Modelling

33

service the whole region. There is a geometrical relationship that develops from the above scenario. The locations of these larger regional centres may be determined in relation to the smaller centres when the trade area is described by hexagons (Figure 2.5). These hexagons are the most spatially efficient way of servicing a region. A circle provides overlap and over-services a trade area, whilst a square trade area leaves certain parts of the region not serviced. Once the location of the regional centre is determined, the next largest centres will occur, in theory, at the midpoints between three of these or at the apexes of their hexagonal trade areas.

Nicholls

Fraser Dunlop Charnwood

Palmerston Mulanggari

Spence

Franklin

Flynn

Crace Kenny

McKellar Lawson

Florey

Higgins

Giralang

Mit ch ell

Latham

Holt

Melba

Ev at t

MacGregor

Watson

Kaleen

Scullin Page Belconnen

Lyneham Downer

Bruce O'Connor

5 km

0

Turner

LEGEND

Hackett

Dickson Ainslie Braddon

Acton

CBD (Queanbeyan)

City

Reid

CBD (Civic) Campbell Pa rk es

Town Centre - existing Town Centre - proposed Group Centre - existing Group Centre - proposed Neighbourhood Centre - existing Curtin Neighbourhood Centre - proposed

Russell

Yarralumla Capital Barton Hill Deakin

Forrest

Kingston Griffith

Fyshwyck

Figure 2.5 Loschian Modifications (after Davies, 1976) in Christaller’s Hexagonal Trade Areas and the Northwest Retail Hierarchy for Canberra, Australian Capital Territory For the progression of trade areas, the nesting principle is that the single largest trade area, incorporates three at the next level, then 9, 27, 81 etc. In a hexagon, it may be seen that the largest trade area encloses one plus one third of the six adjoining lower level trade areas, which, on summation, is the equivalent of the trade area of three of the next level trade areas. Such a system of the stratification of urban centres in a hierarchy, based on the number of centres and surrounding trade areas, is a K=3 proportionality system defined by Christaller. He made subsequent modifications to this nesting principle by considering the role of transportation and the impact of administrative boundary lines through K = 4 and K = 7 systems (Davies, 1976). Such relationships have promoted further research (for example, Walmsley and Weinard, 1989; Dennis et al., 2003) and there is still interest in investigating central place theory (K=3 systems) with spatial interaction modelling (for example, Openshaw and Veneris, 2003) and global city-city networks (Derudder and Witlox, 2003).

34

C hapter 2

The model of a nested hierarchy of retail centres was applied to Canberra and the Australian Capital Territory (ACT) in its planning scheme of the first decade of the 20th Century. The planned hierarchy of shopping centres consisted of Civic as the CBD; Town Centres; Group Centres; and Local Centres and this structure has been incorporated into the Territory Plan. This reflected the work of Berry and Garrison (1958) who proposed an extension of an urban hierarchy to include an intraurban hierarchy along the lines of the corner store, neighbourhood shopping centres, community shopping centres, regional shopping centres and the central business district (CBD). They observed hierarchies developing in areas of uneven population and unequal purchasing power. They also introduced two important important concepts, namely: 1. Threshold: This is the minimum level of demand necessary to support a business activity or a condition for market entry; and 2. Range of Good: This defines the maximum distance consumers are prepared to travel to a centre or firm. According to Berry and Garrison (1958), the interaction of these two concepts allows for the hierarchical differentiation between centres but not a formal arrangement of hexagonal trade areas. The reason is that some centres or firms may earn excess profits, whilst others can have diminished profitability and the economic uniformity assumption is no longer valid. The planned retail hierarchy in the Territory Plan for Canberra was defined accordingly, as follows: 1. Civic (CBD): This is the centre for commercial, entertainment and tourist facilities as well as a range of major community facilities. 2. Town Centres: This is main focus for the district population for shopping, community facilities, entertainment and recreation. Belconnen, Woden and Tuggeranong are defined for this role in the hierarchy. 3. Group Centres: These are designed to provide opportunities for major weekly shopping and other retail and personal services, primarily for people in adjacent suburbs. Examples include Dickson, Curtin and Kippax. 4. Local Centres: These lower order centres underpin the hierarchy and are designed to provide a focus for convenience shopping and for community and business services to meet the daily needs of neighbourhoods. An example of the north and northwest part of this hierarchy is seen in Figure 2.5 showing Civic, the Belconnen Town Centre (with a Westfield regional planned shopping centre) and surrounding Group and Local Centres. The hierarchy was not perfect, with many lower order centres established in inner north and south Canberra,

Introduction to R etail and C onsumer Modelling

35

independent of the planned hierarchy and the establishment of a major retailing centre at the industrial suburb of Fyshwick and the semi-rural area of Pialligo. However, it aimed to follow the Christallian idea that each level of a hierarchy provides all functions found in lower levels of the hierarchy, plus additional functions characteristic of that particular level of the hierarchy. The Local Centres provided the surrounding neighbourhood with a small independent supermarket, post office, food, convenience and personal service in the 8 to 15 shops in the precincts. However, a problem emerged for planning authorities in the 1990s, after the ‘de facto’ deregulation of shopping hours, where many of the Local Centres experienced a decline in turnover. Shopping trip destinations had seemed to have fundamentally changed. The Retail Policy for Canberra (1996) states: “Social changes such as increasing numbers of two income families and increased personal mobility have changed shopping patterns with more shopping done at the large centre.”

The trade areas were being distorted by changing socio-economic status and the deregulation of hours had apparently focused purchasing power on Town Centres rather than lower order centres in the hierarchy. Such possible distortions were recognised in Central Place Theory by Losch (1954), who made further modifications to this model by reviewing the possibilities of alternative trade areas and seeking to find the average location of all conceivable trade area arrangements of firms possessing the good or service. He found, among other things, that the density of centres are variable across space, where there can be considerable concentrations not found in the Christaller model, in discrete sectors of the trade areas of the higher order centre (Figure 2.5). Further, the hexagon trade areas could be distorted through variations in purchasing power, reflecting the impact of socioeconomic status on market area analysis. This was the case in Canberra, where shopping hour liberalisation had further distorted the hierarchy to create a ‘vacant shop problem’. There had been a decline in the viability of these small centres by 1995, where 13 out of 44 Local Centres had over 10% vacant shops, three years after the ‘de facto’ deregulation of hours (ACT Planning Authority, 1995). For example, the results of this structural change were assessed by fieldwork (Vallard, 1998) at three Local Centres (Aranda, Scullin and Page) that had listed vacancy rates of 14%, 3% and 15%, respectively (ACT Planning Authority, 1995; Figure 2.5). The results are presented in Table 2.1. In the three years since shopping hour liberalisation, two of the three independent supermarkets had closed. All three butchers had shut. There had been a decrease in the retail amenity with an invasion of non-retail functions (Accountant, Art Dealer, Architect) and second–hand goods retailers. This re-structuring was the spatial implication of changes in consumer preferences, apparent in 1996-97 household surveys (Table 2.2). These surveys suggest that there has been a loss of patronage to higher order centres. This has had its ramifications in the restructuring of the Local Centres, an on-going vacant shop problem and a high turnover of small business.

36

C hapter 2

Aranda (ACT) Survey 5/5/98: 14% vacancy in 1995 Present Activity Past Activity Second Hand Computer Shop

Estate Agency

Hairdresser

Art Dealer

Butcher

Restaurant Vacant

Restaurant Supermarket

Scullin (ACT) Survey 12/5/98: 3% vacancy in 1995 Present Activity Past Activity Vacant Printer Manicurist Manicurist Hairdresser Hairdresser Accountant Pizza Shop Second Hand

Architect/Post Office

Chemist Printer

Chemist Butcher

Cafe Baker Supermarket

Newsagency/Post Office Agency Baker Supermarket

Page (ACT) Survey 12/5/98: 15% vacancy in 1995 Present Activity Past Activity

Comments Computer shop for 6 years and originally a post office for 15 years (till 1991) Hairdresser left 2 years ago (1996) after staying for 15 years Left centre 3 years ago (1995) after long tenancy No change Closed 2 years ago

Comments Relocated in complex Present for 6 years No change Accountant present for 2 years Pizza shop for 8 years Architect 1991-97 and previously a post office No change After a tenancy of 18 years, butcher departed in 1997 Café for 2 years No change No change

Comments

Vacant

Supermarket

Closed 1998

Tavern

Amusement Centre

No change

Restaurant

Restaurant

No change

Pizza Shop

Butcher

Butcher closed in 1992

Vacant

Real Estate Agency

Vacant two years

Leather Goods

Post Office

Post office closed 1995-96

Hairdresser

Hairdresser

No change

Cafe

Architect

Café opened Architect 1993

in

1997

and

Table 2.1 Survey of Tenancy Changes in Local Centres, Canberra, 1998 (Source: Vallard, 1998) The Canberra Household Shopping Preference Survey (1996-97) was instigated by the Australian Capital Territory (ACT) Government to evaluate a policy move to restrict the trading hours at regional planned shopping centres (defined as ‘Town

Introduction to R etail and C onsumer Modelling

37

Centres’) over lower order district (‘Group Centres’) and neighbourhood centres (‘Local Centres’). The Government’s aim was to re-invigorate Local Centres, by restricting large centre night shopping at Town Centres using retail trading hours (7am-7pm Monday to Thursday; 7am-10pm Friday; 7am-7pm Saturday; and 7am5pm Sunday). The aggregate trading period over seven days (85 hours) was still over double that in 1992 (with no Sunday trading). The policy was repealed within 12 months, in 1997, after a concerted effort by the major supermarkets and retail property corporations. The statistics of the household surveys in Table 2.2 show, in particular categories, the decline in local shopping and overall patronage, despite the restrictions introduced by the ACT Government. For example, local trips to the ‘Baker’ declined from 21.5% to 15.7% from 1996 to 1997 and also declined in total from 27.3% to 22.1%, whereas supermarket shopping remained approximately the same over the period (Table 2.2). Similar declines are recorded in the categories ‘Butcher’ and ‘Newsagent’. The explanation is that increasing numbers of households are no longer shopping locally at the neighbourhood ‘Newsagent’ and ‘Baker’ over the 12 months, but rather travelling to the higher order Group Centres that have major supermarket chains and undertaking multi-purpose shopping on their floorspace. This is a confirmation of the model prediction described in Figure 2.4 and Equation (2.4). Category Baker Bank Butcher Bank Hairdresser Newsagent Petrol Station Post Office Supermarket Takeaway Video Other

Total 1996

1997

Town 1996

1997

Group 1996

1997

Local 1996

1997

27.3 11.6 12.5 28.5 8.3 38.7 17.8 20.8 74.5 18.8 16.3 24.3

22.1 10.1 7.5 28.7 6.9 32.7 16.6 14.3 73.8 18.5 16.2 17.9

31.8 58.6 23.6 49.9 16.7 37.1 38.7 44.5 96.6 33.2 21.4 96.8

16.4 35.1 6.8 25.1 2.3 34.3 18.2 26.0 80.7 33.1 9.7 43.8

42.9 28.6 19.7 51.3 12.2 68.4 35.1 35.1 94.8 31.7 30.7 55.1

39.4 31.2 15.8 53.1 11.7 62.2 28.4 32.0 94.5 25.1 31.0 38.0

21.5 2.4 9.1 19.1 6.3 28.5 10.4 14.1 65.8 13.3 10.9 8.4

15.7 0.6 4.3 19.3 5.2 20.4 11.9 6.8 65.4 15.1 10.8 8.7

Table 2.2 Shopping Centres used in the Canberra Household Shopping Preference Survey 1996, 1997 (Australian Bureau of Statistics) Central Place Theory is underpinned by the minimum levels of consumer demand necessary to sustain retail functions and the greatest distance these consumers would be prepared to travel. Under trading hour deregulation, do these assumptions lead to supermarkets with the longest trading hours being located within the highest order centre? In other words, is there a central time hierarchy and does this match the central place hierarchy? Within Canberra, Coles Supermarkets trade 6.00am to midnight Monday to Sunday, except at Manuka (2km southeast of Capital Hill: Figure 2.5) which trades 24-hours. Manuka, in terms of time accessibility for

C hapter 2

38

shopping, is the highest order centre, yet spatially, it is the equivalent of a Group Centre. For Sydney, 24-hour Coles supermarkets (Woolworths does not trade 24hours) were plotted in February 1996 for Sydney (Figure 2.6). Of Coles 49 supermarkets, 13 traded 24-hours six-days-a-week (except Sundays) and, once again, the time hierarchy was quite different to the central place hierarchy. Most of these supermarkets were located on major arterial roads out of Sydney and not one was located within a higher order regional centre. Walmsley and Weinard (1990) studied a subset of this central place hierarchy in south central Sydney. They calculated the ratio of central places nested according to a 2:15:42 scheme (for regional, community and neighbourhood centres). The 24hour Coles supermarkets were located, not in the higher order centres within this hierarchy, but in what they classified as neighbourhood centres, namely, Ramsgate (on General Holmes Drive, a major arterial road for Sydney south) and Kareela (on the Princes Highway, the other major southern route). These supermarkets aimed to capture late night time-poor professional and shift workers on their way to and from work. The demand threshold comes from passing trade rather than home-based trips. However, this relationship appears to have weakened substantially, because in February 2003, only 2 of the 13 supermarkets were still trading 24 hours (Bondi Junction and Lindfield) and both Ramsgate and Kareela now closed at 12 o’clock midnight. R ich m on d W es t

W arriew o od

W in ds or

A sq uith H o rn sb y Pa

c if

T urra m urra

ic

Dee Why

C a stle H ill H

W es t P en n an t H ills

M a nly

C h atsw o od G ra ce B ro s.

L an e C o ve W es t R yd e

M t D ruitt W es te rn

L in dfield

C h atsw o od

N o rth R ock s B la ck tow n

S t M ary s G re at

M a nly V a le y

E pp ing

K in g s La ng ley D o on side P en rith

w

N e utral B ay

P arra m a tta

H igh w ay

M e rrylan ds G u ildfo rd

K in g s C ros s A ub u rn

B on d i Ju nc tio n

B urw o od A sh fie ld

F airfie ld 1 0 km

Highway

R o sela nd s

H u rs tville

M a rou bra Ju nc tio n

R a m sg ate

C a sula m Hu

P ag e w oo d

E arlw oo d

B an ks tow n

L iv erp oo l

2 4 -h o u r C o le s S u p e rm a rk e t

R a nd w ic k

y

5

Hw

0

R e dfern

e

M a cq ua rie Fie ld s K are ela

S ylva nia M iran da

M into

Pr

in

ce

s

O th e r C o le s S u p e rm a rke t

R e g io n a l C e n tre E ng a dine

Figure 2.6 The Location of 24-hour Coles Supermarkets in Sydney 1996 The time hierarchy is therefore very different to a central place hierarchy and appears to be influenced by the transport network rather than a hexagonal

Introduction to R etail and C onsumer Modelling

39

arrangement of centres serving local areas. The centres with supermarkets operating 24-hours in 1996 served a wide area of Sydney and it may be argued that time has, in fact, been substituted for floorspace in terms of market penetration. For example, a 2000 sq m supermarket trading 24-hours may have the equivalent trade area of a 5000 sq m supermarket trading 12 hours per day. Time and space may be substitutable, but not equivalent in terms of space and time accessibility. Central Place Theory looks at the efficient allocation of populations to the centres of distinct sizes. It has been used by planning authorities, such as, in the Australian Capital Territory, as a model for the assignment, spatially, of retail functions. What has happened since shopping hour liberalisation is that smaller order centres have declined in the range and type of retail functions offered, unless they are located on major arterial roads. There, supermarkets can dominate the time landscape by capturing threshold population from passing motorists. This means that new developments can locate away from town centres and, by trading seven days-aweek, draw trade away from the traders in the main street of the town. A central place hierarchy is not equivalent to the time hierarchy. The space-time substitution is not linear but curved by time lines. This is shown by the time cone in the spacetime convergence (Figure 1.5). Likewise, the equivalence of the gravity coefficient β to a squared term of trip frequency (a mathematical statement in Equation 2.3 of this convergence) also suggests a time curvature from trip activity per unit time. Therefore, this non-equivalence between space and time hierarchies and high household mobility are major reasons for the significant commercial development opportunities located away from town centres. 2.5.3

MODELLING THE UTILITY OF SHOPPING HOURS AND PLACES

How do we model the level of satisfaction consumers have in their ‘when’ and ‘where’ shopping decisions? There are a number of ways this can be done using the idea of utility as a definition of the level of shopping satisfaction for individual and aggregate consumers. We will look briefly at the neo-classical ideas of utility as indifference curves in a trade-off between the number of visits and time spent shopping. The ideas will then be interpreted empirically. A utility model for centre scale will then be developed that could account for the polarisation of shopping centres towards community or regional malls in Australia in the 1990s. The extension of trading hours can be modelled by analogy to the neo-classical consumer choice model. Consumer preferences are described by a set of indifference curves of constant satisfaction (or utility) constrained by the consumer’s budget. It is assumed that a consumer can decide on a trade-off between the number of visits to a centre and the time spent shopping. A graphical representation of this situation is presented in Figure 2.7. The consumer equilibrium is at a point, where they spend (to) hours per week visiting a centre on (Vo) occasions within a budget constraint line (Y ) for the shopping week. What is spent over the shopping week ( price multiplied by

40

C hapter 2

time, p*t) and what is spent per visit (price multiplied by the number of visits, p*v) cannot exceed this budget line per week. The slope of the budget line is negative and defined by the ratio of expenditure over the shopping week relative to the visits per week. All points on the indifference curve (U3 ) cannot be attained, whilst on (U1) they are not chosen because the consumer can afford a position that is preferable to these. The equilibrium point is called a ‘point of tangency’, meaning that the slope of the indifference curve equals the slope of the budget line at that point. The slope of the indifference curve is termed the marginal rate of substitution which describes the rate at which a person is willing to trade shopping with other activities. The slope of the budget line describes the rate that a person is able to trade with other activities at current market prices. It is interesting that the intercept of the graph defines the trading period per week, where it is assumed that all of the person’s budget (Y ) is spent over that week. When prices vary, new equilibrium are established in terms of the number of visits to the shopping centre. When there is a price rise, the budget line moves from (YV ) to (YVK ) (see Figure 2.8, top). The intercept (V ) indicates the largest amount of shopping visits that the consumer could afford if all their income was spent on visiting the centre. When prices rise, the number of visits that the person could make diminishes. Conversely, when there is a price fall, the person can now afford to make (YVL ) visits and the slope of the budget line decreases, allowing for more shopping trips. The consumer’s demand curve can be plotted for the influence on prices with visits (see Figure 2.8, bottom) and as prices rise,

T im e(T ) Trading W eek

S hopping Tim e

T1 Y 1 T0 Y 0 t1 t0 U 3 Indifference U2 C urves U1 V0 V1

V N u m b e r o f V is its to C en tre

Figure 2.7 Consumer Equilibrium Analysis for Shopping Time and Visits to a Shopping Centre

41

Introduction to R etail and C onsumer Modelling

Time(T)

Y

P

O

VK

Q

V P ri c e R is e

VL

V

P r ic e F a ll

Num ber of visits PV

Price

PV PV PV

A 1

B

0

C 2

Dem and curve V1

V0

V2

Visits

Figure 2.8 Impact of Price Shifts (top) and Demand Curve Formation (bottom) consumers will make less visits. The responsiveness of the consumer’s demand to price is measured by the demand elasticity, which is the ratio of the percentage change in visits to the percentage change in prices. The higher the elasticity, the more responsive the consumer is to price changes. When the trading hours in Figure 2.9 are extended from (T0 ) to (T1 ), the assumption is that more of the individual’s total income can be allocated to the shopping budget and there is a complementary shift in the budget line. The scenario is what every large retailer publicly promotes, that the extension of hours would invite more of the discretionary income to be spent shopping, allowing the consumer to attain a higher level of satisfaction shopping (U2 to U3 ). The consumer would spend more time shopping and undertake more shopping visits to that centre. A qualification becomes apparent when we construct the consumer’s response to changes in income. These are shown by the parallel shift in the budget lines in

42

C hapter 2

Figure 2.9. The income curve connecting all the equilibria in the classical theory is termed ‘trading hour consumption’ (Figure 2.9). Notice that the positive slope of this curve infers two possibilities, namely, that an extension of trading hours allows higher income households to make more visits to the centre and/or, more money can be allocated from discretionary expenditure from all households for visiting the planned shopping centre.

Extension of Trading Hours

T im e(T ) T2

B udge t S hift (Y 0

T1 T0 P

Q

Y 1Y 2)

T ra ding H our C ons um ption C urv e

0

V 0V 1V 2

A

B C

V

N u m b er of V isits to a C entre Figure 2.9 The Trading Hour Consumption Curve for Shopping Centre Time-Visit Allocations The (V0t0, V1t1,..) points in Figure 2.9 can be plotted for income versus number of visits and the resulting curve is called an Engel curve (or in this case, the trading hour consumption curve). If this curve is upwards-sloping, it indicates the response of a normal good, in that rising incomes would be translated into more visits to the shopping centre. Conversely, if the Engel curve is downward-sloping, it means as income rises there are less visits to the centre. The responsiveness of ‘demand changes’ to ‘changes in income’ is measured by the income elasticity. A higher positive value means that the consumer is sensitive to income changes relative to their centre visitation. If an extension of trading hours follows the income effect outlined above, there would be a corresponding trading hour consumption curve (see Figure 2.9) and a trading hour elasticity defined by the percentage change in visitations per week divided by the percentage change in trading hours:

e = (Δv/v)( ΔT/T)

or

43

Introduction to R etail and C onsumer Modelling

e = (Δv/ΔT)(T/v)

(2.5)

A high positive trading hour elasticity means that the consumer would be very responsive to more trips to a shopping centre with an extension of trading hours. The corollary is that large regional shopping centres would have a higher trading hour elasticity than lower order centres, since they would be more likely to attract high income consumers. This means that the longer hours allow more money to be spent with more trips to the centre. Furthermore, this income curve is also a trading hour consumption curve as affluent consumers demand more time to shop. This high income-extended hour hypothesis was tested in the 1995 Armidale Retail Study when 209 late night shoppers (7.00 pm to 9.00 pm) at Coles and Woolworths, were surveyed over a week (Figure 2.10). The results show that it is the more mobile and affluent sub-populations who take advantage of these shopping times. Armidale is a university town which accounts for the high percentage of students within the samples. ‘Double income no kids’, ‘double income with kids’ (DINKS and DIWKS) and ‘single income professional’ households are the major beneficiaries of this policy (Baker, 1995a). The extended hours appear to be less important for other single income and retired households. It essentially provides shopping convenience for the affluent, where the fundamental construct is at least one professional income. O th e rs R e tire d S e lf-E m W h ite C o lla r B lu e C o lla r P ro fe s s io n a l D IW K S D IN K S S tu d e n ts 0

5

10

15 20 P e rce n ta g e

25

30

35

Figure 2.10 Percentage Composition of Socio-economic Status of Late Night Shoppers (Armidale, NSW, November, 1995) In the South Australian Inquiry (1994), Sexton Marketing presented a socioeconomic breakdown of the 20% of respondents who wanted extended trading hours (see Figure 2.11). Both ‘professional’ and ‘para-professional’ had the highest ranking, followed by ‘aged 15-19’, then ‘plant operators’. These categories are very

44

C hapter 2 N e t P r e f e r e n c e B y

f o r

E x te n d e d

H o u rs

S e g m e n t

M a le s F e m a le s 1 5 -1 9 2 0 -2 4 2 5 -3 4 3 5 -4 4 4 5 -5 4 5 5 -6 4 O v e r 6 4 M a n a g e rs P r o f e s s io n a ls P a ra - p ro fe s s io n a ls T r a d e s p e rs o n s C le r ic a l S a le s P la n t O p e r a t o r s L a b o u r e r s P e n s io n e r s W e lf a r e R e c ip ie n t s N o t s ta te d F a r m O w n e r s M a rr ie d S in g le - s te a d y S in g le w it h f r ie n d s S in g le li v e a lo n e L iv e w it h f a m il y R e t a il o w n e r R e ta il e m p lo y e e F a m i ly o w n s r e t a i l F a m ily w o r k s r e ta il M e t r o p o li t a n S .A . C o u n try

-1 0

0

1 0

2 0

3 0

4 0

P e rc e n ta g e

Figure 2.11 Shopping Preferences for Extended Hours by Socio-economic Groups (Source: South Australian Trading Hour Inquiry, 1994) similar to the surveyed socio-economic groups who were using late night trading in the Armidale Retail Study (Figure 2.10). In South Australia, those that were against extended hours, as a net aggregate, were ‘pensioners’, ‘over 64 years’, ‘retail employees’ and ‘family-owned retailers’. Therefore, it appears reasonable that the (To) to (T1) extension of hours has attracted the higher income households to undertake late night shopping. This is the expectation of the neo-classical model framed above. The idea of a parallel income and trading hour consumption curve seems a reasonable theoretical advance. This micro-economic model applies to individuals and underpins much of the economic rationalists’ arguments for the deregulation of shopping hours. Is there any empirical justification for such a model for a community of consumers, because retail policy should be in the interests of the community rather than individuals? For this question to be answered, a measure of the level of shopping satisfaction (or utility) needs to be constructed. How can this be undertaken, since an individual’s internal representation of the world may differ markedly from objective reality?

Introduction to R etail and C onsumer Modelling

45

Rather than cognitive distance, it was the notion of attractiveness of shopping places that could underlie a measure of the utility of shopping at a mall. Downs (1970) was the pioneer in this approach and he used a bipolar rating scale (semantic differentials) and factor analysis to identify what is fundamental to centre image. These are listed below in descending order of importance: •

service quality

•

price

•

structure and design

•

shopping hours

•

internal pedestrian movement

•

shop range and quality

•

visual appearance

•

traffic conditions

A review of ten studies of important attributes considered in previous studies of consumer behaviour are summarised in Table 2.3. I have aggregated these into three groups in descending order of importance. • Group 1 Distance from Home, Price, Product Quality, Product Variety • Group 2 Service, Parking, Convenience, Design, Store Variety • Group 3 Trading Hours, Product Reliability, Cleanliness, Product Freshness, Accessibility, Number of Shops, Atmosphere, Reputation, Promotion

46

C hapter 2 Dist

Downs (1970) Burnett (1973) Hudson (1974) Pacione (1976) Spencer (1978) Patricios (1978) Schuler (1979) Blommesstein (1980) Hudson (1981) Timmermans (1982) Thang & Tan (2003)

Price

Service

Quality

♥

♥

♥

♥ ♥

♥

♥

♥

♥

♥

♥

♥

♥

♥

♥

♥ ♥

♥

♥

♥

♥

♥ ♥

♥

Acc ess

♥ ♥

♥

Design

♥ ♥

♥

Parking

♥

♥

♥ ♥

Variety

♥ ♥

♥

♥

♥

Table 2.3 A Review of the Important Attributes Considered in Studies of Consumer Behaviour Shopping centre perception is, therefore, a complex weighted interplay of an individual’s subjective assessment of the shopping environment. Baker (1991) constructed a measure of this perception by selecting the five attributes of accessibility, variety, price discounts, parking and design and asked respondents, sampled on the floorspace of the mall, to rate these attributes in a Likert scale of ‘very poor’, ‘poor’, ‘ok’, ‘good’ and ‘very good’. As a control, the respondent was then asked to rank their perception of the shopping centre overall. Each attribute then received an average rating out of five, which can be aggregated to a measure out of 25. This measure has been used as a proxy for the level of satisfaction of shopping in a particular centre (Table 2.4). Parking

Design

Price Discounts 3

Variety

Accessibility

Centre 4 3 3 5 Level of Satisfaction Total: 18/ 25 Rating scale is from 1 = Very Poor; 2= Poor; 3 = OK; 4 = Good to 5 = Very Good

Overall 4

Table 2.4 An Individual’s Assessment of their Level of Shopping Satisfaction (Utility)

Introduction to R etail and C onsumer Modelling

47

The socio-economic underpinnings of the neo-classical utility model could be tested for aggregate responses (necessary for public policy formation) using three hypotheses outlined below. The testing was undertaken using samples in the Sydney Project, Australia, 1988/89 and 1996/98. The results have to be put into context, since they use averages, rather than absolute measurement for aggregate versus individual consumption, and that samples were taken at different malls within the Sydney retail hierarchy. Nevertheless, the outcomes were surprising. Hypothesis 1: HDI consumers have higher shopping centre utility than LDI High and low disposable income (HDI and LDI) consumers were determined from an index of occupation (based on partner’s work, the rank of their job, divided by the number of dependent children under 15 years). On average the HDI consumers (greater than or equal to 4) had a utility level of shopping within the malls of 18.0, whilst LDI consumers (less than 4) also rated the centres slightly more at 18.1 out of a possible 25. For the top and bottom quartiles, this spreads to 18.1 for HDI shoppers, whilst for the bottom quartile the result was 18.6. For the 1996/98 samples, there had been a net loss in average utility for both groups (HDI = 16.9 and LDI = 17.0). The neo-classical model suggests that HDI consumers would have a higher level of shopping satisfaction compared to LDI consumers. This appears not to be the case. Hypothesis 2: HDI consumers have higher number of shopping centre visitations than LDI consumers A corollary of the model is that HDI consumers should visit planned shopping centres more often that LDI consumers and that an aggregate reduction in utility over the decade should result in less trips and less time shopping there. The results are the opposite to this expectation in the Sydney Project. In 1988/89, the mean centre visitation rate was 1.45 visits per week for LDI shoppers but only 1.20 for HDI shoppers. This relative difference is replicated in 1996/98 where LDI shoppers visited, on average, 1.86 times and HDI shoppers undertook 1.55 trips. Less advantaged households shop more often than busier, more advantaged households. Further, rather than a decrease in trip frequency, with lower utility levels over the decade predicted by the neo-classical model, there has been a rise in trips to centres for both groupings. There appears to be a more dominant explanatory variable that is overriding group perception of planned shopping centres. Whilst there has been a rise in trip frequency, there has not been an increase in shopping utility. Hypothesis 3: HDI consumers spend more time shopping than LDI consumers In 1988/89, the length of shopping time for LDI consumers was 102.5 minutes compared to 83.8 minutes for HDI. This had remained relatively the same for LDI respondents over the decade (100.0 minutes in 1996/98 samples), but for HDI respondents, the average time had contracted to 75.5 minutes.

48

C hapter 2

Low income households therefore visit planned shopping centres more often and spend more time shopping per trip than higher income households. They could have, in the lower quartile range, a higher level of shopping satisfaction. This is the expectation of the neo-classical model. However, over the decade, with the extension of shopping hours at the sampled centres, the prediction of higher levels of satisfaction across the socio-spectrum is not supported. Both groupings had a decline in perceived shopping utility at their particular centres. This did not transfer into lower trip frequencies, but rather the reverse happened, in that the number of trips increased over the week. However, the length of the shopping trip within the centre decreased slightly for LDI and more noticeably for HDI respondents (as anticipated by the model). Therefore, there are some cases when the neo-classical model appears relevant, but not in the expected way, because it is the lower socio-economic consumers that have the higher level of utility, shopping frequency and length of visit. What does all this mean? The neo-classical model suggests that if the trading hours in Figure 2.7 are extended from To to T1, the assumption is that more of the individual’s total income can be allocated to the shopping budget (and hence there is a shift in the budget line outwards). This is what is promoted by the pro-trading hour deregulation lobby, whereby an extension of retail trading hours allows more discretionary income to be spent on shopping. Consumers can therefore attain a higher level of utility. Did this happen from the deregulation of shopping hours in the Sydney Project? HDI consumers spent 100.4 hours per week shopping in 1988/89 (f × p) and 117.0 minutes in 1996/98. They would be expected to allocate a higher proportion of discretionary income to shopping at these centres. Further, the model predicts an increase in shopping utility. The empirical evidence suggests that this is not the case. Likewise, LDI consumers spent 148.6 minutes shopping in 1988/89 and this increased to 184.1 minutes in 1996/98. This increase is also not reflected in a rise of shopping utility. There could be another reason for the inconsistency in the results, since the sample of centres in the Sydney Project differ in their floorspace from community to regional PSCs. If the percentage of HDI consumers in the 1988/89 samples are plotted against the corresponding trip frequencies for this sub-population, there is a clear non-linear relationship (R-squared equal to 0.53) very much dependent on the scale of the centre (Figure 2.12). The suggestion from this data is that the income effect at large PSCs is ‘normal’ for high income consumers (higher income patronage results in more trips) whereas for smaller centres the relationship is ‘inferior’. What could be a reason is that the model is only appropriate to affluent ‘time-poor’ consumers, who increase their spending with the shift in the budget constraint with extended hours. However, these consumers are still a minority, particularly in smaller centres and so, when the data is aggregated for the total population, this relationship is lost in the averages.

Introduction to R etail and C onsumer Modelling

49

A . E m p iric a l BSA WBAP 14

BSM

Large , Centre

,

12

AMA

, Small Centre

WBMP

10

,

HIGH DISPOSABLE INCOME

16

WBMCP 8

MLM

WCM AMM

6 .7

.8

.9

1 .3 1 .1 1 .2 F R E Q (H D I)

1

1 .4

1 .5

1 .6

1 .7

B . G ra p h ic a l

, Small Centre Behaviour

, Large Centre Behaviour Inferior

,

Normal

,

In c o m e (I)

R e g io n a l

Y2

Q

Y1 Y0

PI

P

OI

O S u b -re g io n a l

VO

V1

V2

V 1I

V OI

V N u m b e r o f V is its to C e n tre

Figure 2.12 Analogous Engel Curves Relative to the Size of the Centre (a) Percentage of HDI Respondents in the Sample Plotted with HDI Trip Frequency and (b) a Model showing Normal and Inferior Engel Lines with Centre Scale These ideas, however, do not explain why trip frequency has increased over the decade, whilst shopping centre utility has declined across the socio-economic spectrum. Therefore, the underpinnings of the micro-economic model, when aggregated, are not obvious in the testing of the ideas against aggregate data. This might be a problem of the assumptions or proxies used to test the relationships.

50

C hapter 2

Remember, we are using averages (although from large samples of over 1000). Therefore, the model may be ideologically relevant to a section of the population, but not justified for the community as a whole. This does not mean that the idea of utility in a shopping model is not useful. For example, in Sydney, the aggregate level of utility was aggregated for each centre (nine samples at five malls surveyed in 1988/89), averaged and then assigned to the type of centre (for example, community sub-regional, regional malls). The centre size was defined by the number of shops. What was surprising is that the level of utility increased with the mean trip distance, yet this is a function of the retail amenity of shopping at a regional mall, outweighing the length of the journey (Figure 2.13). The results are also suggestive that the slope of the regional centre distribution is steeper than the community and sub-regional centres. The smaller centres are less elastic or responsive to distance than the regional centre, presumably because of the higher number of lower order centres at these levels in the central place hierarchy. The regional centre also lies on a higher indifference line, implying that beyond a 3.5km threshold, a consumer would prefer to travel further to the regional centre than a lower order centre, if that alternative existed. The utility values are consistent with this inference, since the regional centre’s utility values ranged from 19.5 to 20.0, whilst the lower order centres averaged from 16.5 to 19.0 (where the average range is defined from very poor =5 to very good =25). The idea of size-distance trade-offs in the context of indifference lines seem to be relevant to shopping trip behaviour.

