The Proceedings of the 1972 Meeting of the IGU Commission on Quantitative Geography 9780773594111

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PROCEEDINGS of the 1972 Meeting of the IGU Commission on Quantitative Geography

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PROCEEDINGS of the 1972 Meeting of the IGU Commission on Quantitative Geography

Edited by Maurice Yeates Professor of Geography Queen's University

McGill—Queen's University Press Montreal and London 1974

© McGill-Queen's University Press 1974 International Standard Book Number 0 7735 0168 1 Library of Congress Catalog Number 74 75799 Legal Deposit second quarter 1974 Bibliothbque nationale du Qu6bec

Design by Carl Zahn Printed in Canada by Pierre Des Marais Inc.

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Contents

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The Process Method Versus the Hypothesis Method: A Nonlinear Example of Peasant Spatial Perception and Behavior RICHARD W. WILKIE page 1

The Nearest-Neighbor Statistics for Testing Randomness of Point Distributions in a Bounded Two-Dimensional Space SHIN-YI Hsu AND J. DAVID MASON page 32

Parametrization of the Gamma Probability Density Function for Examining and Modeling Space Use BARRY S. WELLER AND GERALD J. LACAVA page 55

Representation of Spatial Point Pattern Processes by Polya Models ARTHUR GETIS page 76

Use of Two-Stage Least Squares to Solve Simultaneous Equation Systems in Geography DAVID R. MEYER page 101

Optimum Urban Population Densities EMILIO CASETTI page 113

Environmental Quality and Urban Population Density KINGSLEY E. HAYNES AND MILTON I. RUBE page 121

Deducing Psychic Transport Costs from the Transportation Problem MARK D. MENCHIK page 136

A Model for Location of Service Facilities in a Non-Western Urban Environment P. D. MAHADEV AND K. R. RAO page 164

Foreword

Between August 9 and 17, 1972, the 22nd International Geographical Congress convened in Montreal in conjunction with the biennial meeting of the International Geographical Union. Over 2,200 persons attended the conference which embraced all aspects of geography. The IGU Commission on Quantitative Geography held two paper sessions during the period of the conference, and these sessions were extremely well attended by representatives from many delegations. This volume contains lengthy versions of papers presented at the two sessions, and as such it is to be regarded as a collection of articles relating to the development and application of quantitative methods in spatial analysis. The papers have been arranged for publication in a sequence that appears to reflect the general interests of most geographers concerned with the use of mathematical models. The first, by Wilkie, is methodologically oriented, though his argument is couched in terms of a specific example. He neatly summarizes the work of the past few decades and indicates possible paths for future development. This is followed by a series of articles concerned with the development of spatial probability models. Hsu and Mason deal specifically with the boundary problem as it relates to the nearest-neighbor technique based upon the Poisson model. The work of Wellar and LaCava on the gamma probability density function is interesting in that it not only describes the derivation of the parameters of the model, but also indicates how the model may be used in specific spatial situations. The last article in this group, by Getis, focuses upon Polya models. This paper is notable, not only because of the clear presentation, but also because Getis includes a discussion of process as it relates to the interpretation of spatial probability models that appear to describe various distributions. The article by Meyer contributes to our understanding of the structure of geographic interdependencies and uses two-stage least squares procedures applied to an urban residential location model to describe and assess the technique. This urban application theme is continued by Casetti who develops a model for urban population densities that maximizes a locational welfare function. Haynes and Rube use a different approach to incorporate social differences and physical deterioration into urban population density models, though it is to be noted that the derived utility function does not imply a specific density distribution. The last two articles offer a change of pace as they are concerned with the application of mathematical programming models to spatial behavior. Normative vii

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models of this type are extremely attractive in specific geographical situations where behavior can be regarded as rational within the evaluative and perceptual limits of the subject. Menchik discusses the nature of these limits within the framework of the Transportation Problem and develops an entertaining interpretation of "psychic transport costs" with respect to space preferences. At a more practical level, Mahadev and Rao use the model to examine the degree of optimality in the distribution of service facilities in Mysore, India. The reader may care to fit these articles into the developmental model of Human Geography presented in Figure 3 of Wilkie's article. It is suggested that, taken together, they represent an increase in our level of understanding as well as scholarship over the quantitatively oriented contributions of a decade ago. It is recognized, of course, that this latter comment is based upon a qualitative assessment which depends upon the perception of the individual reader ! MAURICE YEATES,

Kingston December, 1972

The Process Method versus The Hypothesis Method : A Nonlinear Example of Peasant Spatial Perception and Behavior RICHARD W. WILKIE (University of Massachusetts, Amherst)

How valid are multivariate statistical techniques of analysis based on linear assumptions? Is it possible that such techniques can at times misdirect the analysis in such a way that they may hide some relationships that exist in the real world? Many of those who work with quantitative data would agree that at times such techniques of analysis do cover up vital relationships between variables that exist only at certain levels within the data. Many others, however, seem to ignore this fact and thus fail to see countertendencies within various levels of data (e.g., subculture groups) that are canceled out when lumped together in aggregate linear statistical analyses. Therefore, it is the purpose of this article (a) to examine briefly the trend against exclusive use of linear analyses; (b) to point out that what this writer has called the Process Method is more effective than the Hypothesis Method for isolating the occurrence of nonlinear relationships in changing and evolving communities; (c) to demonstrate why spatial studies of the processes of change are necessary and that they are part of a behavioral period in geography that is already underway; and (d) to develop several lower-level techniques for examining and illustrating nonlinear relationships in a case study. Illustrations throughout the article focus on the relationships among the following variables collected for each peasant family in the village of Aldea San Francisco, Entre Rios, Argentina: (a) social-class position; (b) spatial perception of distances between and populations in communities within the region; (c) frequency of regional movement; and (d) sources of communication entering each household (local versus cosmopolitan media inputs). As will be shown below 1

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in analysis of data on peasant spatial perception and behavior collected in the study area, linear analyses of research materials often failed to show the complexity of opposing processes at work in the community. In contrast, many nonlinear analyses yielded greater insights into different spatial perceptions, structures, and decision-making behavior of various population subgroups in the village. One reason behind this occurrence is that different forms of human behavior arise out of different combinations of forces influencing decision-making within subculture groups, even within an apparently homogeneous lower-class peasant community. Two variables may correlate and have significance within one subgroup and have no significance among other subgroups. Further, it is the particular combination of interaction among many variables (some high, some medium, and some low) that produces a given form of behavior, not just the factors that have high correlation. Most studies which use linear forms of statistical analysis, however, do not isolate these different subgroups, but lump them together. Not only are the subgroups hidden within the aggregate totals, but the results often become misleading—not for those relationships that are linear at a given level, but for the nonlinear relationships that are judged to be insignificant and are discarded. In all probability, the main problem between the use of linear and nonlinear techniques relates to the scale at which the data are analyzed. The close examination,of a nonlinear relationship of aggregated data at the national or regional level often leads to the discovery of a subgroup where linearity actually exists between these same variables, but only within that particular subgroup. Thus it appears that the analysis should move through a series of steps beginning at the aggregate data level and slowly working down to smaller and smaller subgroups. At each step, nonlinear patterns can tell the researcher almost as much as the linear patterns, except that beyond a certain point the pursuit of nonlinearity will not prove to be useful. Somehow this problem of data scale is ignored, perhaps because of the view that higher-level laws of human behavior that cut across all groups are more desirable. Is it not more logical, however, for social and behavioral scientists to isolate first the laws of human behavior within the various levels of a given culture? The similarities and differences between subgroups can then lead us to higher and more encompassing behavioral laws as they are compared cross-culturally, but this cannot be done adequately if important intermediate steps are left out of the analysis by discarding behavioral laws for individual subgroups before they are even discovered. It was the use of nonlinear techniques which helped this author give meaning to social class in Aldea San Francisco. For example, aggregate linear analyses showed that the variable, social class, did not correlate significantly with aspects of migration or with more dynamic forms of spatial perception and behavior. Once the social-class subgroups were isolated, however, it became obvious

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through a combination of linear and nonlinear analyses that a group in the middle class stood out as having more accurate spatial perception and more dynamic behavior in most of the measured relationships than either the upper- or lowerclass groups.' Figure 1 (Wilkie, 1972A, p. 103) helps to illustrate that, for the upper-class peasant, the decision to migrate is often based on dictates imposed by peer groups, family, or life styles encouraged by the Church, while the moves involve shorter distances and smaller new communities and are much more passively dependent upon others than are the moves by migrants in the middle and lower classes. For one sizable group of middle-class peasants, who are often more "internally directed" by psychological, social, and spatial influences, the decision to move is generally planned to control one's own life and to select carefully the physical and social environment in which one lives. A second smaller group of middle-class peasants also behaves much like the upper- and lower-class peasants. Lower-class peasants tend to be torn between their traditional authoritarian ties to the community and a greater economic need that works to force them out of the community. The lower-class moves, especially those of the lower-class females, are not well planned, do not involve many options, and are more chain-like than the moves of the middle class. Although it appears to be a contradiction, both externally directed peasants (usually upper class) and traditionally directed peasants (usually lower class) look into the community. That is, they look away from themselves and into the community for answers, usually to peers or tradition. Internally directed peasants (usually middle class) turn inward to their own standards and core values for guidance of their behavior; consequently they look outward from the values and traditions of the community. Thus, for very different reasons, the externally directed and traditionally directed peasants hold Aldea San Francisco together as a community. In their eyes the internally directed peasants represent a disruptive force within the community because they do not adhere to custom and peer expectations. This often creates a push from within the community against peasants who are internally directed personally, and national- and world-oriented in general. This group, however, is also more receptive to pull forces from the cities than either of the other groups, and their moves tend to demonstrate a greater degree of planning and dynamic behavior than the other two subgroups. Thus, in spite of the fact that the variable social class did not surface from the aggregate linear analyses as significant, through nonlinear analysis it did turn out to be 1. It should be noted that the use of the terms upper, middle, and lower class relate specifically to social stratification within the peasant community of Aldea San Francisco. Even though each of the three peasant groups personally identify with national upper-, middle- or lower-class norms, these peasants would be ranked by outside observers for the most part as lower class in the national socioeconomic structure. The significance of each of these subgroups within a lower-class peasant community would be lost without making this distinction.