Number of Destinations

N R e g io n a l U tility = [1 9 .5 -2 0 .0 ]

S u b - re g io n a l C o m m u n ity

C

2 .1 k m

U tility = [1 6 .5 -1 9 .0 ]

3 .6 k m

d

M e a n D is ta n c e

Figure 2.13 The Utility of Shopping at a Hierarchy of Malls in Sydney with Trip Distance, 1988/89

Introduction to R etail and C onsumer Modelling

51

As part of the modelling process, these measures of relative utility from these nine samples were correlated with a range of variables in an exploratory data analysis (Step 3 in the modelling process, p.31). This is part of scientific induction where data is explored in the search of hypotheses. As a result of this process, Baker (2000) found that this measure of relative utility was found to be associated with the inter-locational trip frequency k (that is, trips to and from the mall) in the Sydney 1988/89 data set. Consequently, this relationship was formalised by a theoretical guess (Step 4) and was checked statistically (Step 6) by using 26 samples from the Sydney Project (1988/89 and 1996/98). There was therefore a possible space-time relationship between utility U and a combination of inter-locational mean trip frequency k and distance D relative to the number of destinations N (R-squared of 0.53; Figure 2.14), namely: U =

kD N

(2.6)

Figure 2.14 The Relative Utility Distribution with (kD)/N for Sydney Project 1988/89 and 1996/98

52

C hapter 2

Figure 2.15 The Regression of the Gravity Coefficient β and U2N2/MD2 for the Aggregate Sydney Project in 1988/89 and 1996/98 (excluding the regional Bankstown Square samples) It seems reasonable that if you are satisfied with a centre you will shop there more often and/or travel further to it. When Equation (2.6) is substituted into Equation (2.2), we deduce another form of the gravity coefficient, namely:

β =

U 2N 2 MD 2

(2.7)

What we have done is combine two statistically meaningful statements into a formal model in Equation (2.7). Equation (2.7) can be checked against data from the Sydney Project and it is interesting that there is only a significant relationship (R-squared equal to 0.67) when the large regional mall (Bankstown Square) is excluded from the regression, both for the 1988/89 (regulated) and 1996/98 (deregulated) periods (Figure 2.15). This form of the gravity coefficient only appears to be robust for trips to sub-regional and community planned shopping centres. The squared form of this utility function and its relationship to the gravity coefficient is of immediate interest, since this suggests the possibility of positive and negative values in the definition of ‘place utility’. Originally presented by Wolpert (1965), place utility was defined to be the positive and negative level of satisfaction that an individual considers to be obtainable from a given location. Baker (1982) interpreted place utility as a potential function within a differential equation, where

Introduction to R etail and C onsumer Modelling

53

positive and negative oscillations of utility represented paired comparisons for each alternative used for evaluation. The squared form of relative utility means that both satisfaction and dissatisfaction can equally contribute to the assignment of utility in the gravity model. Utility can also affect the propensity to undertake trips to a PSC. A decline in satisfaction will decrease the gravity coefficient β unless there is a trade-off with mean trip distance. This means that for lower order PSCs, a drop in β could not only occur from the deregulation of trading hours (from local consumers shopping elsewhere because of extended hours), but also because they do not like shopping at that particular centre for reasons other than trading hours. This means that perceptions can also change trip destinations and make our problem of disentangling behavioural from space-time structural effects more problematic. 2.5.4 MODELLING SHOPPING TRIP ACTIVITY Shopping may be just one activity within a wider travel behaviour context and as such, consumer behaviour cannot be treated as independent of travel activity. The problem of central place theory and spatial interaction modelling is the exclusive assumption that consumers travel to the nearest centre and undertake distance minimising behaviour. However, even Christaller realised the limitations of this scenario in his original work when, according to Pred (1967), there are two possible divergences from this optimal behaviour scenario, namely: 1. Consumers may attempt to minimise the total travel effort by combining shopping in a multi-purpose trip and so may purchase both low- and highorder goods at a more distant higher order regional centre than the closest lower order centre; and 2. Consumers may travel to a more distant centre if price savings exceed additional transport costs. Pred (1967) suggested that shoppers were likely to have incomplete knowledge of the retail distribution system and will undertake journeys that will not necessarily result in the optimisation of potential opportunities. Consumers are ‘satisfiers’ rather than ‘optimisers’ in a significant proportion of shopping trips. We therefore need to appreciate this distinction when considering sub-populations that frequent shopping centres. Another point from space-time modelling is that household mobility is rising and consumers’ shopping opportunities are increasing as activity patterns are becoming more dispersed. Increasingly, shoppers are drifting up the central place urban hierarchy with rising household mobility at the expense of lower order centres. In the previous section, the empirical evidence suggests that consumers perceive trips to regional malls differently to centres that are lower in the retail hierarchy. There appears to be a difference in shopping expectation, at least in the propensity

54

C hapter 2

for trip-making. This conclusion of distinct behaviour at regional malls in Sydney can be further demonstrated by returning to Equation (2.3), p. 32, in the RASTT model, namely:

β=

k2 M

This states that the slope of the distance decay away from a mall (β) is a function of the trip frequency k squared; divided by the transfer coefficient M. The gravity coefficient is therefore inherently a function of time through the frequency of centre visitation. This is why the traditional interpretation of β as just the spatial impedance is not appropriate. A high number of trips corresponds to large β coefficients (with the converse for a low number of trips) and this fusion of space-time constructs is a significant result. The gravity coefficient is more an indication of influence than impedance and defines aggregate consumer loyalty to their nearest centre. There is a fundamental relationship in the ‘when’ and ‘where’ consumers travel to do their shopping and as such, β must be reinterpreted in the context of Equation (2.3). What is also surprising is that despite the squared trip frequency term, Equation (2.3) is still a linear relationship and explains 53% of trips to the sampled malls. In the Sydney Project (1980-1989), the value of β is higher for the smaller community planned centres, decreases for sub-regional centres and then increases again for large planned regional centres (see Figure 2.16). Likewise, for mean trip frequency to the centres, they decreased from community malls to sub-regional malls, but increased for large regional malls (Figure 2.16). This quadratic relationship for β and k with centre size shows that gradients become positive for regional centres with ~ 146 retail destinations or 45,000 sq m of floorspace. This minimum point of inflection of change in slope occurs when M ~ 2.8. This type of distribution is replicated over a decade later in the 1996/98 samples (see Chapter 4). The impact of hierarchy and the scale of shopping centre makes the shopping trips non-linear and trip behaviour different within the mall hierarchy. This can be used to develop a model of two types of shopping trip behaviour, namely, ‘small centre’ and ‘large centre’ trip behaviour. This will be elucidated theoretically in Chapter 3. The transfer coefficient M within these distributions was calculated separately for the samples undertaken from the Sydney 1980/82 and 1988/89 Project at three shopping centres. When M is high, the β value is lower, meaning that trips to the mall are more widespread and increasing double car-ownership may allow households more access to alternative shopping destinations and less loyalty to one particular centre. You would therefore expect to find a greater proportion of two car households at regional centres. Households without car ownership would have a lower M value, and Equation (2.3) suggests that they would more probably shop at smaller shopping centres within the hierarchy with higher relative β values. So, with this proxy of

Introduction to R etail and C onsumer Modelling

5 55

1.2

Gravity Coefficient

1 .8

.6 .4 .2 Y = 2.005 - .021 * X + 7.143E-5 * X^2; R^2 = .57 0 50

75

100

125

150 175 Centre Destinations

200

225

250

2

Frequency

1.8

1.6

1.4

1.2 Y = 2.593 - .021 * X + 7.265E-5 * X^2; R^2 = .552

1 50

75

100

125

150 175 Centre Destinations

200

225

250

Figure 2.16 Quadratic Distributions of the Gravity Coefficient (top) and Mean Trip Frequency (bottom) with Centre Size (Sydney Project 1980/82 and 1988/89). The Point of Inflections are at N = 147 and 145 Centre Destinations, respectively, for Small (negative slope) and Large (positive slope) Centre Behaviour

56

C hapter 2

aggregate mobility, we have introduced our first socio-economic construct into the analysis. The mobility hypothesis suggests that, with rising aggregate mobility, there will be a reduction in the propensity to shop at neighbourhood centres (with higher β values) and a drift towards larger shopping centres (with lower β values). This hypothesis was reviewed in Baker (1994a) by sampling three shopping centres in the Sydney Project (MarketPlace Leichhardt, Westfield Burwood and Bankstown Square) in 1980/82 and returning there in 1988/89 for a comparative study. The sample sizes ranged from 110 to 253 surveys per hour and each centre was surveyed at least twice during these periods. The mean values for gravity coefficient β, trip frequency k and mobility M for these two periods (1980/82 and 1988/89) were β = 0.72 reducing to 0.59 at the end of the decade, k remaining constant at 1.35 trips per week and the transfer coefficient (mobility) rising from M equals 2.6 to 3.2, respectively. The change in the propensity for trips in the trade areas in the 1980s is exclusively due to the rise in mobility in households. The constant trip generation over the decade supports the hypothesis of the importance of retail hours on trip behaviour (since hours were regulated and constant over the period, trip frequency remained constant). Therefore Equation (2.3) appears empirically relevant for community and subregional PSC samples in the Sydney 1980/82 and 1988/89 data sets (Baker, 1994a and confirmed in samples in Sydney 1996/98; Auckland, New Zealand 2000; see Chapter 4). There is a quadratic relationship for β and k with centre size, where both gradients become positive for regional PSCs with over approximately 146 retail destinations or 45,000 sq m of floorspace. Why does this occur? Why is there an increase in attractiveness in regional shopping malls? A reason is that the increments of population are increasing to these centres, independent of distance, since choice can no longer be defined solely in terms of distance minimisation. At the regional centre, choice is internalised, where they offer a wider range of goods and services within one centre and where the price savings more than offset the economic and social cost of longer travel distances. It is more time efficient, if you are comparison shopping (for such things as clothing or electrical goods), to travel further to a regional centre. ‘One-stop’ shopping allows for the inspection of a number of like-shops in the decision-making process. For this reason, the population increments become larger towards a regional shopping centre at corresponding distances and this is reflected in the increase in the β coefficients in the data of Figure 2.16. Choice is no longer just a statement of distance minimisation at this level of the hierarchy and this is why the definition of utility changes, in Equation (2.7), to include both positive and negative values. This trip preference also has implications for the frequency relationship (through Equation 2.3), because consumers will more likely travel to higher order centres because of this attractiveness of choice. This has allowed a classification of shopping behaviour as

Introduction to R etail and C onsumer Modelling

57

distance minimisation strategies (or ‘small centre’) behaviour and choice maximisation strategies (or ‘large centre’) behaviour. This effect of clustering shopping opportunities together is termed the ‘agglomerative effect’ (Fotheringham, 1982; Fotheringham, 1985; Baker, 1994a). Such agglomerative effects destroy the linearity of the space-time correspondence in shopping trip behaviour and there are distinct types of behaviour dependent on the time of day and year as well as the size of the centre. The differences in the relationships are summarised in Table 2.5 (Baker 1994a). Equation (2.3) seems more appropriate to ‘small centre’ shopping behaviour where consumers periodically visit the centre, via the grocery trip, and the trip frequency distribution follows a normal distribution centred on once-a-week shopping. Conversely, large centre behaviour is characterised by more complex interaction reflected in a substantial underestimation from a normal distribution for once-a-week shopping and low frequency visits (from the Fourier transform of the spatial distribution). Therefore, it was concluded that aggregate shopping trips to a regional PSC does not follow a simple periodic shopping trip hypothesis (Baker, 1994a). Indeed, theoretical research in Chapter 3 suggests that it is underpinned by a completely different shopping strategy. These fundamental differences in the size of the shopping centre, suggest that the concept of a centre hierarchy is relevant and counters the claims by Beavon (1977, quoted in Warnes and Daniels, 1979) that hierarchies have been more readily recognised than have been justified by the data and that intra-metropolitan size distribution of central places is a continuum. However, in this analysis, lower order community centres and higher order regional shopping centres have distinctive and discrete characteristics. Distance minimisation is no longer the appropriate behavioural construct at a regional centre and, as Christaller (1966, quoted in Warnes and Daniels, 1979) recognises: “In practice, heterogeneous influences generate more complex landscapes than those produced by a single locating principle (p. 385).”

Shopping trip modelling is much more than just using distance minimisation assumptions to generate the gravity model. Equation (2.2) shows that retail change occurred before trading hour deregulation within Australia in the 1980s, possibly because of rising household mobility. This has important policy implications (Step 8 of the modelling process, p. 26). Further, the trip distributions are going to be affected by the size of the shopping destination and the RASTT model shows that two types of trip behaviour can be defined. This can therefore form a framework to study shopping change which is a fundamental aim of modelling in this study.

58

C hapter 2

Small Centre Behaviour *There are high R-squared values from the gravity hypothesis. *The trip frequency distribution follows the normal distribution of the Fourier hypothesis and therefore shopping behaviour is simple harmonic. *Patronage is concentrated in the local region and the centre exerts a strong spatial influence upon regular shopping behaviour. *It may be characterised by grocery trips and may be patronised by less mobile households (particularly in the morning). *There is no change in the gravity distributions from morning to afternoon samples. *It is defined by an aggregate transport coefficient M < 2.8. *The gradient of the gravity coefficient to centre scale is negative.

Large Centre Behaviour

*It occurs for shopping centres with the number of outlets N>146. *The gravity hypothesis is still relevant, but with lower R-squared values. *The trip frequency distribution does not follow the normal distribution of the Fourier hypothesis, where there is substantial underestimation of once-a-week and low frequency visitation and as such, shopping behaviour is not simple harmonic. *Patronage is dispersed over a wider area throughout the day and consumers are less ‘bonded’ to that particular centre. *The centre is patronised by more mobile consumers in the afternoon. *The gradient of the gravity coefficient to centre scale is positive. *It is defined by an aggregate transport coefficient of M > 2.8.

Table 2.5 Characteristics of ‘Small’ and ‘Large’ Centre Behaviour (Source: Baker 1994a) 2.5.5

MODELLING THE MULTI-PURPOSE TRIP

Traditional theory has assumed that consumers make single purpose trips (in particular, within spatial interaction modelling), but consumers combine shopping purposes to reduce the time and cost of travel. Multi-purpose shopping (MPS) trips are a subset of combined-purpose (CP) trips where, in the latter, shopping is joined with a whole range of other possible activities (shopping multiplied by work, recreation or socialising) whereas MPS is restricted to just shopping-shopping interactions. Why do consumers undertake multi-purpose trips? It could be argued that an extension of rational behaviour is multi-purpose shopping (MPS) where, by undertaking such a shopping event, consumers reduce the time and cost of travel. The examination of this process has involved the application of the many different theoretical themes to this specific problem and these are outlined in a source matrix of literature associations in Table 2.6 (Baker, 1996). The emphasis of research has been concentrated, firstly, in the probabilistic interpretation of the structure of shopping trips (for example, O’Kelly, 1981) or secondly, the analysis of the spatial econometrics of consumer behaviour (for example, Eaton and Lipsey, 1982).

Introduction to R etail and C onsumer Modelling

59

Baker (1996) developed a simple model based on the relationship between MPS and the inter-locational trip frequency (k), and discovered a significant relationship using exploratory data analysis of Sydney 1988/89 (R-squared equal to 0.47 for p= 0.041 (Figure 2.17). This relationship is replicated in the Sydney 1996/98 and Auckland 2000 data sets (Chapter 4). Whilst this seems to be a simple equation, it appears to be very important when explaining the nature of shopping trips in PSC trade areas. Hanson (1980) argues that the rationality of multi-purpose trips implies that a set of locations will be selected, minimising the total effort to obtain the desired set of goods and services. This principle of least effort is deeply embedded into central place theory (Thill and Thomas, 1987). Indeed, Kohsaka (1984) argues that multi-purpose shopping is a fundamental dimension of change, namely:

“By introducing multi-purpose shopping trips into the central place system, a metropolitan dominance will occur in which high-level centres become prosperous and lower-level ones decline.” (p. 251) This is supported by Ingene and Ghosh (1990) who argue that multi-purpose shopping lowers travel costs and shifts demand to agglomerated centres. Exclusive multi-purpose shopping will only occur in areas relatively far from retail outlets. Bacon (1984) identifies socio-economic reasons for this by defining sub-populations such as, ‘newly married couple with both partners working’ as being under a time constraint and so will be more likely to make infrequent multi-purpose trips to higher order centres. While most of the emphasis has been on the relationship between MPS and large centres, West (1993) uses MPS to differentiate stores in shopping hierarchies of planned and unplanned centres. ‘Small centres’ (< 19 outlets) cater for convenience shoppers of high frequency trips to m stores, such as supermarkets, whereas ‘large centres’ (>50 outlets) are characterised by MPS trips to m stores and comparison shopping at C stores. This suggests the hypothesis that there are at least two types of MPS occurring at planned suburban shopping centres (PSCs), namely: 1.

MPS constructed around infrequent shopping to higher order (regional) shopping centres; and

2. MPS formed around convenience, supermarket or comparison trips

60

C hapter 2

S.T. S. T.

R. U.

E.M.

C. P.

S. E.

I.

B .

C.P.

S.E.

I.

B .

Bacon (1984) Ingene & Ghosh (1990)

O’Kelly & Miller (1984); Baker (1996)

Arentze & Timmermans (2001) Oppewal & Holyoake (2004)

Horowitz (1980) Arentz et.al. (1992)

R. U.

E. M. M. C.

M.C.

O’Kelly (1983) O’Kelly (1981) West (1993)

Eaton & Lipsey (1982) Kohsaka (1984) West et al. (1985) Mulligan (1983) Thill (1985) Bacon (1995)

Mulligan (1987)

Ghosh & McLafferty (1984)

Hanson (1980)

Table 2.6 Source Matrix of Literature Associations for Multi-purpose Shopping. The diagonal of the matrix represents authors using only the one approach. The right-hand vacant off-diagonal elements represent areas not yet explored. The left hand elements are included for ease of use (Source: Baker, 1996). Source Papers: S.T. - Space-Time Modelling (Hagerstrand, 1952); C.P.- Central Place Theory (Christaller, 1966); S.E.- Spatial Econometrics (Curry, 1979); I.-Inventory Modelling (Bacon, 1971); B.- Behavioural Modelling (Pred, 1967); R.U. - Random Utility (McFadden, 1974); E.M.- Entropy-Maximising (Wilson, 1967); M.C.Markov Chains (Golledge, 1967)

Introduction to R etail and C onsumer Modelling

61

Figure 2.17 The Relationship between Trip Frequency and the Percentage of Multipurpose Shopping (Sydney Project 1988/89)

Bacon (1984) has raised two important questions in the application of spatial demand theory to a model defining the frequency and location of an individual consumer’s purchases for a given hierarchy of shopping centres, namely: 1. Under what circumstances would a consumer prefer multi-purpose trips to separate single-purpose trips? 2. Why should frequencies be different? Bacon argues that the frequency of purchasing may be a way of defining a more realistic consumer behaviour beyond the ‘single-purpose’ and ‘nearest centre’ hypotheses. This approach, while producing an explanation for the evolution of a shopping hierarchy, is based on the interplay of inventory costs versus the travel costs of shopping. This is very difficult to measure empirically. However, the themes of ‘choice over time’ and ‘choice over location’ are fundamental and are inferred by the RASTT model. The equation defines change and rates of change of ‘where’ and ‘when’ consumers shop and is solved by defining both spatial and temporal choice as equal to the trip frequency. The difference is that this space-time model is framed in terms of variables and constants that can be measured (trip distance, frequency, length of trading period per week, centre loyalty or bonding and household mobility). How then do we introduce multi-purpose shopping into the model?

62

C hapter 2

The variable of the number of multi-purpose shoppers per 100 respondents ϕ was introduced inductively from an exploratory data analysis with the above variables (Baker, 1997). The data from the Sydney Project (1988/89) suggests a simple but significant linear relationship between the standardised number of multi-purpose shoppers and the mean inter-locational trip frequency k, namely:

ϕ = hk

(2.8)

This means that the higher the number of trips, the greater the propensity to multipurpose shop. This was a surprising result, considering Hanson’s (1980) idea that rationally, consumers try to minimise the total effort in trip-making by multipurpose shopping. It would be expected that consumers multi-purpose shop to reduce the number of trips, but in this data set, it is not the case. The relationship was once again replicated in 1996/98 at the same centres in Sydney (and Auckland, New Zealand), so it appears to be underpinned by more that just a statistical anomaly. This apparent contradiction will be explored in Chapter 4. This multi-purpose trip frequency relationship is then substituted into Equation (2.3) to yield:

ϕ = ± g βM

(2.9)

The interest in this expression is that MPS has a positive and negative state in spatial interaction. Equation (2.9) can be rewritten, namely:

β=

ϕ2 H 2M

(2.10)

where g and H are constants. This relationship was tested by the regression analysis of the nine samples in the Sydney 1988/89 data set (Figure 2.18). The result had a strong linear R-squared value of 0.83 (p = 0.006) for β as the independent variable. Therefore, we have derived, empirically, a third relevant form of the gravity coefficient, this time related to the propensity of MPS in the sampled population. When a quadratic regression is fitted for MPS and centre destinations, there is a similar significant relationship, as was the case with β and k previously (Figure 2.19).

Introduction to R etail and C onsumer Modelling

63

Figure 2.18 The Relationship between the Gravity Coefficient and the Percentage of Multi-purpose Shopping Squared Divided by the Transfer Coefficient (Sydney Project 1988/89)

Figure 2.19 The Distribution of MPS changing with Centre Size (Sydney Project 1988/89). The Point of Inflection is at N = 167 Centre Destinations for Small (negative slope) and Large (positive slope) Centre Behaviour

64

C hapter 2

There are clusters of points exhibiting both small and large centre behaviour (minimum point of N = ~167 ) and an agglomerative effect (or dampening) of the MPS equation with centre scale (N < 167 ). Therefore, the reason for the ± sign in Equation (2.9) is not fortuitous, but rather suggests that there are two types of MPS, one related to small centre behaviour at lower order shopping centres and the other, ‘large centre’ behaviour, found at regional shopping malls. This shows the value of mathematical modelling identifying two possible types of behaviour and the merit of statistical testing to show that quadratic regression is a more meaningful approximation of the data than a linear regression and that agglomerative effects are meaningful to understand multi-purpose shopping. 2.5.6

CLASSICAL SPATIAL INTERACTION MODELLING

The gravity model has been central to the synthesis of spatial interaction modelling ever since Reilly (1931) first studied the relative retailing pull of two competing cities, from an analogy to a physical gravitational law (Davies, 1976). In the context of shopping behaviour, it is assumed that the number of customers expected to travel from a given origin will be inversely proportional to the distance to the retail destinations, until a distance threshold is attained, limiting the demand for goods and services. This idea was expressed using an analogy from Newtonian physics:

Ta Pa § db · = ¨ ¸ Tb Pb ¨© d a ¸¹

2

(2.11)

where Ta and Tb is the proportion drawn to cities a and b, Pa and Pb the population sizes of cities a and b, and da and db the distance from an intermediate town from cities a and b. Ellwood (1954) modified Reilly’s original gravity formulation in order to make it more suitable for studying the impacts of proposed shopping centres. The square footage of each retail centre was substituted for population as a measure of attractiveness, whilst travel time between retail centres was substituted for physical distance, to enable the break-point from the proposed centre to each of the existing centres to be calculated.

Davies (1976) states that spatial interaction modelling is different to central place theory for three reasons, namely: 1. It defines the constraints between retail centres and their market area rather than an explanation for the growth of retailing at one place over another. It is a theory of movement rather than a theory of location. 2. It has no rigid underlying assumptions that are contained in central place theory. It revolves around a range of mathematical models, based on the

Introduction to R etail and C onsumer Modelling

65

gravity concept analogous to Newton’s law in physics. The fundamental model of spatial interaction theory is termed the ‘gravity model’. 3. This gravity model has been applied to many different problems in spatial interaction modelling (for example, traffic flows, population flows, shopping movements), whilst central place theory is just concerned with the evolution of centres and markets. The evolution of the gravity model is intimately related to the rise and fall of the quantitative revolution in geography in the 1960s and 1970s. It was the lynch-pin, in many ways, to the evolution of quantitative methods and the way ideas from coevolving areas were applied to it. For example, Huff (1963) developed a significant contribution to the study of consumer behaviour by introducing the concept of utility and probability into the gravity model. The perception of the utility of a shopping centre by a consumer in the Huff analysis is influenced significantly by the effect and the expense involved in travelling to alternative centres. Furthermore, market areas are defined in terms of probability contours by the expression: P( c ij ) =

Sj

ªSj Tijλ « λ j =1 « ¬ Tij n

¦

º » »¼

(2.12)

where P(cij) is the probability of a consumer at a given point of origin i, travelling to a given shopping centre at j; Sj the square footage of selling space devoted to a particular class of goods at shopping centre j; Tij is the travel time involved in getting from a consumer’s travel base i to shopping centre j and λ is a travel constant. In this model, gradients of probability around a shopping centre could be calculated and market area boundaries defined as points of equal probability. In his analysis, Huff identified a major problem with the gravity model, namely that it possessed very little theoretical content and did not explain why regularities in shopping behaviour occur. Schneider (1959) supported this view when developing a gravity base for the intervening opportunities model of Stouffer (1940). This model states that the probability of a trip located at a destination in a region is proportional to the number of destination opportunities contained in that region. A trip is preferred to be as short as possible with evenly distributed destinational opportunities. A method of deriving the gravity model was therefore necessary in order to establish a degree of explanation of the empirical data. Distance has been shown to be a major determinant in the number of urban spatial interactions, yet the question remained as to what the underlying processes were that produced the regularities in behaviour. The answer was sought by Wilson (1967, 1970) in deriving production-constrained

66

C hapter 2

gravity models from maximising entropy (or disorder) in a complex information system. The Newtonian analogue was replaced by entropy as the basis of gravity interaction. Wilson’s original model was applied to the interaction between the journey-to-work of a number of workers in a residential zone to the number of jobs in an employment zone. As well, the relationship between potential levels of expenditure in residential and shopping zones could be examined. The traditional gravity model was generated by calculating the probability of a distribution of trips proportional to the state of the system which satisfies the constraints:

¦T

= Oi

(2.13)

¦T

= Dj

(2.14)

ij

j

ij

i

¦¦ T c

ij ij

i

=C

(2.15)

j

where Tij is the number of work trips; Oi is the total number of work trip origins in i ;

Dj is the total number of work trips; and C the total cost. The solution is of the form:

(

Tij = Ai B j Oi D j exp − β c ij

)

(2.16)

where Ai and Bj are the balancing constants for each origin and destination. The cost of travel cij is introduced to replace the distance function as a general measure of travel impedance. Gould (1972) provides an excellent overview of understanding this technique. The retail form of the gravity model states that the most likely state of the system is where the trips between home and a shopping centre is directly related to the number of shoppers (Oi), the number of shopping destinations (Dj ) constrained by Lagrangian balancing factors, Ai and Bj, and the total expenditure C , available as a function of the negative exponential function of cost cij , controlled by a third Lagrangian balancing function β (defining the expenditure available to the system). The problem is in estimating the values Ai and Bj and the total expenditure available to the retail system being studied. The solution requires repeated iterations to obtain the best fit of Ai and Bj for the shopping data and, because the Lagrangian balancing factors depend on each other, this iteration procedure affects the nature of shopping expenditure in the system. Whether this parallels the real situation becomes problematic.

Introduction to R etail and C onsumer Modelling

67

The Wilson model provided a theoretical basis to gravity interaction and has been applied to a wide range of studies of aggregate movement. However, while this approach provided good empirical fits, the explanation for consumer spatial behaviour were negligible. Many subsequent papers have taken up this theme. For example, Gordon (1976) argues that the stochastic derivation of the gravity model is not a denial of individual rationality, but a recognition of the impossibility of accounting for the diverse characteristics of individual preferences. Curry (1978) and Sheppard (1978) have questioned this when arguing that entropy maximisation is a method of making minimally prejudiced statements in the face of incomplete information. Sheppard (1978) argues that Wilson’s derivation is based on the paucity of information, where there are two obvious constraints on interaction patterns in space, namely information and cost. He contends that the questions raised by Huff (1963) of why regularities occur has not been solved by entropy-maximising methods, and argues that the orientation of research should be the search for the theory of individual behaviour that can be aggregated into patterns of group interaction, consistent with the gravity hypothesis. In subsequent research, Wilson (1971) argued that there are a number of advantages in using entropy maximising methods for modelling spatial interaction. The technique generates models of complex phenomena that can satisfy a larger number of constraints. These constraints can then help interpret the terms of the model. For example, the term λ is the Lagrangian multiplier associated with the cost constraint, the value of which decreases as the distance of the trip origin to the city centre increases. Sheppard (1978) agrees that the gravity model, although actually explaining very little, paradoxically continues to provide good empirical descriptions. Nevertheless, there is a problem linking micro-behaviour and macro-behaviour using entropy maximising methods. As Hudson (1976) notes, a distribution of trip behaviour is most probable at the macro-level and implies a corollary of maximum uncertainty at the individual level. The Wilsonian view is therefore that random consumers make chance decisions in a chaotic world. One of the mysteries of the gravity model is how it can take the form of a negative exponential function to a base constant e. It has the amazing mathematical property of self-generation under differentiation. No matter how many times we break it into parts, it still has the same form, never changing, because the exponential e (or exp) is a constant under an infinite amount of cutting. More formally, it is represented by the limit of a binomial series as n increments approach infinity. The binomial series, namely: n

1· § ¨ 1 + ¸ as n→ ∞ and 1/n → 0 n¹ ©

can be summed to produce for the first ten terms

68

C hapter 2

1+1+

1 1 1 1 + + + ............ + = exp = 2.71828152 2 3 4 r

The exponential e is then a constant, principally defined by the first terms of the series expansion. In a retailing context, there is a limited supply of centres (and floorspace) and a descending series of opportunities defined by the distance to the residence. The ratio in this assignment process is this constant ‘e’. It is the most likely assignment if consumers minimise distance x. This assignment depends on the constant β, the gravity coefficient and, in physics, it has the interpretation, as Gould, 1972 states, as something to do with the energy available to the system. The gravity coefficient β affects the slope of the distribution (Figure 2.20). In the RASTT model, Equation (2.3) states that β is fundamentally related to the time available to households. This is why the trading hours T of shopping centres affects the spatial trip distributions, since β1 > β2 means that trips to the mall are more dispersed when the shopping hours of the centre are longer ( β ≈1/ T hrs per week).

L2

S h o p p in g C e n tre

β1

V

Population Density

φ

L1

β2

β2 β1

L2

L1 D i s ta n c e

d

Figure 2.20 The Gravity Model as a Negative Exponential Distribution away from a Shopping Centre for β1 > β2 Why then do we get the negative exponential distribution in the gravity model? The model states that, in terms of a population of shoppers, there would be a continuous decrease in patronage away from the shopping centre, with each decrement of distance proportional to the magnitude of the number of people undertaking the trip. Why do less people make such trips in this way? An explanation is that the probability of selecting an alternative shopping destination (termed ‘intervening shopping opportunities’) of the same size increases with distance away from that particular retail centre. The probability of such alternative opportunities facing an individual may increase with distance for a particular level of a central place hierarchy,

Introduction to R etail and C onsumer Modelling

69

according to a positive logarithmic function (the inverse of a negative exponential function; Figure 2.21 and Fotheringham, 1986). There is, then, an equivalent trade-off in the supply of destinations and the number of trips (or demand) to the destination. This equality holds if there is a constant number of destinations and if consumers

1

Prob(Alt)

.8

8 7 6 5 4 3 2 1

P ro ba b ility o f A lte rn a tiv es

Prob(Pop)

1 0 .9 .8 .7 .6 .5 .4 .3 .2 .1

.2 0

C S ho ppin g C e ntre

5 km

P ro ba b ility o f S ho pp ing at C e n tre "C "

d D istan ce

Figure 2.21 The Trade-off between Constant Logarithmic Supply of Destinations and Constant Negative Exponential Demand for a Shopping Centre C , if Consumers Minimise Trip Distance minimise their trip distance to the centres. Therefore, if the supply probability of a choice of an alternative destination increases in this way, then the probability of a spatial demand function depreciates in a similar way to achieve the market equilibrium for the most likely state of the system. Both supply and demand curves are like the mirror images of each other. For example, in Figure 2.21, there would be a probability of 0.2 at 5km of an alternative centre not being found. Therefore, the 0.2 weighting of a constant population would still probably travel to shopping centre C . Conversely, there would be a 0.8 probability of the consumer seeking an alternative destination from an intervening opportunity. This assumes that all destinations are the same size. This weighting assignment is important in determining the proportion of household expenditure in trade areas. If trading hours are extended (β1 > β2 ), more consumers 5km from the shopping centre can visit that centre and there would be less probability of visiting other local centres. This is why Baker

70

C hapter 2

(2002) argued that there is a global vacant shop problem in traditional shopping precincts and central places. The negative exponential function occurs because of the assignment between constant populations at residence (demand) with constant destinations (supply), if those consumers minimise distance. The time available for both consumers and destinations will affect the nature of the trip distributions in space. The gravity model seems to go through a cycle of situations every decade (the socalled wheel of retail gravitation, Brown, 1992b). For example, a direction in mathematical modelling of spatial behaviour emerged at the end of the 1970s, where the gravity model was derived from potential theory (Curry, 1978; Sheppard, 1979, 1980; Tobler, 1981; Baker, 1982). This was not surprising since previous works had incorporated the idea of potential functions in the mainstream discussions of the gravity model. For example, Huff (1963) used market and sales potentials, Wilson (1971) population potential and Gordon (1976) an economic potential function. Curry (1978a) argues that pair-wise choice in the framework of Luce (1959), while obeying Smith’s (1975) axioms, is not only consistent with gravity formulations but, more importantly, is consistent with the concept of potential. He formulates a potential theory of random preferences explicitly in terms of Markov processes (following on from Golledge, 1970 and Burnett, 1974). Spatial ordering is engineered from a theory of random walks and potentials are expressed in terms of probability. However, the problem with this interpretation is that, due to a lack of information, the consumer prefers a decision distribution that is as random as possible and has the smallest expected loss. This ignores a preference structure in determining consumer wants and describes, empirically, only a small percentage of shopping behaviour at a PSC. Sheppard (1979a) takes a more general view of potentials, where he sees them as an integral concept in spatial interaction, requiring no analogy to social physics. He argues that if the ‘intervening opportunities’ model of Stouffer (1940) is conservative, then a potential function exists for it. The concept of ‘potential’ provided a unifying definition of influence that is applicable to spatial behaviour. Nevertheless, such ideas have not progressed the model substantially. There is always an undercurrent of physical analogy which makes the widespread adoption of this and other ideas problematic. The efforts by Huff (1963) and Wilson (1967) still remain the reference point for subsequent disaggregate choice modelling and aggregate flow modelling, respectively. Brown (1992b) states that, in the 1980s, there were several attempts to synthesis aggregate and disaggregate approaches to consumer behaviour (such as, Fotheringham, 1986; Nijkamp and Reggiani, 1990) and search for a universal model of spatial interaction (Alonzo, 1978), as the efforts to operationalise and test the models (Step 6 in the model-building process) proved problematic (for example, Miller and O’Kelly, 1991). Further, there is substantial literature developed on the calibration procedures and problems, such as the modifiable area problem (Openshaw and Taylor, 1979; Okabe and Tagashira, 1996). More recent work has looked at the

Introduction to R etail and C onsumer Modelling

71

gravity model and trade areas (O’Kelly, 1999) and statistical processes (Bavard, 2002). The gravity model has a rich tradition and has been applied to central place theory, rent-bid theory and spatial competition. Nevertheless, Brown (1992b) pointed out that the late 1980s were also a time of growing disenchantment with the nature and direction of gravity modelling. The comment made by Openshaw in Section 2.4 on ‘Model-building and its Weaknesses’ could specifically apply to the gravity model. Brown (1992) views that the wheel of retail gravitation has turned from the dissatisfaction of the 1980s to the practicalities of the 1990s, where the original simple model is increasingly used to satisfy private sector demands for ‘good numbers’. This quest for the ‘good number’ has well-served the political aspirations of both the private and state organisations. Further, the introduction of geographical information systems and the expressions of distributions in maps have helped promote the model as relevant and understandable. The use of the gravity model, in a way, is a good proxy for the state of health of quantitative geography. 2.5.7

INVENTORY MODELLING

An application of inventory modelling for consumer behaviour to central places has been an area of significant research. Each household purchases, stores and consumes stock at regular intervals and commodity types have different frequencies of consumption (Curry, 1962; Nystuen, 1967). Papageorgiou and Brummell (1975) argue that such a hierarchy of commodities has influenced consumers to undertake single and multi-purpose trips to a range of different shopping centres at different time periods. This approach has been tied to rational maximising behaviour in the context of stock supply and demand by Eaton and Lipsey (1982). Bacon (1984) argues that storage costs are an important influence over the friction effect of distance upon shopping trips. The choice of shopping centres offering many commodities for fixed shopping frequencies is deduced by cost-minimising behaviour. He found, empirically, that some journeys are multi-purpose and others are single purpose, the nearest centre postulate does not hold and that journeys to work and non-shopping activities reinforce the attractiveness of large shopping centres. Kohsaka (1984) assumes that purchasing follows a Poisson distribution, where the aim of the consumer is to minimise shopping costs. A shopper seeks to optimise the configuration of multi-purpose trips in a central place hierarchy within this aim. The problem with this approach is that there is no utility maximising axioms and no reasons for the development of hierarchies. Cost potentials are nonconservative (Baker, 1989) and their allocation to spatial choice can only be framed in probabilistic terms. Inventory modelling has also been important in developing a space-time context for shopping travel demand. More recent work has looked at the optimal frequency of periodical inventory taken within a time period (Sandoh and Shimamoto, 2001) and shows the increasing recognition of the importance of time in supermarket management.

72

2.5.8

C hapter 2

STOCHASTIC PANEL-DATA MODELLING

A major development of the 1980s has been the application of stochastic panel-data models to consumer purchasing behaviour. Chatfield, Ehrenberg and Goodhardt (1966) introduced the negative binomial distribution (NBD) model for brand purchasing. The NBD is a two dimensional stochastic model that describes and predicts many regularities in consumer purchasing behaviour and it has been applied to shopping centres by Wrigley and Dunn (1984); Dunn, Reader and Wrigley (1983); Dunn and Wrigley (1985); and Dunn, Reader and Wrigley (1987). The model is based on a number of assumptions, namely: 1. the number of purchases of a product, at a particular store, by a single consumer, in successive equal time periods of arbitrary length, are independent and follow a Poisson distribution; 2 the long-run average purchasing rates P P Pn across different consumers 1,2,....N can be described by a Gamma distribution; and 3 it follows on from (1) and (2) that the frequency distribution of purchases are aggregated to time periods of a week or longer.

Halperin (1985) argues that the major advantage of the NBD is that it provides theoretical norms for a wide range of indices of consumer spatial behaviour. These provide anchor points to compare empirical results. This is the same argument that is used in this study. The theoretical results are the reference curves to actual empirical findings and the task is to try to identify reasons for any significant deviations between these observed and expected results. Kau and Ehrenburg (1984) have developed the NBD into a more general Dirichlet model and Dunn and Wrigley (1985) have applied this to investigate the complex patterns of multi-store and multi-centre purchasing revealed in urban shopping behaviour. Market shares and store loyalty in the product field can be defined for each individual store within the Dirichlet distribution. A problem with both models is that they are both purchase incidence models and so explanatory socio-economic, demographic, attitudinal and locational variables must be added to the model arbitrarily. Furthermore, the question remains about the stationarity assumptions. Wrigley and Dunn (1984) note that there is empirical evidence that the propensity for consumers to make shopping trips varies both with the day of the week and with the time of day. This is a major weakness. Halperin (1985) points out that a further source of controversy with this approach is the evaluation of the ‘goodness-of-fit’. The tables obtained using these models contain no formal measures of significance, but rather only a series of theoretical norms to which observed values can be

Introduction to R etail and C onsumer Modelling

73

compared. It is the regularity of the results that are significant, with no statistical overtones. More recent research looked at alternatives to this type of modelling because of its weaknesses. For example, Reader (1993) compared a mass-point model to the Dirichlet distribution of shopping trips to retail centres and Popkowski, Leszczyc and Timmermans (1996) introduced a risk hazard model of consumer store-choice dynamics as being less restrictive in duration effects on the negative binomial and Dirichlet models. There has been some work undertaken in this area over the last five years (for example, Ehrenberg et al. 2004). 2.5.9

SPACE-TIME MODELLING

One of the characteristics of space-time modelling has been that it has appeared in various guises in the previous modelling approaches, namely:

Physical Analysis Physical space-time behaviour analysis has its geographic origins in the work of Morrill (1968) who studied the idea of spatial diffusion of a population and Hagerstrand (1970) who focused on paths through space-time and conceptualised a set of constraints which confine activity to travel behaviour within a space-time prism. This second approach was applied by Lenntorp (1976) to calculate the shortest path length over a sequence of space-time activities. A third approach was introduced by Curry (1978a) who viewed the random sequencing of movements as a random walk problem and introduced the stochastic solutions of the diffusion equation to be appropriate to geographic problems. Inventory Analysis A major area of research where space-time modelling became significant in the 1980s is in the application of inventory analysis. Sheppard (1979b) used this approach to describe a theory of travel demand, based on the demand for goods and the decision on the frequency of travel. Eaton and Lipsey (1982) assume that at fixed and regular intervals, each household checks its stocks. Shopping takes place to cover consumption until the next check and individuals select retail outlets by the commodity type and to minimise transport costs. As a result, a consumer may have differing travel patterns in lengths, content and spatial setting. This has been put into a space-time context by Lentnek, Harwitz and Narula (1981); Harwitz, Lentnek and Narula (1983); and Narula, Hartwitz and Lentnek (1983) where storage costs, leisure time and commodity type are put into a utility maximising framework. The Narula, Hartwitz and Lentnek (1983) algorithm is expressed qualitatively as flow charts whose output is a vector of consumption and work time, that is applied to a pattern of shopping trips in space and time. Therefore, Thill and Thomas (1987) argue that it is necessary to conceive a framework for trip behaviour that combines

74

C hapter 2

both spatial and temporal aspects of travel choices and that considers multi-purpose and multi-stop behaviour.

C entral Place Analysis The frequency of purchasing a range of commodities has been developed as part of central place theory by Losch (1954). Curry (1962) and Nystuen (1967) describe consumer demands of shopping centres in space as part of the satisfaction defined by commodities of different purchasing frequencies. Bacon (1971) and Eaton and Lipsey (1982) consider potential shopping activities in space and time as individual demand functions, and purchasing frequencies as exogenous variables. The combination of purchases with the frequency of purchasing has been the centre of work by Bacon (1984); Ghosh and McLafferty (1984); Ingene (1984); and Bacon (1995). Stochastic Processes The trips of consumers can be modelled stochastically in space and time. Central to this development has been the idea of Markov chains that has been applied to sequences of stop purposes (for example, Wheeler, 1972; Kitamura, 1983). A set of random variables forms a Markov chain if the probability of the next value or state depends on the current value or state and not upon any previous values. There are a number of problems with the Markov approach to consumer trip behaviour. These have been summarised by Thill and Thomas (1987) as: 1. the evolution depends on only the last state occupied and therefore assumes memoryless behaviour by consumers; 2. the transition matrix is stationary (that is, it does not change over time) and is independent of the characteristics of sub-populations and environments; and 3. the duration of stay in each state does not vary over time. One method has been to derive the elements of the transition matrix from the random utility model (Horowitz,1980). However, these problems remain with the approach and therefore limit its applicability to consumer trip behaviour.

Spatial Interaction An approach to aggregate trip cycles has been developed by Baker (1985) and Baker and Garner (1989) that uses Fourier analysis to compare theoretical and empirical frequency distributions to shopping malls. This developed into the RASTT model of this study. The decomposition of time into periods of recurring shopping behaviour could then be empirically investigated. The important development here was that these periodic solutions were generated from a space-time differential equation, similiar to the one used by Hotelling (1922), to investigate the migration of people

Introduction to R etail and C onsumer Modelling

75

in the USA. Consumers can either space-discount their shopping opportunities to the first or second order ( ∂/∂ x, ∂ 2/∂ x 2 ) or time-discount these to the first or second order (∂/∂ t, ∂ 2/∂ t 2 ). The advantage is that these trips are part of the solution of differential equations when market boundaries are applied. The destinations n are then allocated according to the frequency of consumption to a centre or commodity in the time solution, whilst the space solution is the negative exponential gravity model of travel distance to the centre ( Baker, 1994a; 2002). Socio-economic factors are incorporated into the coefficient and these characteristics affect the space-time distributions. The RASTT model, used extensively in this book, is the result of this type of space-time modelling. By using differential equations with operators, it is arguably a powerful method at looking at shopping trip change over space and time. 2.6 A Vision for Dynamic Trip Modelling Baker (2003) states that one of the great attractions of modelling is that, by capturing past and present relationships, there is an ability to make predictions of the future based on any changes in these relationships. This possibility of glimpsing the future provides an incentive to try and build models that use the correct underlying variables and approximate the reality being observed. It also underwrites the value of a model for policy formation, because it may anticipate problems that may be avoided if the relationships within are understood. It may then be a cornerstone to rational planning (Longley, 2000). However, this may be a difficult goal to achieve because of the problems of untestable assumptions and the way statistics can be manipulated to achieve the desired outcomes. The motivation to produce ‘good numbers’ from ‘bad models’ is strong within the commercial world of retail development. This chapter has endeavoured to introduce and give a brief overview of modelling. Good models produce relevant questions. The problem with many retail models are that they were proposed either from analogy to physics or use economic assumptions that are devoid of reality. The aim should be to set up an initial model with the minimum number of assumptions that can be then tested against robust data. Once the most appropriate model is selected, it should then be applied to new situations and broader contexts. The results may only be conditional generalisations that can be expressed, at best, in terms of probabilistic distributions. This is not a failure of the model as a tool, but a recognition and understanding of the complexity that is being studied. The results may only be conditional and expressed by a degree of explanation, but if they are significant statistically, then they provide a reference point to define a plethora of other behaviours relative to the conditional generalisation. We need models to be benchmarks to provide a basis to understand why there are outliers that do not conform to the model and its predictions. This introduction to modelling has used practical examples to show the process of model-building and its application to policy. For example, the rapid disintegration of lower order centres in the retail hierarchy of Canberra, from the deregulation of

76

C hapter 2

shopping hours, was predicted by the RASTT model (Baker, 1994b). Further, it is only meaningful if we understand Christaller’s original formulation of a central place hierarchy. A corollary to this model is that it also provides a benchmark to understand the evolution of a time hierarchy for supermarkets trading 24-hours a day, sevendays-a-week. In terms of policy, the ACT Government tried to limit the hours of higher order Group Centres in 1997. Their aim was to try and remedy the situation of time accessibility by annihilating neighbourhood centres and preserving a Christallian hierarchy. It failed because of a concerted effort by major retail and shopping centre corporations. So, even then, ‘good numbers’ from ‘good models’ were still unsuccessful in providing the government with the resolve to resist those parties. Such is the nature of ‘Storewars’ in Australia.

Retail modelling had its origins in the 1930s where Reilly introduced the law of retail gravitation and set the course for the evolution of the gravity model. Many ideas were borrowed from other disciplines to develop this model and the success of modelling was questioned periodically at end of the 1970s and 1980s. The advent of geographical information systems (GIS) has re-focused modelling and there are a number of guidelines proposed by Baker (2003; Baker and Boots 2005) for the future of the model in geography. Firstly, a retail model has to be more relevant, testable and reproducible. Secondly, the data gathered must be robust, where inherent bias must be understood. Thirdly, there is no set way to create a good model and it must be accepted that there can be many different, equally valid, representations of observed phenomena. Fourthly, modellers have to respond to criticisms positively to make their models more meaningful. There should be a place for both quantitative and qualitative methodologies, elucidating generality and individuality in retail situations. Fifthly, modellers must understand the mathematical language that is being used. Sixthly, there has to be an increasing realisation of the role that the geographical model has in public policy. Seventhly, geographic information systems using maps should be valued as part of the ability of the model to communicate with the wider community. A brief overview has been presented of types of relevant shopping modelling. The subsequent chapters will focus on one particular model (the retail space-time trip or RASTT model), because of its ability to model change over space and time at different geographic scales, from convenience stores to the Internet. It will allow for a benchmark to compare retail systems evolving over time and cross-culturally between Australia and New Zealand. It will show the interdependence of ‘where’, ‘when’ and ‘how often’ consumers shop in different retail contexts.

CHAPTER 3 Dynamic Trip Modelling 3.1

B ackground to the RASTT Model

How can we examine changes in consumer trip behaviour within the retail landscape? One method is to construct a calculus model and look at empirical ‘snapshots’ over time to study changes in shopping trip behaviour. This step is necessary to provide a framework to study shopping in a number of space-time contexts. The retail landscape is translated by abstraction. For example, shopping centres become density points of spatial and temporal demand. When we look at consumer trips through the glasses of mathematical abstraction, the space-time assignment of trip strategies creates particular theoretical behaviours within a retail hierarchy based on this demand. Baker (1994a) has shown that ‘large centre’ behaviour has characteristics not found in lower order planned shopping centres (PSCs) and that this behaviour, not only varies with centre size, but over time. Further, lower order centres may assume the characteristics of a higher order PSC in the pre-Christmas rush. What this means is that time can substitute for floorspace and that behaviour at PSCs is dynamic. There is a time-space connectivity and, implicitly, the existence of a temporal pattern in each and every spatial pattern (Janelle, 1968; 1969). Planned shopping centres are therefore not just points of spatial demand in a classical economic sense, but rather there is a time demand imbedded within the synthesis of what defines ‘retail floorspace’. The corollary of this proposition is that a change in the time boundary will fundamentally affect spatial demand and its allocation by consumers within a retail hierarchy. This is what underlies the retail aggregate space-time trip (RASTT) model that will be developed in this chapter. The RASTT model was introduced previously to demonstrate the art of model building. Now, we want to look at this model more formally, by reviewing ways of deriving it and checking the robustness of the assumptions. Since the model is based on space and time operators, Equations (2.1) and (2.2) can apply to an individual trip to a shopping centre, to ten thousand shoppers travelling to a mall or millions using the Internet. The scale and order of magnitude of demand are different, but the distributions are very similar. Why? All involve the negative exponential or a gravity function, which does not change because of the invariant nature of the spatial operator (w /w x). The difference is that the Internet involves ‘very weak’ gravity interaction and this should be reflected in very low values of the gravity co-efficient E in Equation (2.3). Therefore, a significant advantage of using this type of model is that it can apply equally to trips to neighbourhood stores, planned shopping centres or to the flow of transactions globally on the Internet. The latter applications will be developed in this study.

77

78

C hapter 3

The RASTT model treats a shopping centre as a retail laboratory and uses measurable variables. Spatial and temporal variables can be averaged per sample and there are constants such as ‘floorspace’ and ‘trading period per week’. The model is made robust from ensuing empirical relationships rather than from the postulation of normative behavioural assumptions (such as utility maximisation). The hypotheses should be testable and reproducible. Therefore, the advantage of the RASTT model is that it has physical descriptors with minimum assumptions that can be measured through direct observation or a survey instrument. The data comes from large samples of ‘face-to-face’ interviews, generated relative to stated behaviour on the floorspace of PSCs, rather than longitudinally at residences. The respondents are then aggregated into a relative density function, enabling a space-time differential equation to be constructed in space and time for each centre. The method is another physical view of retail modelling and is an alternative to the entropy-maximising approach of trip assignments. The advantage of the RASTT model is that the assignments are dynamic, that is, there is a direct relationship between trip timing and the trading hours of the mall. There are other approaches, such as economic modelling, which can be used to look at how trading hours impact on retail markets. Rouwendal and Rietveld (1998) argue that with heterogeneous firms there are situations where restrictions of opening hours increase welfare. They assume time is a continuous variable and that the relevant time intervals are determined exogenously by the kind of product under consideration. We consider time, not as continuous and approaching infinity, but allocated relative to a cycle of events (Monday to Sunday) with a maximum boundary of 168 hours per week. The RASTT model allows for the study of the impact of changing this time boundary on spatial shopping patterns. Ferris (1990) has linked the time spent shopping with the store’s location in an economic model and recognised that a household’s ability to take advantage of particularities of time and place is a function of the length of time that the stores stay open for business. Using a spatial model of monopolistic competition, Ferris argues that by maximising utility in the households’ trade-off between discrete time and inventories of traded goods, the households’ demands for retail output will increase with the hours of operation and decrease with longer distances to the nearest store. Further, there is a space-time connectivity, where distance and opening hours are substituted through their effect on shopping time (Ferris, 1990, 183). This same thesis underpins the RASTT model, but here we approach the modelling process by assuming that aggregate consumer shopping time is averaged within the boundary of the fixed centre hours, rather than fixed just over a representative sample of the population. The RASTT model deals with relative densities of shoppers (once again sampled from the floorspace of the mall) and not the assumption that there are a fixed number of consumers (as in Rouwendal and Rietveld, 1998). The RASTT model therefore provides an alternative view to economic modelling and its applications in this study are instructive of its strengths (and weaknesses).

Dynamic Trip Modelling 3.2 3.2.1

79

Space and Time-discounting Shopping Trips INTRODUCTION

Space-time modelling had its origins in the work by Morrill (1968) who studied the idea of spatial diffusion of a population and Angel and Hyman (1970) who introduced urban velocity fields and non-linear spatial diffusion. Hagerstrand (1970) was also a pioneer, who focused on travel paths through space-time and who conceptualised a set of constraints that confined activity behaviour within a spacetime prism. These new directions, at the time, were a response to a conclusion by Chorley and Haggett (1967) in their seminal work on Models in Geography which stated: Deterministic study of change over space and time is thus difficult to treat within the same model and it would seem almost inevitable that we should separate the two types of changes and treat spatial evolution within a model system rather than within one single model (p. 57 0). Both Hagerstrand and Morrill provided a response to this pessimism, to a point where a number of books and articles appeared in the 1980s firmly integrating spatial and temporal processes within the one model (for example Wilson, 1981; Griffith and Mackinnon, 1981; Baker, 1982, 1985; Griffith and Haining, 1986). This study also tries to integrate space-time processes into a model where the fundamental questions are: ‘Where’, ‘When’ and ‘How often’ consumers shop and ‘How are these questions related’? The way this relationship is developed is through the use of differential equations, looking at how an unknown population variable (φ) is changed by (∂φ /∂), and the rate of this change (∂2φ /∂2) over space and time. These rates of change are known mathematically as derivatives. The symbol ∂ represents an operator on the population variable and this is much like the relationship that a verb has with the subject in a sentence. It describes a process of change acting on the population variable. The differential equations looked at in this study are usually partial differential equations, because the population variable is a function of the process of two variables (space and time). There are two basic partial differential equations in the RASTT model formed from combinations of operators underpinning the distance and time-minimisation shopping strategies. We are interested in how the changes in trips impact upon spatial interaction structure. The solutions to these equations can give a description of behaviour separately in space x and time t, meaning that we can look at spatial interaction structure (including shopping centres) dynamically over time. The equations of the RASTT model are special cases of a general linear secondorder partial differential equation for trip behaviour, namely:

80

C hapter 3

J [φ ] = A

∂ φ ∂ φ ∂ 2φ ∂ 2φ ∂ 2φ B C + Fφ +E +D + + 2 2 ∂t ∂x ∂t ∂x ∂t ∂x

(3.1)

where A, B, C, D, E and F are coefficients of x and t (but not of the population). This can be used to classify equations (see Bleecker and Csordas, 1996). For example, if B 2 − 4 AC = 0

(3.2)

then, Equation (3.1) defines the classical diffusion equation ∂ 2 u ∂u =0 − ∂x 2 ∂t

(3.3)

B 2 − 4 AC > 0

(3.4)

Likewise, if

then the form of Equation (3.1) is the wave equation ∂ 2u ∂ 2u − =0 ∂x 2 ∂t 2

(3.5)

The space-time operators can therefore lead to a classification of these types of equations into a matrix that will be referred to in this study (Table 3.1). Distance minimisation strategies are a solution of the supermarket equation, whilst timeminimisation strategies are a solution of the diffusion equation. Variable φ ∂/ ∂t

∂/ ∂x Continuity Eq

∂2/ ∂x2 Diffusion Eq

∂2/ ∂t2

Supermarket Eq

Wave Eq

∂2/ ∂t2 +∂/ ∂t

Internet Eq

∂2/ ∂x2+∂/ ∂x

Telegrapher’s Eq

Table 3.1 A Classification of Relevant Equations using Space-Time Operators Therefore, the two basic strategies of distance and time-minimisation are termed, ‘space-discounting’ and ‘time-discounting’ behaviour. Previous studies have focused on time-discounting behaviour (Baker, 1994a, 2000a), where the impact of time boundaries on gravity-type interaction can be studied. This behaviour uses a particular differential equation (Equation 2.1; the so-called ‘supermarket equation’,

Dynamic Trip Modelling

81

see Baker, 2000a), its solution (Equation 2.2) and subsequent relationships (Equations 2.3 and 2.4). It can show the relationships between trips through space to malls, shopping within malls and the timing of these trips relative to the malls’ shopping hours. The supermarket equation defines regular trip behaviour, where distance decay constrains periodic spatial interaction. The weekly grocery trip to a supermarket is an example of this type of model. It can also be applied to shopping on the Internet. It is constrained by the time-space convergence of origin-destination pairs (Figure 1.5). Space-discounting behaviour, conversely, is more likely in shopping centres without supermarket anchors and in retail parks or power centres. Increasingly, it is becoming more important in all types of trip behaviour with the increasing number of ‘timepoor’ households in 21st century society. Behaviour does not have to be regular and the resulting classical gaussian interaction fits into an inverted cone of space-time convergence (rather than the distance decay of a time-space convergence). It is this flexibility of dealing with different types of trip strategies that is a major strength of this model. Therefore, the characteristics of these two types of behaviour will be summarised below before we proceed to develop the model theoretically. 3.2.2

TIME-DISCOUNTING TRIP BEHAVIOUR

Within this general space-time differential equation there are four derivatives essential to understanding space-time behaviour. The first type of trip behaviour is defined as time-discounting behaviour, where the first order space operator and the second order time operator defines what Baker (2000a) called the ‘supermarket’ equation, namely: ∂ → Population Variable ( φ ) ≈ Population Variable = c × exp( − x ) ∂x → Gravity Model ∂2 → Population Variable ( φ ) ≈ Population Variable = ∂ 2t c × sin (t ) or cos (t ) → Periodic Model

(3.6)

The time-discounting in Equation (3.6) results from our interest in the dynamics of the ‘when’ question, namely, studying the time rate of change of consumers moving through a cycle of time situations in space. Such populations, according to the above operator definition, distance-minimise their trips to shopping centres. This is why there is a negative exponential distance solution in the model and much research has been undertaken on time-discounting behaviour because the spatial solution is a version of the traditional gravity model of trips. Such ‘space-poor’ behaviour of distance minimisation occurs from less mobile groups, such as the aged or when consumers are undertaking the traditional weekly grocery shopping trip (Baker, 2000a). The solution also has time-based demand waves, where the periodicity in time (in the trigonometric solutions of sine and cosine) is a corollary of distance

82

C hapter 3

minimisation. Such trips are also found in lower order shopping malls (community to sub-regional malls up to 45,000 sq m in floorspace; Baker, 1994a). Time-discounting behaviour states that a distance constraint is a major determinant of spatial interaction and cross-correlates with particular socio-economic groupings, trip types or centre size. 3.2.3

SPACE-DISCOUNTING TRIP BEHAVIOUR

Within this general space-time differential equation there are two derivatives essential to understanding a second type of trip behaviour. The first order time operator and the second order space operator define space-discounting behaviour (Baker, 1994) analogous to classical diffusion. These are the operators defining space-discounting behaviour, namely: ∂ → Population Variable ( φ ) ≈ Population Variable = c × exp( −t ) ∂t → Time Delay ∂2 → Population Variable ( φ ) ≈ Population Variable = ∂2x c × sin (x ) or cos (x ) → Periodic Spatial Demand

(3.7)

Space-discounting occurs in relation to the ‘where’ question, namely: where is a population in space relative to the rate of its change as it moves through time? In terms of time management, such populations, according to the above operator definition, minimise the time spent on their trips to shopping centres. This is why there are negative exponential time solutions in the model, and Baker and Garner (1989) state that such behaviour is more likely to be found at large shopping centres, where more ‘time-poor’ households are likely to shop. These operators define classical diffusion with applications found in a whole range of physical problems. Classical diffusion theory is based upon two fundamental concepts, namely: 1. the net transfer of consumers j across a unit surface is proportional to the gradient of the population density perpendicular to the unit area; and 2. that consumers are conserved on their trips, so that diffusing units are neither created nor destroyed. These two concepts are summarised by Equations (3.8) and (3.9), respectively: j = −K

∂φ ∂x

(3.8)

Dynamic Trip Modelling

∂φ ∂j + =0 ∂t ∂x

83

(3.9)

where K is a diffusion coefficient describing the transport environment. The market boundaries are imposed in terms of space. 3.2.4

CLASSIFYING SPACE-TIME SHOPPING BEHAVIOUR

Therefore, by using these operators, we can define two specific types of trip strategies, namely: space and time-discounting trips and a number of retail situations (Baker and Garner, 1989; Baker, 1994a, 1996) which are described as: 1. Behavioural Space-discounting is a result of time-minimisation and irregular trip behaviour, whereas time-discounting is a product of distance minimisation and regular behaviour. 2. Hierarchical These strategies have been linked, hierarchically, where space-discounting behaviour is more likely found at lower order shopping centres (less than 45,000 sq m), whereas time-discounting behaviour has a higher incidence at large regional shopping malls (greater than 45,000 sq m). 3. C entre-type Those planned shopping centres with supermarket anchors (or a high proportion of supermarket floorspace) would more likely exhibit time-discounting behaviour (such as, in Australia). Conversely, those malls with exclusive department store anchors (such as in the USA) should show space-discounting behaviour. Trips to retail parks and power centres should exhibit similar spacediscounting behaviour. 4. Socio-economic ‘Time-poor’ households, such as ‘two income professional’, would more likely exhibit space-discounting behaviour, whereas less mobile households (such as, the aged) would more likely undertake time-discounting behaviour to malls. 5. Trip Purpose The weekly grocery trip has a higher propensity for time-discounting behaviour, whereas those seeking a gift would more likely undertake space-discounting as a shopping strategy. 6. Time of the Year It is more likely that some large sub-regional shopping centres will exhibit spacediscounting behaviour in periods of peak demand, such as before Christmas. Such

C hapter 3

84

centres will assume the behaviours of larger malls as a significant proportion of shoppers look for gifts rather than regular shopping purposes. The use of these space and time operators therefore allows for the definition of specific types of behaviour in a range of situations. Much of this study deals mostly with time-discounting behaviour, for a number of reasons. Firstly, the gravity model appears as a solution within the model. Secondly, the model has the ability to impose time restrictions on the spatial patterns. Thirdly, the operators allow a scale invariance, so we can look at interaction locally and globally within the same model framework. Fourthly, two major supermarket chains are increasingly dominating the Australian retail landscape and the growth is, in part, attributable to time-discounting behaviour. However, space-discounting is important in particular situations and it will be modelled extensively to identify its features. This is because shopping malls with mixed anchors (such as in Australia) can exhibit both time and spacediscounting behaviour in different situations and socio-economic groups. Further, most malls in the US would have a higher propensity for space-discounting behaviour because supermarkets are, in general, not major anchors in their malls. Both behaviours are therefore the building blocks of understanding and classifying consumer trip behaviour. 3.3 3.3.1

Characteristics of Space-discounting B ehaviour INTRODUCTION

The diffusion equation is underpinned by the space-discounting assumption. This is when consumers use time-minimisation strategies to underpin their shopping trips. A solution to the diffusion equation (Equation 3.3) can be obtained by separating the variables. This can be done simply by equating both space and time derivatives to a separation constant g2. The solution is of the form:

­sin gx ½ ¾ ¯cos gx ¿

φ = A exp− g 2 t ®

(3.10)

The negative exponential function in time assumes time-minimisation as a trip strategy. Further, the spatial solution states that there are peak periods of regular trips from a residence to the mall and back (much like a longitudinal wave). These spatial demand waves converge upon nodes (or shopping centres) regularly. How often do consumers undertake this convergence and what is the nature of the frequency distribution? The standard technique is to take a Fourier transform of this timeminimisation distribution. The time domain can be defined as:

Dynamic Trip Modelling

­°φ exp − g 2 t ; t ≥ 0 , g > 0 ½° φ( t ) = ® 0 ¾ °¯0; otherwise

°¿

85

(3.11)

As such, the Fourier transform is defined as: ∞

ψ ( h ) = ³ φ ( t ) exp − i 2π ht dt

(3.12)

−∞

or evaluating using the conditions in Equation (3.11) ª g 2 − i 2π h º 4 2 2 » ¬ g + 4π h ¼

ϕ( h ) = φ0 «

(3.13)

The graph of the real frequency distribution h is a normal or gaussian curve of both positive and negative trip components (Figure 3.1). There is a second mysterious component, an imaginary frequency distribution as a result of summing all possibilities in the integral in Equation (3.12). What does this idea of imaginary frequency mean? Is it just an abstract mathematical device or is there a meaning that is necessary for us to develop in this context? For example, the idea of a transverse spatial demand ‘wave’ expressed as a trigonometric identity is something that is not observable. We do not see consumers bobbing up and down in their cars in the ‘wave’ trying to minimise time by shopping at a large shopping mall. However, the

φ

Imaginary Real

f

Figure 3.1 The Fourier Transform of exp-g2t for Space-discounting Consumers (adapted from Weaver, 1983)

86

C hapter 3

inference of the demand as a longitudinal wave converging and diverging away from the shopping mall is something more tangible, although the wave characteristics are not directly observable. We may capture this idea by a longitudinal demand wave, expressing the spatial trigonometric solution in Equation (3.10) in the imaginary form, namely: sin gx → exp 2π igx

(3.14)

and the space-time trip distribution as:

φ ( x , t ) = A exp − 4π 2 Dg 2 t × exp 2π igx

(3.15)

The frequency domain of this distribution is defined as:

φ ( h , t ) = A exp − 4π 2 Dh 2 t × ϕ ( h )

(3.16)

The space-time frequency behaviour is determined by the inverse Fourier transform (Weaver, 1983) with respect to the frequency h in Equation (3.16), namely:

S( x ) = A

1 4 Dπ t

exp

− x2 4π t

(3.17)

This is the normal or gaussian space-time distribution: the common solution of any diffusion problem. The evolution of the trip distributions is defined by taking the convolution integral of Equation (3.16) for a trip ξ from the shopping centre: §

· ª exp − ( x − ξ ) 2 º ¸ r( ξ )« »dξ ¸ 4π t ¬ ¼ © 4π Dt ¹

φ ( x , t ) = A¨¨

1

³

(3.18)

For successive time periods, t = 1, 2, 3… the normal distribution disperses around the shopping centre (Figure 3.2). The market for the centre evolves to capture a

Dynamic Trip Modelling

φ

87

T0 V

V

V

Shopping Time T 0 T1 T2 T3 Shopper Density

T1

T2 T3 0

Distance

Figure 3.2 Changing Market Areas for Space-discounting Behaviour for Successive Time Periods (adapted from Weaver, 1983) greater market share of the spatial domain over time, so by the afternoon there is a wider area defining the trip origins. It is interesting that at opening time at t = 0, there is a spatial delta function, a type of discontinuity, as the mall opens. We see this as a vertical line centred at the location of the centre at x = 0 (Figure 3.2). What does this mean? Mathematically, Equation (3.17) becomes an impulse response function, where the delta function is a jump function and the density curve is discontinuous. We will see shortly that this produces a rippling in the trip distribution, presumably from consumers adjacent to the mall or retail park undertaking regular shopping trips to the centre. Whilst this is not a feature of space-discounting, there could still be a neighbourhood effect discernible within the trip distribution in space. This could be why vacant shops occur adjacent to regional malls with shopping hour liberalisation, because the changing retail hours impact on this ‘neighbourhood’ effect. Therefore, gaussian distributions should be associated, in general, with space-discounting behaviour with a time-minimisation strategy. Examples should include large regional shopping centres or retail parks as specific destinations for this type of shopping behaviour.

3.3.2 INITIAL AND BOUNDARY CONDITIONS The above situation has the market evolving over a daily cycle without any boundaries affecting the internal spatial interaction. However, this is not the case. This situation may be looked at by imposing some initial space-time conditions upon the interaction at the shopping centre. This is where we introduce the idea of initial

88

C hapter 3

and boundary conditions. The Penguin Dictionary of Mathematics states that the initial conditions for functions or derivatives set the frame of reference or circumstances for the problem. Boundary conditions are used to obtain a solution to the problem from those defined circumstances. Such a solution is valid for the region specified by the conditions. For a system evolving over time, initial conditions are those that must be satisfied by the solution. What are the conditions at the time of consumer sampling? This idea can be investigated by studying the general situation and then, in the next section, looking at specific circumstances of opening hours. Suppose the population function φ at a shopping centre located at x = 0 varies according to a function λ1 (t) and by a function λ1 (t) at the trade area boundary for the centre at x = L. Suppose the initial population distribution is defined by φo(x). The initial and boundary conditions for the shopping centre distribution are (after Weaver, 1983):

Initial C onditions

φ ( x ,0 ) = φ o ( x ) φ o ( 0 ) = λ1 ( t )

(3.19)

φo ( L ) = λ2 Boundary C onditions

φ ( 0 , t ) = λ1 ( t )

(3.20)

φ( L ,t ) = λ 2 ( t )

What we want to do now is find out the rate of change of the periodic trips to the centre. This is done by taking a finite Fourier sine transform of the space-discounting differential equation (Equation 3.3), namely: dΦ ( x ,t ) 2 2 π n ªπ n º =α Φ ( n ,t ) + ( −1 )nΦ ( L ,t ) − Φ ( 0 ,t )» « dt L ¬L L L ¼

(3.21)

n = 1, 2 , 3...

and applying the boundary conditions yields: 2

[

]

dΦ ( n ,t ) 2απ n §π n · + α ¨ ¸ Φ ( n ,t ) = λ1( t ) − (− 1)n λ2 ( t ) 2 dt L L © ¹

(3.22)

n = 1, 2 , 3,...

The general equation can be solved (see Weaver, 1983) to yield the spatial domain:

Dynamic Trip Modelling ∞

ª

n=1

¬

89

ª §π n ·2 º § π n· º π nx ∞ 2απ n +¦ 2 exp «− α ¨ ¸ t » × ¸t» sin L n=1 L ©L¹¼ ¬« © L ¹ ¼»

φ ( x,t ) = ¦An exp «− α ¨ t

ª

2

º

³ [ λ ( t ) − ( −1) λ (t ) ] exp «α §¨©πLn·¸¹ t» dt sin πLnx n

1

2

«¬

0

(3.23)

»¼

At the commencement of the shopping measurement, φ (x,0) = φo(x). Therefore, ∞

π nx

n =1

L

φ ( x ,0 ) = φ0 ( x ) = ¦ An sin

(3.24)

where An , the population density coefficients are defined as: An =

2 L π nx φ o ( x ) sin dφ o ³ 0 L L

(3.25)

Equation (3.25) at first, does not seem consistent with the boundary conditions, since x = 0 at the centre when φ (0, t) = 0, not the λ1(t) that is required. Weaver (1983) argues that the solution lies in the fact that there is a jump discontinuity at the shopping centre at x = 0. What does that mean in the context of the space-discounting assumption? This will be investigated next when we use more specific boundary conditions for the opening of the shopping centre. 3.3.3 MORNING AND AFTERNOON SHOPPING TRIP DISTRIBUTIONS Now the aim is to try to approximate, more closely, a shopping situation using the above analysis (after Weaver, 1983). Suppose that the period of measurement occurs over a day from opening to closing time. Initially, there are no consumers present at the centre or in transit at the beginning of the day and so φo(x) = 0. The market boundaries are defined when the population density λ2(t) = 0 and there is no discontinuity at this boundary. At any time during the day λ1(t) = φ* . Therefore, this is defined as (after Weaver, 1983):

φo ( x ) = 0 λ2 ( t ) = 0 λ1 ( t ) = φ* ,t . > 0 λ1 ( t ) = φ0 ,t . < 0

(3.26) (3.27)

90

C hapter 3

Since φo(x) = 0 initially, then there are no density measurements possible and therefore An = 0 for all n. With this boundary, Equation (3.23) reduces to: 2 2 t ª ª º 2απ n π nx §π n · º §π n · exp t exp − × − t dt» sin α φ α « ¨ ¸ » « ¨ ¸ • 2 L © L ¹ »¼ 0 ©L¹ «¬ «¬ »¼ n =1 L ∞

φ ( x,t ) =¦

³

(3.28)

which upon integration within the summation yields: ∞

φ ( x ,t ) = ¦ n =1

2φ • πn

2 ª π nx § π n· º «1 − exp− α ¨ ¸ t » sin L © L ¹ »¼ «¬

(3.29)

For various equally spaced shopping periods over the day, Equation (3.29) is represented in Figure 3.3. Initially, during the first shopping period, the patronage distribution closely mirrors a gravity-type model of spatial interaction, but with every succeeding period, the gravity-type relationship weakens to approach, approximately, a straight line. Therefore, if a survey of consumers was undertaken at the beginning of the morning period, the model predicts a gravity-type interaction, but if it occurred in the afternoon, there would be a distinct increase in the linearity of the population distribution. The gravity coefficient  would decrease. This suggests that there is a greater proportion of longer trips in the afternoon, whilst morning activity concerns those with immediate access to the centre from the surrounding neighbourhood. This ‘afternoon’ effect of the weakening of gravity-type spatial interaction is an indicator of the space-discounting assumption of time-minimisation. In other words, ‘timepoor’ households have a greater propensity to travel to large centres in the afternoon. The ‘afternoon effect’ should be associated with shopping trip distributions at retail parks or planned shopping centres with only department stores or category killers as anchors. However, it can still be found in shopping centres with supermarket anchors (such as, in Australia), if there is a substantial number of time-minimisation shoppers in the sampled population. Such a case was observed in the Sydney 1988/89 data set, where a sub-regional centre (Westfield Burwood) exhibited such behaviour in the pre-Christmas period (Figure 3.4). This is compared to a community planned shopping centre (Ashfield Mall) in the same data set in the week before Easter, where the distributions closely follow the expectation of time-discounting behaviour, that is, where there is no change between morning and afternoon distributions (Figure 3.4). The afternoon effect can be detected in the pre-Christmas rush to this sub-regional centre in the line plot, the anti-clockwise shift in the gravity regression line and the

Dynamic Trip Modelling

91

bias in the cumulative percentile curves comparing both morning and afternoon trip curves (Figure 3.4). The gravity coefficient  reduces from 0.53 (in the morning) to 0.46 (in the afternoon). This also suggests an expanded market area in the afternoon, but this measure is very sensitive to the leverage in the regression from population density points determined near the market boundary L.