Richard W. Wilkie 5

extremely important, as vastly different migration processes were at work in each social-class group based on its own set of priorities, desires, dreams, and fears. One of the most interesting aspects of this analysis is that such great internal variation exists between population subgroups in what was thought to be a relatively homogeneous community that is lower class at the national level. The Hypothesis and Process Approaches to the Laws of Human Behavior Nonlinear relationships within the data are often ignored because most young scholars are trained to use the Hypothesis Method, which is considered the scientific approach. Other scientific approaches exist, however, and in gathering data for a study of migration in Argentina the standard Hypothesis Method was avoided because it usually permits testing only a few variables. Rather, a method was designed to generate enough data about human affairs in a peasant community to examine many complex and interrelated factors. Since the purpose of this approach was to measure a universe of variables involved in the various processes of change in Aldea San Francisco, the method was called the Process Method by the present author (Wilkie, 1968). Regarding the Process Method, this author states elsewhere (Wilkie, 1972A, p. 80): Such an approach stands in contrast to the so-called "scientific method" in which a limited hypothesis is set forth to test the relationship of a few variables which, a priori, are thought to be important. Studies of the processes of change demand an open-ended approach because seldom do only two or three variables relate independently from a maze of man's interactions, perceptions, and attitudes. In short, we are interested, for example, not only in finding out if A is related to B as might be hypothetically presumed, but, more importantly, how variables, A, B, C, D, E, F, G, H, etc., are interrelated to each other (A, B, and C may not be important in themselves but only when they combine with F).

Cattell (1966, p. 12) in contrasting the multidimensional and bivariate approaches supports this view. He has stated that the psychologist who pursues the multivariate approach may actually be proceeding more wisely than his far more exact bivariate experimentalist brethren. The important point is that he is attempting to bring into a single experimental field of reference all the variables necessary to detect and define the concepts that need to be employed for scientific understanding and without which it may not be possible to arrive at any lawful relationship. By contrast the bivariate experimentalist often starts out with such a meaningless fragment of the totality that it is impossible to encompass any lawful relation or construe the conceptual sentence. There is no particular reason why one should expect to find a simple and clean lawful relationship between any two of the two thousand variables that could be measured in a given situation. The lawful relation is more likely to exist between two or more abstracted concepts, each of which could be an underlying factor representable and measurable only by perception of a weighted combination of many variables.

Having suggested the existence of different approaches to discerning lawful relationships, let us explore the reasons why the Hypothesis Method tends to focus primarily on linear analysis, while the Process Method includes both linear and nonlinear forms of analysis.

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The very nature of the Hypothesis Method' has led scholars to examine only the linear relationships between variables. Digman (1966, p. 460) states that: Typically, in an analysis of variance, an investigator is examining the plausibility of a hypothesis of some kind of relationship versus none. In this sort of analysis, the question of non-linearity is irrelevant, particularly where the experimental variables are not ordered, i.e., where they consist of qualitative categories.... In the case of a two-way or more complex design, interaction should always be of interest to the researcher. Unfortunately, the discovery of significant interaction is all too frequently regarded as an annoyance, which implies that investigators, looking for simple answers to questions, are disappointed to find complex ones.

The result of this method, all too often, is that many scholars examine only the linear relationships within the data, thus avoiding the problem of what to do about nonlinear patterns. Digman (1966, pp. 459, 475) states, however, that "linear methods are usually regarded as useful in the determination of first approximations of relationships ... [yet] investigations in the field of multivariate psychology have been characterized by an almost complete dependency on linearity and additivity." Gould's argument (1970, p. 441) that "most of us seem to be stuck in a linear rut" is consistent with the view of other observers (Baken, 1954; Camilleri, 1962; Lindquist, 1953; Lubin, 1961; Morrison and Henkel, 1969; Plackett, 1960; Rozeboom, 1960; and Sidman, 1952) who also raise the issue. For example, Morrison and Henkel (1969, p. 136) state that this approach often leads "to an emphasis on prediction or `variance explained' in an acuarial sense rather than to concern with developing explanatory concepts and theories." Gould (1970, pp. 440-41) carries this argument against the exclusive use of linear analysis one step further by pointing out that many of the classical parametric statistical methods developed during the nineteenth and early twentieth centuries were founded on random distributions and used for "plausible descriptions of contemporary physical data and experiments, [which] are frequently quite meaningless as descriptions of social and behavioral data." In the study of human behavior the significant relationships may only be seen by consistency within whole patterns of interrelated variables which, if considered in isolated components, might not pass the sacred criteria of 95 percent confidence. The implications of a research perspective which subordinates logical significance to statistical significance are noted by Lubin (1961, pp. 815, 817): "Scientific inference is not identical with statistical inference. Tests of significance are certainly part of scientific methodology, but they are neither necessary nor sufficient for scientific induction.... [Thus the] most important questions arising out of a statistical finding of significant interaction are non-statistical ones of cause and control." 2. It has been suggested that it is unfair to set the Hypothesis Method up in the straw-man position, but that is not the purpose of this discussion. Since hypotheses can be useful in almost any research design, criticism is directed at the overstatement and overapplication by most (but not all) researchers who tend to claim that it is the only scientific approach.

Richard W. Wilkie 7

Debates in the social sciences about the role of statistical significance tests in assessing substantive significance have increased considerably since the late 1950s. In sociology, for example, Taylor and Frideres (1972, p. 464) mention agreement "that substantive significance is not statistical significance, and that it is a mistake to confuse the two." Morrison and Henkel (1969, p. 139) state this case strongly: Alas, statistical inference is not scientific inference. To have the latter we will have to have much more than the facade that claims of significance provide. But how is scientific inference possible if significance tests are of little help? This question leads us beyond the scope of this paper, but we have offered some hints: replication over diverse samples as well as internally, the use of abstract concepts, and the incorporation of such concepts in deductive theories with the conditions of their validity specified. There are, of course, no computational formulas for scientific inference: the questions are much more difficult and the answers much less definite than those of statistical inference. In the absence of such computations ... [we] will have to use [our] brains. We agree ... that science will not suffer. Not only is the researcher overly concerned with linear models, but as Cattell (1966, p. 13) points out, the hypothesis-testing researcher seldom makes major scientific breakthroughs in theory: Research need not begin with the hypothesis at all, and in its true life setting, a finished hypothesis is rarely the real germinal point of research action. It can begin with noticing a curious and intriguing regularity.... A statistical count of fruitful researches of any really frankly written history of science will show that the real turning point in major scientific theoretical advances, in a quite substantial and noteworthy proportion of cases, has been the noticing of just such intriguing regularities or irregularities as those in observed phenomena or tables of data gathered for some other primary purpose or out of sheer curiosity.

It should be noted again, however, that hypotheses in themselves are not to blame; rather, the method of using them is often abused. Instead of pursuing a "sacred ritual" where the entire analysis revolves around the central hypothesis, the Process Method lets hypotheses evolve out of the data that have been collected, as the nature of the relationships begin to take shape during the analysis.' Thus, the Process Method does not exclude the testing of hypotheses, but they are not central to the analysis. When the researcher is trying to discover cause and effect networks of relationships, which are different for each subculture group, the testing of hypotheses for every two-way relationship is often superfluous. This does not mean that a researcher should enter into a study without pursuing welldefined goals, but it is important that he not let those predetermined goals limit spontaneous creativity. The researcher who uses the Process Method is in a much freer position to allow the results of the relationships within the data to shape both the kinds of process hypotheses to be posed and the theoretical laws of human behavior which will result. Contrary to popular belief, it is not much more difficult to study a small group in great depth than it is to study a large group with only a few variables and a 3. Cattell (1966, pp. 15-17) refers to the creation of hypotheses from the data as the "inductivehypothetico-deductive spiral."

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hypothesis, except that one has to be more broadly trained. Nor is the selection of variables in the Process Method random, since great care must be taken in the construction of the universe of questions to be asked. Quantitative precision in the collection and handling of the variables is demanded on the same level as those who would test only several variables around a central hypothesis. In short, it is not logical to confine ourselves only to the analysis of linear relationships and to rely for the determination of significance only on statistical tests, yet the standard Hypothesis Method leads researchers to do just that. It becomes too easy to forget or to overlook the fact, for example, that within the processes of change one force often works to counter another force, thus resulting in nonlinear relationships in the behavior of population subgroups. Studies which attempt to interpret these nonlinear relationships as well as the more obvious linear relationships are essential if our research is to reflect reality rather than to develop an artificial view of reality. Linear Models and the Overemphasis on Studies of Structure Part of the problem of dependence on linear techniques is an overemphasis on structural studies as opposed to studies of the processes of change. Change and the processes which bring about change are constant and part of reality. Some processes of change are rapid, others slow; but the constant flux makes change more normal than nonchange. Thus, change may be defined as the primary reality, and studies of the processes of change considered fundamental to any theory about human social existence and interaction.' In contrast, structure is a more formal concept—an abstraction made from reality for one moment in time, often as an idealized model of what the system is under perfect conditions. Structural analysis is useful as a conceptual tool for understanding complexities; but since structures evolve constantly, those structures which are analyzed at one moment in time do not offer views of reality, but rather momentary truths. Most truths fade fast on rapidly moving research frontiers. The average life of a truth in the physical sciences is considered to be about ten years. Are we to assume that a truth in the social and behavioral sciences should last longer than a truth in the more precise sciences, or are we to recognize that with constant change we must constantly redefine new structures (the new truths)? This is not to say that all laws or truths are to be discarded at regular intervals. Rather, we must learn to think in terms of defining processes of change on an equal basis with our efforts in defining structures. Too many researchers, however, stop with Bergmann (1957, 1962) argues that process knowledge is the ideal in the sense of being the ultimate goal of all science, and to possess process knowledge is to know (1) the conditions of closure for the study of controlled boundary conditions, (2) a complete set of relevant variables, and (3) the process laws. 4.

Richard W. Wilkie 9

structure, and base theory on these findings in a rather deterministic manner. This form of determinism leads many researchers in each discipline to look for causes only in the structure of their own core specialty. Nonlinear Models and Complicated Truths Because of such problems, it seems more reasonable to base theory on a changing multidimensional system that attempts to explain the change dimension in a given discipline (e.g., spatial change) rather than to base it on the more static system which centers on only the structural components in that one discipline (e.g., spatial structure). Much of the widespread dissatisfaction with existing theories in the social sciences may reflect this preoccupation with structural patterns and a neglect of small-scale generating processes which invariably involve the interaction of structural components of several disciplines. If analysis of changing and evolving systems gives a more valid understanding of reality, theories should be based on forces that bring about change and the processes of change.' Thus, the study of process is a more unifying concept for the social and behavioral sciences and humanities than studies of structure; studies of process encourage scholars in various disciplines not only to make their work more relevant to others but also to search those other disciplines for important variables of explanation for use within their own component of concentration. To infer that structural studies have not outlined change over time would be somewhat misleading. Many social scientists have studied what they call process, or change over time, but in most cases these studies have represented a second stage in structural analysis (see Figure 2), or perhaps an intermediate step between structure and process. These historical base-line studies only tell us in what way the systems have changed from one moment in time to another; generally they do not tell us how the change occurred. Figure 2 illustrates how these three approaches or stages fit together to create a kind of spiral staircase of ongoing understanding within a given disciplinary component. Each of the three approaches represents a supporting column for the study of change, just as each of the three represents a step on the staircase. Step 1 (structure) outlines what the systems are at any one time. Step 2 (structure of change) is built on the first step, and shows comparatively how the structures have changed from one time period to the next. Step 3 (process of change) is built on the first two stages, and analyzes the interaction of many variables in an attempt to understand and specify the causes (and laws) by which the structures 5. For example, we cannot understand spatial change only by studying spatial structure and organization; rather we must examine how spatial structure and organization interact with psychological core values as well as with social, economic, political, temporal, philosophical, ecological, and other structural systems to give us greater insights into an evolving spatial order.

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Figure 2 Simple Structure and Process Interaction Diagram showing research approaches in the study of human behavior

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Richard W. Wilkie 11

change and evolve. Finally, knowledge of the laws that determine the processes of change then allows us to reset the definitions of the structures at a more sophisticated and higher level. This entire analytical progression is ongoing, each step built on the others. All three steps are of equal value and dependent upon the other two steps to reach a higher level of understanding. Step 2 (structure of change) can be bypassed initially in many studies, but eventually it is a vital step in the analysis as it yields insights into the forces of change themselves.' The processes of change, however, cannot be understood without first establishing what the structures are that are changing, nor can we understand the laws behind the structural systems without studying the processes and forces that are shaping and changing them. Although the traditional Hypothesis Method could be used to explore Steps 2 and 3, it usually focuses on Step 1—outlining structure, while the Process Method tends to focus on Step 3—outlining the process of change. Structure and Process Trends in Human Geography Perhaps it is too early to suggest that the trends in human geography in the twentieth century that are outlined in Figure 3 will accurately predict the 1970s and 1980s. If the need for understanding processes of change shown in Figure 2 can be taken to predict short-term direction, spatial processes and behavioral geography in general will continue to expand. This does not mean that the behavioral approach will or should be the exclusive emphasis in geography. On the contrary, a diversity of approaches should be encouraged as necessary for the total development of the field as outlined in Figure 2. In fact, while it does not show up clearly in Figure 3, all three approaches discussed here—culturalhistorical, structural, and behavioral—are interdependent. All three methods need to be strengthened so geography can move into the structure-process period (D in Figure 3) where each is recognized as being of equal value and each builds on the other two (spiral in Figure 2). Until that time, the three approaches will most likely be concerned with maintaining their own independence and traditions while maintaining an air of superiority. The placement of the cultural-historical approach on a lower level of understanding is not an evaluation of that school of geography in a qualitative sense. 6. Not only is Step 2 (structure of change) essential in understanding how structures and processes within various disciplinary components change over time, but the structure of change provides longitudinal studies of keys to another level of understanding. While Step 3 (process of change) specifies how interrelated factors lead to change, it is not supposed that these dynamics remain constant. Therefore, once the processes of change have been initially established in Step 3, longitudinal studies of changing structures and processes themselves will provide laws governing these forces. Additionally, philosophers of science will move beyond explaining the structures, structures of change, and processes of change in various disciplinary components (e.g., social, political, spatial, etc.) and will want to explain the laws behind the structures, structures of change, and processes of change themselves. Consequently all three steps are of equal value in the ongoing understanding process of human behavior.

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The point being made is not that one approach is more important or reaches a more meaningful level of understanding than the others, but that each provides a base of support which opens up new insights and possibilities for the other methods. Without the base of strength that the analytical-descriptive period of geography provided, the structural period could not have evolved as rapidly as it did. Additionally, the behavioral period is building from the already established bases of the other two approaches, especially the more recent structural (spatial organization) period. As the interaction between geographers in each of these major trends gains momentum, it will become clearer that the ultimate strength of each depends on strength in all three. Ideally, geography should reach a period of more harmonious equality (spiral in Figure 2) in the structure-process period (D in Figure 3). This means that a high priority should be given to the development of the behavioral process approach to geography. Until it is more fully developed, it will remain as a major roadblock holding up the ultimate advancement of all of geography. To make this needed breakthrough in behavioral geography, however, there must be a second quantitative revolution, this time to develop techniques of analysis for process. With certain exceptions, the quantitative techniques used and developed for understanding the more static structures will not accomplish what is needed in order to understand dynamically changing relationships. This paper does not have a solution to this dilemma, but it does provide several lowerlevel techniques that may help other researchers who are seeking insights into the complexity of diverse forms of subgroup spatial behavior. Isolating Levels and Subgroups within the Data Isolating the unit of analysis for a study of the processes of change has always proved to be difficult. The different behavioral processes operating within subgroups of society only come to light when the data are divided into levels with natural breaks between. One of the least complicated methods for discovering these natural levels is through the analysis of both data arrays and bivariate scatter diagrams. This approach requires close visual examination of the data frequencies and diagrams, as well as close personal familiarity with the data from the interviews through to the analysis. Instead of forcing divergent groups into a continuum, the researcher is able to distinguish, for example, between groups which show great growth and change and those which reflect relatively static positions. As Rosenberg (1971, p. 56) notes: "One of the major errors psychology continues to make is that it treats the human organism as a closed system, a mechanical model of fixed inputs and predictable responses. I would like to suggest that the human being is an open system, capable of change and endless growth. We should abandon the mechanical model for what might be called the process model." One technique—mathematical distortion—has ignored these differences, yet it

Figure 3 Structure and Process in Human Geography in the Twentieth Century

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STRUCTURE—PROCESS PERIOD structur . realization toot a nd process of chang v lue ,ae softructure change are o

HOPEFULLY BY THI 19D0's

LEVEL OF UNDERSTANDING

DUE IN THE IDDLE AND LATE 1970's 1

PER~OD , © BEHAVIORAL(P SPATIAL

CHANGE

ACUUT

behavioral descriptive

1?77

OU EVOITA1SNE or PROFESS f LATE 1950's Washington Sweden

that is of alue

structural descriptive

OD, RI A © STRUCTURAL TPE

OUANTISTRUC1URE 1REC REVOLUTION for

descripti ve traditional Cul tural•h1storical

DESCRIPTIVE I a ANALYTICAL PERIOD

WOO

I

l eun-lend eessl et Dro micro: re9lØl (initial atteFeDts

2R0 J: 1045

SCHOLARSHIP OVER TIME

RICHARD W. WILKIE February 1972

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is commonly used by researchers for ease of analysis. This distortion, which allows linear analyses of the data, does not allow the researcher to isolate readily the various subgroups which exist or to permit easy identification and interpretation of the particular levels in variables at a later stage in the analysis. If we are ever going to meet the needs and provide meaningful options for the various subgroups in society (even subgroups within subgroups), it is important to know that they exist, where they exist, and what their needs and aspirations are. This can never be accomplished if they are lumped together in ways that often cancel each other out of having any statistical significance. It is true that ultimately most meaningful relationships are linear within any behavioral subgroup if they are to be significant. But the question is: How does one isolate the behavioral subgroups before one tests for linearity or nonlinearity to discover significance? This author found that close analysis of data arrays and scatter diagrams for nonlinear patterns in the aggregate totals helped to isolate the existence of these subgroups. To have depended exclusively on linear tests for significance at the aggregate level would have been grossly misleading for those variables which were linear only within one behavioral subgroup. Therefore, the analysis should move through a series of steps beginning at the aggregate data level and slowly working down to smaller and smaller subgroups. At each step one can determine what pattern exists—linear or nonlinear. If nonlinear, it can tell the researcher almost as much as the linear pattern. It should be stressed, however, that a number of variables are needed in order to give consistency between nonlinear patterns. The lack of detailed data for isolating levels and dissatisfaction with the techniques of analysis has led many researchers who are working on structural studies to feel a sense of incompleteness in their work. Such a feeling of frustration has been well described in a self-revealing novel by an economic historian, Williams (1960, pp. 9-10): It is a problem of measurement, of the means of measurement, he had come to tell himself. But the reality which this phrase offered to interpret was, he could see, more disturbing. He was working on population movements into the Welsh mining valleys in the middle of the nineteenth century. But I have moved myself, he objected, and what is it really that I must measure? The techniques I have learned have the solidity and precision of ice-cubes, while a given temperature is maintained. But it is a temperature I can't really maintain; the door of the box keeps flying open. It's hardly a population movement from Glynmawr to London, but it's a change of substance, as it must have been for them when they left their villages. And the ways of measuring this are not only outside my discipline, they are somewhere else altogether, that I can feel but not handle, touch but not grasp. To the nearest hundred, or to any usable percentage, [a move like my own] is indifferent, but is not only a relevant [move] : without it, the change can't be measured at all. This quote underscores the basic differences between research projects which are bound up in limited studies of structure and studies of process which examine the forces that together create an action pattern of human behavior.