Jump discontinuity

φ

Ripple effect

Shopper density

T0 - Morning shopping T1 - Afternoon shopping

T1 Straight line relationship between φ and x Ti

m

e

T0

0

Distance away from shopping centre

x

Figure 3.3 Changing Trip Distributions over a Day for Space-discounting Shopping (adapted from Weaver, 1983)

C hapter 3

92

Line Chart

16

5

Wes tBM PopD ens

4

Wes tBA PopDe ns

A shM P opDens M

12

3 2 1

A shM P opDens A

10 8 6

4

0 -1

Line Chart

14 Y Variab les

Popu lation De nsity

6

2 0

1

2

3

4

5

6

7

8

9

0

10

0

1

2

3

Regression Plot

3

1 0

Wes tBM

-1

Wes tBA

Popu lation De nsity

Popu lation De nsity

2

-2 -3 -4 -5

5

6

7

8

7

8

Regression Plot

2

A shM A

1

A shM M

0

-1 -2 -3 -4

0

2

4

6 8 Distan ce

10

12

-5

14

0

1

2

3

4

5

Y = . 973 - .455*X ; R 2 = . 85

Y = 2. 551 - .934*X ; R 2 = . 928

Y = 1. 514 - .531*X ; R 2 = . 909

Y = 2. 647 - .958*X ; R 2 = . 911

Norma lis ed Cumula tiv e Pe rce ntile

Norma lis ed Cumula tive Pe rce ntile

1.2

.8 .6 .4 .2 0 -2 -2 0 2 .4 .6 .8 1 1.2 Pop. Den sity Morn. Norm alise d

Pop. Den sity Afte r. No rm alise d

1

6

Distan ce

1.2

Pop. Den sity Afte r. No rm alise d

4

Distan ce

Distan ce

1 .8 .6 .4 .2 0 -2 -2 0 2 .4 .6 .8 1 1.2 Pop. Den sity Morn. Normalise d

Figure 3.4 Changing Morning and Afternoon Distributions for Westfield Burwood and Ashfield Mall for pre-Christmas Space- and Time-discounting Trip Behaviour, respectively (Sydney Project 1988/89) The jump discontinuity and ‘rippling’ in the population density in the neighbourhood of the centre is a result of consumers adjacent to the centre who regularly shop at the centre. This is termed the ‘neighbourhood’ effect and only occurs in consumer trips in

Dynamic Trip Modelling

93

the local area. Therefore, space-discounting means that there can still be regular fluctuations in spatial demand from a ‘neighbourhood’ effect and an ‘afternoon’ effect, where it is more likely ‘time-poor’ consumers will undertake a trip to the mall. 3.3.4

GAUSSIAN TRIP AND FREQUENCY CHARACTERISTICS

A characteristic of space-discounting behaviour is found in the trip frequency distribution [namely, f/4 (once-a-month), f/2, f, 2f, 3f to 6f (six times-a-week) shopping]. A normal or gaussian spatial distribution of trips surrounding a shopping centre will also have a normal frequency distribution. The Fourier transform of a gaussian space function is also gaussian (see Weaver, 1983), namely:

φo ( x ) = exp( −hx 2 )

ϕ( k ) =

π h

(3.30)

(

exp − π 2 k 2 h

)

(3.31)

When h = π, there is a self-reciprocity, where:

(

)

(

ℑ exp − π x 2 = exp − π k 2

)

(3.32)

In the example in the previous section, Westfield Burwood in the pre-Christmas rush appeared to exhibit this negative exponential self-reciprocity between the trip frequency and distance distributions (Figure 3.5), if the f, 2f, 3f to 6f trip frequencies are used (that is, once-a-week, twice-a-week,…). Indeed, these seem also to be a feature of some samples with a predominance of time-discounting behaviour. When the frequency distribution for f, 2f, 3f, to 6f are plotted for each distance zone (1km concentric circles) from Westfield Burwood, it is surprising that even with this selfreciprocity, the distributions are different. Zone 1 (0 to 1km) from the centre, has an even distribution right across the different trip frequencies, whereas in Zone 2 (1 to 2 km), the distribution is much more peaked over the once-a-week trip and twice-aweek trip. This feature is summarised by a frequency model described in Figure 3.6. The corollary is that the average shopping time in Zone 2 should be greater than in Zone 1. Further, this ‘neighbourhood effect’ suggests impulse shopping is more common than regular periodic shopping. This corollary is tested from the Westfield Burwood pre-Christmas rush afternoon sample (15/12/88). The results show that for Zone 1, the median time per trip was only 40 minutes, whereas for Zone 2 it was 110 minutes (Figure 3.7). The wide range in the whiskers and outliers is a further feature of this peak period of shopping before Christmas in a sub-regional centre under shopping hour regulation. There appears to be a ‘just-in-time strategy’ from consumers limited by time accessibility to regional PSCs.

C hapter 3

94

10 9 8

6 5 4 Zone 2

P. O. Distance

7

3

Zone 1

2 1 0

0

1

2

3

4

5

6

7

F re q u e n c y

Figure 3.5 The Negative Exponential Self-reciprocity between Trip Frequency ( f, 2f, 3f to 6f) and Post Office Distance for Westfield Burwood (Sample: 15/12/88A), Sydney Data Set, 1988/89

Z o n e 1 (0 -1 k m )

ψ (s)

φ (t)

t

Z o n e 2 (1 -2 k m )

φ (t)

fO

f

fO

f

ψ (s)

t

Figure 3.6 The Fourier Frequency Assignment Ψ ( f ) with Shopping Time (t) (adapted from Richards and Williams, 1972)

Dynamic Trip Modelling

Zone 1 (0-1 km) 350

Zone 2 (1-2 km)

Box Pl ot

350

300

300

250

250

200

200

150

150

100

100

50

50

0

95

Box Pl ot

0

Dur at i on

Dur at i on

Figure 3.7 Testing the Shopping Time Hypothesis in Figure 3.6 with Distance Zones 1 and 2 in the Westfield Burwood pre-Christmas Rush Sample (15/12/88A) with Box and Whisker Plots 3.3.4 SPACE-DISCOUNTING AND THE TWO CENTRE TRIP Convolution is a concept in mathematics where functions are rotated about an origin and then swept to positive and negative infinity (Weaver, 1983). This idea can be applied to consumers making shopping trips when they combine two centres within their shopping activity space. How do multi-stop trips affect the space-time distributions? Suppose that there are two adjoining shopping centres within one market area, competing for patronage. Define a gaussian distribution r(x) for Centre 1 as exp 2 2 (-β x ) and for centre two, g(x) as exp(-β x ), respectively. Consumers who shop at both centres combine shopping opportunities and such coupling may be described by a convolution definition r(x)* g(x), namely: ∞

r( x ) * g ( x ) =

³ r( ξ )g( x − ξ )dξ

(3.33)

−∞

Therefore, consumers shopping at two centres within the shopping journey have a combined function h(x), namely:

[

] [

h( x ) = exp − α x 2 * exp − β x 2

]

(3.34)

96

C hapter 3

Applying the gaussian result of Equation (3.33) yields:

(

)

§−π 2 f π exp¨¨ α © α

2

(

)

§−π 2 f π exp¨¨ β © β

2

ℑ exp − α x 2 =

· ¸ ¸ ¹

(3.35)

· ¸ ¸ ¹

(3.36)

and ℑ exp − β x 2 =

Theorem 3.5 in Section 3.5.3 states that the convolution of the two shopping centres is defined as: H ( f ) = ℑ[h( x )] =

π 2 (α + β ) f π2 exp − αβ αβ

2

(3.37)

The combining of two centres means that there will be a greater spread of trip frequencies in the trip distribution. Alternatively, the spatial distribution function is found by taking the inverse Fourier transform of this result, namely: h( x ) = ℑ −1 [H ( f )] =

π α+β

exp −

αβ x 2 α+β

(3.38)

This means that the combining of two shopping centres together is going to give a greater gaussian spreading over space, meaning greater market penetration, where such space-discounting consumers will travel further to shop at this combined opportunity. This can be directly applied to retail parks, where there is a grouping of more than one retail unit together on the outskirts of towns. The more units there are, the greater the likelihood of town shoppers pursuing a time-minimisation strategy in the retail park. This is an example of an agglomerative effect, where shoppers can be attracted away from town centres by the grouping of ‘category killers’. This example shows that not only can changing the time characteristics affect gaussian behaviour, but the scale of shopping opportunities can have a similar effect. This coupling effect has been reported by Baker (1994a) as an explanation as to why a sub-regional centre (Westfield Chatswood), in a morning sample (23/3/89), possessed features comparable to the larger regional centre Bankstown Square, on the same day, with twice the floorspace. The expectation was that Westfield Chatswood would have similar features to another sub-regional, Westfield Burwood, also surveyed the same day in March (Table 3.2). This was not the case.

Dynamic Trip Modelling Centre (morning samples)

Floorspace (sq m)

Trip Distance (km)

Gravity Coefficient

Westfield Chatswood Westfield Burwood Bankstown Square

27,701

4.9

0.40

Trip Frequency (Trips/week) 1.16

31,424 67,000

3.4 4.2

0.69 0.47

1.29 1.31

97

Table 3.2 Time-space Characteristics of Three Malls in the Sydney 1988/89 Data Set Westfield Chatswood (27,701 sq m) had ‘large centre’ characteristics (mean distance 4.9km, β = 0.40 and mean trip frequency of 1.16 trips per week) more akin to the regional centre Bankstown Square (67,000 sq m) with parameters of mean distance 4.2km, β = 0.47 and mean trip frequency of 1.16 trips per week. Another sub-regional centre, Westfield Burwood (31,424 sq m) was used as a control. It had exhibited large centre characteristics and space-discounting behaviour in the pre-Christmas rush five months previously (mean distance 4.4km, β = 0.46 and mean trip frequency of 1.37 trips per week). On the same day as the Westfield Chatswood survey (Thursday before Easter), Westfield Burwood had more typical sub-regional spacetime characteristics (mean distance 3.4km, β = 0.69 and a mean trip frequency of 1.29). Why did Westfield Chatswood display these ‘large centre’ characteristics? Baker (1994a) proposed that there was a coupling effect of Westfield Chatswood with another sub-regional planned shopping centre (Chatswood Chase; ~30,000 sq m) only 500m away. The combined effect of both centres could explain the spacetime statistics for Westfield Chatswood. This means that the combining of two shopping centres together would give a greater market penetration. The gaussian spreading over space is indicated by the mean trip distance and gravity coefficient, where such space-discounting consumers will travel further to minimise time to shop at this combined opportunity. This conclusion is supported in the survey where 26.2% of shoppers at Westfield Chatswood indicated that they also shopped at Chatswood Chase. 3.4 3.4.1

The Time-discounting Model DERIVING THE BASIC MODEL (AFTER GHEZ, 1988)

The RASTT model defines ‘when’ and ‘where’ consumers enact aggregate shopping behaviour. The previous section looked at shopping trip behaviour from spacediscounting opportunities where populations undertake a time-minimisation strategy. Spatial interaction usually forms a normal or gaussian distribution around the centre and this can change over time. In this section, the analysis looks at time-discounting behaviour, where consumers minimise trip distance as a strategy to reduce the effort in shopping. It also allows for the imposition of trading hour boundaries and allows for the study of its impact on the gravity model of spatial interaction. It is very much

98

C hapter 3

appropriate to the study of supermarket shopping and is applicable to shopping malls with a substantial percentage of supermarket floorspace (such as, in Australia). The so-called ‘supermarket equation’ is also constructed around a differential equation of spatial and temporal operators acting on a population function φ, but now space is differentiated once and time twice (Equation 3.6; Baker, 1985; Baker, 1994a; Baker, 2000a). This is a new differential equation with very little, if any, study in the mathematical literature. However, it is still a special case of a general linear secondorder partial differential equation for trip behaviour (Equation 3.1) and is fundamental to the study of retail change. The space-discounting differential equation was simply postulated in the previous section by an analogy to the derivation of the heat equation (see Section 3.3). Here, an alternative way will be investigated of deriving the supermarket equation by assuming a time-based random walk along a time line. It has an advantage of being applicable, not only to shopping trips, but also to Internet traffic, so it suits our purpose to introduce it for time-discounting trip behaviour and illustrates the probabilistic underpinnings of this type of modelling. Here, we assume consumers order retail destinations along a shopping time line. The observer can only probabilistically estimate this choice sequence. As such, the derivation of the constitutive relationships follows the usual procedure of a one-dimensional random walk, but now through time (see Ghez, 1988). Consider a time line with an arbitrary origin (residence) and a series of shopping destinations positioned by integers i = 0, 1, 2, 3,.... A population of Φi consumers selects each shop and these consumers can move to adjacent identical shops with a frequency f defined by the number of destinations visited m in a shopping centre with a total N outlets per unit distance. The movement is forward through time from shop i to shop i +1 per unit distance. This trip is constrained by the opening or closing hours of the day of the week and the visitation rate to each destination or shop is defined by the rate of f× Φ. In the context of trip behaviour to shopping centres, the time boundary is weekly (168 hours per week) and consumers face the same situation every week: when do they undertake the regular shopping trip (such as, for the weekly groceries)? For this shopping trip problem, the journey through space along the time line is assumed to be forward through the shop destinations at i −1, i, i+1. The flow of consumers out of shop i is therefore defined as: Ei + 1 = f × ( Φ i − Φ i +1 )

(3.39)

and from i-1 into the shop i, the flow is namely: Ei −1 = f × ( Φ i −1 − Φ i )

(3.40)

Dynamic Trip Modelling

99

The change in the distribution of consumers in space, into and out of the i th shop from an origin (such as a carpark or residence), is given by defining all possible transitions for a sequence of shops visited on the time line: dΦ i = −( f × Φ i + f × Φ i + 1 ) − ( f × Φ i − f × Φ i −1 ) dx

(3.41)

The space-discounting equation (or rate equation) in terms of the distribution of consumers in and out of the i th shop in a shopping centre is (by collecting terms):

dΦ i = f × (Φ i +1 + Φ i −1 − 2Φ i ) dx

(3.42)

This exchange rate of consumers considers only nearest-neighbour destinations around shop i. This assumes specific categories of shops are clustered together (food and non-food retailers). Further, this assumption defines consumers only undertaking distance minimisation behaviour and that the trips are independent of a retail hierarchy. We need these assumptions to keep the model simple and linear at this stage. We are not trying to describe all shopping trip behaviours. This assumption of only having trips to the nearest shopping destinations can be expressed simply in terms of the exchange rate between destinations (using Equations 3.39 and 3.40), namely: dΦ i = − (E i + 1 − E i − 1 ) dx

(3.43)

This is a definition of a conservation law (that is, consumers are assumed not to be lost between shops). Equations (3.39) to (3.43) define the evolution of a time line between destinations in space where the time distance between points and the hierarchical network in the allocation to shopping destinations is not relevant, rather, what counts is the ordering of destinations in the journey. Equation (3.43) states a conservation law and the change in population is simply defined by shoppers in (E i- 1) and shoppers out (Ei+1). Our shopping behaviour system has some further assumptions to simplify proceedings. Firstly, the frequency of transfer f is assumed constant; secondly, it does not depend on the consumer distribution in the trade area of shop i; thirdly, the shops are nearest neighbours only within the configuration of shopping opportunities; and fourthly, the ordering of destinations is undertaken by using a distance minimisation strategy. The shopping trip time to and within centre p is assumed to be equal to the time travelled between the origin (such as the carpark or residence) and the i th shop which has the co-ordinate of t i = i p on a time line. The population density continuous function

100

C hapter 3

š

φ ( x , t ) is assumed to interpolate the previous function φi (x) by the following assumption: š

φ ( xi ,t ) = φi ( x )

(3.44)

at a shop located at t = ti. The assignment of this population density function around a group of adjacent shops (such as, a shopping centre) at ti can be expanded by a Taylor series (omitting the spatial function, for the moment): š

š

φ i ±1 = φ ( t i ) ± p

š

š

∂φ ∂t

ti

+ p2

∂ 2φ ∂t 2

ti

+ terms of order p 3

(3.45)

We assume that the flow of consumers into and out of a shopping centre are the same as for an individual shop. The shopping centre with a population density φ (x,t) is then: š 2š ∂φ 2 ∂ φ + terms of order p 3 (3.46) φi ± 1 = φi ± p +p 2 ti ti ∂t ∂t The flow of shoppers in and out of this shopping centre follows the same rule for a shop as in Equation (3.42). When our expression for a shopping centre is substituted into this condition, the results are the basis of the retail space-time trip (RASTT) model: š

š

∂φ ∂ 2φ = f × p 2 2 + terms of order p4 ∂x ∂t

(3.47)

The rate of exchange between destinations (the shopping centre and the carpark or š residence) is only continuous and conservative for the interpolating function φ as (Ghez, 1988): š

š

E( x , t ) = − f × p

∂φ ∂t

(3.48)

Continuous exchange between a shopping centre and carpark or residence can be calculated when the consumer density at the centre is known. The continuous conservation law for a shopping centre (analogous to the discrete system for individual shops in Equation 3.43) is then: š

š

∂E ∂φ + terms of order p4 = −p ∂x ∂t

(3.49)

Dynamic Trip Modelling 101 What is being assumed in the conservation law is simply that no shoppers disappear in the centre and all find their way home. Now, if the standardised population š density φo for a shopping centre is equal to φ0 = φ / p where p is equal to the average time spent shopping in the centre and f equals the average number of visits per unit trip distance, then the average centre transport coefficient M for the centre (that is, the transport infrastructure available to facilitate movement, such as highways, public transport or parking) can be introduced by the definition: M = f × p2

(3.50)

This definition can only be confirmed by empirical testing of M and states that the transport co-efficient M of the shopping centre is equivalent to the number of trips to the centre per unit distance, multiplied by the square of average time spent shopping. The rate of exchange, the conservation law and supermarket equation become respectively:

R ate of Exchange E( x , t ) = −

2 1 ∂ φo M ∂t 2

(3.51)

The equation states that the rate of exchange between a shopping centre and residences is a function of the transport constant surrounding the centre and how the rate of changing population density varies over time (the shopping week). For example, the spatial distributions would vary from different exchanges on Mondays compared to Fridays.

C onservation Law ∂φ o ∂E =− ∂x ∂t

(3.52)

This equation defines the continuous conservation law where no consumers disappear in the exchange between the shopping centre and residences.

Supermarket Equation 2 ∂φ o 1 ∂ φo = ∂x M ∂t 2

(3.53)

C hapter 3

102

Equation (3.53) is the second order differential equation defining consumers moving along a time line to, from and within a shopping centre with a constant frequency of exchange. This is termed the ‘supermarket’ equation within the RASTT model (Baker, 2000a). The reason for its name will soon become apparent. The solution to this equation can be stated in a number of basic forms related to the methods used to solve it. There are two solutions of immediate interest, namely: 1. The Periodic Gravity Form This method solves the equation by separating the variables. It involves negative exponential solutions in space where the constants may be real or complex. This defines not only the so-called ‘gravity model’ in space, but also involves periodic waves in time (that can be expressed as a complex variable, including imaginary time). This solution can only apply to particular shopping centres.

2. The Gaussian Form This form uses the initial value problem method of taking the Fourier transform of the population density distribution along the time line of the shopping paths. This solution is a normal distribution and underpins the statistical nature of trips. It becomes very useful when studying the flow of Internet traffic in Chapter 5. Whilst both forms are solutions, they are not mathematically equivalent. For example, the gaussian form is not a product solution of the differential equation and cannot be a linear combination of a product solution (Bleeker and Csordas, 1992). This means that the gaussian form is basically an aggregate view of the trip processes, whereas the periodic gravity form can apply to individuals as well as populations of consumers (because the linear combinations are also solutions). The model in Equation (3.6) can therefore deal with the problem of the scale of interaction. The focus in the next section will be the periodic gravity model in the context of traditional spatial interaction. Both forms will be used in Chapter 5 when we apply this to the modelling of the Internet. 3.4.2

THE PERIODIC GRAVITY MODEL

The supermarket equation (Equation 3.53) can be solved by separating the variables. In the particular product solution in space and time, the gravity model of trip distance D and periodic shopping are the model constructs, namely: §

φ o = A exp¨ − ¨ ©

· ª sin( kt ) º k2 D¸ « ¸ ¬cos( kt )»¼ M ¹

(3.54)

Dynamic Trip Modelling 103 where k is the inter-locational trip frequency (ITF); A, the consumer density on the shopping centre floorspace; D the exchange distance to residences for a population density φ (or x - xo, where xo = 0 is assumed to be the location of the shopping centre); and β is the gravity coefficient (β = k2/M). The time-based trigonometric functions simply state that the spatial demand of the gravity model of trip distance (residence to shopping centre) is repeated on a regular basis and the most likely category for such repeated behaviour is the weekly grocery trip. As such, the solution to the time line of regular visitation to a shopping centre is constrained by the gravity model of spatial interaction and termed the ‘supermarket’ equation (Baker, 2000). The ITF is introduced arbitrarily as the separation constant to solve the differential equation, where M(1/X)∂ X/∂ x = (1/T)∂ 2T/∂ t 2 = - k 2 for X,T functions of space and time, respectively. It is defined by the average number of times per week that a population of consumers travel to a particular shopping centre. This solution is stating that the undertaking of regular or periodic trips (sin kx) to a shopping centre is discounted by how far away consumers are to a centre (where exp - β x is the discounting function), since there are increasing opportunities to shop elsewhere the further they live from this particular centre. The wave-like nature of the trips is simply shoppers moving backwards and forwards through space between a residence and mall. The periodicity is constrained by the distance decay effect from the increasing numbers of potential alternative centres. This assumes all the centres are the same size. The nature of shopping trips per week in this model is also impacted by the hours that the shopping centre trades per week. This is a major advance for spatial interaction modelling because spatial patterns are shown to have imbedded within them, time paths. Shopping centres therefore represent nodes for the convergence of space and time. The trading hours can be imposed through boundary conditions on the differential equation. The opening of the shopping centre can be simply defined when there are no people present on the floorspace, that is, when population density (φ) is φ = 0 at t = 0. Further, when the centre closes at T, there would be no consumers left on the floorspace and this would be defined when φ = 0 at t = T. For regular shopping trips, with a shopping trip frequency k, assume that sin kt = 0, and for defined trading boundaries:

k = npπ / T

(3.55)

where n is assumed to be the number of shopping stops consumers make in an average of p hours per trip (n = 1 one stop; n = 2 two stops…). The fundamental frequency k0 is defined by for n = 1 stops as: f =

p , where k = 2 π f 2T

(3.56)

104

C hapter 3

where f is the fundamental intra-centre assignment frequency. This defines the assignment of shopping time per stop within the centre.

The average number of minutes (p) Sydney 1988/89 data set to be related an R-squared of 0.96 (Figure 3.8). frequency (p/2T) and mean number frequency) is simply: f =

spent shopping has also been shown in the to the mean number of shops visited (m) with The relationship between the intra-location of shops visited per week (mw = m × trip

mw p ≈ 2T 2T

(3.57)

where 2T is retained for conceptual reasons. The regression of this relationship from the 1988/89 Sydney data is linear with a significant R-squared value of 0.74 (Figure 3.9). There is not the non-linearity from ‘small centre’ and ‘large centre’ behaviour evident in the distribution and the cluster of ‘large centre’ samples conforms, surprisingly, to the line of best fit. This supports the assumptions underpinning the model. Therefore, the number of destinations visited (stops) m per week is proportional to the time spent shopping p. 10

9

8 Destinations

7 6 5 4 3 2

1

Y = -19.179 + .464 * X - .002 * X^2; R^2 = .963

0 60

70

80

90 100 Duration (m)

110

120

130

Figure 3.8 The Relationship between Mean Shopping Duration (p) and Destinations Visited m (1988/89) showing the Shift Right towards ‘Large Centre’ Behaviour

Dynamic Trip Modelling 105 1.4

1.2

Dur/2*TradHr

1 .8 .6 .4

.2

Y = .339 + 6.903 * X; R^2 = .744

0 0

.02

.04

.06 .08 Dest*Freq/2*TradHr

.1

.12

.14

Figure 3.9 The Regression of Intra-centre Shopping Frequency ( f = p / 2T ) and the Mean Shopping Duration p per Trading Week T ( f = m× k /2T) showing the Shift Right towards ‘Large Centre’ Behaviour However, the trip is divided between the journey to the centre and behaviour within the centre. A way to relate both components is to assume that the first stop of the trip is the car parking site, train station or bus stop. Should this be counted in the normal mode solutions above (Equation 3.57)? Arrival at the centre is the first stop of the shopping trip. Since we are trying to relate the trip to the centre and behaviour within the centre, the counting of the parking stop before walking in the centre is consistent with the theory. This means that there are at least two stops for a shopping trip (that is, n = 2 for car park plus shopping centre stop) rather than one, and that f = p / T rather than f = p / 2T. Further, the trip assignment of stops within the centre must be adjusted to the form of 1+ m w (where the number one, is the carparking stop and mw is the number of shops visited). This inclusion allows for browsing within the centre, where such consumers are calculated as part of the population density, yet still do not enter any of the shops. They simply park and browse. There is also a further adjustment. The opening and closing hours assignment for the shopping centre includes both the journey to the centre as well as the return trip to the residence. This introduces a double counting in the f = mw /2T estimate. This, therefore, has to be halved to only include trips to the centre for theoretical validity. A modified expression of carparking and the trip to the centre for the fundamental shopping frequency is then: f =

1 § m + 1· p ¨ ¸= 2 © 2T ¹ T

(3.58)

106

C hapter 3

The mean shopping hours per week for an ensemble of consumers are therefore estimated as: T=

m+1 4f

(3.59)

where m is the mean number of shops visited and f the average intra-centre shopping assignment frequency. A solution of the retail aggregate space-time trip (RASTT) model therefore allows an estimate of the mean trading hours of a centre relative to the spatial demand for destinations at the centre. Further, it provides an estimation as to whether there is an undersupply or oversupply of trading hours at a shopping mall. Are these equations justifiable? Baker (2000a) tested these theoretical results from the Sydney 1988/89 data. Whilst the regression between f = p / T and m / 2T is meaningful, the m/2T values are substantially over-estimated by a factor of two (Figure 3.10). As indicated previously, the classical definition (m/2T ) of the fundamental shopping centre frequency should therefore be halved ( m/4T ). The opening and closing hours of the centre are defined only for the positive domain from 0 to T and it is only the journey to the centre that is constrained by the hours of operation. Figure 3.10 shows that (m /4T ) provides a better fit in the Sydney 1988/89 data to the time-based definition of fundamental shopping frequency ( p/T ), simply because we are only considering the journey to the shopping centre in the calculation of its mean trading hours. (t/T) / week (m/2T)/ week (m/4T) /week

.14 .12

p/T

.1 .08 .06 .04 .02 0

Shopping Centres

Figure 3.10 Comparison for the Theoretical m /2T and m/4T Values with the p/ T Empirical Estimates from the Sydney 1988/89 Data Set (Baker, 2000a)

Dynamic Trip Modelling 107 Is there any empirical justification for including the carparking, train station or bus stop as the first destination in the stopping rule defined by f =1+ m/4T ? This comparison is made in Table 3.3. Whilst the results were not conclusive in the Sydney 1988/89 data set, the (m +1) estimate produced less convergent results for the ‘large centre’ samples, yet more convergent estimates for ‘small centre’ samples (Table 3.3). Therefore, it appears that the (m+1) estimate produces the best estimate for ‘small centre’ behaviour, whilst just the m values seem more appropriate for ‘large centre’ behaviour. Yet the question remains, why would counting the carparking stop be more appropriate for smaller rather than larger shopping malls? Perhaps it is that smaller PSCs are more likely to exhibit time-discounting behaviour. The trading hour estimates relative to centre size can be defined as: T=

m 4f

T=

m+1 ‘small centre’ behaviour (including the carparking stop) (3.61) 4f

general trip behaviour

(3.60)

Therefore, there is some empirical confidence in the relevance of the solutions to the supermarket equation to shopping centre dynamics. The (nπ /T ) form can therefore be substituted into Equation (3.54) to yield the spacetime shopping patronage for a shopping trip lasting p hours to a planned centre as: nπ φ = A exp( − β x ) sin( t) (3.62) T

1 BS 23/3/89M 2 BS 23/3/89A 3 WC 23/3/89M 4 WB 15/12/88MP 5 WB15/12/88AP 6 WB23/3/89MOP 7 AM 23/3/89M 8 AM 23/3/89A 9 ML 8/12/88A

Empirical k0= p/T .052 .058 .035 .035 .057 .035 .043 .044 .034

Theoretical k0=m/4T .055 .067 .037 .039 .053 .033 .040 .051 .028

Theoretical k0=m+1/4T .060 .072 .042 .044 .059 .038 .045 .056 .033

Table 3.3 Various Estimates of the Intra-centre Shopping Frequency (per week) in the Sydney 1988/89 Data Set (M = Morning sample; A = Afternoon sample; P = Peak period; OP = Off-peak period. Shopping Malls: Regional, BS-Bankstown Square; Sub-regional, WC-Westfield Chatswood; WB-Westfield Burwood; Community, AM-Ashfield Mall, ML-MarketPlace Leichhardt)

108

C hapter 3

All terms approach zero as x→ ∞ and the gravity coefficient β is defined as (nπ/T )× (1/M). The magnitude of spatial interaction in the gravity (negative exponential model) depends on the number of shopping centres n in the consumer’s activity space. The more shopping centres (as n increases) the more rapid is the decline in trip-making to a particular shopping centre (that is, the gravity coefficient β is steeper) because of the spatial competition. Likewise, the propensity to undertake a trip to more than one centre is a function of the distance to that centre. The rates of trips to shopping centres are a function of the population φ and population gradients over time are proportional to the number of centres n, since the gradient of sin (nπ /T ) by differentiation is: π nπ t (3.63) n × cos T T Is trip behaviour to the shopping centre related to time behaviour within the centre? Baker (2000a) showed that the intra-centre shopping frequency ( f ) is related to interlocational frequency (k) as: f = ± Mk

(3.64)

Is this relationship equivalent to the previously defined f = mk/2T ? This can be tested from data from the Sydney Project (1988/89) and (1996/98). The results are very encouraging (Figures 3.11 and 3.12). For 1988/89, there was a significant R-squared value of 0.48, but this improved to 0.75 in the 1996/98 samples. However, over the decade, there was a substantial change in the slope of the relationship (0.060 to 0.025) and this can be directly attributed to the doubling of shopping hours. However, the regression of Equation (3.64) for the combined data set produced an Rsquared value of 0.134 (p = 0.0655 > 0.5) which is not significant. Therefore, the results are not replicated empirically and the question is still open as to whether there is a significant relationship between trips to and within a shopping centre. Equation (3.54) can be plotted for values determined by sampling consumers on the floorspace of community and regional planned shopping centres for the average shopping time p and for one centre (n = 1) for the trading week T with no carparking (and therefore p ≈ m). The model is then, simply: mπ φ = A exp( − β x ) sin( t) (3.65) T Baker (1997) simulated Equation (3.65) for values determined from sampling consumers on the floorspace of community and regional planned shopping centres in the 1988/89 Sydney Project data (MarketPlace Leichhardt and Bankstown Square). The 17,500 sq m MarketPlace Leichhardt (or MPL) was surveyed on the afternoon of

Dynamic Trip Modelling 109 .14

Dest*Freq/2*TradHr

.12 .1 .08 .06 .04 .02

Y = -.034 + .06 * X; R^2 = .477 0

0

.5

1

1.5

2.5 2 Sq.R(Freq*M)

3

3.5

4

4.5

Figure 3.11 Regression showing the Relationship between Two Forms of the Intracentre Shopping Frequency ( f = mk/2T )) and ( f = Mk ) for the Sydney Project 1988/89 .14

Dest*Freq/2*TradHr

.12 .1

.08 .06

.04

.02 Y = -.005 + .025 * X; R^2 = .751

0 0

.5

1

1.5

2 2.5 Sq.R(Freq*M)

3

3.5

4

4.5

Figure 3.12 Regression showing the Relationship between Two Forms of the Intracentre Shopping Frequency ( f =mk/2T )) and ( f = Mk ) for the Sydney Project 1996/98 (Note the change in slope over the decade from trading hour deregulation.)

110

C hapter 3

8/12/88 whilst the 66,000 sq m Bankstown Square (or BS) was sampled on the morning of 23/3/89. The model was then simulated to examine the impact of what happened when the opening and closing times of these centres were extended to 70 and 100 hours per week (from 49.5 hrs per week). The results are shown in Figures 3.13 and 3.14. Equations (3.66) and (3.67) define the regulated pre-1992 distributions for these samples, where centres were allowed to trade T = 49.5 hours over six days (144 hours/week).

14.8 exp (-.84x) sin (.352) for MarketPlace Leichhardt (m =5.5 dest/wk) (3.66) 2.2 exp (-.47 x) sin (.689 t) for Bankstown Square (m =10.87 dest/wk)

(3.67)

The MarketPlace Leichhardt sample for ‘small centre’ behaviour (Figure 3.13) shows average shopping behaviour for six days-a-week. When the hours are regulated to 49.5 hrs/week with no Sunday trading (total hours equals 144 hrs/week), there is a mid-week lull in trading for the centre. Demand at the centre is strong on Mondays and Tuesdays, there is weak demand on Wednesday, but Thursday and Friday return to peak periods. This has ramifications for sampling design, since the nature of the gravity distribution changes also between Tuesday and Wednesday. The Bankstown Square sample for ‘large centre’ behaviour showed quite a different pattern (Figure 3.14). The simulation suggested that, for regulated conditions of 49.5 hrs/week and no Sunday trading, there would only be a period of regular behaviour at the beginning and end of the week from the lower trip frequency. This periodicity would, however, be masked by the greater proportion of infrequent and random patronage not recorded in the model and simulation. Once again, the day of the survey may produce variations that may just be part of the dynamics of that particular shopping centre. The impact of the extension of trading hours on these ‘small’ and ‘large’ shopping centres may be simulated for 70 and 100 hours per week under deregulation (now including Sunday trading). The centre operates seven days-a-week over a period of 168 hours. The gravity co-efficient β and the shops visited m are assumed constant for the purpose of this simulation. For MarketPlace Leichhardt, the impact of the 70 hours with Sunday trading sees consumer demand shift towards the end of the week. Patronage on Mondays and Tuesdays are now substantially reduced, whilst for Bankstown Square, there is now only one regular shopping period towards the end of the week. This is a significant result because it shows a fundamental change to the structure of the shopping week when the hours are deregulated to include Sundays.

Dynamic Trip Modelling 111

Figure 3.13 Simulation of the Time-space Shopping Distributions for 49.5, 70 and 100 Trading Hours (per week) for ‘Small Centre’ Behaviour (Baker, 1997) What happens to these two types of trip behaviour if the aggregate number of hours is increased from 70 to 100 hrs/week with the other coefficients remaining constant? The result is surprising (Figures 3.13 and 3.14), since the distribution at the regional shopping centre approaches that of the community shopping centre. In other words, ‘large centre’ behaviour, in terms of frequency and periodicity, mirrors shopping trips under ‘small centre’ behaviour. This is a remarkable prediction because it means long trading hours at a regional shopping centre will see an increase in trip frequency there. This prediction underpinned the decision of the Australian Capital Territory (ACT) government to restrict the trading hours of regional centres in the Canberra shopping hierarchy in 1997. These distributions will be checked with actual data from the 1996/98 sampling of Sydney planned shopping centres.

112

C hapter 3

Figure 3.14 Simulation of the Time-space Shopping Distributions for 49.5, 70 and 100 Trading Hours (per week) for ‘Large Centre’ Behaviour (Baker, 1997) 3.4.3

THE FREQUENCY CONTENT OF A PERIODIC SHOPPING EVENT

In the previous section, we have looked at trip behaviour for m visits to shops. In this section, the nature of the trip from the residence to one shopping centre will be studied (that is, for an n = 1 trip). This involves examining the structure of the trip assignment across a trading week. In the previous section, the trip distributions were constructed around the assumption that the expected distribution for a single trip would last, on average, p hours within the centre. Shoppers can undertake less than,

Dynamic Trip Modelling 113 or more than, a once-a-week trip depending on a range of socio-economic and product factors. This section looks at how to deal with this assignment across the trading week within the periodic trip behaviour postulated by the time solution in Equation (3.62). Baker (1985) has developed a method of studying the periodic shopping trip through the analysis of frequency components of this shopping wave, namely: g( t ) =

η

¦B

f

sin 2π f t

(3.68)

f =1

If the amplitude (in this case, a population index or density) and the frequency of trips to the shopping centre can be measured within the sine or cosine solution of Equation (3.62), then its functional behaviour can be determined. The higher frequency terms help define the finer details of the function, whilst the lower frequency components contribute more to the overall shape of the function (Figure 3.15). If one frequency component is eliminated, then the shape of the shopping wave will change, dependent on the size of f (Weaver, 1983).

Individual S1 S2 S3 S4

S5

Aggregate S

Figure 3.15 The Aggregation of Population Demand Waves (Sf) of Different Frequencies (adapted from Weaver, 1983)

114

C hapter 3

A more general situation is given by linearly combining both sine and cosine functions, namely: η

Φ ( t ) = ¦ A p cos 2π f p + B p sin 2π f p t

(3.69)

p =1

The resulting longitudinal wave will have its own unique shape, which will not resemble any of the trigonometric functions used to form it. A longitudinal wave means that shoppers trip regularly to a shopping centre from a residence and return there in some type of spatial oscillation of demand. The peak period is a time of higher traffic flows and, possibly, congestion. We include both sine and cosine solutions because of the possibility of 24-hour trading and that some demand might occur at lower frequencies than once-a-week. Note that each sine or cosine function contributes to the shape of Φ (t) and this is dependent on the amplitude. The consequence is that a shopping demand wave can exist and can vary in shape depending on the component frequencies. Therefore, it can change according to the time of day, the day of the week or during peak demand times, such as before Christmas. Thus, this aggregate behaviour depends not only on what shopping frequencies are present, but on the proportion of each component frequency. The purchasing behaviour at the pre-Christmas rush involves an increase in the number of low frequency patronage or once-a-year trips by consumers looking for gifts which will affect the shape and behaviour of the population demand wave at that time. When the shopping centres are in a hierarchy, the interaction patterns are more complex and this simple synthesis is less appropriate. This function Φ will have a Fourier series frequency content if the following Dirichlet conditions are met: 1. Φ is periodic with period T, that is, Φ (t+T ) = Φ (t); 2. Φ is bounded; and 3. in any one period the function may have, at the most, a finite number of discontinuities and a finite number of maxima and minima. These conditions are assumed in this simple synthesis of regular shopping behaviour. A consumer can shop periodically according to a set time finite period T of trading. The formulas for Af, Bf, and f for the time component of Equation (3.62) can be defined as (Weaver, 1983): f = p/T ,

p = 1, 2, 3,…

(3.70)

Dynamic Trip Modelling 115

The pure cosine frequency terms are given by: Ak =

2 T

³

T/2 −T / 2

F ( t ) cos

2π ft dt T

(3.71)

and the pure sine frequency content terms by: Bk =

2 T

³

T/2 −T / 2

F ( t ) sin

2π f t dt T

(3.72)

Bo = 0 Within the shopping situation, the interest lies in functions over a finite domain and what happens outside this domain does not concern this model. Therefore, a function can be periodic within this domain and have a Fourier series, but a consumer can still undertake non-periodic behaviour outside this domain. Within a discrete time period of a < t < b, the consumer may shop periodically and so the periodic function can be protracted for this event. The problem is that there is a possibility of introducing discontinuities into the protracted function. This leads to a Gibbs’ phenomenon or ‘rippling’of demand about the discontinuity. The mathematical explanation for this event is that an attempt has been made to force a series of uniformly continuous functions to converge to a discontinuous function. This is contrary to the basic truth that a series of uniformly continuous functions must converge to a function that is also uniformly continuous (Weaver, 1983). Therefore, the imposition of shopping hours upon a mall could produce a discontinuity around the opening and closing hours. This means fluctuations in population densities are possible, such as, from traffic congestion from accessing or departing the parking lot. Another example of this fluctuation is from impulse shopping from households adjacent to the centre (the ‘neighbourhood’ effect). Such events are considered as jump functions, rather than an aggregate continuous trip function from more distant residences. A function Φ is called even, if and only if Φ (-t) = Φ (t) and similarly is termed odd, if and only if Φ (-t) = -Φ (t). Odd and even functions are centred in the middle of their period (traditionally zero) and the limits of the integration in Equations (3.71) and (3.72) refer to the values of -T/2 and T/2. This property of Fourier series has yet to be fully understood in shopping analysis. In the shopping case, it seems reasonable only to consider positive time periods. There is empirical evidence for this position, because if both positive and negative domains are included, there is a double counting (see Figure 3.10). However, the trip cycle still involves a return component to the residence.