Richard W. Wilkie 15

In order to illustrate the Process Method developed here, the following section discusses the interrelationships of four variables which have been selected from a comprehensive analysis of interaction and change in an Argentine peasant community presently being developed by the author. Illustration of Nonlinear Analyses in the Process Method The argument developed here to support the use of nonlinear forms of analyses involves (a) showing a need for the use of nonlinear techniques; (b) developing a way to examine and illustrate nonlinear relationships; and (c) analyzing the relationships among four variables: social-class position, accuracy of regional spatial perception, total regional movement during one year, and primary source of communication information for each peasant household group. These four variables were selected from 136 variables collected on all 290 inhabitants (58 household groups) of Aldea San Francisco, Entre Rios, Argentina. In an intensive study of this community, between 300 and 600 items of data were collected for each family. These items were grouped into 136 composite variables that cut across the major components within the community, including demographic, economic, psychological, social, spatial, political, historical, and ecological components (Wilkie, 1968). The Nature of a Nonlinear Relationship: Social Class and Regional Spatial Perception When data on social class' in Aldea San Francisco are tested against regional spatial perception (distance between and populations of communities within the region), most researchers probably would decide that no relationship exists because the correlation coefficient of 0.06 is neutral. A scatter diagram of the same data, however, reveals in Figure 4 that a pattern exists which helps us to understand how social class in Aldea San Francisco is related to many forms of spatial and social behavior. In the first stage of analyzing the diagram, there is little the researcher can do beyond noting that more accurate regional perception builds to a peak in the middle class and then drops sharply within the upper class. Analysis of my field research notes, however, facilitates preparation of a modified diagram (Figure 5) which corrects two regional perception cases that seem to be out of position. One case of low regional perception, circled X in Figure 4, is an elderly widow who spent most of her life in the upper class and was forced to sell off land possessions in order to obtain funds after the death of her husband. 7. Social-class groups were determined by scaling the average score (0 to 9) given by each head of household for every other family in town on the "good life." Since each respondent had to define the "best possible life" as well as the "worst possible life," the scores reflected definitions set by the respondents rather than the researcher. These good-life rankings proved to be highly correlated with traditional measures of social class (Wilkie, 1968 and 1972A). These rankings had a correlation of 0.88 with economic stratification and 0.76 with respondents' scores of interpersonal confidence with every other family.

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Figure 4 Scatter Diagram: Social Class and Regional Spatial Perception (correlation 0.06; n = 31 families & 155 inhabitants)

8 x

7

high 6

x

5

REGIONAL

x

SPATIAL medium

x

x

x

x

X x

x

x

x

X

x xx

x

x

4

PERCEPTION

x

x

x

3

x

lOw 2

0

1 21

3

4 lower

5

6

middle

7 8 upper

SOCIAL CLASS For this reason she is now ranked in the middle class although her spatial perception remains consistent with the upper class. The second case, X in broken circle in Figure 4, represents a peasant who is one of the poorest in the community, yet is honest, well liked, and ranked higher in social-class status than his economic position warrants. If these two cases are changed and placed in corrected positions according to economic rank, the curvilinear pattern, as seen in Figure 5, is even more obvious. For the rest of the analysis, however, these two cases will remain where they are in Figure 4, as the other peasants perceive them. A next step in the nonlinear analysis of the variables social class and regional spatial perception is to investigate the nature of the relationship within the various natural levels of the data. This can be accomplished in a number of ways. First, one may run a separate correlation analysis of these variables for each social class. By social-class level, the correlation coefficients are 0.57, 0.04, and —0.87 for lower, middle, and upper classes, respectively. This tells us a great deal more than does the correlation coefficient of 0.06 for the three groups combined. The significant rise in the linear correlation among the lower class, the nonexistence of any relationship among the middle class, and the strong negative linear relationship among the upper class helps to show the curvilinear nature of the relationship.

Richard W. Wilkie 17

Figure 5 Nonlinearity Between Social Class and Regional Spatial Perception

8 7 high 6 REGIONAL SPATIAL PERCEPTION

medium 5 4 3 low 2

3 4 lower

5

6

middle

8 upper

7

SOCIAL CLASS correlation 0.57 coefficient by social class

0.04

-0.87

Another way of viewing this relationship is by using a cross-tabulation matrix presenting the total number of individuals falling into each of the possible combinations. Figure 6 illustrates a three-by-three matrix case. Both Digman and Lubin suggest that this technique helps to uncover more sophisticated levels of interaction in the data by making the patterns within the data visually apparent. Lubin (1961, p. 815) expresses the point of view (in a medical context) that it is "far more important to determine the form of the equation relating the treatment effect to the block effect than to make accurate statistical inferences about the variance of the difference between two means." Thus, he is saying that treatment of the patient varies on the variables examined by where the patient falls in the matrix. There are nine different medical approaches to a person with a given illness rather than only two—yes or no—based on linear statistical significance of the variance from the means. Digman (1966, pp. 460-61) uses a visual representation of such data in graph form, as illustrated in Figure 7, where the height of each bar is equal to the number

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Figure 6 Possible Combinations in a Two-way Matrix of Three Levels Each

B3 HIGH VARIABLE B

B2 MEDIUM

B1 LOW

low—A HIGH—B

MEDium—A HIGH—B

HIGH—A HIGH—B

low—A MEDIA= —B

MEDium—A MEDium—B

HIGH—A MEDium—B

MEDium—A low—B

HIGH—A low—B

low—A low—B Al LOW

A2 MEDIUM

A3 HIGH

VARIABLE A

Figure 7 Interaction in a Two-way Analysis of Variance: Social Class and Regional Spatial Perception

Ai

A2

SOCIAL CLASS LEVELS

A3

Richard W. Wilkie

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REGIONALSPATIALPERCEPTION

Figure 8 Peasant Regional Spatial Perception by Social Class Level

lii ,h

high

50

medium

38

low totals: percent 100% number

7 lower

10 0 % 16 middle

medium

low

100% 8

n=31

upper

SOCIAL CLASS

of cases in each of the nine units. This visual presentation, however, lacks the precise percentages that fall into each box of the matrix, even though the researcher can see that linear relationships exist in row B1 and column A 2, and curvilinear relationships exist in rows B2 and B3 and columns Al and A 3. Figure 8 illustrates that another way to view the matrix is to show the percentages in each social class that fall into the categories of high, medium, or low regional spatial perception. Both the lower and upper classes have their highest proportions (72 percent and 50 percent, respectively) in the medium regional spatial perception level, while the middle class has its highest proportion (50 percent) in the high regional perception level. By adding gray background densities to the percentages, there is an additional visual element that makes the nature of the curvilinear pattern even more obvious. Either the use of Zipatone or photographic use of line screens yields the desired gray densities. The researcher may find these options helpful in discovering nonlinear patterns operating within the data as well as in illustrating select patterns to a wider audience who is less familiar with the data. In addition, it is helpful to examine social class and regional spatial perception with reference to the percentage of lower-, middle- and upper-class peasants in each level of regional spatial perception. Figure 9 shows that of those peasants

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Figure 9 Peasant Social Class by Regional Spatial Perception Level totals: percent

ö high M F a U W

—►

► o å

number

10

80

10

100%

10

medium

36

43

: : : 21 : : :

100%

14

low

::::14 .:::

29

57 ty:>. r

100%

7

0

å-1 ra -W

å cn

lower

middle

upper

SOCIAL CLASSES

n=31

with high regional spatial perception, four out of every five (80 percent) are in the peasant middle class, while of those with low regional perception, slightly more than half (57 percent) are in the upper class. Thus, it is obvious that a much larger group of peasants within the middle class has a good grasp of distances between and the sizes of regional centers than do the peasants in either the lower or upper classes in Aldea San Francisco. This ability to understand the spatial network in which the peasants operate daily ultimately leads to a more dynamic group of out-migrants. Of 99 out-migrants studied using this variable, only one in five (22 percent) of those who moved one time came from families with high regional spatial perception. This figure rose to two-fifths (41 percent) for two moves, and for three or more moves three out of five (59 percent) came from families with high regional spatial perception (Wilkie, 1972B). Analyzing the Relationships among Four Variables The discovery of a nonlinear pattern that appears to be significant may provide very little insight into the analysis unless other variables which have been collected also back it up. The real meaning behind the pattern will show up through a logical consistency in the relationships from one set of variables to the next. By adding two new variables for each family—regional movement and sources of communication—we are able to enrich the interpretation of the analysis. The relationship between social class and regional movement (see Figure 10), for example, shows that the relationship is nonlinear except for one subgroup of upper-class peasants which has the highest frequency of regional movement in spite of the fact that the upper-class peasants have the lowest mean scores in accuracy of regional spatial perception. Thus it appears, at least among this one subgroup, that a greater frequency of regional movement to other centers does

Richard W. Wilkie 21

Figure 10 Scatter Diagram: Total Regional Movement and Social Class Position

(correlation coefficient: 0.47) n = 39

SOCIAL

CLASS

xx x

X X

upper class

x x x x x x x x x x x x xx x xxx x xx xxx

middle class

x

x

xx

lower class

0

25

50

100

75

REGIONAL MOVEMENT (number of trips a year)

Figure 11 Regional Movement by Regional Spatial Perception Level

high

REGIONAL MOVEMENT medium (number of trips a year per household) low

39

14

totals: percent 100% number

100%

100%

13

8

7

low

medium

high

REGIONAL SPATIAL PERCEPTION

n=28

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not necessarily lead to an increased perception of community sizes and distances. Surprisingly, it may be noted in Figure 11 that nearly half (47 percent) of those peasants with low spatial perception have high regional movement, while over one-half (57 percent) of those peasants with high spatial perception have only medium regional movement. The final variable added to the analysis concerns the sources of communication (information) entering the household. At one extreme, orientation of the peasant toward the national and urban life for sources of information is used to designate the peasant as a "cosmopolite" (Merton, 1957, p. 18). At the other extreme, dependency on only local sources for information classifies the peasant as a "localite." Interaction between source of communication and each of the other three variables is found in Figures 12 and 13 (social class), Figure 14 (regional movement), and Figure 15 (regional spatial perception).

Figure 12 Sources of Communication by Social Class Levels

COSMOPOLITE communication

transitional

LOCALITE communication totals: percent number

100%

100%

8

16

lower

middle

100% 7

n=31

upper

SOCIAL CLASS

In analyzing the relationship between source of communication and social class, observe in Figure 12 that the middle class is highest in cosmopolite communication (50 percent), the upper class is highest in the middle transitional category (57 percent), and the lower class is highest in localite communication (50 percent). By examining only the localite level in Figure 12, we see that there is a decline in the percentage of localites as the social class level goes up. This trend

Richard W. Wilkie 23

Figure 13 Peasant Social Class by Source of Communication Level totals:

COSMOPOLITE

9

communication

—s transitional

LOCALITE communication

::: 28 :::

percent

number

: : : 18:::

100%

11

36

100%

11

100%

9

36

>: 44 :

lower middle upper

n=31 SOCIAL CLASS

Figure 14 Regional Movement by Levels of Communication Orientation totals:

percent ,_ ■

COSMOPOLITE communication

—~ transitional

: : : 22 .::

36

::::28::::

number

::::22.:::

100%

9

36 .

100%

11

100%

8

LOCALITE communication low medium high REGIONAL MOVEMENT (number of trips a year) per household

n=28

does not lead to a cosmopolite orientation in the upper class, but to a transitional orientation where a mixture of both rural and urban sources of communication is found. A cosmopolite orientation is found most often in the middle class (50 percent). Several observations can be made by analyzing Figures 12 and 13 together. First, although only one-half of the middle-class peasants are cosmopolites, they

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make up nearly three out of every four (73 percent) of all cosmopolites in the community. This is partially explained by the fact that 45 percent of the community is middle class, 30 percent lower class, and 25 percent upper class. Second, more than one-half (57 percent) of the upper-class peasants are transitional, yet they make up only slightly more than one-third of the community members in that category. Finally, in the localite level one-half of the lower-class peasants are localites, and they make up slightly under one-half (44 percent) of all localites in Aldea San Francisco. Since the largest group of cosmopolites falls into the middle class and the middle class moves about in the region less than the upper class, we may hypothesize that cosmopolites also have lower regional movement. This in fact is the case. Figure 14 shows that higher regional movement does not correspond highly with cosmopolite orientation, just as it does not lead to greater spatial perception of the region. The largest group of cosmopolites (56 percent) is found in the medium level of regional movement, while of those with high regional movement, only about one peasant in four (22 percent) is a cosmopolite. No clear pattern emerges in the transitional communication level, as it is evenly divided between low, medium, and high regional movement. In the localite level, however, nearly two out of every three peasants (63 percent) have low regional movement. Thus low movement corresponds with localite, but high movement does not correspond with cosmopolite—medium movement does. In fact, among medium regional movement types, the percentage at each level of source of communication increases monotonically as we move up from localite (13 percent) through transitional (28 percent) to cosmopolite (56 percent). This pattern is reversed among low regional movement types, and both of these patterns help to accentuate the curvilinear nature of the relationship. At this point in the analysis we can ask an important question: Why do those peasants who have (a) a cosmopolite orientation to information and (b) a high spatial perception of their own region have less regional movement than many others? Here we can hypothesize that since they know more about their own region and the outside, they plan more carefully the trips that they do make, and their trips tend to be less random in frequency compared with those of other groups who seem to travel more on whim than on rational planning. Using the Process Method (which does not exclude the testing of Process Hypotheses but lets the results of the data help to generate hypotheses), it is possible to turn to a number of other variables that may interrelate with frequency of regional movement, such as planning and life reflection, attitudes of parents toward child exploration, trust versus mistrust of the environment, ambition versus resignation, and/or many others. In fact many of these variables do support this hypothesis (Wilkie, 1973), but it is not feasible to go into them here. If the standard Hypothesis Method had been used from the beginning, the hypothesis most likely would have been that higher regional spatial perception and interest in cosmo-

Richard W. Wilkie 25

polite sources of information would lead to higher regional spatial movement. Having discovered that this is not the case, the researcher would probably have discarded the variable and missed the real implications of the data. Furthermore, items relevant to the problem would not have been collected. Figure 15 Scatter Diagram: Regional Spatial Perception and Communications Orientation (correlation coefficient: 0.71) n = 31

x x

COSMOPOLITE communication x

x

x

x

x x x

x

x

x

X X X X

transitional

x

x

LOCALITE communication

x low

medium

high

REGIONAL SPATIAL PERCEPTION All of the relationships examined to this point have been nonlinear, yet have been quite revealing. One last analysis of the relationship involving the original four variables, that of source of communication and regional spatial perception, remains to be made, however, and Figure 15 illustrates that the relationship is linear and has a correlation of 0.71. Thus, the more a peasant understands and perceives properly the spatial system in which he functions from day to day, the more he is also oriented to communication from the outside world. Knowledge of one's own habitat and confidence in that knowledge seems to create cosmopolites; and, in addition, as one's interest in the outside world increases, there is an increased awareness of one's immediate environment. These influences can become circular and part of the behavioral system for that particular subgroup. A major purpose of this analysis has been to develop techniques for isolating the different behavioral systems in Aldea San Francisco. We have seen that both nonlinear and linear analyses serve to enlighten our understanding of these systems. Figure 16 is a simplified pictograph of the way these four variables fit

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Figure 16 Behavioral Families Related to Peasant Social Class Levels (Aldea San Francisco, Entre Rios, Argentina) UPPER SOCIAL CLASS OF PEASANTS

I

r 1

1

I

i JL _I L i

'

4

I

I

1

MIDDLE SOCIAL CLASS OF PEASANTS

REGIONAL MOVEMENT

LOW REGIONAL SPATIAL PERCEPTION

(BOTH HIGH A LOM)

TRANSITIONAL COMMUN CATION ORIENTATION

LOWER SOCIAL CLASS OF PEASANTS

R.t

I

t

HIGH COSMOPOLITE MEDIUM REGIONAL COMRUNI CAT ION REGIONAL SPATIAL ORIENTATION MOVEMENT PERCEPTION

I i

MEDIUM REGIONAL SPATIAL PERCEPTION

LOCALITE LOW COMMUNICATION REGIONAL ORIENTATION MOVEMENT

together into a behavioral family for each of the peasant social-class levels in the community. Behavioral Family A (the upper-class peasantry) is associated primarily with low regional spatial perception, a transitional orientation to communication, and either low regional movement or high regional movement (two groups). Behavioral Family B (the middle-class peasantry) has predominately high regional spatial perception, a cosmopolite orientation to communication, and medium regional movement. Finally, Behavioral Family C (the lower-class

Richard W. Wilkie 27

peasantry) has medium regional spatial perception, a localite orientation to communication, and low regional movement. These groupings are not meant here to be final, but they may give some behavioral insights into a small peasant community in Argentina. Limitations of This Case Study Clearly the major problem in the example of Aldea San Francisco is the limited number of cases with which to determine the relationships. Ideally, a total sample for every variable of every member of a small group would be the best approach in a process study. While lacking that complete data for a number of variables that later turned out to be important, this author found that the results from about half the households in Aldea San Francisco were entirely consistent with the patterns established in those variables where every member of the community was included. For example among the 243 out-migrants, similar nonlinear patterns continue to show up in the data, especially with regard to the variable social-class position. Figure 17 shows the mean migration behavior by social class of 116 females who left the community between 1920 and 1967 (Wilkie, 1972, Figure 17 Female Migration Behavior by Social Class (Aldea San Francisco, Entre Rios, Argentina; n = 116) Percent as OutMigrants

Social Classes UPPER

67.4

MIDDLE

21.2 78.9

UPPER

Number of Moves 1.47

MIDDLE

167

18.4

\ 86.1

LOWER

Social Classes

Distance in Kilometers

Age

270

N 19.8

Percent "Active" Migrants 18.5

179 Urban Size Levels* after 1st move latest move 4.65

45.5

1.57

4.31 6.91

7.18 %

LOWER

1.38

32.0

5.20

5.40

*The mean urban size levels are as follows: 4= 5001 to 10,000 population, 5= 10,001 to 20,000 population, 6= 20,001 to 50,000 population, 7= 50,001 to 100,000 population, and 8= 100,001 to 500,000 population.

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p. 98). In each case, with the exception of the percent of females as out-migrants, the middle-class females were the most dynamic of the three groups. In contrast, with the exception of average number of moves, the upper-class females were the least dynamic. It is this insight into the consistency of the majority of the relationships tested in this study that strengthens the interpretation (Brodbeck, 1968; Cattell, 1966; Morrison and Henkel, 1969; Olsson, 1969B; Turner, 1968), and it shows that decision-making processes are vastly different within each of the population subgroups in Aldea San Francisco, especially by social-class level. This is clearly a case where substantive inference reveals more than statistical inference. A second problem is that only four variables have been presented out of 136 that were collected in the field. Obviously, the more variables that are included in the analysis the more complex, yet more revealing, the interpretation becomes. The inclusion of more variables into the analysis in this paper, however, is not possible because the major purpose has been to illustrate that most nonlinear relationships add as much insight into the behavioral interpretation as do the linear relationships. The purpose of the Process Method is to examine the particular sets of relationships (some high, some medium, and some low) that lead to different kinds of spatial behavior. Conclusions This illustrative case has been used to (a) show that at times dependency on only linear analyses obscures relationships that actually exist in the real world, but only among certain subgroup levels, (b) develop a lower-level technique for analyzing nonlinear patterns that result from different subgroup behavior, and (c) give theoretical insights into the processes which are changing and evolving the spatial behavior of social-class subgroups in a small peasant community. All too often the linear rut pushes us into almost total acceptance of correlation coefficients as the sole criteria of importance which may further obscure some relevant but nonlinear phenomena that need to be explained. Utilizing data from Aldea San Francisco, Entre Rios, Argentina, this study has shown that significant and complex nonlinear relationships exist in five out of the six relationships tested for the entire community. These relationships most likely would have been ignored in simple linear analyses using aggregated data. Through the use of techniques devised to analyze and illustrate nonlinear relationships, consistent patterns within the data are revealed which help to isolate the subgroups of similar behavior where tests of linearity for significance do prove meaningful. In the sixth case, a linear relationship of aggregated data for the entire community is found which logically supported the conclusions derived from the nonlinear analysis at that level. Thus the Process Method, utilizing both nonlinear and linear forms of analysis, leads to an understanding of the way in which many variables interact to bring about change. Key to this discussion was the question of what

Richard W. Wilkie 29

role the data plays in the development of a body of theory and laws concerning spatial behavior. Since change is constant and evolving structural systems are normal, it is essential to include the process dimension of study within the social and behavioral science disciplines. The Process Method, which involves the development of Process Hypotheses, is quite different from the traditional Hypothesis Method which tends to focus only on linear correlations while testing a limited number of variables involved with studies of structure. Both studies of process and structure are important, however, because the study of the processes of change is not possible without, as a first step, carefully establishing the nature of those structures which are changing. If we are to base theory and development policy only on past structures (most often established through aggregate linear analysis) and to ignore the processes that change the structures and the behavior within different subgroups and levels of society (often discovered in nonlinear analysis), then our planning will not only be several steps behind current and future reality but also may at times be wrong. The statistical canceling out of many different forces does not necessarily negate their reality. Acknowledgments The author would like to acknowledge criticism offered by Surinder K. Mehta, David R. Meyer, James W. Wilkie, and Jane R. Wilkie. Any shortcomings, however, are solely the responsibility of the author. References BAKAN, D. (1954). "A Generalization of Sidman's Results on Group and Individual

Functions, and a Criterion." Psychological Bulletin, 51:63-64. BERGMANN, G. (1957). Philosophy of Science. Madison: University of Wisconsin Press. BERGMANN, G. (1962). "Purpose, Function, Scientific Explanation." Acta Sociologica, 5:225-38. BRODBECK, M. (1968). "Models, Meaning, and Theories." In May Brodbeck (ed.), Readings in the Philosophy of the Social Sciences. New York: MacMillan, pp. 579-600. BURroN, I. (1963). "The Quantitative Revolution and Theoretical Geography." Canadian Geographer, 7:151-62. CAMILLERI, S. F. (1962). "Theory, Probability and Induction in Social Research." American

Sociological Review, 27:170-78. CATTELL, R. (1966). "Psychological Theory and Scientific Method." In Raymond Cattell (ed.), Handbook of Multivariate Experimental Psychology. Chicago: Rand McNally, pp. 1-18. DIGMAN, J. (1966). "Interaction and Non-Linearity in Multivariate Experiment." In

Raymond Cattell (ed.), Handbook of Multivariate Experimental Psychology. Chicago: Rand-McNally, pp. 459-75. GOLD, D. (1969). "Statistical Tests and Substantive Significance." American Sociologist, 4:42-46.