116

C hapter 3

3.5 The Fourier Transform and Aggregate Periodic Trips (adapted from Weaver, 1983) We will now return to consider how to describe aggregate shopping trip behaviour for frequency of trips k to a shopping mall. Suppose that we study a periodic function F(t) = sin 2π ft for a period T of one week under the assumption of timediscounting shopping opportunities. When f is equal to an integer m, then the frequency components can be calculated as (Weaver, 1983): 12

Bk = 4 sin(2π mt ) sin(2π ft) dt

³

(3.73)

0

It is important to understand that sin 2π ft is an odd function and so we only have Bk terms. The orthogonality equation means that this integral has a value when f = m and so this function has only one frequency. Each trip has a discrete frequency for an individual consumer (for example, the weekly grocery trip on Thursday at 5.00pm). Note that we have limited the time boundary to positive values defined by the integral. How do we then describe a population of consumers and the trips they undertake? The supermarket equation applies not only to an individual but equally to a population of consumers undertaking trips of different frequencies. The trip frequency k is a mean value for the population. However, once or twice, or... six times a week shopping trips are still discrete events for this population. The question that must therefore be answered is how both aspects can be incorporated into the analysis. The complex form of the Fourier series for periodic shopping f within a trading week T for a particular good is defined for a population function by φ (t) as: ∞

φ( t ) =

¦C

f

f = −∞

exp

i 2π ft T

(3.74)

The constant C f measures the amount of discrete frequencies ( f = p/T ) that are combined to represent the periodic function φ (t) as:

C

f

=

1 T

T 2

³ φ (t )

−T 2

exp − i 2π ft dt T

(3.75)

Dynamic Trip Modelling 117 As the shopping period increases, the fundamental frequency ( fo = 1/ T ) decreases and so the discrete frequencies come closer together until as T→∞ they equal a continuous spectrum, namely: lim f a = f , a = −1, 0 , 1...

T →∞

and

Δf a = f a +1 − f a =

a+1 a 1 − = T T T

(3.76)

With this assumption, multiply both sides of Equation (3.75) by T : T 2

TC

fa

³φ (t )

=

−T 2

and as T→∞

ϕ ( f a ) = lim ( TC T →∞

exp − i 2π f a t dt T

(3.77)

∞

fa

³

) = φ ( t ) exp − i 2π f a t dt

(3.78)

−∞

Likewise, Equation (3.74) produces:

φ( t ) =

∞

¦ TC

f = −∞

f

§ i 2π f t· 1 exp ¨ ¸ © T ¹T

(3.79)

As T → ∞, then T C a= ψ (f), fa = pa /T = f and Δf = df. This summation may approach an integral to yield: ∞

φ ( t ) = ³ ϕ ( f a ) exp(i 2π f a t ) df a

(3.80)

−∞

Equations (3.78) and (3.79) define a Fourier transform pair of a frequency spectrum for a population of consumers. We have used the imaginary part of the distribution as a mathematical tool, which we will discard in the empirical analysis (because shopping trips are in real time). In the end, it simply means that we are double counting the frequency distribution. The good thing is that this result can now be used for functions that are not required to be periodic. Now φ (t) is the inverse Fourier transform of ψ (fa ), and ψ (fa) is the Fourier transform of φ (t). This links frequency formally into the time analysis of periodic and non-periodic shopping behaviour. This

118

C hapter 3

procedure demonstrates the change that occurs under aggregation where the mathematical function of summation is replaced by an integral, and discrete frequencies of the individual are replaced by an aggregate frequency spectrum. However, our limits are not to infinity but to 168 hours per week, so the question is: how many consumers do we need to construct the aggregate demand wave to assume the integral? The shopping period T s for trading in Sydney before 1990 is assumed to be 49.5 hours per week where the shops are allowed to operate, although this is now more variable, with shopping hour liberalisation. As such, fa= 1/50 or 0.02 (where the mean shopping time is assumed to be one hour) and aggregate periodic shopping behaviour under such small frequency increments may be assumed to be continuous. The Fourier transform has a number of properties defined by a set of theorems which are of interest and will be stated here without proof (see Weaver, 1983). Theorem 3.1 If both functions r(t) and g(t) have Fourier transforms given by Ψ ( f ) and G( f ), then the function h(t) = a r(t) + b g(t) has a Fourier transform given by aΨ ( f ) + b G( f ). Theorem 3.2 If the function r(t) has a Fourier transform given by Ψ (f ), then the Fourier transform of the function r(t-a) is given by Ψ (f ) exp-2Ψ ifa. This theorem indicates that under aggregation, a phase shift in the shopping time from a range of individual shopping periods gives rise to a sinusoidal type of modulation in the frequency domain. Such a modulation under aggregation produces group behaviour where the frequency of trips could be greater or less than individual trips. This suggests that aggregation can produce externalities (such as parking and traffic congestion) that can change the overall demand function. Group and individual demand are not identical. Theorem 3.3 If the function r(t) has a Fourier transform given by Ψ ( f ), then the function g(t) = r(t) exp- 2Ψ iat has a Fourier transform defined byΨ ( f-a). The duality of the transform pairs, indicated here for a sinusoidal type modulation in the time domain, results in a phase shift in the frequency domain. Therefore, a change in the shopping hours over a week will produce a shift in the trip frequency for a commodity or patronage at a shopping centre. This will be an important hypothesis to test in the empirical work in Chapter 4.

Dynamic Trip Modelling 119

Theorem 3.4 If the Fourier transform of r(t) = Ψ (f ), then the Fourier transform of rΨ (at) is given by {1/|a| Ψ ( f/a)} for ‘ a’, a real number not equal to zero. This may be significant if r(at) is defined in terms of travel times. The ‘a’ coefficient represents a scale change in the frequency distribution. The concept of convolution is found throughout most physical sciences and, in systems theory, it is central as the impulse response integral. It is determined by rotating a function about the origin and then sweeping it from +∞ to −∞ Weaver (1983) defined convolution as a law of composition that combines two functions to yield a third. The convolution of two functions r(x) and g(t), r(t)*g(t) is defined as: ∞

r ( t )* g ( t ) =

³ r (ξ ) g (t − ξ )dξ

(3.81)

−∞

Convolution is associative and commutative. The Fourier transform of the convolution of two functions is defined by two theorems, namely: Theorem 3.5 If both r(t) and g(t) have Fourier transforms given by Ψ ( f ) and G ( f ) respectively, then the Fourier transform of h(t) = r(t)*g(t) is given by Ψ ( f ) G ( f ). Theorem 3.6 If both r(t) and g(t) have Fourier transforms given by Ψ(f) and G (f) respectively, then the Fourier transform of the product of these two functions h(t) = r(t)g(t) is given by H( f ) = Ψ ( f )*G( f ). The symmetry of the Fourier transform is also of interest in this application to spatial behaviour and this is summarised in the following theorem: Theorem 3.7 Assume that the function r(x) has a Fourier transform given by Ψ(f), then: (i) if r is a real function Ψ ( f ) = Ψ *(-f ); and (ii) if Ψ is a real function, then r (t) = r*(-t). The Fourier transform of any real function is Hermitian if Ψ (f ) = Ψ *( -f ) and, as such, the amplitude spectrum is always even. For this reason of symmetry, the negative frequency range of the Fourier transform contributes simply a factor of 2 to the analysis and, in the case of shopping trips, the results have to be divided to only capture the positive contribution.

120

C hapter 3

The convolution of r and g is the area under the product curve and shows the degree of overlap between the two distributions. This seems relevant to the study of behaviour in a retail hierarchy when market boundaries can overlap. A second concept is the cross-correlation of the two functions r and g. This is obtained by first taking the complex conjugate of the second function g and then displacing it with an amount -t. Unlike convolution, the cross-correlation function is not commutative, namely:

r(t) # g(t) ≠ g(t)#r(t)

(3.82)

This means that if r(t) is the forward journey and g(t) is the return journey, the cross correlation between both parts of the journey are not equal, even if they follow the same spatial path. The mathematical reason is that time is complex with an imaginary component. The translation is that the return journey along the time path to the residence is not a physical time reversal but relative to the cycle of situations to the time boundary of Sunday. Within a cycle of situations per week, shoppers are returning to a time boundary of Sunday to start the week. Time is therefore relative and this return to the boundary each week is also imaginary, in the sense that it is repetitive, rather than contravening some physical law of the arrow of absolute time. This relationship is perhaps easier to grasp when we look at flows of the Internet, globally, through different positive and negative relative time zones. When a function is cross-correlated with itself, it is defined to be autocorrelated and the integral is given as: r (t ) # r( t ) =

∞

³ r( ξ )r * ( ξ + t )dξ

(3.83)

−∞

The calculation of the Fourier transform of a function that is the cross-correlation of two other functions, is defined by the following theorem: Theorem 3.8 If both functions r(x) and g(x) have Fourier transforms given by Ψ ( f ) and G( f ), respectively, then the cross-correlation function h(t) = r(t) # g(t) has a Fourier transform given by Ψ ( f )G*( f ), where G*( f ) is the complex conjugate of G( f ). Therefore, when r is autocorrelated, the Fourier transform is defined as:

ℑ[ r( t )# g( t )] = Ψ ( f )Ψ * ( f ) = Ψ ( f )

2

(3.84)

Dynamic Trip Modelling 121 The squared density function eliminates the negative frequency components, which may make it suitable for application to a probabilistic interpretation of interaction. We do not have to deal with the idea of negative frequency distributions in our shopping trip. This is similar to what happens in quantum mechanics with the square modulus of the wave function as a probability density function to locate the position of an electron. The cycle of situations of the trip may therefore be complex, but we can postulate that this square modulus is a probability density. We can use mean values of quantities and measure how long consumers shop in a mall. Hence, such numbers must be real and this population density function is also real. The trip cycle can be formalised to locate the consumer within the shopping week as : = P([ 0 ,T ], x ) T

=

³

T

³

(3.85)

2

P( t , x )dt = Ψ ( t , x ) dt

0

0

We can normalise this condition (that is, there is a probability equal to one of a consumer shopping for food and groceries) by this expression. This is independent of space. In other words, consumers will shop for food no matter the location of their trip origin. The normalised time demand function is then: ∞

³

∞

∞

−∞

−∞

³

P( t , x )dt = Ψ ( t , x )*Ψ ( t , x )dt =

−∞

2

³ Ψ ( t, x ) = 1

(3.86)

For an individual consumer, the normalised time demand function can be obtained for a shopping period from 0 to T from the time solution of the supermarket equation, namely:

φ n = A sin

nπ t T

(3.87)

and using Equation (3.86) 2

∞

T

§ n π t· ¸dt © T ¹

2 2 ³ Ψ ( t , x ) = A ³ sin ¨

−∞

0

= A2

T 2

(3.88)

122

C hapter 3

and if

A= 2 T

φn =

2 nπ t sin T T

(3.89)

The time demand function for n = 3 states that an individual consumer will have equally likely visits to three centres over the trading week (Figure 3.16).

ψ 3 (x)

I ψ 3 (x) I 2

Figure 3.16 The Demand Wave φ3 showing the Equal Likelihood of Visiting Three Equally-sized Centres (n =3) over the Trading Week We have treated an individual time demand wave but how do we deal with the aggregate wave to a shopping centre? This is done by using the Fourier transform of how aggregate consumers φo construct their shopping week in terms of trips under the time-discounting assumption, namely: T

Ψ ( f ) = ³ φ o t ) exp( −i 2π f o t ) dt

(3.90)

−T

For a general shopping wave defined by φ =φo cos 2π fo t, that is independent of opening and closing times through a full cycle of situations for complex time, the Fourier transform is: T

Ψ ( f ) = ³ φ o cos 2 π f o exp( −i 2π f t ) dt −T

which upon evaluation yields

(3.91)

Dynamic Trip Modelling 123

Ψ( f )=

φ o ª sin 2π ( f o + f )T sin 2π ( f o − f )T º + « » 2 ¬ π( fo + f ) π( fo − f ) ¼

(3.92)

The negative frequency range is not necessary for the application to shopping trips to malls, so it is omitted from the model. Therefore, the trip frequency distribution over the week for time-discounting consumers is:

Ψ ( f ) sin 2π ( f o + f )T = φo 2π ( f o + f ) or

ϕ( f ) = φ o sin c[2( f o + f )T ] where sin c =

sin π t πt

(3.93)

The graph of Equation (3.93) is shown in Figure 3.17. The greatest amplitude is at fo equal to the once a week shopping trip f1. This distribution can fit into a gaussian wave packet over time and we can interpret the distribution probabilistically by squaring ψ ( f ) (and eliminating the negative frequencies). In other words, the greatest probability density for period shopping is the once-a-week trip to buy food and groceries.

ϕ (f) sinc x

ϕ (f) for T = 1.0 T

/ ϕ (f) / 2 for T = 1.0

ϕ (f) for T = 0.5

-3 -2

-1 0

1

2

3

f

f1/4 f1/2

f1

f2

f3

f7

Figure 3.17 (a) The Graph of a sinc Function and the Gaussian Wave Packet (dotted line) and; (b) the Gaussian Wave Packet for T = 0.5, 1.0 and the Probability Density Function for T = 1.0 (dotted line) (adapted from Morrison, 1990)

124

C hapter 3

The aggregate solution for the supermarket equation (Equation 3.51) can be rewritten, if the time function is a complex variable (Baker, 2000a), as:

φ = φ o exp( fx − iat )

(3.94)

The group wave is constructed from an infinite number of individual demand waves with infinitesimally differing wave numbers. The amplitude function φ (a) determines the mixing of the time demand waves in the group. The normalised function in terms of probability distribution is:

Ψ (t,x ) =

1 A

T

³ A( f ) exp i(ifx − at) df

(3.95)

0

where A( f ) represents the Fourier transform of this group time demand wave (Morrison, 1990). The function is characterised by its centre, namely, fo, the once–aweek shopping trip. The individual demand waves have different wave lengths and consequently, there should be constructive interference in the longitudinal wave, noticeable during peak times (such as, a gridlock in carparking). The extent of the interference increases as⏐t – T» increases, so the population density (the amplitude) decreases with increasing⏐t – T» . This means that if trading hours are increased then there should be a reduction in the population density for the once-a-week food and grocery shop. A Fourier transform of the time-based solution of this equation is a gaussian wave packet of the form: p2 φ ≅ φ o exp− (3.96) 2T With increasing shopping hours, there is a shift towards higher frequency trip behaviours (2fo, 3fo , . , 7 fo) per week (Figure 3.18). This result can be tested for the Sydney Project from samples taken under trading regulation (1988/89) and deregulation (1996/98) at the same centres on the same relative days of the year (Figure 3.19). For aggregate samples, this predicted shift was observed over the decade, where there was a greater propensity for higher frequency trips in 1996/98 as trading hours increased from 49.5 hrs to an average of 63.6 hrs (see Chapter 4). There is less once-a-week shopping and more infrequent trips (once-a-fortnight and once-a-month).

Dynamic Trip Modelling 125

/ ϕ (f) / 2

/ϕ ( f) /2 f o r T = 5 0 h r s (re g u la te d )

/ϕ (f) /2 fo r T = 6 0 h rs (d e re g u la te d ) /ϕ ( f) /2 f o r T = 7 0 h r s

f

f1 / 4

f1 / 2

f1

f2

f3

f7

Figure 3.18 The Probability Density of the Weekly Grocery Trip as a Gaussian Wave Packet with the Shift Towards More Frequent Trips with Extended Shopping Hours

40

1988-89 1996-98

Percent

30

20

10

0 0

1

2

3

4

5

6

7

F re q u e n c y

Figure 3.19 The Higher Frequency Shift with the Extension of Shopping Hours in the Sydney Project from Regulated Hours in 1988/89 to Deregulated Hours in 1996/98

126

3.6

C hapter 3

Estimating Shopping Centre Hours

One of the important applications of the solution to the supermarket equation is an estimation of the mean trading hour boundary for sampled shopping distributions. This is only a relative measure, but it should give an indication of whether there is a current undersupply or oversupply of hours relative to actual aggregate consumer spatial demand for shopping floorspace. Alternatively, the time boundary can also be deduced by finding numerical solutions of T from the Fourier transforms of frequencies developed in Baker (1994a). How comparable are these estimates, since the possibility of two methods of calculation should strengthen the decision-making process? The original idea, derived from Fourier’s theorem, was to use the theoretical distribution, based around once-a-week shopping, to assist in the assessment of whether a shopping sample showed periodic behaviour. The central tenet of this physical analogy is that any periodic function of time can be decomposed into constituent frequencies made up of integral multiples of a fundamental shopping frequency f (assumed to be once-a-week shopping). The frequency distribution is then f/4 (once-a-month), f/2, f, 2f, 3f to 7 f (seven days-a-week) shopping. This coincidence is therefore evidence for shopping periodicity. The theoretical distribution was only calculated for the positive frequencies (Baker, 2000a) by: § sin 2π f t · ¸ ¸ © 2π f t ¹

Φ ( f ) = φ ¨¨

(3.97)

where Φ ( f ) is the patronage index component (for example, once-a-week or once-amonth shopping) and φo is the maximum population intercept from the gravity distribution, both standardised for the number of consumers (per hundred). The matching of the maximum values of the Φ ( fo )index for once-a-week shopping, with the φo index, could be based on the mathematical result that the Fourier transform of a gaussian function is in itself gaussian (Weaver, 1983). If the frequency distribution follows a normal distribution, there is a corresponding gaussian time distribution of exp(-βt2). The self reciprocity is assumed to apply between distributions based on Φ ( fo) and φo(t). In other words, the closer the theoretical and empirical distributions, the more likely the aggregate distribution is periodic and the stronger the timediscounting behaviour. For example, Figure 3.20 shows the theoretical and empirical intra-centre frequency f distributions for samples at MarketPlace Leichhardt (8/12/88A) and Bankstown Square (23/3/89M). The community planned shopping centre (MPL) substantially showed a normal distribution of trip frequencies and a close approximation between predicted and sampled distributions (a Chi-square test showed possible statistical correspondence). Conversely, the planned regional

Dynamic Trip Modelling 127

shopping centre (BS) showed a marked difference between the theoretical and empirical distributions (although still approximating a normal distribution), particularly underestimating the once-a-week shopping and the lower frequency trips to the centre. The statistical conclusion was that Bankstown Square did not possess significant periodicity in the aggregate distributions to gain sufficient self-reciprocity assumed by the Fourier transform. This suggested that there was substantially less time-discounting behaviour at the regional shopping mall in the 1988/89 data set. In Baker (1994a), the relationship between the inter-locational and intra-centre frequencies had not been fully understood and the Fourier testing in the Sydney Project assumed f = 1/50 or 0.02 for all centres (namely, for the once-a-week visit, the average time spent shopping was 60 minutes from the 49.5 hrs/week period). It has since been realised that f = 0.02 was not a constant and that it increased with centre scale. Fortunately, this does not affect the theoretical distribution because of the nature of the sine function. For example, in the calculation for MarketPlace Leichhardt (8/12/88A), the theoretical frequency components, using f = 0.02 in Baker (1994a), produced a once-a-week φ (1) value of 32.6, whereas the actual mean value of f = .034/week yielded a near-identical 32.5. This correspondence is similar for the other samples, so the ranking in Table 2 in Baker (1994a), has basically not changed. The only cautionary note is that the calculation is done in degrees not radians. The estimation of the trading period relative to consumer demand can now be calculated by Equation (3.60) from the supermarket equation (SE) and compared to the Fourier analysis (FA) values (Baker, 2000a). The formulas used in this comparison are in the following form:

SE Estimation § η · ¸ Ta = ¨¨ ¸ © 4 fa ¹

FA Estimation

Sample

§ 1 · ¸ + ¨¨ ¸ © 4 fa ¹

Parking

§ Φ ( f1 ) · ¸2π f a sin 2π f a T = ¨¨ ¸ © φo ¹

(3.98)

(3.99)

128

C hapter 3

MarketPlace Leichhardt 12/8/88 Afternoon

35

33.1

32.6

30 25 20

16.3

16.3

16.3

13.1

15 10.8 10

8.1

8.1

5.4

6.9

5.6

6.2

5

5 0 Theoretical

Empirical Frequency

Bankstown Square 3/23/89 Morning

45

40.1

40 35 30 25

21.2

20

10

10.6

10.6 7

7

9.1 5.3

5 0

15.2

14.2

15

4.6

4.1

3.5

0.6 0.25

0.5

1

2 3 Theoretical

4

5

0.25

0.5

1

2 Empirical

3

4

5

Frequency

Figure 3.20 The Theoretical and Empirical Frequency Distributions for MarketPlace Leichhardt (8/12/88A) and Bankstown Square (23/3/89M) (Baker, 2000a) Those samples in the Sydney 1988/89 data set that exhibit periodic behaviour from comparisons between the theoretical and empirical distributions should also show some degree of correspondence in both trading hour estimates. For non-periodic samples, the estimates should diverge, since Equation (3.98) is no longer an appropriate approximation. For example, MarketPlace Leichhardt (8/12/88A) yields SE and FA trading hour estimates per sample: § η · ¸ Ta = ¨¨ ¸ © 4 fa ¹

Sample

§ 3.7 · Ta = ¨ ¸ 4 × . 023 © ¹

§ 1 · ¸ + ¨¨ ¸ © 4 fa ¹

Sample

Parking

1 · § +¨ ¸ 4 × .023 ¹ ©

Parking

= 40.2 + 10.9 = 51.1hours/week

Dynamic Trip Modelling 129 § 33.1 · sin 2π × .034T = ¨ ¸ 2π (.034 ) © 37 .5 ¹

T = sin (.2161T) = .1907 = 50.9 hours/week1.

This matching from both methods is very encouraging and should not be surprising since the community centre MarketPlace Leichhardt (8/12/88A) does show periodic behaviour in the Fourier definition and so we should expect some agreement between both estimates. The SE and FA values are listed in Table 3.4 with a periodicity ranking according to a matching Chi-square statistic (Baker, 1994a). Those samples exhibiting significant periodicity and ‘small centre’ behaviour had reasonable concurrence in the estimates, but diverged as the incidence of ‘large centre’ behaviour increased from either centre scale or a pre-Christmas rush effect. The regional Bankstown Square 23/3/89 Morning sample scored quite highly in the periodicity test, but this can be accounted for by the high incidence of grocery shoppers (26%) in the sample. The SE estimate was fairly consistent, in that the demand for extended hours was greater at regional PSCs over community PSCs, presumably because there is greater comparison shopping undertaken at larger centres necessitating a longer shopping period. Periodicity Rank Sydney Samples 1988/89 AM 23/3/89M ML 8/12/88A WB 23/3/89M Critical Value (Periodicity) BS 23/3/89A WB 15/12/89A BS 23/3/89M AM 23/3/89A WC 23/3/89M WB 15/12/88M

FA Estimation Trading Hours/week 56.0 51.1 56.5

SE Estimation Trading Hours/week 50.2 50.9 56.7

63.5 66.6 99.0 56.6 110.0 47.2

65.1 51.1 58.1 65.0 61.7 62.9

Table 3.4 SE and FA Estimates of Mean Trading Hours for the 1988/89 Sydney Data Set (Baker, 2000a) Note, there is no new information provided by converting η and fa to weekly values in the SE estimation, since the inter-location shopping frequency used for this task is cancelled in both the numerator and denominator. This is why fa is 0.023 in the SE estimate and 0.034 (0.023 ×1.5 trips/week) for the FA calculation. 1

130

C hapter 3

There is no socio-economic basis in this estimation. The exception in the consistency of the SE estimates occurred at the sub-regional Westfield Burwood pre-Christmas rush sample in the afternoon. However, Baker (1994a) concluded that this centre on that day exhibited a much higher proportion of space-discounting behaviour as shoppers tried to minimise shopping time in gift-seeking. Therefore, the ‘supermarket’ equation was not the appropriate differential equation for this sample and this may account for the deviation in the estimate. 3.7

Two Dimensional Space-time Modelling

The analysis and discussions so far have concentrated on space-time behaviour in one dimension. The extension into the two dimensions of geographic space follows basically the same mathematical procedures. Whilst the empirical testing will be undertaken in one dimension for simplicity, this section is included purely to show how to deal with the supermarket equation and Fourier analysis in more than one dimension. This involves interpreting population assignments in rectangular grids or through concentric aggregation with polar co-ordinates. The space-discounting equation in geographical space can simply be written for a population function for a two-dimensional grid as: § ∂ 2φ( x , y , t ) ∂ 2φ( x , y , t ) · ∂φ ( x , y , t ) ¸ = D ¨¨ + ¸ ∂t ∂x 2 ∂y 2 © ¹

(3.100)

Shopping patronage is assigned to a market area from an initial population function φ(x,y) at a mall. It is assumed that the market boundary is represented where there is zero population patronising the centre. Therefore, suppose that these market boundary conditions L and Q are defined as (Boas, 1966):

φ (x,0, t) = 0 φ (x,Q, t) = 0 φ (0,y, t) = 0 φ (L,y, t) = 0 φ (x,y, 0) = f(x, y), f(0, y) = f(L, y) = f(x, 0) = f(x, Q) = 0

(3.101)

The resulting differential equation is solved by taking the finite sine transform of both sides of this equation with respect to x, applying the boundary conditions f(0,y,t) = f(L,y,t) = 0 and then doing the same for y. The solution of this equation is then:

Dynamic Trip Modelling 131 ∞

ª πn 2 πq 2 º π q y π nx ) +( ) » t sin sin ; Q ¼ Q L ¬ L

∞

φ ( x , y , t ) = ¦¦ Anq exp − α «( n =1 q =1

(3.102)

n = 1,2...; q = 1,2...

This then describes the population function at t within the market boundaries and expresses the space-discounting assumption in two dimensions. The properties of a Fourier transform can be extended in two dimensions for the analysis of behaviour in geographic space. For the most part, the properties for twodimensional analysis carry directly over from the previous one-dimensional theorems in Section 3.5. The Fourier transform pair of a space-discounting function f(x,y) in geographic space is defined as:

Ψ ( q,r ) =

∞ ∞

³ ³ φ( x , y ) exp− 2π i (qx − ry ) dxdy

(3.103)

− ∞− ∞

φ( x , y ) =

∞ ∞

³ ³Ψ ( q , r ) exp− 2π i(qx − ry ) dxdy

(3.104)

− ∞− ∞

In empirical work, the population density may be calculated either in rectangular or circular units. The unit rectangle is defined by:

rect a,b (x,y) = 1, -a < x < a, -b < y < b = 0, otherwise

(3.105)

This function can be written as the product of two population demand pulses, namely:

rect a,b (x,y) = pa(x)pb(y)

(3.106)

which yields a Fourier transform

Ψ ( q , r ) = 4 ab sin c( 2π aq ) sin c( 2π br )

(3.107)

For circular two-dimensional functions, these radial symmetric functions can be expressed in terms of polar co-ordinates. The function φ (x,y) can be converted from Cartesian co-ordinates into polar co-ordinates φ (τ,θ) using the transformations x = τ cos θ and y = τ sin θ where τ is called the radius from the shopping centre and θ the travel angle. The circular function φ (x,y) = x2+y2 is radially symmetric since it is independent of the angle θ. Therefore, the Fourier transform of a radially symmetric function is determined by first converting the transform variables q and τ into polar

132

C hapter 3

co-ordinates by means of the transformation equations q = f cos ζ and τ = f sin ζ . Using the (x,y) and (q,τ) transformations and Equation (3.102) yields: ∞ π

Ψ ( f ,ζ ) = ³ ³ φ ( τ ,θ ) exp− 2π ifτ (cos ζ cos θ + sin ζ sin θ ) τ dθ dτ

(3.108)

0 −π

It is assumed that φ is a radially symmetric function around the shopping centre and can be written solely in terms of τ. As such (Equation 3.108) can be rewritten as: ∞

π

0

−π

Ψ ( f ,ζ ) = ³ τφ( τ ) ³ exp− 2π ifτ cos( θ − ς ) dθ dτ

(3.109)

For this problem, the integral definition of a zero order Bessel function is used, namely: Jo( x ) =

π

1 2π

³ exp− ix cos( θ − ζ ) dθ

(3.110)

−π

and the above equation can be rewritten as: ∞

Ψ ( f ) = 2π ³ τφ( τ )J o ( 2π f τ ) dτ

(3.111)

0

The zero order Bessel function is a special case of the nth order Bessel function whose series expansion (for integer values of n) is given as: Jn( x ) =

∞

(− 1)q

§x·

¦ q! (n + r )! ¨© 2 ¸¹

n+ 2q

, n = 1,2...

(3.112)

q =0

The Bessel functions are oscillatory and are like damped trigonometric functions (see Figure 3.21). It should be noted that these functions vanish at an infinite sequence of values of x. In shopping centre analysis, the positive values of x for which Bessel functions vanish are called zeros of the functions. These are the trading limits of the daily shopping cycle. Unlike those of trigonometric functions, the zeros of Bessel functions are not equally spaced along the x-axis. This becomes a problem in concentric aggregation in determining a population density function because we assume one kilometre concentric bands surround the mall. Note that in Equation (3.111), ψ is only a function of the radius r and thus the transform ψ is also a radially symmetric function. This special case of the two-dimensional Fourier transform is known as the Hankel transform. The inverse transform is defined as:

Dynamic Trip Modelling 133 ∞

φ ( τ ) = 2π ³ fΨ ( f )J o ( 2π fτ ) df

(3.113)

0

J1(X) 1

0.5

J0(X)

Y1(X) Y0(X) 0

0 5

5

10

10

-0.5

-1

Figure 3.21 Bessel Functions of Order Zero (left) and One (right) Therefore, Hankel transforms can be calculated for circular functions of shopping density around a shopping centre. For example, a circular function is defined as:

circ(τ) = 1, t < T 0, otherwise

(3.114)

In Cartesian co-ordinates this function is expressed as:

circ(x,y) = φ0, 0,

x2 + y2 < T2 otherwise

(3.115)

Using Equation (3.111), the Hankel transform of this function is defined as: a

Ψ ( f ) = 2πφ o ³ τ J o ( 2π fτ ) dτ 0

which can be evaluated to give

Ψ ( f ) = φo

TJ 1 (2π f T ) f

(3.116)

134

C hapter 3

The result is a first order Bessel function (Figure 3.21) that defines the population density for that frequency at a unit radius from the shopping centre. This area of work in two-dimensional analysis is still an active area of inquiry, particularly in determining ways of dealing with the assignment of empirical data with concentric aggregation. It has the advantage of being able to overlay ‘region by radius’ methods from GIS packages of census material. The challenge is still to determine a way to correctly assign the width of the zones according to the above two-dimensional treatment. 3.8

Estimating Market Penetration with an Extension of Shopping Hours

Norris (1990) states that there has been little development since the 1960s in the techniques used (such as, in gravity modelling) for forecasting the impact of retail developments. However, the time-space convergence within the gravity coefficient suggests that any changes in time boundaries have immediate effects on spatial behaviour. Furthermore, the non-linearity within some of the socio-economics of planned retail hierarchy means that the impact could be uneven between different order centres. The implication of the space-time convergence at a shopping centre is that a substantial extension of trading hours is equivalent (in terms of market penetration) of adding new floorspace to planned shopping centres with supermarket anchors. The corollary of this outcome is that an impact assessment should also occur for any significant extension of shopping hours. The supermarket equation and its solutions allows for the development of a dynamic market area analysis. The extent of the primary trade area of a planned shopping centre is a function of its trading hours. This model, then, allows for the estimation of market penetration if the hours were extended by varying degrees. This is a major advance and very relevant to decision-making by centre management in a deregulated trading hour environment. The next step is to show, with an example, how to put into practice the theoretical and empirical relationships and link them to population census and household expenditure data. An appropriate form of the gravity coefficient for this task can be formally stated (for β = c f 2/N; Baker, 1997, 2000a) as:

β=

cm 2 16 NT 2

(3.117)

where N is the number of destinations at the PSC, m the mean destinations visited per week and c a constant. This expression permits a time adjustment to be made for β and allows for a forecast of the impact that a substantial extension of trading hours could have on the surrounding market area. It is a statement of the dynamic nature of the market areas of planned shopping centres. Emerging questions are: ‘To what

Dynamic Trip Modelling 135 extent will the primary trade area expand from extending shopping hours?’ or ‘What hours are necessary to capture high disposable income households currently beyond the primary trade area?’ For example (Baker, 2000a), MarketPlace Leichhardt in the 1988/89 Sydney data set, on the pre-Christmas 8/12/88 Afternoon sample, had β = 0.84 and a mean trip distance D = 2.6km for uniform trading hours of 49.5 hours per week (Baker, 1994a). It is assumed that the mean trip distance to the shopping centre represents the boundary of the primary trade area. Using the ‘region by radius’ command of the Supermap Census 91 program, this concentric market area can be mapped and household characteristics aggregated from the Australian Population Census (1991). Since the population densities and gravity coefficients are calculated in the Sydney 1980/82 and 1988/89 data sets by concentric aggregation in one kilometre bands, it is assumed that the modifiable area problem in translating census data is not significant. Accordingly, there were, in the MarketPlace Leichhardt primary trade area of radius 2.6km, 36,289 households, including 5,529 households (or 14.9%) with income greater than $A60,000 (substantially lower than the Sydney average of 17.9%). Suppose centre management wants to ascertain what impact there would be if MarketPlace Leichhardt extended its mean shopping hours from 49.5 to 60 hours per week under trading hour deregulation. Equation (3.117) yields a time-adjusted gravity coefficient for an extension of 10.5 hours to T= 60 hours per week:

β=

111,153 × (4.7 16 × 7

)2 2 0 × (60 )

= 0.61

where ‘small centre’ behaviour with carparking is assumed, that is, n = 3.7 + 1 = 4.7 ; and c = 111,153 from β N / f 2 or 0.84 × 7 0/(.023)2. This produces a mean trip distance of D = 3.7km (Figure 3.22). The N = 70 destinations would have to be adjusted for shop vacancies (for example, a 10% vacancy in the shopping centre would amend β to 0.68 and D to 3.4km). If there is 100% occupancy, then a drop in the gravity coefficient from 0.84 to 0.61, increases the market penetration of the primary trade area from an average of 2.6km to 3.7km from an increase in mean trading hours from 49.5 to 60 hours per week. The interesting point is that this community planned shopping centre, with time now substituting into the centre floorspace, has trip distance characteristics of the larger sub-regional centre, Westfield Burwood (N = 90 destinations), which averaged β = 0.62 and D = 3.9km (Baker, 1994a). Therefore, a 10.5 hour extension of average trading hours could be equivalent to the addition of 20 new shops in terms of market penetration. This translates to 5,789 sq m of equivalent floorspace (Figure 3.23) and represents a predicted additional $A26.8 million in turnover. For MarketPlace

136

C hapter 3

Leichhardt, aggregate turnover was estimated at $A97.6 million (mean turnover to floorspace ratio of $A4,625 per sq m in 1996/97 dollars), so the 10.5 extension of hours is equivalent to a 27.5% increase in turnover. This is the best case scenario, since it assumes that MarketPlace Leichhardt has no competition from the surrounding retail hierarchy and such estimates have to be treated with caution (Norris, 1990). These figures are still illustrative but there is a prediction that extended hours would substantially increase turnover at MarketPlace Leichhardt. 1.4

Gravity Coefficient

1.2

1 .8 .6 .4 .2

Y = 1.324 - .191 * X; R^2 = .785 0

0

1

2

3 4 PO Distance

5

6

7

Figure 3.22 The Relationship between the Gravity Coefficient and Mean Trip Post office Distance in the Sydney 1980/82, 1988/89 and 1996/98 Data Sets

The time-adjusted gravity coefficient of 0.61 is equivalent to a mean trip distance D = 3.7 km. Using the ‘region by radius’ option of Supermap on Census 1991, this increase in primary trade area from a 10 hour extension of hours accesses a further 33,783 households, including 5,548 households with income over $A60,000 (16.4%). This forms the potential time ‘doughnut’ for the market area in this ‘what-if ’ analysis (Figure 3.24). Affluent households have a greater propensity to travel from the 3-4km zone to a community planned centre (as well as other regional centres) and management should note that the proportion of such households in the ‘doughnut’ was still 1.6% below the Sydney average. Potential household expenditure analysis could then be undertaken and the HDI districts targeted by advertising for promotions and sales. With the potential of doubling the number of high income households, the extension of shopping hours at the centre would seem to be an appropriate strategy.