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(1970). "Is Statistix Inferens the Geographical Name for a Wild Goose?" Economic Geography, 46 (Supplement): 439-48.

GOULD, P.

JOHNSTON, R. J. (1970). "Grouping and Regionalizing: Some Methodological and Technical Observations." Economic Geography, 46 (Supplement): 293-305. LINDQuIsT, E. F. (1953). Design and Analysis of Experiments in Psychology and Education. New York: Houghton-Mifflin. A. (1961). "The Interpretation of Significant Interaction." Educational and Psychological Measurement, 21:807-17.

LUBIN,

MERTON,

R. (1957). Social Theory and Social Structure. Glencoe, Ill.: The Free Press.

MEYER, D. (1971). "Factor Analysis Versus Correlation Analysis: Are Substantive Interpretations Congruent?" Economic Geography, 47 (Supplement) : 336-43. MORRISON, D., and R. HENKEL (1969). "Significance Tests Reconsidered." American Sociologist, 4:131Ø. OLSSON, G.

(1969A). "Trends in Spatial Model Building: An Overview." Geographical Analysis, 1: 219-24.

OLSSON, G.

(1969B). "Inference Problems in Locational Analysis." In K. Cox and R. G. Golledge (eds.), Behavioral Problems in Geography: A Symposium. Evanston: Northwestern University Studies in Geography, No. 17, pp. 14-34. PLAcKSrr, R. L. (1960). "Models in the Analysis of Variance." Journal of the Royal Statistical Society, 22:195-217. ROGERS,

E. (1962). Diffusion of Innovations. Glencoe, Ill.: The Free Press.

ROSENBERG,

B. G. (1971). "Psychology Through the Looking Glass." Psychology Today,

Vol. 5. ROZEBOOM, W. W. (1960).

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"A Note on Functional Relations Obtained from Group Data." Psychological Bulletin, 49:263-69.

K. W., and J. FRIDERES (1972). "Issues Versus Controversies: Substantive and Statistical Significance." American Sociological Review, 37:464-72.

TAYLOR,

TURNER, M. (1968).

Philosophy and the Science of Behavior. New York: Macmillan.

R. W. (1968). "On the Theory of Process in Human Geography: A Case Study of Migration in Rural Argentina." Unpublished Ph.D. dissertation, Department of Geography, University of Washington.

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WILKIE, R. W. (1972A). "Toward a Behavioral Model of Rural Out-Migration : An Argentine Case of Peasant Spatial Behavior by Social Class Level." In Robert N. Thomas (ed.), Population Dynamics of Latin America: A Review and Bibliography. East Lansing, Mich.: CLAG Publishers, pp. 83-114.

WILKIE, R. W. (1972B). "Urban-Rural Relationships in Migration from an Argentine Village." Paper presented at the IBP V General Assembly (International Biological Programme), Seattle, Washington, August 31, 37 pp.

Richard W. Wilkie 31

R. W. (1973). "Selectivity in Peasant Spatial Behavior: Regional Interaction in Entre Rios, Argentina." Proceedings of the New England-St. Lawrence Valley Geographical Society, 2:10-20.

WILKIE,

WILLIAMS,

R. (1960). Border Country. London: Penguin Books.

WINCH, R., and D. CAMPBELL (1969). "Proof? No. Evidence? Yes. The Significance of Tests of Significance." American Sociologist, 4:140-43.

The Nearest-neighbor Statistics for Testing Randomness of Point Distributions in a Bounded Two-dimensional Space SHIN-YI HSU (SUN Y-Binghampton)

and J. DAVID MASON ( University of Utah)

The nearest-neighbor technique was originally developed based upon the Poisson distribution of points in space by astronomers in the late nineteenth and early twentieth centuries (Hertz, 1907). In the late 1940s and early 1950s this technique was reviewed in the same mathematical framework by ecologists and applied to ecological studies (Skellam, 1952; Moore, 1954; Clark and Evans, 1954). Since the middle 1950s geographers have borrowed this technique from ecologists and employed it in geographic research. However, the application of the nearestneighbor technique in geography has generally been invalid. The sources of error are threefold. First, an error was introduced in the estimation of the mean (hence the variance) of nearest-neighbor distances arising from using the totality of points as centers of origin for measuring distances because they are not randomly selected (Kendall, 1963). In addition, when there are reflexive neighbors, the same distance is used twice, hence the rule of summing independent random variables (distances) is violated, and a biased estimate is obtained. Second, the Poisson model of point distributions in space assumes that the phenomenon under study 32

Shin-yi Hsu and J. David Mason 33

is unbounded, whereas geographic research is undertaken almost always in areas having definite boundaries. This means that the boundary effect exists in reality but is neglected by the model. And lastly, the Poisson model requires the assumption of infinite population size, whereas geographic population, such as urban places in a given state, is finite and usually small. Therefore, the Poisson model should not achieve a good approximation. The purposes of this paper are: (1) to examine the error in the estimate of the mean of nearest-neighbor distance under the Poisson point process and to propose a method to obtain an unbiased estimate; (2) to examine the problems of boundary effect and population size; (3) to develop the nearest-neighbor statistics in a bounded point set to solve these problems; and (4) to compare this new model with the Poisson model.

Error in the Estimation of the Mean Distance The error in the estimation of the mean and variance of the nearest-neighbor distances is derived from violation of one of the assumptions in the Poisson model of point distributions in space. This can be understood better by examining the assumptions and the derivation of the Poisson model. There are four primary assumptions by which the Poisson model, or the formula P(k) = e-'i(2)k/k! (to be explained in equation 1), of point process in time and space is derived. These four assumptions of Poisson spatial process are given by Dacey (1964, p. 44) as follows: (A) Statistical equilibrium: the probability that a region of area A contains n points is the same regardless of (1) the location of the region and (2) the shape of that region. (B) Independence of events: the probability that one point occurs in any region dA is )4A, and the probability that more than one point occurs in that region is of smaller magnitude than AdA. (C) Differentiability with respect to A. (D) Boundary conditions: denote the probability that a region of area A contains exactly n points by P(n, A). In the physical situation, it may be taken that A is nonnegative and for n 0 0 P(-n, A) = 0 P(0,0) = 1 P(n, 0) = 0

In applying the Poisson model to test whether a given distribution is significantly different from Poisson random, two additional assumptions are required (Parzen, 1962, pp. 31-32). Consider a group of points distributed in a space S, where S is a Euclidean space of dimension _ > 1. For each region R, let N(R) denote the number of points (finite or infinite) contained in the region R. The array of points is said to be a Poisson distribution with density p > 0 if the following assumptions are fulfilled : (i) the number of points in nonoverlapping regions are independent

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random variables; more precisely, for any integer n and n nonoverlapping (disjoint) regions R1 , R2, . , R„, the random variables N(R1 ), ... , N(R„) are independent; and (ii) for any region R of finite volume, N(R) is Poisson distribution with mean p V(R), where V(R) denote the volume of the region R. By following these two assumptions, the classical Poisson model is applied in spatial distributions as follows. Let p represent the mean density of the total region, which is the total number of points (N) divided by the total area (A), that is, p = N/A. Then from a random distribution of points on a plane, the probability that a randomly chosen unit area will contain exactly k points is the Poisson function derived from the above-mentioned four primary assumptions. (1) P(k) = e-2(2)k/k! (2 is mean density). Equation (1) can be used in the quadrat method to calculate Poisson frequencies against which the observed frequencies are compared according to y2-test of "goodness-of-fit” (King, 1969, p. 81). The nearest-neighbor technique is also derived from equation (1) (Clark and Evans, 1954). Let the specified area be a circle of radius r. If p is the number of points per unit area, pare is the mean density. Then the probability of finding exactly x points in this circle is: P(x) = e-Ø(pnr2)x/x! (A = pnr 2). (2) Consequently, the probability that no points will be contained in this circle is: P(0) = e-Pa'. (3) If the specified area is a circle about a randomly chosen point, the probability that the circle will contain no other point with a distance r of the chosen point is also e-P„'2. Therefore, the probability of distances of nearest neighbor less than or equal to the distance r is:

P(r) = 1 — e-Q1°2. Then, the probability density of r is:

(4)

dP(r) = 2pnre-P"2 dr. (5) The expected (mean) value of r can be obtained by applying the general equation of E(r) which is f rg(r) dr as the following: CO

0

E(r) = f 2par2 a-' dr = 1/(2‘5)

(6)

By comparing equation (1) and equation (4) and relating them to the two secondary assumptions, it can be understood that the random variables in the quadrat method are the number of points, k's, whereas those in the nearest-neighbor technique are distances, r's. The assumption (i) requires that k's and r's be independent variables. For k's to be independent, take points in nonoverlapping regions (quadrats) ; for r's to be independent, we should sample centers of origins for measuring nearest-neighbor distance. More precisely, sampling the centers of

Shin-yi Hsu and J. David Mason 35

origins with replacement and using only one of the reflexive distances can assure that all of the distances are independent random variables. These two principles are, however, violated by geographers in the calculation of mean distance and its standard deviation. In practice, we treat the total population as the centers of origin and measure the distance from each point to its nearest neighbor. The mean of the distance obtained can be expressed as follows: robs. = E rtmal/Naoln1 •

(7)

Therefore, to correct the error, an unbiased estimate of the mean distance should be obtained as follows:

N [~~

.ampto. effective

runbiascd = k=1

Nanmplc, cflcctivc

(8)

rIJ/

where Nsample, effective = sample points — invalidated endpoints of the pair. This means that an unbiased estimate should be obtained by sampling randomly the centers of origin, and by counting the paired distance only once, and invalidating the endpoints of the pair at the same time. We need to prove that runbiased is an unbiased estimator of the expected nearest-neighbor distance E(r) in equation (8) (Hsu, 1969). To show that equation (8) is correct, let g(r) = Ç as indicated in equation (4) and (5), for r > 0. The ru are independent random variables with density g. Let f be the characteristic function of g, i.e., f(u) —

g(x) etas dx

(9)

f 0

where

—oo< u < co, and i= f'(u) = f ixg(x) elan dx

(10)

hence —if (0) = f xg(x) dx = E(r). 0 N

To show E(r,,,,blased) = E(r), letfN be the characteristic function of

E r u/N, and k=1

the random variables r i/N are independent, we have (12) IN (u) — (f(u/N) )N because the characteristic function of a sum of independent random variables is the product of the characteristic function of the summands. Let h be the characteristic function of Funblased •

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Then h(u) = E(eØ,.obta.")

• sample size k=0

iu£r,iik

where I,,(x)= 1,if x e A =0, ifxA

hence, h(u)) =

C''azmple s;ze E( eiuEr'1ik )P( erce N tive = k))

k=0 • sample size k=0

c

k ( P(Netteetive = k=0samplc size (f(u/k)) = k)

(13)

since

—ih'(0) = E(runbiused). We want to show —ih'(0) = — /'(0), i.e., h(0) = f0)

since h'(u ) =

C`sI ampie eiu k(f(u/k))k— if (u/k)(1/k)P(Neaectivc = k) V

kLL =

then, h'(0) = ~` sample k=

(14)

/ ~y size (f(0))k-1J /+(u/k)( 1 /k)P(Nerceetne = k)

E

= J(v(lO) k= 1sampiesize/1)k— 1P(Nefrccsivc = k)

l

= f(0)(Esampie size P(N \ effect;ve = k) — k/= = f(0)

0

P(Neff ective

= O) )

(I — P(NcffectIYC = 0))

= f (0)(1 — 0) = f(0).

(15)

Equation (15) proves that the mean of the observed Funbiased is equal to the theoretical mean. Therefore, unbiased proposed in this paper is an unbiased estimator of the expected nearest-neighbor distance E(r) for a random sample from a very large population size such as samples from a certain species of plant within a vegetation formation, and so on.

Problems of Population Size and the Boundary Effect The nearest-neighbor technique still has some drawbacks regarding its application in geographic research in addition to the above-mentioned problem. The reasons are (1) it neglects the boundary effect, and (2) the number of points in a geographic distribution is generally not large enough for the Poisson model to achieve a good approximation (King, 1969, p. 102). These problems stem from the fact that the

Shin-yi Hsu and J. David Mason 37

derivation of the nearest-neighbor statistics requires the assumption of an unbounded point set and an infinite number of points. This is clearly indicated in equation (1) where x is the number of points in space S, and it takes on integers 1, 2, 3, and so on, up to infinity, and in equation (6) where the expected value of the distance is calculated by allowing the radius of the region to reach infinity in the integration of rg(r) dr. Geographic distributions, however, have definite boundaries, such as the boundary of a given state, and have a small population size, such as urban centers in the state. The boundary effect, moreover, is even greater when the radius of nearest-neighbor distances extends beyond regional border (Figure 1). With respect to the population-size problem, geographers have used the totality of points as centers of origin because sampling is not realistic for a small population size. This procedure not only cannot solve the problem, but creates more bias in the estimation of the mean distance as well. Figure 1 Boundary effect

r,

•

MODEL I

MODEL II

Geographers have recognized boundary problems for years and tried to solve them by means of (1) discarding the points near the boundary (Dacey and Tung, 1962), and (2) mapping the area (square) into torus (Dacey, 1965). These methods, nevertheless, cannot solve the fundamental problem (King, 1969, pp. 91-102). The Nearest-Neighbor Technique in a Bounded Point Set In order to solve the boundary problem, the authors have proposed to use the nearest-neighbor statistics in a bounded point set. The theoretical distribution is derived from certain assumptions. We consider a finite number of points distributed independently in a circle with unit area in such a way that the probability of a point being in a given region is equal to the area of the region, independent of the shape of the region. The sampling statistics we derived from all N effective

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points, which equal the total points minus half of the reflexive neighbors. Hence, it is not the average of independent random variables (distances). However, we will assume that they are pairwise independent in the calculation of sample variance. This assumption is realistic because (1) mathematically that is similar to approximating sampling without replacement by sampling with replacement, and (2) numerically it can be supported by Monte Carlo simulations performed by the authors. Assume that the points are randomly placed in a circle of unit area. Given that P1 is selected randomly as the center of origin for measuring the distance to its nearest neighbor P2. The question then becomes what is the probability that only P2 falls inside the circle with radius > r. This probability function is constructed in the following manner. Let P1, ... , P" denote the n points that are randomly placed in the circle. Let ri denote the distance Pi to its nearest neighbor, 1 < i < n. Let P(r) denote the probability that r1 < r and let P(r/v) denote the conditional probability that r1 < r given the value of v, where v is the distance from P1 to the center of the circle. The form of P(r/v) depends on whether or not r + v < 1/ f (Figure 1). We have P(r/v)— -

{1—(1 —P(,grr,fv))^ -2, if1/f — v5r5 1/,gr+v 1 —(1 — mar 2,if05 r5 1/f —v

(16a)

where I — P(x,y) = (1 — X2)/2 — (x2 — 1 — y2)(1 — (x2 — 1 — y2)2/4y2)112/2yir — (1/n)sin-'((x2 — 1 — y2)/2y) + (x2 — 1 + y2)(x2 — (x2 — 1 + y2)2/4y2)1/2/2yir + (x2/r)sin-'(x2 — 1 + y2)/2xy). The relation 1 — P(x, y) is derived by Graney (1969, p. 25). Equation 16a will be called Model I, which takes into consideration the boundary effect. Model II neglects the boundary and is given by P(r/v) = 1 — (1 — ar2)"- 2, if 0 5 r 5 1 / f . (16b) Since v has a density equal to 2nv over (0, 1/ f) and equal to zero otherwise, the probability density function, f(r), of P(r) is found to be Iw.r—•

1.r

f Qi(r, v)dv + j Q2(r,v)dv,if05r51/f i •. f(r) =

IJa

Q2(r,v)dv, if1/f 5 r5 2J

(17a) Model l

Shin-yi Hsu and J. David Mason 39

where

Q1(r, v) = 47t 2v(n — Q2(r,

v) =

2)r(1 — 7Cr2)"-3

2nv(n — 2)(1 — P(,r,,v))"- 3 ä (Vi r, Vito)

and f(r) = (n

— 2)(1 — rtr2)n -3 2nr, if0 S r S 1A/n

(17b) Model II.

The mean, standard deviation, skewness, and kurtosis of f(r) can be calculated from the related general equations. Table 1 summarizes the characteristics of the population distribution Model I. Table 1 Characteristics of the Population Distribution of Model I N

Mean

100 500 1000

0.05239 0.02290 0.01614

Standard deviation Skewness 0.02843 0.01204 0.008372

0.8574 0.7854 0.07738

Kurtosis 4.0157 3.7047 3.6269

Sampling Distribution of Mean Distance in Model I and Critical Regions To achieve statistical inference or hypothesis-testing by the use of Model I, it is necessary to determine the sampling distribution of the population mean. Let r"

denote ( 1 /n

E ri 1. From the knowledge of f(r) (Table 1) the first four central i=t moments of r" were obtained for n = 100, 500, 1000, and hence, the skewness and kurtosis of r" were found in Table 2.

Table 2 Characteristics of the Sampling Distribution of Population Mean (r,) N

Skewness

Kurtosis

100 500 1000

0.0086 0.0016 0.00077

3.010 3.001 3.0006

Then using Pearson's K-criteria (Kendall and Stuart, 1969, p. 151): 2 ß1(ß2 + 3) K= 4(2ß2 — 3ß1 — 6)(4ß2 — 3ßi)

(18)

where ßl = skewness, and ß2 = kurtosis. Beta distributions were found to fit the first four central moments because the K's were less than zero. However, it is not realistic to employ the Beta functions as the desired sampling distribution because there is essentially no difference between

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the obtained Beta distributions and a truncated normal distribution. Thus, the normal distribution is determined to be the sampling distribution of fn. From the above discussion, the test statistics for determining whether a given distribution is significantly different from random is the standard normal variate. Table 3 lists the mean and standard deviation of r for the purpose of statistical inference. To compute the test statistics, we use: Z = (robs. stand. — rn) lOr„ where Fobs. S18nd

=fns, /~, and

(19) fn and a:, from Table 3; and

robs. = Ereffective/Neffective where Ereneetive = Er — invalidated reflective neighbor distances NerkeNve =

total points (N) — invalidated end reflexive points.

If Z < —1.96 or — 2.58, we have clustered distribution significant at 0.025 or 0.005 level. If Z > 1.96 or 2.58, we have uniform (equal-distance) distribution, significant at 0.025 or 0.005 level (two-tailed test).

Applications of Model I in Comparison with Poisson Model

To demonstrate how Model I (Appendix A) can be applied in testing point patterns, an example outlined in Figure 2 is given. The entire region is consisted of seven subregions. We are interested in seeing whether or not the point patterns in subregions are constant throughout the entire regions. First, empirical (observed) data regarding radii (r), areas (A), N effective points (Neffective), and observed areas nearest-neighbor distances (fobs) are given in Table 3A. Determination of these point patterns will then be given by Poisson decisions and Model I decisions. The formula of sampling mean and sampling standard deviation of the Poisson model are 1/(2. and 0.26136/ ND, respectively (Clark and Evans, 1954). Table 3B gives the Poisson expected means and its standard deviations for the seven subregions together with decision indicators given by Z = (robs. — fe:p)/Qexo • It has been decided (by the Poisson model) that all the point patterns in seven subregions are not significantly different from random. In utilizing Model I for statistical inference, the sampling mean and sampling standard deviations for different population sizes have to be obtained from Table 3. Therefore, in relation to our example, these figures for n = 17, 23, 33, ... , 77 (Table 3A), are given in lines 2 and 3 of Table 3C. The decision indicators are given by Z = (robs. stand. — fn)/afn as explained in formulas (19). It is concluded in Table 3C that the point pattern in the largest region is significantly different from random (more precisely, it is more clustered than random), and the point distribution in the fifth subregion is very close to a clustered pattern. Consequently, the Poisson model made a wrong decision.