Dynamic Trip Modelling 137

Retail Floor Space ('000 sq m)

70 60 50 40 30 20 10 0 20

40

60

80

100

120

140

160

180

200

220

Number of Retail Outlets

Figure 3.23 The Relationship between Retail Floorspace and the Number of Retail Destinations: Y = – 0.811+0.313X , R 2 = 0.92 (Source: Annual Reports 1995: Westfield, General Property Trust, Stockland, Schroders, National Mutual)

1 km

F iv e D o c k R o z e lle

M a r k e tp la c e L e ic h h a rd t

A s h fie ld

S y d n e y C B D (3 km ) P a rra m a tta R d

N e w to w n D u lw ic h H ill N u m b er of H ig h In co m e H o us e h o ld s

1991 Census of P o p u la tio n a n d H o u s in g

0 18 24 34 46

17 - 23 - 34 - 46 - 73

Figure 3.24 The Aggregate Household Time ‘Doughnut’ for MarketPlace Leichhardt 8/12/88A through an Extension of Trading Hours from 49.5 to 60 hours per week

138

3.9

C hapter 3

Stochastic Space-time Trips (after Kleinrock, 1975)

The question must now be raised as to whether it is possible to relate space and timediscounting shopping strategies within the one theoretical framework. So far, this research has concentrated on aggregate consumer behaviour and the best way to link both strategies is to look at individual trip behaviour to a shopping mall. This will allow us to continue the probabilistic treatment started with the derivation of the supermarket equation in Section 3.4. However, this time the starting point is the space-discounting assumption of time-minimisation in a shopping trip. Before we come to this fundamental question, the background and fundamentals of a disaggregate trip will be reviewed first. 3.9.1

INTRODUCTION

The relationship to individual behaviour can be studied if the shopping trip is viewed in terms of discrete steps rather than as a continuous population function. Each step has a transition probability in space and time. The resulting differential equation is the same as for classical diffusion, so the implication is that a space-discounting consumer can be described using traditional probability theory. Such an approach should be relevant to large centre analysis that has a higher proportion of random patronage and to low frequency trip behaviour, such as gift-seeking consumers. These ideas were first presented by Curry (1964) in his study of settlement evolution. He argued that randomness was one approach to the study of settlement distributions in a central place system. Curry realised that such an evolution fitted into a diffusion process, but preferred to interpret each step in terms of transition probabilities. Such an interpretation is valid for an intra-urban hierarchy such as shopping centres. The reason is that the solution of the conventional diffusion differential equation, has the same result as a consumer undertaking a random walk from a shopping centre with n steps per unit time. Central to this development has been the idea of Markov chains that has been applied to sequences of stop purposes (Wheeler, 1972; Kitamura,1983). A set of random variables forms a Markov chain if the probability of the next value or state depends on the current value or state and not upon any previous values. A Markov process is fully determined by the two functions P1(y1, t1) and P1|1( y2,t2, | y1,t1), where P1|1 is called the transition probability. The whole set can be reconstructed as, for example:

t1 < t2 < t , P3 (y1,t1; y2,t2 ; y3,t3 ) = P2 (y2,t1; y2,t2 ) P1|2 (y3,t3 | y1,y2,t2) = P1 (y1,t1) P1|1 (y2, t2| y1,t1 ) P1|1 (y3,t3| y2,t2 ) (3.118)

Dynamic Trip Modelling 139

Integrating this identity over y2 for t1 < t2 < t3 and dividing both sides by P1(y2,t1) yields the Chapman-Kolmogorov equation:

P1|1 (y3,t3 | y1,t1) = P1|1(y3, t3 | y2 ,t2 ) P1|1 (y2, t2| y1, t1) dy2

(3.119)

This equation is obeyed by the distribution: P1 1 (x 2 t 2 y 2 , t 1 ) =

ª ( x − x 1 )2 º exp « 2 » 2π (t 2 − t 1 ) ¬« 2(t 2 − t 1 ) ¼» 1

(3.120)

for −∞ < y < ∞ and t2> t1. If a Wiener process [p1(y1,0) = d(x1)] for t >0 alone is assumed, then this distribution is reduced to a probability distribution: P1 ( x , t ) =

ª− x2 º exp « » 2π t ¬ 2t ¼ 1

(3.121)

This is a solution for space-discounting behaviour. This process has been applied to the urban retail model by Vorst (1985) to derive a spatial interaction model with a distance decay function. The Master equation is an equivalent form of the ChapmanKolmogorov equation for Markov processes. The Fokker-Planck equation is developed from the Master equation. It takes the usual form: 1 1 ∂P ∂ {P[B(x )]} = {P[A(x )]} + 2 ∂x 2 ∂t ∂x

(3.122)

For the special conditions A(x) = 0; B(x) = 2a, this equation simplifies to the diffusion equation for space-discounting behaviour. Curry (1978b) was the first to introduce this equation into geographical analysis in his study of price random walks in the spatial economy. In a retail context, the extra ∂/∂x term describes the drift of shoppers away from the mall, perhaps to an adjoining traditional main street shopping precinct. The aim, therefore, of mall management would be to stop this drift of pedestrian movement, so expenditure remains within the centre. The above equations are all ‘relatives’ for certain stochastic conditions. What is the difference between a stochastic and random process? Kleinrock (1975) defines a stochastic process as a family of random variables X(t) where the random variables are indexed by the time parameter t. For example, the number of people in a shopping centre is assumed to be a function of time. Meanwhile, a random process describes the transfer of a consumer in some space system defined by three parameters, namely: state space, index time t and statistical dependencies among random variables for

140

C hapter 3

different index time values. For state space, if the number of shops or trip paths are finite and countable, the process is a discrete state and called a chain. If the distribution is over a finite or infinite interval, the process is in a continuous state. Stochastic processes are described by the dependencies among the random variables and this leads to a number of different types of processes. These may be summarised below (Figure 3.25).

1. Stationary Processes A stochastic process X(t) is defined to be stationary if ξx(x,t) is invariant to shifts in time for all values. For example, the location of a shopping centre is invariant over time. 2. Markov Processes A set of random variables (Xn) forms a Markov chain if the probability of the next value or state is xn+1 and depends only upon the current state xn and not on any previous values. The Markovian assumption has been applied to sequences of stop purposes (for example, Nystruem, 1967) or stop locations (for example, Lerman, 1979). In discrete-time Markov chains, state changes are defined by a set of integers, 0 1,2,...n whilst for continuous-time Markov chains, the state transitions occur at any instant. Since the past history is summarised in the current state, it cannot be determined how long the event has been in this state. As such, there are strict interpretations on the time distributions in that state, which must be exponential. Such a distribution is said to be ‘memoryless’. This interpretation creates problems in the application to consumer trip behaviour. Do consumers evaluate shopping centres solely on the boundaries and events of the current trip? For this distribution, past shopping experiences can be defined only in terms of current shopping states. Such an assumption is difficult to justify in terms of human behaviour, for consumers do remember past experiences and this helps them to evaluate current shopping strategies. Furthermore, Thill and Thomas (1987) argue that it is futile to explicitly consider the remote history of trip chaining through nth order Markov models, owing to the computational complexity and the huge data requirements. This problem has been addressed by Nystrum (1967) who described the probability distribution of the journey duration by a binomial process and Kitamura (1983) who identified a constant hierarchial order in sequencing stop purposes. The memoryless property is still a major barrier to the application of consumer trip behaviour. 3. Arrival-Departure Processes This is a special class of Markov chains in which the defining condition is that transitions take place between neighbouring states only. Such a process limits discrete state space as a set of integers where Xn = a, then Xn+1 = a–1, a , a+1 with no other integers. This is a further assumption behind space-discounting, namely that time is minimised only when trips occur to neighbouring states. The corollary to this

Dynamic Trip Modelling 141 is that the larger the number of destinations within a mall, the greater the likelihood of time-minimisation within the trip.

4. Semi-Markov Processes This is when the Markov process is generalised to include the times between state transitions to obey the arbitrary probability distributions and is not restricted to exponentially distributed times in a state. Semi-Markov processes relate to trip behaviour beyond the space-discounting assumption of a time-minimisation strategy. 5. R andom Walks A random walk is defined as a consumer moving between states in some state space where the emphasis is on location. This approach has immediate appeal to geographic research and was taken up by Curry (1964) for the analysis of settlement patterns. The major characteristic is that the next position that the process takes is equal to the previous position plus a random variable, whose value is drawn independently from an arbitrary distribution (which does not change the state of the process). The emphasis is placed on the position after n transitions, rather than the distribution of time intervals. These transitions are more important than the time it takes to complete the shopping trip. Gift shopping is more dependent on the sequencing of the shops to be visited than the time it takes to visit each shop. 6. R enewal Process This is related to a random walk, except that the idea is not to follow a consumer from state to state, but to count the transitions that take place as a function of time. It describes the time between the (n-1)th and the nth transitions, rather than a random walk, where the interest is in the state of the process. S e m i-M a rk o v p ro c e s s a r b itra r y a r b itra r y

P ij fr

R a n d o m w a lk

M a rk o v p ro c e s s

P fr

a r b itra r y m e m o ry le s s

ij

= qj - i a r b itra r y

B ir th - D e a t h p ro c e s s

P fr

ij

= 0 f o r | j- i | m e m o ry le s s

V

P ij fr

1

P o is s o n p ro c e ss λi = λ

P u r e B i r th p ro c e s s μi = 0

R enew al proc es s q1 fr

= 1 a r b itra r y

Figure 3.25 A Venn Diagram showing Set Relationships among Selected Random Processes (Source: Kleinrock, 1975)

C hapter 3

142

The relationship between these processes for discrete state systems can be seen in Figure (3.25) in the form of a Venn diagram (after Kleinrock, 1975). This figure shows that arrival-departure processes form a subset of Markov processes, which themselves form a subset of a class of semi-Markov processes. Similarly, renewal processes form a subset of a random walk process, which also are a subset of semiMarkov processes. When the common distribution for Xn is a discrete distribution, then there is a discrete-state random walk, where the transition probability pij of going from state i to state j will only depend on the difference in indices j-i (which can be denoted qj-i). A random walk that has the following characteristics is of special interest in our attempt to link space and time-discounting behaviour. This special case is referred to as the Poisson process. It has two relevant features:

1.

the distribution of time between transitions t r is a discrete stage and hence memoryless, and qj-i = 0 when | j-i | >1

2.

when qr = 1 and tr is independent of the state (thus giving a constant arrival rate)

Such a random walk lies at the centre of the diagram and enjoys the properties of each process. 3.9.2 HIERARCHICAL CLUSTERING OF SHOPPING TRIPS (AFTER MONTROLL AND WEST,1979) A random walk that is of interest in this research is one investigated by Pearson (1905). The problem involves a consumer returning from a work journey from a city and travelling a distance in a straight line to a residence. From there the consumer can shop in any direction from the residence and travel distance a, in a second straight line. The consumer repeats this process n times. The problem therefore is: what is the probability that after n periods the consumer is at a distance r and r + δ r from the CBD? In other words, what is the nature of this suburbanisation of these shopping trips? Suppose that the shopping trip lengths can vary and are denoted by a1, a2, a3,..........an and a shopping trip can be specified by these numbers and the angles θ1, θ2 , θ3 ,......θn between the successive trip directions and the x axis. The distance traversed after n periods is the magnitude r of a complex number: z( θ 1 ,θ 21 ,...θ n1 ) = r( θ 1 ,θ 2 ,...θ n1 ) exp iϕ ( θ 1 ,θ 2 ,...θ n )

(3.123)

The distribution of distances traversed in n periods is that of r(θ1,θ2 ,....θn ) which results from allowing all θj’s to take on all values (with equal probability) between 0 and 2π. The first step to mathematically determine the distribution of r, is to obtain

Dynamic Trip Modelling 143 the joint probability density function P(x,y) of x and y in geographic space. This function is the double Fourier transform of the characteristic function. As such: P( α , β ) = exp i(α x + β y ) ∞

=

³ ³ P( x , y ) exp i(α x + βy )dx

dy

(3.124)

−∞

Using the fact that the required average is obtained by averaging over all the values of θj in the interval (0,2π) with the identities:

and

x (θ 1 ,......... θ η ) = a j cos θ

j

y (θ 1 ,......... θ η ) = a j sin θ

j

(3.125)

allows for the evaluation of Equation (3.124). The result is the probability of a consumer being between r and δ r after n steps, P(r)δ r as: ∞ ­° n 2π rP( x , y )∂r = r∂r ρ ®∏ J o ( ρa j −∞ ° ¯ j =1

³

½° )¾J o ( rρ )dρ = P( r )∂r °¿

(3.126)

where for geographic space, polar co-ordinates have been introduced as:

ρ = (α 2 + β 2)1/2

r = (x 2 + y 2)1/2

(3.127)

and J0 being the Bessel function of order zero. For a space-discounting consumer, the Fourier transform of a gaussian distribution is defined as:

p(k,t) = exp (-D t k2)

(3.128)

This equation obeys the chain condition of a stationary stochastic process defining the transition between any two shopping states k as:

p(k,t) = p(k,t-t1) p(k,t1)

(3.129)

A more general transform that satisfies the chain condition is given as:

p(k,t) = exp (-b|k|α t)

which yields the probability distribution:

0< α ≤ 2

(3.130)

144

C hapter 3

P( x , t ) =

³ exp[− b k

∞

1 2π

α

]

t − ik x dk

−∞

(3.131)

Equation (3.130) defines a Levy distribution and when α = 2, it simplifies to a gaussian distribution. So the space-discounting trip behaviour is a special case of a more general distribution. Now, returning to a Pearson random walk, suppose that the aim is to calculate a shopping distribution for two dimensions (n=2) with a general Levy distribution of trip lengths. From (3.124) the shopping trip can be defined as: ∞

ϕ (t) =

³J

o (at)λ (a)da

(3.132)

−∞

which can be evaluated to be in gaussian form as: ∞

³

ϕ (t) = 2 J o ( at )(2aσ ) 2

−1 2

0

§ 1 = Io ¨ 2 2 © 4t σ

§ − a2 exp¨¨ 2 © 2σ

· ¸¸ da ¹

1 · · § ¸ exp¨ − 2 2 ¸ 4 t σ ¹ © ¹

(3.133)

where Io(m) is the zero order Bessel function of purely imaginary argument. The general Levy distribution of λ(a) is evaluated using Parseval’s theorem for the relationship between the integrals of a product of two functions and the integrals of their Fourier transforms as: V2 ( t ) =

1 2π

−1 2

t

³ (t

−t

2

−k2

)

p( k )dk

(3.134)

When this is evaluated, a shopping trip with a Levy distribution of trip lengths p(k) has the form:

p(k) = 2 exp ( - b |k| α)

0 4.26), but there is autocorrelation in the DW statistic (d= 2.708 for 1.454 < d < 2.546). This correlation suggests the strong possibility of an equivalent correspondence, but the DW statistic suggests a missing variable or that we have not constructed the relationship correctly. What we notice is that trip distance is per unit distance in the probabilistic definition so it should be divided by the average trip distance to standardise the result. Further, we have only included the shopping time (ps) within the centre in the regression. There is also a trip time from the residence (pt) to the centre and a return journey (2pt). The transport coefficient M needs the correct physical version of shopping time p, namely: p = ( ps + 2 pt ) / 60

(4.6)

This definition can now be substituted into the probabilistic form of M. Stage Two (The Second Approximation) If we add the travel time to the centre (ranging from 10.9 to 16.6 minutes in 1988/89 and 11.2 to 17.7 minutes in 1996/98), doubling it for the return journey, and standardising the mean trip frequency per unit distance, the resulting regression between the deterministic and probabilistic definitions of M is still significant but less robust, with an R-squared value of 0.52 (F = 26.10). However, the DW statistic of d = 1.682 is now within the range (1.454 < d < 2.546) suggesting, with these corrections, that there is now no positive or negative autocorrelation within the residuals. The autocorrelation in the previous step alerted us to a definition problem and we have been able to correct it. However, we notice that the statistical relationship has shifted to the left, suggesting that there is a double counting within the deterministic definition. The squared trip frequency term needs to be halved in order to define an average trip frequency between positive or negative k. The result is that M should be β = ½ k2/M when equating it with the probabilistic form.

204 C hapter 4

Stage 2 Second Approximation

12

12

10

10

M = Freq^2/Grav

M = Freq^2/Grav

Stage 1 First Approximation

8

6

4

2

8

6

4

2 Y = 1.311 + .766 * X; R^2 = .711

Y = .371 + 2.132 * X; R^2 = .492

0

0 0

2

4

6

8

10

12

0

2

4

M= (Dur)hr^2*Freq

6

8

10

12

M=(Dur+2TripT)/60^2*Freq/km

Stage 3 Third Approximation 12

M=0.5Freq^2/Grav

10

8

6

4

2 Y = .185 + 1.066 * X; R^2 = .492

0 0

2

4

6

8

10

12

M=(Dur+2TripT)/60^2*Freq/km

β=

k2 k ⎯ ⎯→ = ; p = ( p s + 2 p t / 60 ) M 2 p2

Figure 4.13 The Stages in the Evolution of a New Definition of the Gravity Coefficient (using 1988/89 and 1996/98 data): First Approximation Regresses the Deterministic and Probabilistic Forms of the Transport Coefficient M; Second Approximation Regresses a Revised and Standardised Form of M Eliminating the Autocorrelation; Third Approximation Corrects the Double Counting in the Deterministic Form of M. The Line of Perfect Match between both Forms is at 45 Degrees from the Origin (0,0)

Empirical Testing of the R ASTT Model in Time and Space 205

Stage Three (The Third Approximation) With this correction, the regression shows that there is an equivalence between both forms with an R-squared value on 0.48, no autocorrelation within the residuals, and an improvement in the line of best fit in approaching the 45 degree line of perfect match. (The coefficient in the dependent variable improves from 0.76 in Stage 1 to 1.006 in Stage 3, suggesting a better approximation between both forms than at Stage 1.) The result is that a new form of the gravity coefficient can be derived exclusively on the arguments of shopping time and frequency, namely: M =

k2 = kd × p 2 2β

(4.7)

where p = ( ps + 2 pt ) / 60 and

β=

k 2 p2

(4.8)

This equation allows for an aggregate measure of the gravity coefficient by using time and frequency averages. This is a major advance in this type of modelling and allows the RASTT model to assess aggregate spatial interaction from the time line of trips. The result is an independent estimation of the gravity coefficient, by-passing the problems of partitioning and estimating coefficients from regression. The regressing of the empirical results of the gravity coefficient from the Sydney Project with both the raw and standardised probabilistic form (Figure 4.14) supports the robustness of the new relationship. This is reassuring, where the nonstandardised data yields an R-squared value of 0.66 (F = 47.53) and DW statistic of d = 2.015, but on standardising the probabilistic variables in the equation, the Rsquared value improves to 0.84 (F = 123.76) and there is still no autocorrelation in the DW statistic of d = 2.345 (1.454 < d < 2.546).

206 C hapter 4

1.4

Gravity Coefficient

1.2 1 .8 .6 .4 .2 0 0

.25

.5

.75

1 1.25 Freq/(Dur)hr^2

1.5

1.75

2

1.4

Gravity Coefficient

1.2 1 .8 .6 .4 .2 0

0

.03

.05

.08

.1 .13 .15 .17 .2 (Freq)hr/2 (Shop T + 2Trav T)^2

.23

.25

Figure 4.14 Regression between Raw (top, R^2 = 0.666) and Standardised (bottom, R^2 = 0.84 ) Probabilistic Forms of the Gravity Coefficient. The Dotted Lines are the 95% Confidence Lines from the True Mean of the Regression

Empirical Testing of the R ASTT Model in Time and Space 207

This time underpinning of M in Equation (4.7) is now extremely helpful in understanding what we are measuring in the model. This coefficient could be alternatively defined as the propensity for the assignment of shopping and travel time to a particular centre. When this coefficient M is plotted with the number of shopping destinations within particular malls at the time of sampling, there is a striking change in the nature of the M distribution (Figures 4.15 and 4.16). In the 1988/89 data set, there is a linear increase in M with centre scale. In other words, consumers simply assign more time shopping at the larger centre. Further, the time allocated to shopping increases at a constant rate of substitution up the retail hierarchy towards the regional shopping centre. This is a significant relationship (R-squared value of 0.45; F = 5.856 > 5.59 at the 0.05 level). However, in the 1996/98 data set, the line of best fit is now a quadratic regression (R-squared value of 0.77; F = 23.03; Figure 4.16). The time assignment has become non-linear. Why? A reason for this change is that consumers have become polarised when assigning their shopping time, either devoting more time at smaller community or larger regional PSCs than the midranked sub-regional PSCs in Sydney. This supports the shopping time data in Table 4.8, where there were spectacular increases in the amount of time spent shopping at Bankstown Square per week (for example in the afternoons from 183 minutes in 1989 to 230 and 310 minutes in 1997 and 1998, respectively) and smaller increases at the community PSCs (for example, Ashfield Mall in the mornings increased from 128 minutes in 1988 to 146 minutes per week in 1998). Westfield Burwood (the subregional mall) showed little change in time allocation, indeed, there was significant decline in afternoon pre-Christmas shopping in one equivalent sample (196 minutes in 1988 to 102 minutes in 1996). It should not be surprising therefore that Westfield Burwood was redeveloped into a regional mall in 2000. There is an increasing propensity for consumers to spend more time at the bigger regional PSCs over the decade. The rate of trading hours-floorspace substitution increases with centre scale and consumers were spending more of their shopping time at regional rather than sub-regional PSCs in the retail hierarchy. The time increases at the community PSCs are more modest. The corollary then is that consumers are shopping slightly more at community PSCs and much more at regional PSCs, but allocating less time shopping at sub-regional PSCs and traditional centres (and therefore less household expenditure at these destinations). Whilst the answer is not within this data set, it is reasonable to conclude that prime candidates for market capture are trips to the lower order traditional centres and sub-regional planned centres in the surrounding trade area. This could be one reason why the vacant shop problem has manifested itself particularly in neighbourhood and town centres and accounts for the ACT Government in Australia trying to restrict the shopping hours of regional PSCs in an attempt to encourage neighbourhood shopping.

208 C hapter 4 5

M=0.5Freq^2/Grav

4

3

2

1 Y = 1.053 + .004 * X + 2.525E-6 * X^2; R^2 = .446

0 50

70

90

110

130

150

170

190

210

230

Centre Destinations

Figure 4.15 Aggregate Transfer Mobility (M) with Centre Size (Number of Shopping Destinations) for the 1988/89 Sydney Data Set 5

M=0.5Freq^2/Grav

4

3

2

1 Y = 4.098 - .043 * X + 1.848E-4 * X^2; R^2 = .768

0 50

70

90

110

130

150

170

190

210

230

Centre Destinations

Figure 4.16 Aggregate Transfer Mobility (M) with Centre Size (Number of Shopping Destinations) for the 1996/98 Sydney Data Set

Empirical Testing of the R ASTT Model in Time and Space

209

5

4.5

M=0.5Freq^2/Grav

4 3.5 3 2.5 2 1.5 1 .5

Y = 4.644 - .055 * X + 2.236E-4 * X^2; R^2 = .623

0

50

70

90

110

130 150 170 Centre Destinations

190

210

230

Figure 4.17 The Aggregate Curve for Trading Hours Regulated (1980/82 and 1988/89; 15 samples) and Deregulated (1996/98; 17 samples) Data from the Sydney Project When the data is aggregated for three periods of the Sydney project (1980/82, 1988/89 and 1996/98; Figure 4.17), the resulting distribution is non-linear (Rsquared value of 0.62; F = 23.2) with some autocorrelation in the DW statistic of d = 1.391 (1.563 < d < 2.437). The 1980/82 samples were included to try and give equal weighting between regulated (15 samples) and deregulated data (17 samples) at the sampled centres (Figure 4.15). The resulting distribution suggests the propensity for shopping time assignment is now nearly twice as great at regional than community shopping centres. The minimum trade-off in M with centre scale is 123 destinations and this equates with a critical M value of 2.44. This is close to the value assigned to be the boundary between ‘small’ and ‘large’ centre behaviour (M = 2.8; Baker, 1994a). 4.5.4 HOW HAS THE GRAVITY COEFFICIENT CHANGED OVER TIME? Apart from direct analogy, we have previously stated that there are two principle ways of deriving a gravity model of trip distributions; firstly, by optimising the assignment of trip origins to destinations using entropy maximising techniques from statistical mechanics (Wilson, 1967); and secondly, by developing a second order differential equation (the retail aggregate space-time trip or RASTT model) based on timediscounting shopping opportunities. The solution yields the gravity distribution of trips in space and periodic behaviour over time (Baker, 1985). In the latter method, space and time appear fundamentally intertwined and the ability to derive both space

210 C hapter 4 and time estimates of the gravity coefficient is a fundamental corollary of this conclusion. The empirical estimate of the gravity coefficient is not so simple, particularly in spatial aggregation, because of the problems of how we partition space and standardise data to obtain the discounting rate of distance decay (namely, the gravity coefficient). The new derivation of a time-based gravity coefficient will be extremely useful in comparing its value with its spatial determination from the regression of population densities. This new derivation means that we avoid issues such as the influence of the way we assign spatial units to determine the gravity coefficient and we can at least use it as an independent check of our spatial estimate. The estimation of a gravity coefficient from spatial partitioning therefore requires some rules of engagement. There are a number of factors to consider, namely: Geometric Structure There are two alternatives in the assignment problem. Firstly, we could just individually allocate the distance travelled by each respondent to a frequency distribution of distance decay. Secondly, we could superimpose a geometric structure upon the data and assign the trip distances according to bands rather than individually. If we take the latter option the coefficient is determined relative to geometric structure. In the Sydney Project, we imposed concentric zones away from the location of the shopping centre and aggregated according to these circular bands. The reason was that this allowed for a comparison with population census data, using a ‘region by radius’ mode relative to the centroids of the collection units. Individual data is not accessible from the Australian Census and it was thought concentric aggregation was the best compromise to link both data sets. Standardisation The assignment needs to be standardised, firstly, because of the differences in the population samples of respondents and secondly, as the area of each band increases, it is more likely to capture trip origins because of a larger spatial net. This would create an anomaly in the distance decay estimation. The easiest solution to the variable sample size is to construct a patronage index of ‘per hundred shoppers’ and so make a relative measure. The second problem requires this index to be converted into a density measure (per unit area) and this, in the case of concentric aggregation, is calculated by dividing the index by the band area. Spatial Partitioning What size do we make the band width for aggregation? Does it matter if we use 1.0km or 1.5km widths? The problem is that by assigning the data in 1.5km rather than 1.0km annular bands, the number of points for the regression decreases and the correlation with the line of best fit usually increases because we are, in fact, spatially smoothing the data.

211

Empirical Testing of the R ASTT Model in Time and Space

These issues can be investigated by using a morning sample taken at the regional mall, Bankstown Square, on the 3rd November 1980 (in Baker 1994a, Table 1 p.346). The post office distances to the shopping centre are aggregated as an index per hundred shoppers and plotted for a concentric band width of one kilometre (Figure 4.18). How do we test for distance decay? One way could be to fit a quadratic regression to the data and the curve shows this effect (Figure 4.18, top left) with a degree of explanation (R-squared) of 0.83. What does this mean? The R-squared value represents the proportion of the sum of the squares of deviations in the Y-axis (population index) about their mean that can be attributed to a quadratic relationship between the population index and distance. It is also the square of the correlation coefficient R. This is a more comprehensive measurement of correlation because it is not affected by positive and negative slopes such as those found in a quadratic relationship. So we have, in the above example, an R-squared value of 0.83, close to a perfect correlation of 1.00. However, this high correlation does not necessarily mean a causal relationship, but only a distance decay trend may exist for the population visiting the shopping centre. It is also important to note that R can change with the sample size and therefore we have to test this relationship relative to sample size. 3.5

30 Y = 30.212 - 5.072 * X + .214 * X^2; R^2 = .83

3

Y = 4.056 - .421 * X; R^2 = .802

25 20

Ln(Pop Index)

Pop Index(/100)

2.5

15

10

2 1.5 1 .5 0

5

-.5 -1

0 0

2

4

6 Distance

8

10

0

12

1

2

4

8

10

12

1 Y = 1.509 - .593 * X; R^2 = .885

Y = 1.552 - .572 * X; R^2 = .885 0

0

-1

-1

Ln (Pop Dens)

Ln(Pop Dens)

6 Distance

-2

-3 -4

-2

-3 -4

-5

-5

0

2

4

6 Distance

8

10

12

0

1

2

3

4 5 6 7 Distance (1.5km Bands)

8

9

10

Figure 4.18 (top left) Quadratic Regression of Population Index and Distance Decay from Bankstown Square 3/11/80 (1km Bands); (top right) Log-Linear Regression of Population Index and Distance Decay from Bankstown Square 3/11/80 (1km bands); (bottom left) Quadratic Regression of Population Distance and Distance Decay from Bankstown Square 3/11/80 (1km bands); (bottom right) Quadratic Regression of Population Density and Distance Decay from Bankstown Square 3/11/80 (1.5km bands)

212 C hapter 4 So we may feel happy with this result. However, the problem is that this form is nonlinear and means that it is only of limited value because the RASTT model is a linear differential equation. A theoretical modelling rule is to keep away from non-linear systems, if it is at all possible, but unfortunately many real world systems (including shopping trips) can be non-linear. Nevertheless, this distribution can be made linear by a simple transformation, by taking the logarithm to the base e (Ln) of the population function (Figure 4.18, top right), namely: Ln (Pop Index) = 4.06 – (0.42 × Distance)

(4.9)

If there is distance decay, the points should approach a straight line. We can add a random error term to this relationship, but because of the RASTT model we are only interested in the deterministic part. The deterministic distance decay effect is simply the linear regression model, namely: Y

=  o – 1 X

(4.10)

This relationship can be tested for significance relative to the sample size and the nature of the model either by an F (Fisher)-Test or the probability p of it lying inside the normal random distribution of events at 5% (0.05) or 1% (0.01) in the tail of the F-distribution. The F-test is defined by the mean square due to the regression (MSR) divided by the mean square due to error (MSE). The MSR on the numerator for a linear regression has one degree of freedom for one independent variable (distance) and N-2 degrees of freedom for MSE (where N is the number of points) because both estimates  o and 1 in the sum of squares are due to deviations from the regression line. For the quadratic regression, there are two degrees of freedom for MSR and N-3 degrees of freedom for the MSE. Therefore, the determination of the critical boundary for significance would differ for a linear and a quadratic regression. The denominator of the F-test (MSE) will increase if there is more variability about the estimated regression line and therefore the higher the F-statistic, the more likely it is to reject the null hypothesis (namely, the relationship is purely random). Therefore, the logarithmic transformation into a linear regression produced an R-squared value of 0.80 (compared to 0.83 for the quadratic regression; see Figure 4.16 top left) and an F-statistic of 36.45 which is well beyond the critical value of 5.12 in the F-distribution. The F-statistic for the quadratic regression was 19.52 > 4.46 (the critical value in this case). Both representations show a relatively strong and significant statistical relationship, so why undertake the logarithmic transformation? The answer is that it allows us to link the empirical analysis with the mathematical negative exponential solutions of the time-discounting RASTT model, since the negative exponential function is the inverse of a logarithmic function. An equivalent statement can be made of this relationship, namely: Pop Index = 57 .97 exp (– 0.42 × Distance) where exp (4.06) = 57 .97 is the constant. The 1 constant 0.42 is just the gravity coefficient  used in the spatial interaction measurement.

Empirical Testing of the R ASTT Model in Time and Space

213

Whilst this method is statistically relevant, it is not a physically correct picture of the geometric structure we have imposed upon the data. The bands further from the area could potentially distort the gravity coefficient because they cover a wider area. In the Bankstown Square morning sample, 3/11/1980, the 4-5km band has a substantial jump in the index of 19.1 compared to the 3-4km band of 7.9 (Table 4.14). Is this jump just an artefact of the way the data has been aggregated? This possible misspecification can be corrected by dividing the population index by the area of the band to yield a population density and its logarithm can be plotted against distance (Figure 4.18, bottom left). The resulting R-squared value is 0.89 with an F-statistic of 69.27, suggesting a stronger relationship because we have reduced the variability of the Zone 4-5km band. It is also a more theoretically correct picture of the problem and the gravity coefficient  is now 0.57 (Baker, 1994a). This method is what basically underpins the determination of the distance decay effect in this book. Distance .5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5

Pop Index (/100)

Pop Density

24.7 17.8 7.9 19.1 9.7 8.2 1.7 0.4 0.8 1.7 0.4

2.18 1.18 0.37 0.69 0.29 0.20 0.04 0.01 0.01 0.03 0.01

Table 4.14 Population Index and Density Assignments for 1km Bands: Bankstown Square Morning Sample, 3/11/1980 How robust is this measurement of distance decay? Once we know how  was computed, we can make estimations of bias, if necessary, based on the concentric aggregation. For example, in concentric aggregation, do we use mid-points or boundaries and what is the magnitude and nature of the modifiable area problem? For example, when the Bankstown Square morning sample data, 3/11/1980, was recalibrated using 1.5km bandings (with mid-points 0.75, 2.25, 3.75 …). The regression of the logarithm of population density and the distance with these larger bands produced a gravity coefficient of  = 0.59 (an increase of 3.5% over the 1.0km bandings of the data; Figure 4.18). Whilst there was no change in the R-squared value of 0.89, the F-test statistic for the 1.5km band dropped to 38.42 which, while still significant, suggests that the fewer regression points allow individual fluctuations to depreciate the F-value.

214 C hapter 4 This modifiable area problem and what we assign as the distance co-ordinate in the regression analysis was explored further in Baker (2000b), where empirical experiments were set up for Bankstown Square over all the samples taken in 1980/82 and 1988/89. Originally, this regional centre was of interest because there was a problem with the Bankstown postcode assignment from respondents. The centroid of this suburb lay outside the first 1km ring, so it was decided to make the first ring at 1.5km and then construct subsequent 1km annuli from this first zone. This raised many questions on the degree to which the researcher, in defining the partitioning and spatial units, can influence the gravity coefficient. Two of these questions are explored below. Boundaries or Mid-points? What difference is there for a gravity coefficient if there is a partitioning of the first band at 1.5km, then successive 1km annular rings, if we use the boundaries of the rings (1.5, 2.5, 3.5…) as the distance co-ordinates against the mid-points (0.75, 2.0, 3.0…)? The results are not startling. Where computed  values for boundaries and mid-points were 0.47 and 0.46, respectively, for the Bankstown 1988/89 morning samples and 0.54 and 0.52 for the afternoon samples, there was an error apparently of less than 4%. This was not, therefore, considered a major problem. The distance co-ordinates could reasonably be framed in terms of mid-points or boundaries without substantially changing the regression  coefficient. 1.0km or 1.5km Band Widths? The 3/11/1980 sample at Bankstown Square has already been examined. What is the pattern for the other three samples in the 1980/82 data set? The aim was to try to see whether there is any pattern in the error range for a  calculation (Baker, 2000b). Procedure 1 was set up assigning the first distant point as the 1.5km boundary and subsequent one kilometre annular boundaries at 2.5, 3.5, ... kms. How would the gravity coefficient compare for the annular assignment for the entire distribution as a 1.5km per band (rather than 1km) and taking the mid-point (rather than the boundaries) for each as the distance co-ordinate for the regression? This is termed Procedure 2. This is a rather simplistic example, but is interesting to see to what degree the annular size of each band, and the way the distance co-ordinate is constructed, impacts on the regression and the calculation of  . The results are listed in Table 4.15. The four samples are recalculated using each assignment procedure. The drift to lower  values over the decade at this centre is seen to reflect the growing dominance of regional planned centres in the Sydney hierarchy and socio-economic changes already noted could underpin them rather than aggregation problems. The problem with the 1.5km bands is a reduction in the number of regression points and often a rise in the serial autocorrelation of the residuals (the so-called Durban-Watson statistic). Therefore, the  values for the

Empirical Testing of the R ASTT Model in Time and Space

215

aggregation using Procedure 2 (using midpoints of 1.5km annular bands) was lower on two out of four occasions than the values calculated using Procedure 1 (that is, the first band at 1.5km and subsequent 1.0km bands) with the range in values from +3.5% to -11.9%. The greatest discrepancy was a  value ranging from 0.59 to 0.52. Contrary to expectation, the larger the aggregation units, the higher the R-squared value increased on only two of the four samples. There was no pattern to the fluctuations in the measurements for this centre. This experiment shows how the value of  can vary depending on the decision in spatial partitioning and assignment made by the researcher. The gravity coefficient can vary up to ~12% depending whether 1.0km or 1.5km bands are used, but there appears to be much less of a problem for taking the mid-point over the boundary as the distance co-ordinate. Therefore, the partitioning will affect the value of  and an increase to 1.5km will decrease, on average, the gravity coefficient by -0.02 at Bankstown Square. Sample BSM 89 BSA 89 BSM 80 (Sample 1) BSM 80 ( Sample 2)

Procedure Procedure 1 Procedure 2 Procedure 1 Procedure 2 Procedure 1

0.47 0.48 0.54 0.49 0.59

R-squared 0.78 0.84 0.84 0.92 0.93

DW 2.64 1.88 2.52 2.57 1.35

Number of Zones 11 8 11 8 10

Procedure 2 Procedure 1

0.52 0.57

0.87 0.89

2.07 1.56

8 11

Procedure 2

0.59

0.89

2.54

7



Table 4.15 The Estimation of the Gravity Coefficient for Bankstown Square using Two Assignment Procedures (DW = Durban-Watson Statistic); BSM- Bankstown Square Morning Sample, BSA- Bankstown Square Afternoon Sample (Source, Baker 2000b) There is another major problem to consider in the estimation of the gravity coefficient. There is pressure to maximise the number of points for the regression, but to do so, means the inclusions of the outer concentric bands. These peripheral bands are very susceptible to random fluctuations and the resultant collective densities can introduce systematic variation in the residuals. This form of correlation is known as serial autocorrelation and occurs when there is a correlation of the absolute values of successive residuals. This sequencing of residuals (or runs r) gives a measure ranging from 1.0 to –1.0. If there is no correlation in the residuals, then r = 0. This is one measure that can be used to assess this boundary effect. Another simple measure of autocorrelation is the Durban-Watson (DW) statistic. Like time series analysis, we are ordering the data, which in our case, are concentric zones away from the shopping centre. The mid-points are used as the location co-ordinates. This simple geometric structure allows for a direct application of this statistic. The

216 C hapter 4 DW statistic is the ratio of the sum of the squared differences in successive residuals to the residual sum squared. It is based on the estimated residuals from the regression. A number of assumptions underlie the DW-statistic, two of which are very relevant to our analysis (Gujarati, 1988). Firstly, the explanatory variables are non-stochastic or fixed in repeated sampling. Therefore, the use of a sequential scheme of aggregation zones fulfils this condition. Secondly, the DW-statistic makes no allowance for missing data. This is perhaps why there is sometimes a boundary problem in the aggregation because fluctuations in densities may occur in isolated outer bands introducing gaps in the sequence of data points. Unfortunately, there is not one unique critical value for the DW-statistic to accept or reject the null hypothesis of autocorrelation, but there are lower and upper boundaries, such that, if the computed statistic lies outside these critical values, a decision can be made for the presence or absence of positive or negative serial autocorrelation. We use these boundaries to help decide what band mid-points to include in the regression, but still there are two problematic cases out of twenty-six in the aggregate data set from 1988/89 and 1996/98 that require further scrutiny. For example, this boundary effect is shown in the Ashfield Mall 1998 afternoon sample (Figure 4.19) where there is one point at 13.5km providing the leverage for a DW-statistic of 0.758 (less than the lower boundary of 1.356) indicating strong evidence of positive autocorrelation within the residuals. The resulting  coefficient is 0.52 and an R-squared value 0.74 of this regression with 10 points. When this 13.5km point is dropped from the regression, the R-squared value increases to 0.88 and the DW- statistic becomes 1.38, which is just within the boundary of accepting that there is no autocorrelation of the residuals. 3

3

2

2

Y = 1.136 - .52 * X; R^2 = .737

0

Ln(Pop D)

Ln(Pop D)

Y = 2.038 - .761 * X; R^2 = .875

1

1

-1

0 -1

-2

-2

-3

-3

-4

-4

-5

-5

0

2

4

6

8 Distance

10

12

14

0

2

4

8

6

10

12

14

Distance

Figure 4.19 The Gravity Trip Distribution for Ashfield Mall 23/3/98 (Afternoon) for (left) 10 Zones with DW-statistic of 0.758 and (right) 9 Zones with an Improved DW-statistic of 1.380 Is this a sufficiently robust measurement of distance decay from this community PSC? A time-based estimate of the gravity coefficient may help us in our decisionmaking. Using the formula derived in the previous section, namely:

Empirical Testing of the R ASTT Model in Time and Space

X =

kd 2( ti + 2t j )2

217

(4.11)

where kd is the mean trip frequency per unit mean distance and ts and tt are the mean shopping time and trip times per hour, respectively. The empirical space connectivity equation for Sydney (from Figure 4.14 bottom) is defined as:

β = 0.386 + 2.7 25 X

(4.12)

or ª

º kd 2» «¬ 2( ti + 2t j ) ¼»

β = 0.386 + 2.7 25 «

(4.13)

The estimate for the Ashfield Mall 1998 afternoon sample can be computed as: ª

º 0.689 2» 2 ( 7 2 . 2 + 2 × 12 . 2 / 60 ) ¬ ¼

β = 0.386 + 2.7 25 «

(4.14)

β = 0.7 5 This is quite close to the estimate of distance decay from the nine point model above and therefore is further independent evidence that β = 0.76 rather than 0.52 from the initial regression with positive autocorrelation. The point at 13.5km is therefore considered to be a random fluctuation and not part of the deterministic model. Before proceeding with a comparison between the regression and time-based estimate of the gravity coefficient, it is important to mention further considerations that may underpin the determination of the gravity coefficient. According to Okabe and Tagashira (1996), the number of 1km concentric zones also warrants a correction for aggregation of up to 5.1% for β equal to 3.0. In our case, with β less than 1.0 and a minimum number of eight zones, the correction factor is only ~1.0%, and the coefficient is always flatter ( β underestimated) than the true coefficient (Okabe and Tagashira, 1996). Another issue is whether there is homoscedasticity (or equal variance) or heteroscedasticity (unequal variance) across the regression points, particularly with the time-space connectivity of trip behaviour. Homoscedasticity is the assumption that underpins the linear regression and if there is heteroscedasticity the variance is variable and the consequential t-test and F-test for significance are either over or underestimated. Heteroscedasticity is more likely to be common in cross-sectional rather than time series data (Gujarati, 1988). In the case of the serial assignment of a population index or density to concentric bands, it should not be a problem. However, for such variables as trip frequency, where origins closer to the centre will have accessibility, then the expectation is that the variance will change with each successive distance band. For example, the frequency variance for each of the

218 C hapter 4 distance bands for a sample of 2276 respondents for the Sydney 1996/98 data set is shown in Table 4.16. Band (km) 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8

Mid-point 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5

Mean Trip Frequency 2.73 1.97 1.47 1.18 1.29 0.91 1.03 1.02

Variance 4.56 3.05 1.90 1.26 1.65 0.90 1.23 1.41

Table 4.16 Mean Trip Frequency and Variance per Concentric Band: Sydney 1996/98 Data Set A simple way to test for heteroscedasticity is to do a Park Test on whether the variance to some function is correlated to the explanatory variable (trip distance), namely: Ln( Variance ) = A + αLn( Dis tan ce )

(4.15)

If  is statistically significant, it suggests that heteroscedasticity is present within the data. The results are plotted in Figure 4.20. An F-statistic of 31.38 > 5.99 and p = 0.0014 at the 0.05 level suggests, not surprisingly, that there is heteroscedasticity in the trip frequency variance and distance travelled in the Sydney 1996/98 data set. 1.6 Y = 1.158 - .536 * X; R^2 = .839

Ln (Frequency Variance)

1.4 1.2 1 .8 .6 .4 .2 0 -.2 -1

-.5

0

.5 1 Ln (Distance)

1.5

2

2.5

Figure 4.20 The Park Test Regressing the Logarithm of Frequency Variance and Trip Distance and Testing for Significance

Empirical Testing of the R ASTT Model in Time and Space

219

For the time-space connectivity relationship between the gravity coefficient and mean trip frequency squared, there is also a possibility of heteroscedasticity in the relationship. The reason is that we are testing the relationship for a hierarchy of planned shopping centres (Bankstown Square, regional; Westfield Burwood and Chatswood, sub-regional; and MarketPlace Leichhardt and Ashfield Mall, community malls). The variance could increase systematically, the larger the mall. The results are presented in Table 4.19 and the Park Test is used to see whether the coefficient  is statistically significant for the 1996/98 data set. The results are suggestive, but with the regression yielding only an R-squared value of 0.018 with an Fstatistic of 0.271 < 4.49, is not significant and there is a case for no heteroscedasticity in the time-space connectivity between the gravity coefficient and the mean trip frequency. There also seems to be an increase in the variance in trip frequency from the 1988/89 to the 1996/98 data set (possibly from the deregulation of shopping hours), therefore limiting the ability to aggregate both data sets. This was also checked with a Park Test yielding only an R-squared value of 0.071 with an F-statistic of 1.85 < 4.25. This suggested  is again not significant and there is no case for heteroscedasticity in aggregate time-space connectivity. Are the populations different in the samples over the decade? A Mann-Whitney test with a z-value of -0.68 > -1.96 suggested there was no significant difference between both populations. Therefore, evidence suggests, at this cursory treatment, that the assumption of no heteroscedasticity is valid when comparing data from time periods a decade apart. We have dealt with some of the fundamental issues that underpin the determination of a robust gravity coefficient in the distance decay assignment from the shopping centres. We used a method based around checking that there is no autocorrelation in the log-linear regression of the population density and the mid-points of concentric aggregation. We are particularly interested in random fluctuations in distance bands from the shopping centre that may provide sufficient leverage to underestimate significantly the  gravity coefficient. The method devised was to reduce the number of points in the regression until there was no positive or negative autocorrelation in the residuals. The qualification is constrained by the aim to have the maximum number of points within the regression. This decision-making was assisted by the Durban-Watson d-statistic and serial autocorrelation r–statistic. The results have been used to test whether there had been any changes in the  gravity coefficient over the decade. The values presented previously for both 1988/89 and 1996/98 are a result of this method (Baker, 2002). There are still cases when the inclusion or omission of points, near statistical boundaries, are the decision of the researcher. This has been a lurking problem of the RASTT model up to this point because there is always a suspicion that changes in the  gravity coefficient could just as likely be caused by the way we have partitioned the data. Consequently, individual decisionmaking could also affect the estimate by decisions on what points to include and what points to omit from the regression.

220 C hapter 4 Centre Bankstow n Squar e Morning Afternoon Westfield Burw ood Morning Peak Morning Off-peak Afternoon Peak Afternoon Offpeak

1988

1989 1996 1997 1.72 2.23

2.15

Westfield Chatsw ood Morning

1.41

Ashfield Mall Morning Afternoon

1.70 2.18

MarketPlace Leichhar dt Morning Afternoon

2.86 3.65

1.41

1.75

1.61

1.59 1.88

1.53 2.71

2.43

1998

3.77

2.99 1.48

3.27 2.65

1.36

3.93 3.28

3.12 2.10

Table 4.17 Comparison in Variance in Trip Frequency for 1988/89 and 1996/98 Samples in the Sydney Project for Equivalent Samples (shaded) and Pre-Christmas Samples (bold) This is why the independent time-based estimate of the  gravity coefficient is such an advance because it allows for a further independent check of the spatial determination of  . Using the formula defined previously, the time-based  gravity coefficients have been calculated in brackets against the spatial estimates using autocorrelation configurations. (Table 4.18). It is interesting that the time-based method is a good approximation for the deregulated trading hour data set (1996/98), but for 1988/89, there is a substantial underestimation in the off-peak 1989 Westfield Burwood morning sample and for the morning and afternoon samples for community PSCs (MarketPlace Leichhardt and Ashfield Mall). Such centres exhibit timediscounting behaviour and the restriction of trading hours may mean that there could be a more deterministic structure to shopping trips not captured in the time-based

Empirical Testing of the R ASTT Model in Time and Space

221

random walk assumptions. However, with shopping hour liberalisation, a time-based random walk estimate of the gravity coefficient corresponds closely with the spatial estimate in the 1996/98 data set for 13 out of the 17 samples (except for the 1997 Bankstown Square afternoon and the MarketPlace Leichhardt samples). The reason for this discrepancy is unclear, but it warrants a further look at the decision-making surrounding the spatial estimates. Centre

1988

1989

1996

1997

1998

Bankstown Square Morning

0.47 (0.46)

0.52 (0.49)

0.47 (0.52)

Afternoon

0.54 (0.47)

0.38 (0.50)

0.41 (0.46)

Westfield Burwood Morning Peak Morning Off-peak Afternoon Peak

0.66 (0.54)

0.53 (0.48)

0.66 (0.61)

0.69 (0.55) 0.58 (0.50)

0.46 (0.45)

0.65 (0.60)

0.70 (0.61)

Afternoon Off-peak

0.52 (0.51)

Westfield Chatswood Morning

0.40 (0.46)

Ashfield Mall Morning

0.96 (0.79)

Afternoon

0.93 (0.64)

1.00 (0.91) 0.73 (0.75)

0.76 (0.75)

MarketPlace Leichhardt Morning Afternoon

0.84 (0.68)

0.95 (0.80)

0.90 (0.81)

0.71 (0.57)

0.79 (0.69)

Table 4.18 Comparison in the Spatial and Time-based (in brackets) Gravity Coefficients for 1988/89 and 1996/98 in the Sydney Project, Equivalent Samples (shaded) and Pre-Christmas Samples (bold)

222 C hapter 4 We will investigate further two of the more debatable estimations in Table 4.20. Firstly, in the 1997 Bankstown Square afternoon sample,  was determined at 0.38. The population density for the 13.5km band was included in the regression from the above autocorrelation assumptions because the DW-statistic of D = 1.49 >1.320 and r = 0.48 suggested that its inclusion did not lead to autocorrelation in the distance decay estimate (Figure 4.21, left). However, if the 13.5km point is removed from the regression, the resulting  is recalibrated at 0.50 with a DW-statistic of d = 2.01 >1.32 and r = -0.087. The time-based estimate of  is 0.50 (Figure 4.21, right) and therefore, the question arises whether we should have, in fact, excluded the 13.5km point from the regression. The 1998 sample was spatially regressed at  = 0.41 (d= 2.13 >1.43 and r = -0.181) with the 13.5km point included and the time estimate of  = - 0.46 was much closer to the spatial estimate. Therefore, we are more confident that Bankstown Square in 1998 increased its dominance over its trade area, but for the equivalent 1997 sample we are not so sure, since the decrease in the gravity coefficient was only 0.02. 1.5

1.5 1

Y = .795 - .375 * X; R^2 = .763

.5

.5

0

0

-.5

Ln(Pop D)

Ln(Pop D)

1

-1 -1.5

-.5

-1 -1.5

-2

-2

-2.5

-2.5

-3

-3

-3.5

-3.5

-4

Y = 1.364 - .497 * X; R^2 = .864

-4

0

2

4

6

8 Distance

10

12

14

1

2

3

4

5

6 Distance

7

8

9

10

11

Figure 4.21 The Bankstown Square 1997 Afternoon Gravity Regression (left) Including and (right) Excluding the 13.5km Point There is a similar problem with the 1996 Market Place Leichhardt afternoon sample. If the 11.5km point is included in the regression, the gravity coefficient is 0.55, but the DW-statistic d = 1.27 < 1.32 suggests that there is positive autocorrelation. The r-statistic, r = 0.155, is less conclusive. This point was therefore omitted from the regression and the subsequent  value was 0.71. This is what was recorded in Baker (2002). However, the time-based estimate of the sample yielded 0.57, namely: ½ 1.22 / 3.14 2¾ ¯ 2[7 7 .2 + 2 × 12.9 / 60 ] ¿ ­

β = 0.386 + 2.7 25 ®

(4.16)

This was close to the 0.55 spatial estimate (including the 11.5km point) that was rejected because of possible positive autocorrelation in the regression. Should we

Empirical Testing of the R ASTT Model in Time and Space

223

have rejected the initial regression? This dilemma is helped by the equivalent sample we took at MarketPlace Leichhardt twelve months later. The initial regression for this second control sample, including all points (up to the 9.5km band), gave  = 0.70, but once again d = 1.26, suggested positive autocorrelation in the residuals. The consequence was that the 9.5km point was dropped from the log-linear regression. The new nine-point regression (0.5, 1.5,… to 8.5km bands) yielded  = 0.77, but this time with d = 1.86 > 1.32, there was no positive or negative autocorrelation in the residuals. This value was accepted for subsequent analysis. The time-based estimate produced  = 0.69, namely: ­ ½ 1.47 / 2.68 β = 0.386 + 2.7 25 ® (4.17) 2¾ ¯ 2[7 1.2 + 2 × 11.6 / 60 ] ¿ This was once again within 0.01 of the initial spatial estimation of , whereas for the revised spatial regression, the discrepancy was 0.08. For the 1988 MarketPlace Leichhardt afternoon sample, the initial nine-point regression yielded  = 0.84 with an excellent DW-statistic (d = 1.815 >1.32) and r = -0.077. This value was accepted as robust, although the time-based estimate produced a divergent  = 0.68. Likewise, other samples at Ashfield Mall and Westfield Burwood (outside the pre-Christmas rush) produced similar robust spatial estimations and substantially divergent timebased estimates. This collective anomaly suggests that the problem lies with the time-based estimate because of possible interference of consumer time behaviour with regulated trading hour boundaries (with the normality assumption of the random walk derivation of ). It is for this reason that the time-based estimate is not a good proxy to look at changes in  for samples within regulated trading hours such as in the Sydney samples (1988/89). This was not such a problem for samples taken under deregulated trading hours (1996/98). Therefore, for comparison purposes, the spatial estimate of , using autocorrelative measures, is the best tool to look at changes in  between 1988/89 and 1996/98, yet the results cannot be stated with confidence. We have just gone through an extensive exercise to check how robust our measurement of the gravity coefficient is in the log-linear model of trip distance to the shopping centres. The conclusion is that the spatial estimates of  from concentric aggregation of population densities is probably the best method of testing the hypothesis: that shopping hour liberalisation leads to a dispersion of trip origins further from planned shopping centres. This is because the spatial determination of  appears to be applicable independent of whether the centre hours are regulated or deregulated. This method will be used to compare coefficients over the decade, despite the previously discussed problem of partitioning and autocorrelation between the boundary mid-points.

224 C hapter 4 4.5.5

IS THERE A RELATIONSHIP BETWEEN THE GRAVITY COEFFICIENT OF TRIP ORIGINS AND SHOPPING CENTRE TRADING HOURS?

What then is the theoretical relationship between the gravity coefficient  and trading hours 0 to T? Is it empirically relevant? The answer comes simply from the timespace convergence relationship at the shopping centre and the appropriate model is framed by substituting k= n π /T into k = β M ) (and eliminating k ) to yield:

n 2π 2 (4.18) MT 2 The hypothesis is that the extension of trading hours (T) increases the attractiveness and market domination of the PSC (inferred through falling  values) within the trade area. This is a similar conclusion to an applied discrete choice model for product differentiation (Rouwendal and Rietveld, 1998, 119). The T-squared term suggests that this change will be substantial (Baker, 1994b). Further, the squared terms (n and T) suggest that there would be a non-linear relationship with .

β=

The trading hours for the PSCs in 1988/89 were uniform for the centre (at 49.5hrs), but for 1996/98, the supermarket anchors traded much longer hours than other retailers in the centre. Therefore, we had to calibrate the hours of the supermarket and other retailers relative to their percentage of floorspace. The trading hours therefore vary from centre to centre. With this data, we can now test this time hypothesis, using spatially-determined gravity coefficients only (and not the timebased estimates, for obvious reasons). The regression of Equation (4.18) relating the gravity coefficient to the trading hours per week is not linear in the 1988/89 but quadratic (Figure 4.22), with an R-squared value of 0.45 which is not significant (F = 2.50 < 5.14 at the 0.05 level of significance). This lack of statistical significance aside, the result suggest that as (n/T ) decreases through increased hours, there is an increasing dominance of trip origins further from the planned centres (with falling  values). However, there is a declining return to hours traded until there is a maximum of hours for the centre (65.2 hrs/week) beyond which there is no further capture of trip origins. However, in 1988/89 with centre hours regulated at 49.5hrs, these extra hours are meaningless and suggest that this ‘extra’ shopping time would be spent locally. For 1996/98, at the same centres (except Westfield Chatswood), there was still a quadratic relationship (R-squared value of 0.78) which was now significant (F = 24.79 < 5.14 at the 0.05 level of significance). There has, however, been some significant change in the nature of the relationship with a dramatic bending of the curve with deregulated hours (Figure 4.23). This has three ramifications. Firstly, the minimum point of inflection has shifted from 0.034 to 0.015. This means that centres will increase the market dominance of trip origins within their trade area with shopping hour liberalisation. Secondly, the hours for maximum trading has increased to 79.8 hours per week (from 65.2 hours). This is calculated by determining when X or (n2/T2 ) is minimum by differentiating the

Empirical Testing of the R ASTT Model in Time and Space 225

quadratic equation and setting the result to 0. Then, substituting  2/M(X)=0.03385, the resulting calculation is:

π 2 × ( 4.82 )2 2.4 × 0.015

= T = 7 9.8 hrs

(4.19)

where M = 2.4 and n = 4.84. Beyond these hours, there is no market benefit. It provides a possible explanation for the decline in the number of major supermarket chains trading 24 hours per day in Sydney from thirteen in 1995 to two in 2003. This suggests that 24-hour trading was adopted by Coles supermarkets in 1994 for other reasons. The RASTT model suggests that Sunday trading, rather than 24-hour trading, has had a more substantial impact on the re-structuring of the shopping week. Thirdly, the slope of the quadratic curve has increased where the rate of substitution of trip origins has increased substantially with extended hours. This means consumers living further away are far more likely to travel to these planned centres and there has been an aggregate loss of the equivalent of 30.3 hours per week of shopping time (or 18%) from neighbourhood business in local shopping centres. Less time shopping there equates to reduced household spending. A simple explanation for the decline of neighbourhood and town centres is that consumers are spending less time shopping there and more time, under shopping hour liberalisation, at more distant planned shopping centres. 1 .9

Gravity Coefficient

.8 .7 .6 .5 .4 .3 .2 .1

Y = 1.079 - 35.579 * X + 549.307 * X^2; R^2 = .454

0 0

.005

.01

.015 .02 Dest^2/TradH^2

.025

.03

.035

Figure 4.22 Quadratic Regression of the Gravity Coefficient-Trading Hour Hypothesis for the Sydney Project 1988/89

226 C hapter 4

1 .9

Gravity Coefficient

.8 .7 .6

1988/89

.5 .4 .3 .2 .1

Y = 1.138 - 103.933 * X + 3518.859 * X^2; R^2 = .78

0 0

.005

.01

.015 .02 Dest^2/TradH^2

.025

.03

.035

Figure 4.23 Quadratic Regression of the Gravity Coefficient-Trading Hour Hypothesis for the Sydney Project 1996/98 (This can be compared to the 1988/89 regulated trading hour regression.) 4.5.6 HOW HAS DEREGULATED HOURS IMPACTED ON ‘SMALL’ AND ‘LARGE CENTRE’ WEEKLY ASSIGNMENTS OF SPATIAL DEMAND? Using Bankstown Square and MarketPlace Leichhardt as case studies of ‘large’ and ‘small’ behaviour, respectively, the time-space shopping distributions for the week (now including Sunday) were recalculated according to Equations (3.66) and (3.67), using 1996/98 data (Baker, 2002). This was then compared to six days-a-week trading there on the same day (Thursday afternoon before Easter and three weeks before Christmas, respectively) for 1988/89 (Figures 4.22 and 4.23). The repeat sampling was undertaken to see if the results could be replicated over a 12-month cycle. At MarketPlace Leichhardt, the ratio SD of the supply and demand trading hour estimate (Td /Ts) equals 1.02 (65.7/64.7) for the pre-Christmas sample, indicating near-equilibrium trading conditions for the floorspace (where the trading hour demand Td is calculated using Equation 4.1 and the trading hour supply Ts is the actual hours the centre operates). However, for the 1997 sample at the same time, the demand estimate decreased to 59.6 hours and SD = 0.92. This may not be a random anomaly, since there was also an increase in vacant specialty shops to 10% at the time of the survey. Nevertheless, despite these changes, the integrity of the structural change could be replicated in the 1997 sample and both empirical models are very similar (Equations 4.19 and 4.20).

Empirical Testing of the R ASTT Model in Time and Space

227

6.9 exp(-0.7 1x) sin (0.247 t) MarketPlace Leichhardt Afternoon (1996) (4.19) 8.9 exp(-0.7 9x) sin (0.255t) MarketPlace Leichhardt Afternoon (1997 ) (4.20) At MarketPlace Leichhardt (ML), there was a shift towards ‘large centre’ behaviour, through decreasing  values (0.84 in 1988 to 0.71 in 1996 and 0.79 in 1997) and relative centre densities (10.5 in 1988 to 6.9 in 1996 and 8.9 in 1997). This is a simple indication that, on average, a greater proportion of consumers were travelling further to MarketPlace Leichhardt over the decade from within the trade area and consequently, shopping less locally at traditional shopping centres. There is also a substantial structural change where, by the addition of Sunday (from 144 to 168 hours per week), spatial demand shifted from the beginning of the week (Monday and Tuesday) to between Friday and Sunday (Figure 4.24). Rather than the increase in mean shopping hours at the centre, from 49.5 to 65.7 hours per week, it was the addition of Sunday, extending the relative trading period, that created this significant structural change. Shopping on Sunday affected spatial demand densities for other days, particularly at the beginning of the week. Such a structural change was replicated in 1997 from sampling at the same time at ML (Thursday, three weeks before Christmas). This prediction of a spatial demand shift under deregulation was made in the simulation of the regulated data at MarketPlace Leichhardt by Baker (1997) (see Chapter 3.4.2), where the trading period was increased from 50 to 60 hours and Sunday trading was introduced. This type of structural change has been confirmed empirically and the conclusion is that Sunday trading of major supermarkets (the generators of periodic demand) have fundamentally changed spatial demand over the week. The change in the structure of the shopping week at Bankstown Square was also striking. In 1988/89, there were only two periods of regular shopping observed in the model (at the beginning and end of the week; Figure 3.14). Such episodes were not deemed significant, since there is a greater propensity for infrequent and random behaviour perhaps masking such periodicity under ‘large centre’ behaviour (Baker, 1997). At Bankstown Square, the ratio SD of the supply and demand trading hour estimate for the 1997 afternoon sample (Td / Ts) was 0.88 (54.2/61.6) for the preEaster sample and indicated disequilibrium trading conditions for the floorspace. However, for the 1998 sample at the same time, the demand estimate increased to 59.1 hours (SD = 0.96) and equilibrium conditions were approached. There was no obvious explanation for this under-performance in 1997, although economic indicators there also suggest below optimum trading. From 1995 to 1996/97, there was an increase in vacant shops from 1% to 4% and a decline in sales from $A297.6m to $A289.1m (General Property Trust Annual Reports; 1995, 1996, 1997). The reason, from the RASTT model, was that the mean destinations visited of 5.98 was below expectation in 1997 (compared to 8.22 in 1998), hence the difference in the SD ratio. The economic indicators also improved in 1998 where sales of $A305.3m had returned to 1994 levels ($A302.1m) and vacancies dropped to 2% (General Property Trust Annual Reports; 1994, 1998).

228 C hapter 4

Figure 4.24 Three Dimensional Contour Model of Changing Time-space Behaviour at the Community Centre MarketPlace Leichhardt, from the Regulated 49.5 hours in 1988 to a Supply Average of 64.7 hours in 1998 The structural change in the time-space demand distributions were replicated from 1997 and 1998, although there were substantial differences in the time estimates.

2.2 exp( - 0.38x) sin (0.655t) Bankstown Square Afternoon (1997 )

(4.21)

2.3 exp( - 0.41x) sin (0.856t) Bankstown Square Afternoon (1998)

(4.22)

The deregulated distribution in 1998 was markedly different to 1989 with a substantial rise in periodic shopping episodes over the week from two to five (Figure 4.25), paralleling the rise in mean interlocational trip frequency from 1.49 in 1989 to 1.89 in 1997 and 1.96 in 1998. Indeed, the distribution appeared to have the same structural features as the regulated community PSC (MarketPlace Leichhardt) trading 49.5 hours per week (Figure 3.13). This was predicted under the simulation of this regulated data (Baker, 1997), but the difference is that, empirically, this structural change occurred at 61.7 hours, not at 100 hours per week in the simulation. Such a change was argued to be significant in spatial terms (Baker, 1997, 219). For

Empirical Testing of the R ASTT Model in Time and Space 229

example, since trading hour deregulation, the retail precincts surrounding regional PSCs in Sydney suffered significant rental reductions of prime sites of up to 45% and a halving in capital valuation. At Bankstown Square, there was a 21% drop in prime retail shop values from $A1400 in 1990 to $A1100 in 1993 and remained unchanged until 1996 (Valuer-General’s Office, 1996). In 1996, there were 15% vacant shops in the Bankstown suburb centre, but this improved to 10% by 1997 under a main street rejuvenation program and an expansion of the Vietnamese retail precinct (Stringfellow, 1998). Bankstown Square increased its revenue to $A302.1m in 1994, but stagnated till 1998 in terms of economic performance.

Figure 4.25 Three Dimensional Contour Model of Changing Time-space Behaviour at the Regional Centre Bankstown Square from the Regulated 49.5 hours in 1989 to a Supply Average of 61.6 hours in 1998 Similar trends are apparent in other regional PSCs. From 1993 to 1994 there was a 6.3% increase in moving annual turnover for the top ten PSCs in Australia, without any increase in gross leasable area (GLA), whereas in 1996 a jump in 9.6% in GLA only saw a 0.8% rise in revenue (Shopping Centre News, 1999). Perhaps there was an oversupply of regional PSC floorspace across Sydney in 1996/97 that was compensated by a reduction in the demand for trading hours, such as calculated at Bankstown Square. The important outcomes are, specifically, that the RASTT model

230

C hapter 4

is robust enough to parallel changes in economic indicators (such as vacant shops) at MarketPlace Leichhardt and Bankstown Square in two time-specific and replicative samples. 4.5.7 ARE THE MATHEMATICAL RELATIONSHIPS FROM THE RASTT MODEL ROBUST OVER TIME? A key relationship underpinning the RASTT model is the relationship between the average number of minutes ( p) spent shopping and the mean number of shops visited (m). This relationship from the model is expressed as: f =

mw p ≈ 2T 2T

(4.23)

where (p/2T) is intra-location frequency and mw = m × trip frequency is the mean number of shops visited per week. The regression of this relationship from 1988/89 Sydney data is linear with a significant R-squared value of 0.74 (Figure 3.9). There was a similar strong relationship in the 1996/98 data set with 17 samples yielding an R-squared of 0.84 (Figure 4.26, left). The aggregation of the data sets still produced a significant relationship with an R-squared value of 0.83 and an F-value of 118.87 with no autocorrelation (DW = 1.74 > 1.46; Figure 4.26, right). This relationship is therefore robust and independent of the trading hours of the centre (because 2T can be cancelled from both sides of Equation 4.23). Is trip behaviour to the shopping centre related to shopping destination assignment within the centre? This question was reviewed in the previous chapter. The relationship between the intra-centre shopping frequency ( f ) to inter-locational frequency (k) is defined as: f = ± Mk

(4.24)

This relationship was tested for data from the Sydney Project (1988/89) and (1996/98). As stated previously, the results are very encouraging. For 1988/89, there was a significant R-squared value of 0.48, but this improved to 0.75 in the 1996/98 samples (Figure 4.27 right). However, over the decade, there was a substantial change in the slope of the relationship (0.060 to 0.025) and this can be directly attributed to the deregulation of shopping hours. If both data sets are aggregated, the R-squared value deteriorates to 0.148 which is just not significant (4.18 < F = 4.26) so both regulated and deregulated data in this relationship must be treated separately. Consumers assign shopping destinations differently with their shopping strategies to and within centres because of the effect of trading hours. Therefore, both relationships are empirically significant over time in Sydney.

Empirical Testing of the R ASTT Model in Time and Space 1.4

1.4

1.2

1.2 1 Dur/2*TradHr

1 Dur/2*TradHr

231

.8 .6

.8 .6 .4

.4

.2

.2

Y = .208 + 7.233 * X; R^2 = .837

Y = .187 + 7.994 * X; R^2 = .832 0

0 0

.025

.05

.075 .1 .125 Dest*Freq/2*TradHr

.15

.175

0

.2

.025

.05

.075 .1 .125 Dest*Freq/2*TradHr

.15

.175

.2

.2

.2

.17

.17

.15

.15

Dest*Freq/2*TradHr

Dest*Freq/2*TradHr

Figure 4.26 Linear Regression between ( p/2T ) and (mw/2T) for (left) the 1996/98 Sydney Data Set and for (right) the Aggregate Sydney Data Set (1988/89 and 1996/98)

.13 .1 .08

.13 .1 .08 .05

.05 .03

.03

Y = -.034 + .06 * X; R^2 = .477

Y = -.005 + .025 * X; R^2 = .751

0

0 0

1

2

3 Sq.R(Freq*M)

4

5

6

0

1

2

3 Sq.R(Freq*M)

4

5

6

Figure 4.27 Linear Regression between Inter-location Trip Frequency ( f ) and Intracentre Frequency k, ( f = Mk ), for (left) 1988/89 and (right) 1996/98 Sydney Data Sets 4.5.8 IS MULTI-PURPOSE SHOPPING STILL A FUNCTION OF TRIP FREQUENCY AND THE TWO STATE HYPOTHESIS? One of the relationships presented in Baker (1996) on multi-purpose shopping (MPS) was that there was an empirical relationship between the percentage of MPS in a sample and the mean trip frequency k per week. This relationship was plotted for the 1988/89 and 1996/98 data sets in the Sydney Project (Figure 4.28). This relationship is still robust over the period, with the aggregate regression yielding an R-squared value of 0.70 (F = 48.60 > 4.26 at the 0.05 level; p < 0.0001) and there was no autocorrelation (DW of d = 2.023 for 1.45 < d < 2.55 and serial autocorrelation r = -0.019). If you increase trip frequency to a centre, there is a higher propensity for MPS, even with shopping hour deregulation, when there is more time available for shopping. This was an unexpected result. For MPS, there appears to be a common shift, on average, towards higher MPS across all the centres in the samples from 1988/89 to 1996/98 (Figure 4.28). It appears that consumers, who aim to minimise the total effort in shopping, will partition time in such a way as to increase the bundles of activities engaged in each trip. Indeed, this

232

C hapter 4

propensity appears to have increased with the deregulation of shopping hours. It was expected that with the deregulation of shopping hours, that multi-purpose shopping would decline, since MPS reduces the effort spent in shopping by combining trip purposes to minimise shopping time. The Sydney evidence clearly shows that this is not the case. Further, it would then be expected that the number of shops visited would increase with the substantial rise in MPS. However, once again the unexpected occurred, in that the number of shops visited per trip declined for all centres over the decade with the deregulation of shopping hours. More specifically, between 1988/89 and 1996/98, for the four common centres, the mean destinations visited per trip dropped from 6.4 to 4.8. The median also declined from 5 to 4 destinations over the decade and a Mann-Whitney test (z = -7.76 and p < 0.0001< 0.05) suggested that the populations and their trip behaviour in the category were significantly different. Further, there was a decline in the variance of destinations visited. Consumers were visiting less shops per trip to these shopping centres. What is the reason for this paradox? Why is there a significant jump in multi-purpose shopping (MPS) but a corresponding significant decline in the number of shops visited per trip both in mean, median and variance measures? The answer lies in understanding that the major supermarket chains within the centres are capturing all the purposes of many more consumer trips. Consumers can multi-purpose shop exclusively within the floorspace of a major supermarket anchor and not visit other shops in the centre. Hence, MPS can increase significantly at the same time that the number of destinations visited declines significantly. The diversification of merchandise and the provision of services previously bought at other shops can now be made within the floorspace of the supermarket. This solves the above paradox and why supermarkets may be trading much longer hours than other shops in the centre to perhaps magnify this effect. Is MPS a function of trip frequency in positive and negative states (Baker, 1996)? The percentage of MPS is plotted across the range of shopping centres in the sample according to the number of destinations or shops within the floorspace. For both periods combined into the one regression, there is a quadratic relationship with centre scale (R-squared value of 0.54; F = 13.31 > 3.42 at the 0.05 level, p = 0.0001). There is a negative slope from small to sub-regional centres (Type 1 MPS) and a positive slope for the regional mall samples over the decade (Type 2 MPS). The point of inflection (the boundary between Types 1 and 2 MPS) can be calculated by differentiating the MPS with destinations N and setting it to zero (Figure 4.29), namely: MPS = 112.27 – 1.44N + .005N

dMPS / dN = – 1.44 + .01N = 0

N = 144

(4.25)

This boundary of N = 144 (or approximately 45,000 sq m in floorspace) is the same defined by Baker (1994a) for the division between ‘small’ and ‘large’ centre behaviour.

Empirical Testing of the R ASTT Model in Time and Space

233

50 45 40

MPS%

35

/8 1988

30

9

25

20

6/98 gate 199 re Ag g

15

Aggregate Y = -24.262 + 31.678 x X; R 2 = .669 1996/98 Y = -13.176 + 26.084 x X; R2 = .55

10 5

1988/89

Y = -14.157 = 22.051 x X; R2 = .472

0 1.2

1

1.4

1.6 Frequency

2.2

2

1.8

Figure 4.28 Linear Regression for MPS and Trip Frequency, Sydney Project 1988/89, 1996/98 and Aggregate Regression1 50 45 40 19 96 /9 8

MPS%

35 30 25 te ga re g Ag

20 15

/8 88 19

9

10 5

0 60

80

100

120

140 160 180 Centre Destinations

200

220

240

Aggregate Y = 112.775 - 1.449 * X + .005 * X^2; R^2 = .536 1988/89 Y = 59.478 - .666 * X + .002 * X^2; R^2 = .664 1996/98 Y = 112.884 - 1.388 * X + .005 * X^2; R^2 = .58

Figure 4.29 Quadratic Regression for Percentage of MPS and Centre Scale (Number of Destinations), Sydney Project 1988/89, 1996/98 and Aggregate Regression 1

Bankstown Square sample (23/3/89A) should read MPS at 9.2% rather than 17.2% in Baker (1996)

234

C hapter 4

There is a greater propensity for MPS at smaller centres, presumably because less mobile consumers use it as a trip strategy to minimise the total effort in shopping. The bigger the centre, the less likely it is that these populations will use this type of trip-strategy. Conversely, for the larger mall, the more destinations within the centre, the more likely it is for the more affluent and professional households to use it as a means to combine trip purposes as a time minimisation strategy. This is the simple explanation of the two-state distribution. Can this idea be generalised into a MPS model? With the gentrification of inner Sydney, more affluent households are shopping more often at smaller planned shopping centres (like MarketPlace Leichhardt). Shopping hour liberalisation has made it possible for these centres to be time accessible for these ‘professional’ households, particularly for the supermarket trip. Shoppers therefore may construct the MPS trip at these smaller PSCs around the supermarket visit to minimise the total effort in shopping, both for less mobile and affluent time poor households. This is defined as convenience-based MPS (Type 1 MPS). At the regional mall, this type of MPS is less likely, as there is a greater propensity for the combination of MPS around clothing and gift shopping as part of the trip. This is defined as choice-based MPS (Type 2 MPS). Both types of MPS could have apparently increased at the centres, judging from the bending of the quadratic relationship (Figure 4.29). In other words, the positive and negative gradients for the substitution of MPS in both types of strategies have increased over the decade. The other alternative is that Type 2 MPS is increasingly occurring, not only in regional malls, but also in community PSCs with the gentrification of inner Sydney. We will now investigate this hypothesis by looking at the relationship between MPS and high disposable income (HDI) respondents. If this hypothesis is valid, then we should have a linear relationship. Research into consumer spatial behaviour suggests that higher income consumers travel further to metropolitan centres than average income consumers (Schiller, 1972; Dunn and Wrigley, 1985). For example, Schiller (1972) argues that middle class shoppers make a disproportionate use of regional and metropolitan shopping centres, and also make longer shopping trips to more centres than average. Potter (1977) in his Stockport UK study, states that higher-status respondents used a greater variety of centres at greater distances from their homes than lower-status respondents and that social class differences were more clearly associated with this variation than car availability. However, in the Sydney project, there was no relationship between the mean trip distance and the percentage of high disposable income (HDI) respondents within the samples. Baker (1996) reported a relationship between the percentage of MPS and trip distance (R-squared = 0.70), but when aggregated with the 1996/98 samples, the Rsquared value dropped to 0.12 and was not significant. However, there is one relationship where HDI consumers may help explain the growth in MPS over the decade. Whilst there is a significant relationship between the percentages of MPS and

Empirical Testing of the R ASTT Model in Time and Space

235

HDI respondents within the samples (F= 3.94 > 3.42; p = 0.034), the relationship is quadratic (Figure 4.30) and not linear. This suggests that when HDI consumers are in larger proportions of samples (greater than 29%), there is less likely to be MPS. Presumably, such HDI households undertake single propose trips whenever their shopping time budget allows them. 50 45 40

MPS%

35 30 25 20 15 10 5

Y = -.837 + 2.197 * X - .038 * X^2; R^2 = .255

0 0

10

20

30

40

50

HDI%

Figure 4.30 The Relationship between the Percentage of MPS and HDI Respondents in the Sydney Project (1988/89 and 1996/98) 50

45

40

MPS%

35

30

25

20

15

10

5

Y = 12.064 + .218 * X; R^2 = .518

0 0

20

40

60

80 HDI*k^2

100

120

140

Figure 4.31 The Relationship between the Percentage of MPS and HDI × Trip Frequency Squared (per week) in the Sydney Project (1988/89 and 1996/98) This

236

C hapter 4

This explanation is supported when we use a relationship first proposed by Baker (1996), namely:

MPS = HDI× (Trip Frequency)2

or h = I k2

(4.26)

This relationship is regressed for the aggregate of both sample periods in the Sydney Project and the surprising result was that there was quite a strong linear relationship (R-squared = 0.52, F = 25.74 > 4.26 at the 0.05 level; Figure 4.31) with no autocorrelation (DW of d =1.94; 1.45 < d < 2.55). In other words, the degree of MPS can be better explained by looking at the number of trips per week as well as the proportion of HDI consumers. If HDI consumers are not contributing to the growth of MPS (for samples with over 29% of HDI consumers), then it is most probably from consumers engaging in Type 1 MPS behaviour that can explain the increasing incidence and linearity in the relationship beyond the 29% MPS threshold at both small and large planned shopping centres in the project. Whilst Type 2 choice-based MPS seems to have increased substantially across the PSC hierarchy from the increasing pressure of time minimisation, nevertheless, Type 1 convenience-based MPS within supermarket floorspace is still present within the samples and can increase the explanation of the nature and changes of MPS in Sydney over the period. 4.5.9 HOW HAVE CHANGES IN TIME-SPACE SHOPPING PATTERNS IMPACTED ON THE UTILITY OF SHOPPING AT PLANNED SHOPPING CENTRES? In Chapter 2, it was shown that there was a possible time-space relationship between the level of shopping satisfaction or utility U and a combination inter-locational mean trip frequency k and distance D relative to the number of destinations N, namely: U=

kD N

(4.27)

This was shown to be empirically relevant with an R-squared value of 0.53 for the aggregation of the 1988/89 and 1996/98 Sydney data sets (see Figure 2.14). When these distributions are regressed separately, it is interesting that there has been a halving in the gradient and the utility intercept has dropped from 21.4 to 18.8, yet the relationship has been maintained over the decade (R-squared value of 0.63 in 1988/89 and 0.61 in 1999/98). This suggests that the level of shopping satisfaction has decreased at planned shopping centres within Sydney and part of the reason is a change in the relationship between the time-space parameters. Previously in Chapter 2, the gravity coefficient had been defined for planned shopping centres up to 150 destinations (~45,000 sq m of floorspace) as:

Empirical Testing of the R ASTT Model in Time and Space

β=

U 2N 2 MD 2

237

(4.28)

This relationship has excluded the agglomerative effects of the regional PSC (Bankstown Square) in the data set for both sample periods. The reason why the regional mall does not fit this relationship is still unclear, but the conclusion seems to be that this relationship is robust only for ‘small centre’ behaviour in the lower order centres in the shopping hierarchy. This model was statistically robust for the combined data with an R-squared value of 0.67 (see Figure 2.15). Individually, for 1988/89, there was a significant relationship with an R-squared value of 0.79 (p = 0.0075), whilst for 1996/98, there was an R-squared value of 0.63 (p < 0.00001) for this relationship (Figure 4.32, left and right). What is interesting is that the slope did not change over the decade (3.70 × 10-6 in 1988/89 and 3.70 × 10-6 in 1996/98), suggesting that shopping hour liberalisation has had little impact on the substitutions. 1

1 .9

.8 Gravity Coefficient

Gravity Coefficient

.8 .7 .6 .5 .4 .3

.6

.4

.2

.2

Y = .373 + 4.096E-6 * X; R^2 = .542

Y = .33 + 3.711E-6 * X; R^2 = .789

.1

0

0 0

40000

80000

120000

160000

Utility^2*Dest^2/M*Distance^2

200000

0

40000

80000

120000

160000

200000

Utility^2*Dest^2/M*Distance^2

Figure 4.32 The Regression of the Gravity Coefficient  and U2N2/MD2 for the Sydney Project Excluding Bankstown Square Data (left) 1988/89 and (right) 1996/98

4.6 Application of the RASTT Model to Unplanned Shopping Centres: Armidale in Regional New South Wales, 1995 4.6.1. INTRODUCTION The previous analysis looked at the retail aggregate space-time trip (RASTT) model and how it can be applied to study changes in trip behaviour to a number of planned shopping centres in Sydney over a decade. The question arises whether the model is just Sydney-specific or if it is applicable to a wider context of shopping trip behaviour. This was the motivation to apply the methodology to Armidale, a university town of 23,000 people in northern New South Wales (Figure 4.33). This town’s retail precinct is centred on a pedestrian mall and there were no planned shopping centres present at the time of survey. Further, the shopping hours were deregulated for six days-a-week, but there was still no Sunday trading in 1998 (being one of the last NSW towns for it to be introduced).

238 C hapter 4 A large study was undertaken in November 1995, using the same survey instrument as the Sydney Project. The result was 1308 cross-sectional surveys on different days of the week and at different times of the day. Satisfactory sample sizes were achieved on each occasion from surveying over a two-hour period (Table 4.19). The rejection rates were highest outside supermarkets and lowest in the Richardson’s Arcade (dominated by clothing and softgood shops), indicating the different reactions to the survey, according to trip purpose. Only one respondent per household completed the survey and the sample size was just over 10% of households within the Armidale Trade Area. Tues Morn 193

Tues After 199

Thurs Morn 201

Thurs After 209

Sat Morn 185

Thurs Late Night 7-9.00pm 321

Table 4.19: The Sample Sizes in the Armidale Survey, November 1995 The project also aimed to look at the variability of samples taken over the week. Do time-space variables vary depending on the day of the samples (as well as in the morning and afternoons)? 154OE

Q U E E N S L A N D

B yron B ay

30OS

N E W A rm id a le

OC

EA

N

S O U T H

IFI C

W A L E S PA C

32OS

SYDNEY 34OS 0

50

100

200

K il o m e t r e s

150OE

1 52 OE

Figure 4.33 The Location of Armidale, New South Wales

Empirical Testing of the R ASTT Model in Time and Space

239

Key socio-economic descriptors have been developed in Baker (1995a) to give a quick appreciation of affluence and disadvantage in census districts and market areas. The more advantaged descriptors to be used are ‘double income no kids’, ‘double income with kids’, ‘two and over car households’ and ‘household income greater than $60,001’, whilst the less advantaged descriptors are ‘population aged 65 and over’, ‘no car households’ and ‘household income less than $20,001’. This method was applied to define the Armidale Local Government Area (LGA) and the Primary Trade Areas and compare them both to NSW averages (Table 4.20).

No Car Households Two Car Plus Households Persons Aged 65+ DIWK Households DINK Households Household Income > $60,000 Household Income 4.10; p = 0.0075 < p = 0.05) (Figure 4.49, top). What is surprising is that this relationship significantly strengthens when only samples undertaken with seven days-a-week trading and partial deregulation (excluding Sunday) are regressed. The result is an increase in an R-squared value of 0.60 (F= 35.56 > 4.21; Figure 4.49, bottom). What this suggests is that shopping hour liberalisation encourages consumers, on average, to be more destination-specific within the centre where one, two or three trip purposes now dominate the trip strategy. There may be more multi-purpose shopping, but its range may have been reduced to minimise the total effort in shopping.

Empirical Testing of the R ASTT Model in Time and Space

259

.14

Dest*Freq/2*TradHr

.12 .1 .08 .06 .04 .02 Y = .034 + .016 * X; R^2 = .187 0 0

.5

1

1.5

2 2.5 Sq.R(Freq*M)

3

3.5

4

4.5

.14

Dest*Freq/2*TradHr

.12 .1 .08 .06 .04 .02 Y = -.001 + .026 * X; R^2 = .601 0 0

.5

1

1.5

2 2.5 Sq.R(Freq*M)

3

3.5

4

4.5

Figure 4.49 (top) The Aggregate Regression of Sydney 1988/89, Armidale 1995, – Sydney 1996/98 and Auckland 2000 of f =±√Mk (bottom) Post-1993 Extended Hours Data (Excluding Sydney 1988/89 Points) (28 Samples) showing an Improved R-squared Value of 0.60

260

C hapter 4

The time assignment hypothesis (p/2T = m × k/2T ) There were 38 samples available for regression (including the Westfield Chatswood 1989, 12.00 to 1.30pm sample). The hypothesis is that shopping times and shopping centre destinations are interrelated and the regression shows this strongly with an Rsquared value of 0.72 (F = 91.86 > 4.10; Figure 4.50). It does not seem to matter whether the trading hours of the centre are regulated or deregulated, there is a substantial relationship between the time assignment and destinations visited. This is supported by looking at the deregulated data (post-1993) only where the R–squared value is reasonably close at 0.68 (but there is a decline in the F-statistic to 55.0). The conclusion is, therefore, that the time assignment to destinations within the centre can affect the frequency of trips to the centre, independent of shopping hours. 1.4 1.2

Dur/2*TradHr

1 .8 .6 .4 .2 Y = .209 + 7.312 * X; R^2 = .724 0 .02

.04

.06

.08

.1

.12

.14

Dest*Freq/2*TradHr

Figure 4.50 The Regression of Sydney 1988/89, Armidale 1995, Sydney 1996/98 and Auckland 2000 of p/2T = m × k/2T The trip frequency and multi-purpose shopping hypothesis (MPS = h k) This is another relationship where there is a strong correlation despite changes in shopping times and shopping places. Whether the trips are within a planned shopping centre hierarchy in a city of one million or four million people in different countries or an unplanned pedestrian mall in a non-metropolitan regional centre, the higher the trip frequency, the greater the propensity for multi-purpose shopping. For the Sydney 1988/89, Armidale 1995, Sydney 1996/98 and Auckland 2000 samples (38 points), there was an R-squared value of 0.64 (F = 64.53 > 4.10; Figure 4.51). There was no autocorrelation in this distribution, with d = 1.550 > dv = 1.535, further supporting the robustness of the relationship. The corollary of the condition of time-space convergence (  = k2 /M) is that:

Empirical Testing of the R ASTT Model in Time and Space

MPS = ± Mβ

261

(4.31)

There is firm empirical evidence underpinning the expected two states of MPS (signified by ± in Equation 4.31) and their prevalence depends on the context. Choice-based MPS (Type 2 MPS) is more likely found at sub-regional and community planned shopping centres in Auckland, the unplanned regional centre of Armidale and the regional mall in Sydney. Conversely, convenience-based MPS (Type 1 MPS) has a greater propensity in the deregulated shopping centres of Sydney. Based on this evidence, MPS and its relationship with trip frequency, is a strong candidate for a global construct of time-space shopping trip behaviour. 2.5 2.25 2

Frequency

1.75 1.5 1.25 1 .75 .5 .25

Y = 1.004 + .022 * X; R^2 = .628

0 0

10

20

30 MPS%

40

50

60

Figure 4.51 The Regression of Sydney 1988/89, Armidale 1995, Sydney 1996/98 and Auckland 2000 for MPS = h k The multi-purpose and high disposable income hypothesis (MPS = HDI×k2 ) In the Sydney project (1996/98) there was a strong linear relationship for MPS = HDI×k2). When the other data sets are added to the regression, there is still a strong relationship within the 38 samples (excluding the 1980/82 Sydney samples). The Rsquared value is significant at 0.51 (F = 37.17 > Fc = 4.10; Figure 4.52) and there was no autocorrelation in this distribution, with d = 1.803 > dv = 1.535, further supporting the robustness of the relationship. However, for the post-1993 deregulated time data set (28 samples), there was a substantial weakening in the R-squared value, although little change in the Durban-Watson statistic (R-squared of 0. 26, F = 8.93 > Fc = 4.20 and d = 1.778 > dv = 1.476). What does this mean? The problem does not appear to be statistical with little change in autocorrelation in the data. The conclusion is that high disposable income (HDI) consumers underpin MPS and they

262 C hapter 4 are more likely to engage in this type of activity for choice–based strategies. However, this propensity is substantially weakened when deregulated shopping hours allow these affluent households more time accessibility to undertake single purpose trips. 60 50

MPS%

40 30

20 10 Y = 15.388 + .174 * X; R^2 = .486 0 0

20

40

60

80

100 120 HDI*k^2

140

160

180

200

Figure 4.52 The Regression of Sydney 1988/89, Armidale 1995, Sydney 1996/98 and Auckland 2000 for MPS = HDI × k 2 The time-space and utility hypothesis (U= D × k / N ) The final relationship to be tested is whether time-space constructs influence the level of shopping satisfaction. This utility, relative to centre scale, has declined in Sydney over the decade and shoppers in Auckland appear to be far happier shopping in their planned shopping centres than Sydney in 1988/89 and 1996/98. Overall, for 37 samples, there was an R-squared value of 0.33 (Figure 4.53) with F = 17.298 > Fc = 4.18 and d = 1.417 < dv = 1.502) suggesting a robust relationship, but some positive autocorrelation in the residuals. The quadratic regression of the points improved the d value but there was still possible autocorrelation (d = 1.582 < dv = 1.590 for an R-squared value of 0.427 and F = 12.650). This suggests a possible discounting of utility with the scale of the centre. The persistent autocorrelation could come from the Armidale 1995 samples where the distance estimates for trip origins cover large areas of rural hinterland. When Sydney 1996/98 and Auckland 2000 samples are regressed together (23 points), the R-squared value of 0.13 with F = 3.11 < Fc = 4.28, suggests the relationship is not significant. There is also a strong positive autocorrelation in the residuals (d = 0.981< dv = 1.437). These results could highlight a substantial difference in the way the populations sampled in Sydney and Auckland determine shopping utility. For both the Sydney 1988/89 and 1996/98

Empirical Testing of the R ASTT Model in Time and Space

263

samples, there is still a significant relationship over the decade, but the deregulation of shopping hours has seen the slope halved from 68.8 to 31.5. For Auckland, the linear regression of the six samples produced an R-squared value of 0.63 with F = 6.79 > Fc = 6.61, so this utility relationship could still be relevant for that city. Therefore, this further strengthens the conclusion that consumers perceive planned shopping centres (PSCs) differently in Auckland than Sydney, but this is not a function of the deregulation of shopping hours. Rather, the difference could be that there is no centre scale effect in Auckland that is evident in Sydney from regional PSCs, where consumers in Auckland are happy to drive further, more often, to shop at sub-regional PSCs. 25

Utility

23

21

19

17 Y = 19.109 - 26.63 * X; R^2 = .331

15 0

.02

.04

.06 .08 .1 .12 Dist*Freq/ Centre Destinations

.14

Figure 4.53 The Regression of Sydney 1988/89, Armidale 1995, Sydney 1996/98 and Auckland 2000 for U= D × k / N 4.8.3 SUMMARY The application of the retail aggregate space-time trip (RASTT) model has been tested in a variety of contexts, both in terms of changes over space and time, and the results appear to be robust. Although time does impact on the nature of trips relative to the time-space convergence, the model appears to be applicable and the fundamental relationships robust in different spatial contexts and over time. Even though these contexts are different, the correlations suggest that the distributions are affected by the same trip operators, constrained by socio-economic and behavioural determinants. These same operators will be explored when applied to Internet traffic in the next chapter.

CHAPTER 5 Dynamic Modelling of the Internet 5.1

Introduction

In Chapter 1, the idea was presented that some spatial interaction could be defined as a cone of converging time lines (Figure 1.5). This convergence, connecting origindestination pairs, is defined by the rate of time-discounting (and distance minimisation) and its rate is a function of the technology of transfer. The previous theoretical and empirical work concentrated on origin-destination trips from a shopping mall to residences. However, time-discounting behaviour is not exclusively found at shopping centres. Rather, the time-space convergence means that, at least theoretically, the mathematical operators can be projected beyond this interaction to larger distance scales and smaller time scales. The manifestation of this ‘quantum leap’, in a change of scale, is the Internet. The key question is whether the interaction near the singularity of the convergence (that is, at very small time scales) is the same as our shopping modelling of the previous chapters. If it is, then there should be a type of gravity interaction and a periodic internet demand wave circumnavigating the world relative to the Earth’s rotation. Near this singularity, spatial interaction offers some peculiar features, such as, virtual distance between origin-destination pairs and the ability to go forwards or backwards to different time zones relative to the location of the computer. The RASTT model has been applied extensively, in the previous chapters, to the study of trips to and from point densities (shopping malls) between time-lines, where its time-dependent solutions are relative to centre opening and closing times. In the case of the Internet, the time boundary is a moving day-night transition from the rotation of the Earth and the solutions of the model, with the same operators, should have similar features to our previous work. In this chapter (based on Baker, 2001), the retail space-time trip (RASTT) model will be applied to the Internet, exploring the type of wave developed and the nature of distance decay. The model will be tested using data from the Stanford Internet experiments for the year 2000. The Stanford experiments were undertaken by Dr Les Cottrell at the Linear Accelerator Centre, Stanford, USA. It features 27 global monitoring sites in 2000, pinging transactions every hour to 171 remote hosts distributed around the world. The experiment measures the time taken from pings between origin-destination pairs and the amount of packets shed from congestion on the route. What is meant by ‘packet loss’? When too many packets arrive on an origin-destination trip, routers hold them in buffers until the traffic decreases. When the buffer fills up during times of congestion, the router drops packets. Packet loss is what is being measured here as a proxy of peak demand. The Stanford experiments have been running from 1998 to 2004 with various numbers of monitoring sites and remote hosts. The year 2000 had the greatest connectivity between the number of monitoring sites and remote hosts and presents the best opportunity to test the model. One site in northeast US

265

266

C hapter 5

(Chicago) was selected to test the RASTT model, namely, hepnrc.hep.net.gif because it possessed the greatest number of connected remote hosts. It is with this case study that the model will be tested for spatial and temporal interaction (Figure 5.1).

Figure 5.1 The Location of the hepnrc.hep.net.gif Monitoring Site (top) and the Remote Hosts (bottom)

Dynamic Modelling of the Internet

267

The idea that a partial differential equation can be applied to model the Internet is a significant departure from topological approaches currently being developed. In order to appreciate this work on networks, a brief review of this approach is necessary before we show how the RASTT model can be applied to the Internet (Baker, 2001). The analysis of complex networks can be divided into two major classes based on their connectivity distribution P(m) which defines the probability that a node in the networks is connected to m other nodes (Albert et al., 2000). The first type of networks is characterised by a P(m) that peaks at an average m and decays exponentially for large m. These networks are homogenous in that each node has approximately the same number of links. Exponential networks (such as the random graph model of Erdos and Renyi, 1960) have a connectivity that follows a Poisson distribution peaked at m which decays for m >> m . This is a Type-1 network. The second network type belongs to inhomogenous networks (or ‘scale-free’ networks) where P(m) decays as a power law (or P(m) ~ m-γ ) free of the characteristics of scale. This network has a majority of nodes with only one or two links, but a few large nodes of links guaranteeing that the system is fully connected. This type of model can be visualised by the Internet tree simulated by Cheswick (1999) (Figure 1.4). Examples of this type of network are the World Wide Web and global air networks and both are part of the time-space convergence model (Figure 1.5). Barabasi (2001) uses geographical examples to distinguish both types of networks and these are pertinent to the development of an Internet model. An exponential network is like a road map that has cities as nodes and expressways as links, because most cities are central places located at the intersection of the motorways. Conversely, an airline route map is a Type 2 network, because although most airports are served by a small number of carriers, they have a few hubs (such as, London) from which links emerge to almost all other US or European airports. The WWW is seen as an example of the latter because the majority of documents have only a few links. It appears that Type 2 networks are also hierarchical. They are also preferential, since they contain nodes that have a high probability of being connected to another node with a large number of links. For example, a new Web page is more likely to be linked to the most popular documents on the Web, since these pages are the ones we know about. Research by Faloutsos et al. (1999) has shown that the network behind the Internet also appears to follow the power-law distribution of inhomogenous networks. This means that the physical wiring of the Internet is also dominated by several highly connected hubs. As Barabasi (2001) asks: why do systems as different as the Internet, which is a physical network, and the Web, which is virtual network, develop scale-free networks with a power-law decline in connectivity? This topological

268

C hapter 5

analysis is distinctly aspatial, but is still imbedded in time-dependent variables for the transfer of information. How should the Internet be viewed as a geographical system where both space and time are fundamental to interaction? The RASTT model may provide some insights into the framing of this question. 5.2 The RASTT Model and Internet Transactions (after B aker, 2001) 5.2.1 INTRODUCTION The time-discounting operators of the RASTT model (δ /δ x and δ 2 /δ t 2 ) are not affected by the numbers involved in interaction (applying equally to individuals or populations of billions) and are therefore classified to be scale invariant. These operators are defined relative to time boundaries for movement through physical space. Yet, are these mathematical operators applicable to the Internet, where there is still a real time boundary (the 24 hour rotation of the Earth) defining the movement of transactions? Consumers can also make virtual rather than real trips to sites and therefore the RASTT model offers the scope to explore the movement of demand through virtual space as well as physical space. The immediate problem in the RASTT model is that relative time can be either positive or negative. The idea of negative time in the context of the process of shopping trips was initially thought to be meaningless and the time boundaries were only applied from 0 to T (and the 0 to –T range discarded; see Baker, 2000a). However, the idea of negative values relative to the direction from the boundary for Internet transactions is not as non-sensical as it first appeared. We can model for trips or transactions through space over a 24-hour period (the daily cycle) as well as weekly (168 hours). The spatial origin could be located at a computer at an arbitrary location and the consumer can either go forward or backward along a virtual trip line relative to this 24-hour boundary. For example, if the individual is located at Sydney (33o S Lat and 161o E Long), that person can either go two hours forward in time to a site in Auckland (37o S Lat and 175o E Long) or two hours backwards in time to Perth (32o S Lat and 116o E Long). The RASTT model can be derived for physical trips to a mall and such trips are only viewed positively between time lines. Conversely, virtual trips on the WWW can be defined as either moving backwards or forwards relative to the 24-hour time boundary. This is a radical statement because it gives a plausible example of how relative time can exist as a corollary of a virtual distance and have different properties to physical time. Boulding (1985) states that in the physical sciences, time is assumed to approach infinity in order to focus on spatiallyspecific solutions. Alternatively here, we assume initial spatial locations and produce finite time-specific solutions, including time solutions that can be negative. The study of the Internet as a geographical system therefore provides an opportunity to introduce a new concept to see if it has any further properties of interest. There is a possibility of a convergence of virtual distance into a fixed point (the computer screen) at any time. An important question is therefore: can relative time

Dynamic Modelling of the Internet

269

influence the patterns of space? The idea returns to the concept of a dynamic convergence of locations in the time-space convergence cone. The Internet is perhaps the next stage in the evolution of this time-space connectivity. In the RASTT model, the nature of the time lines in the cone are summarised by a second order time differential (or operator) that can yield positive and negative time-based solutions. This means that unlike physical time, relative time can lead to reversible time-based processes, a truly remarkable possibility. For example, it means that in the election of a US president, polling booths can be closed in the east, yet the proportion of votes counted and reported on TV can feedback simultaneously to voters on the west coast who are still voting (and can change their votes based on the east coast trends). Reversibility of a result is possible within the boundaries of relative time. It is possible to have two simultaneous sites connected by virtual distance on a computer screen in different time zones. The Internet presents a new horizon to geographical systems because we have to now distinguish between relative time and physical time Relative time is measured from a moving periodic boundary (such as, the day-night transition from the rotating Earth). Positive and negative time exists relative to this boundary. This is different to physical time, where in the calculus, time is onedirectional and is taken to infinity to solve the differential equation. Physical time involves real trips between an origin and destination, whereas relative time can involve virtual trips, where the physical location is fixed to a computer screen. Relative time exists because of the speed of transfer brought about by technological change. The Internet allows for the study of time-space convergence near the singularity of the cone from high speed pings in a network. There are some features of the RASTT model that should be expected in Internet dynamics, if it is a correct application. What characteristcis could then be expected from a RASTT model representation of Internet transaction? There are two areas of immediate interest (Baker, 2001). 5.2.2

THE CONDITION FOR TIME-SPACE CONVERGENCE

This condition for time-space convergence in the solution of the supermarket equation (Equation 2.1), when a 24-hour boundary is applied, yields the same relationship between the gravity coefficient  and the square of the mean interlocational trip frequency k divided by the transfer constant M, namely:

β=

k2 M

(5.1)

The inter-locational trip frequency (ITF) defines the average number of trips or transactions undertaken per day by users and, because it is squared, it can be applied to virtual trips either forwards or backwards through relative time. The RASTT model suggests that there would still be gravity interaction of physical distance for Internet patronage, but this would be at least one order of magnitude lower than gravity coefficients computed from shopping trips to malls using concentric

270

C hapter 5

aggregation. Yet we would expect that it would vary for this type of transaction. For example, weekly food orders would have (with k higher) greater  values, indicating the distribution of food would be more localised than for a lower frequency consumption item, such as compact disks. We would therefore expect that a feature of the Internet, as a geographical system, would be ‘very weak’ or long distance gravity interactions, but this would still be relative to the type of transaction and the limits of the distribution system. 5.2.3

TIME-SPACE DISTRIBUTIONS OF INTERNET DEMAND

The type of time-space distributions that could apply to Internet patronage was simulated by Baker (2001) in Figure (5.2) for  = 0.0001, T = 24 hours, x 0 = 0 to x 0 = 10,000 km and an arbitrary population density φo = 10 for a sequence of k values where k = 0.1, 0.2, 0.3, 0.4, 0.5,...,1.0. The simplest distribution of spatial demand for Internet patronage at a site, receiving both positive and negative flows of transactions, is a gaussian-type distribution between k = 0.1 and 0.2 (Figure 5.2). This is not surprising since a gaussian distribution is an equally valid solution to Equation (2.1) for a time-based random walk problem. The solution has some advantages in this probabilistic form, because variables can be expressed as average quantities, such as, ‘distance’ and ‘number destinations per visit’. A gaussian distribution can be defined as probability distribution P(t,x) for a density of transactions at a site φ = φ0. If this site receives na transactions per unit distance d, with total transactions Φ = na d , the probability distribution is defined as: § t2 · 1 φ ¸ = p( t , x ) = exp¨¨ − 1/ 2 ¸ φo 2(π M x ) © 4 Mx ¹

(5.2)

where t is equal to the time for each transaction to travel to the site. The transfer coefficient M can be defined alternatively as: 1 2 nt (5.3) 2 The transfer constant is then the number of transactions per unit distance multiplied by the relative time t taken to reach the site. Equation (5.2) is the type of distribution that has been simulated in the k = 0.1 to k = 0.2 range in Figure (5.2). It is an unbounded gaussian time distribution, where transaction densities can be plotted for 2 ln φ versus t and the slope of the straight line is (4Mx)-1. The average time taken by 2 the transaction is defined by the mean square displacement ( Δt ), namely: M =

Δt 2 =2Mx

(5.4)

The RASTT model can therefore define the possibility of a number of distinctive features of Internet patronage relative to traditional spatial interaction modelling.

Dynamic Modelling of the Internet

271

Figure 5.2 (top) A Range of Possible Time-space Distributions that could apply to Internet Demand are Simulated for  = 0.0001, T = 24 hours, x0 = 0 to x 0 = 10,000 km and a Scaled φ o max = 10 for a Sequence of k Values where k = 0.1, 0.2,...1.0. (bottom) A Three-dimensional Plot Visualising a likely form of the Demand Wave for k = 0.1 (Baker, 2001) Firstly, the gravity model of spatial interaction would have very small  co-efficients compared to a regional shopping mall. Secondly, technology allows for a time-space convergence to occur on the computer screen rather than shopping malls. Thirdly, virtual distance allows for the possibility of near simultaneous connections, both forwards or backwards in relative time. Fourthly, such connections can have implications for activities in different time zones, such as the US presidential elections or stock market activity. Finally, transactions to sites should be represented in their simplest form as time-based gaussian distributions. This type of model (and differential equation) is not found in traditional applications of applied mathematics because of the problem of dealing with a positive and negative time dimension. In the case of the Internet, such difficulty is an advantage because transaction flows

272

C hapter 5

can be modelled globally, relative to a time boundary. It means that relative time has to be viewed differently at this particular scale (defined by the rotation) and has different properties to physical time (such as reversibility). The next step is to look more formally at the derivation of the Internet equation in the context of transaction exchanges between a number of web sites (see Baker, 2001 and Ghez, 1988).

Figure 5.3 The Equal Likelihood of Jumping Forwards in Time to Sites in Auckland or Backwards to Perth from the i th Sydney site defines the Underpinnings of the Type of Differential Equations in Equations (5.16) to (5.18) (Baker, 2001)

Dynamic Modelling of the Internet

273

5.3 Deriving the RASTT Model for Internet Transactions (after Ghez, 1988 and B aker, 2001) The derivation of the Internet version is very similar to a random walk between time lines. We will repeat the basic assumptions and mathematical formulation from Section 3.3.1. Assume a network of remote hosts linked to an arbitrary monitoring site at W, where these sites are designated through integers i = 0, ± 1, ± 2, ± 3,.... For example, a monitoring site at Sydney could have a choice of other sites at i = ± 1 at Auckland or Perth (Figure 5.3). Each monitoring site serves a number of remote hosts at a particular locality and there are Φ i remote hosts linked to each monitoring site i. Assume that each of these remote host’s traffic can hop to adjacent sites with a frequency Γ that does not depend on the characteristics of i. These hops can access sites forwards in time or backwards in time. It is assumed the movement forwards or backwards are equally likely. Therefore, movement from site i to site i +1 per unit distance occurs at a rate of 1 2 Γ Φ i. Likewise, the remote host at site i +1 can reply to the monitoring site at i at a rate of 1 2 Γ Φ i+1 . The resulting rate of exchange is:

Ε i +1 2 =

1 Γ (Φ i − Φ i +1 ) 2

(5.5)

and for i-1 into the site i, the flux is

Ε i −1 2 =

1 Γ (Φ i −1 − Φ i ) 2

(5.6)

The change in Internet traffic into and out of the i th site is given by a definition of all possible transitions:

dΦ i 1 1 1 1 = − Γ Φ i + Γ Φ i + 1 − Γ Φ i + Γ Φ i −1 dx 2 2 2 2

(5.7)

The distance discounting equation (or rate equation) in terms of the distribution of users in and out of the i th site is (by collecting terms): dΦ i 1 = Γ ( Φ i +1 + Φ i −1 − 2Φ i ) dx 2

(5.8)

This exchange rate of Internet traffic is between nearest-neighbour remote hosts around the monitoring site i and this can be expressed in terms of the exchange rate between sites (using Equation 5.5 and 5.6), as:

274

C hapter 5

dΦ i = − Ε i + 1 2 − Ε i −1 2 dx

(

)

(5.9)

The change in the Internet site storage is defined by the difference between flows in and flows out of transactions within the connectivity.

C omments (1) The jump frequency of transactions between sites is constant and it is assumed independent of the site index i and its location in space. The data signal should not change its frequency within the network and does not depend on the location of the computers. This appears a reasonable assumption and agrees with the aspatial nature of the graph theory approach. (2) This frequency of movement does not depend on the distribution of remote hosts or users in the neighbourhood of the i th site. The distribution does not have to be homogenous. This also appears to be a good approximation and parallels the assumptions of graph theory.

(3) The type of transfer network does not influence the process. The only thing that is important is the time-based ordering of the points. The receipt of a transaction does not depend on network structure, but on a timedependent ordering of site hits. This means that the network can have any configuration of sites. Once again, this is a good assumption for the Internet. Equations (5.5 to 5.9) define a time distance between sites where the arrangement of points and the hierarchical network of sites is not relevant, rather, what counts is the time ordering of hits to the site. Equation (5.9) states a conservation law where the transactions in and out define the content of the Internet site. The time distance p between sites is assumed to be equal between the monitoring site and the i th remote host and has the co-ordinate of ti = ip on the routes of transaction flows. The transaction density φ (x,t) is assumed to interpolate the previous function at site i, with the co-ordinate iφ (x) by the following assumption:

φ (t , x ) = Φ i ( x )

(5.10)

Dynamic Modelling of the Internet

275

at destinations located at t = ti, but is arbitrary elsewhere. The assignment of this transaction density function around this web site at ti can be expanded by a Taylor series: ∂φ ∂ 2φ t i + 21 p 2 2 t 1 + terms of order p3 (5.11) φ (t i ±1 ) = φ (t i ) ± p ∂t ∂t and using the condition in Equation (5.8), the expansion becomes, when substituted into Equation (5.10), (noting that we are interested in both forwards and backwards motion relative to the 24-hour cycle of the Earth): ∂φ 1 2 ∂ 2φ = Γp + terms of order p4 ∂x 2 ∂t 2

(5.12)

C omment The condition for this approximation is that p× k 4.17 at the 0.05 level; Figure 5.10). There was some serial autocorrelation suspected in the positive residuals with d = 1.386 30,000) Tamworth 12.5% CBD (1998) 12% (2003)

Dub b o 29 vacant shops CBD (1999) Wagga Wagga 9.3% (1997)

Maitland 19.8% (2004)

District Centres ( 15,000-30,000)

Armidale 6.6% (1998) 9.0% (2004)

Cessnock 16.3% (2000) 19.5% (2003) Grafton 13.8% CBD (2000) 13.1% CBD (2003) Taree 50 vacant shops CBD (1999)

Towns (5000-15000)

Inverell 13.4% CBD (2000) 16.1 % CBD (2003) Gunnedah 13.1% (2004)

Small Towns (1000-5000)

Ob eron 9.4% (2001)

301

Village (