Table 3 Comparisons between Poisson and the Model I Decisions A (1)

(3) Third Smallest Subregion

(4) Fourth Smallest Subregion

(5) Fifth Smallest Subregion

(6) Sixth Smallest Subregion

(7)

51.7 8397.1512 77 0.00916978 4.7351

Empirical Data

Smallest Subregion

(2) Second Smallest Subregion

r(mm) A Nc ff«,;V. D

24.2 1839.8466 17 0.009240 4.4882

27.1 2307.2225 23 0.009969 4.7565

31.8 3176.9116 33 0.01039 4.8303

35.8 4026.4002 41 0.10102 4.7976

39.5 4901.6814 48 0.009793 4.5146

43.3 5890.1544 55 0.009338 4.8382

5.2016 0.6595 -1.0812

5.0078 0.5458 -0.4604

4.905 0.4464 -0.16

4.95491 0.4045 -0.3890

5.0527 0.3812 -1.4116

5.1743 0.3650 -0.9216

5.22144 0.3110 -1.5637 (wrong decision)

0.10463689 0.13432140 0.01856522 -1.5989

0.09902498 0.11350690 0.01337670 -1.0826

0.08569822 0.09324745 0.00908997 -0.8305

0.07560700 0.08299590 0.00722013 -1.0233

0.06448299 0.07631608 0.00611368 -1.9356

0.06304038 0.07170042 0.00540164 -1.5022

0.05167256 0.05949914 0.00377748 -2.1004

robs.

B Poisson Decisions F«P ae,P Z

Largest Subregion

C Model I Decisions Fobs. stand. Fexp Qcsp

Z

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The Boundary Effect It is clear that Model I is superior to Poisson model because the former considers the boundary effect and the latter neglects it. However, it will still be of great interest to compare the Poisson model with Model I and Model II with respect to the critical regions (critical regions = µ ± 1.96 (or 2.58) a at 0.05 or 0.01). Table 4 lists the critical regions of these three models. As seen in this table the Poisson model and Model II determine essentially the same critical regions. However, Model I has significant different critical regions. Another way to assess this "significant difference" is to determine comparable Z-values (critical regions) in the Poisson model when Model I's Z-values equals 1.96 as follows: Let N= 50, A =1 since Z = (fobs. stand. — %tap)/Qoap and let Z = 1.96, we can solve for Tobs. stand. (standardized mean observed distance) robs. stand. = reap R. 1 .96 (Qc,p) = 0.07468204 -T 0.00585619 = 0.06320391 or 0.08616017

Shin-yi Hsu and J. David Mason 43

where reap and oeA , are sampling mean and sampling standard deviation of Model I. And in Poisson model : D = N/A = 50 D = 7.070106781 (D = density, N = sample size, A = area) = 1/2 JD = 0.07071068

= 0.26136/JND = 0.00522720. Thus, / Z = (robs - reap)/aexp = (-1.4361 or 2.9556). This means that the critical values for hypothesis-testing (at 0.05 level) have to be shifted from -1.96 to -1.4361 at one tail and from 1.96 to 2.9556 at the other tail. Similar Z-values can be obtained for N = 100, 200, and so on, as shown in Table 5. It can be concluded that the boundary effect on the degree of disperseness increases as the population size increases up to N = 500. It then starts to decrease, Table 4 Comparison between Poisson Model, Model I, and Model II n

10 25 50 75 100 250 500 750 1000

Model I

Poisson

Model II

µ - I.96a

µ - 2.58o

µ - 1.96a

µ - 2.58a

p - 1.96a

p - 2.58a

.10689 .07952 .06046 .05091 .04488 .02956 .02134 .01757 .01530

.09068 .07304 .05722 .04875 .04327 .02891 .02102 .01736 .01514

.11723 .18446 .06320 .05282 .04614 .03021 .02167 .01788 .01552

.09648 .07688 .05957 .05044 .04440 .02958 .02134 .01768 .01537

.11785 .08209 .06136 .04923 .04519 .02965 .02136 .01759 .01531

.10168 .07560 .05812 .05282 .04357 .02900 .02104 .01737 .01515

Table 5 Comparable Critical Z-Values of the Poisson Model for Different Population Sizes when Z = 1.96 (significant level of 0.05) in Model I Comparable to -1.96 -1.4867

Significant Level (0.14)

50 100

-1.4361 -1.3954

(0.15) (0.16)

200 300

- 1.3612 - 1.3452

(0.18) (0.20)

400

-1.3345

(0.20)

500 600 800

- 1.0476 -1.0666 -1.0938

(0.30) (0.286) (0.274)

n 25

Comparable to + 1.96 3.0988 2.9556 2.8631 2.8014 2.7746 2.7591 2.4666 2.4836 2.5069

Significant Level (0.002) (0.004) (0.005) (0.0052) (0.0056)

(0.0058) (0.0144) (0.0132) (0.0126)

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but the degree is very slight. This disproves the notion that boundary effect decreases as population size increases. Conclusion The problem of the boundary effect and population size associated with the nearest-neighbor technique based upon the Poisson model has puzzled geographers for two decades. This paper has thrown some light on these problems by developing a model which takes into consideration the boundary effect. However, this new model by no means solves all of the problems of boundary effect. For instance, we are still confined in a circular region. Future effort should thus be concentrated on developing the nearest-neighbor statistics for elliptical regions. As of now, we can either employ Model I where the total space is nearly circular, or use the Poisson model for other geometric shapes with some adjustment in the setting of critical regions guided by Table 5.

Shin-yi Hsu and J. David Mason

45

Appendix A Sampling Mean and Sampling Standard Deviation of Model I N

µ

a

N

R

a

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

0.34024100 0.28712140 0.25279852 0.22828971 0.20966451 0.19489455 0.18281273 0.17269252 0.16406221 0.15658169 0.15001984 0.14420745 0.13900429 0.13432140 0.13006944 0.12619196 0.12263446 0.11935689 0.11632391 0.11350690 0.11088136 0.10842676 0.10612529 0.10396237 0.10192704 0.09999554 0.09817201 0.09645506 0.09480816 0.09324745 0.09175523 0.09033848 0.08897983 0.08768097 0.08643967 0.08524264 0.08409746 0.08299590 0.08193532 0.08091381 0.07992858 0.07897826 0.07805687 0.07717173

0.10008660 0.07557810 0.06056226 0.05044736 0.04318232 0.03771730 0.03346028 0.03005293 0.02726191 0.02493966 0.02297663 0.02129328 0.01983899 0.01856522 0.01744299 0.01644811 0.01555650 0.01475676 0.01403346 0.01337670 0.01277769 0.01222925 0.01172588 0.01128037 0.01082945 0.01048308 0.01006249 0.00971327 0.00939198 0.00908997 0.00880650 0.00853841 0.00828716 0.00804976 0.00782461 0.00761300 0.00741144 0.00722013 0.00703853 0.00686567 0.00670112 0.00654408 0.00639522 0.00625132

48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91

0.07631608 0.07548472 0.07468204 0.07390111 0.07314595 0.07241253 0.07170042 0.07100227 0.07033562 0.06968154 0.06904522 0.06842576 0.06782260 0.06723490 0.06666189 0.06650913 0.06555787 0.06502579 0.06450632 0.06399874 0.06350296 0.06301832 0.06254451 0.06208109 0.06162769 0.06118397 0.06074956 0.06032416 0.05990746 0.05949914 0.05909898 0.05870676 0.05832208 0.05794509 0.05757516 0.05721191 0.05685537 0.05650517 0.05616144 0.05582351 0.05549180 0.05516557 0.05484655 0.05453173

0.00611368 0.00598242 0.00585619 0.00573605 0.00561996 0.00550857 0.00540164 0.00530010 0.00519909 0.00510312 0.00501129 0.00492228 0.00483645 0.00475338 0.00467364 0.00450137 0.00452021 0.00444747 0.00437689 0.00430864 0.00424253 0.00417830 0.00411596 0.00405543 0.00399665 0.00393966 0.00388421 0.00383004 0.00377748 0.00372632 0.00367653 0.00362793 0.00358074 0.00353460 0.00348961 0.00344589 0.00340318 0.00336150 0.00332093 0.00328120 0.00346431 0.00320461 0.00316725 0.00313129

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N

µ

a

N

µ

a

92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139

0.05422220 0.05391783 0.05361300 0.05331863 0.05303311 0.05274681 0.05246616 0.05218938 0.05191788 0.05165126 0.05138758 0.05112758 0.05087162 0.05061942 0.05037290 0.05012801 0.04988661 0.04964854 0.04941174 0.04918032 0.04895209 0.04872719 0.04850512 0.04828602 0.04806984 0.04785650 0.04764482 0.04743696 0.04723178 0.04702920 0.04683043 0.04663294 0.04643830 0.04624567 0.04605495 0.04586658 0.04568086 0.04549736 0.04531603 0.04513683 0.04495972 0.04478466 0.04461228 0.04444124 0.04427213 0.04410412 0.04393873 0.04377517

0.00309569 0.00306110 0.00302835 0.00299528 0.00296203 0.00293054 0.00289949 0.00286909 0.00283932 0.00280993 0.00278130 0.00275332 0.00272594 0.00269895 0.00267227 0.00264637 0.00262089 0.00259597 0.00257188 0.00254787 0.00252431 0.00250127 0.00247855 0.00245609 0.00243416 0.00241262 0.00239165 0.00237085 0.00235041 0.00233029 0.00231033 0.00229090 0.00227172 0.00225292 0.00223436 0.00221634 0.00219844 0.00218082 0.00216347 0.00214640 0.00212962 0.00211308 0.00209667 0.00208062 0.00206481 0.00204936 0.00203406 0.00201891

140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187

0.04361340 0.04345340 0.04329544 0.04313887 0.04298397 0.04283072 0.04267884 0.04252878 0.04238050 0.04223373 0.04208826 0.04194426 0.04180172 0.04166054 0.04152084 0.04138252 0.04124571 0.04111008 0.04097564 0.04084263 0.04071090 0.04058042 0.04045118 0.04032279 0.04019595 0.04007030 0.03994581 0.03982246 0.03970023 0.03957912 0.03945909 0.03934014 0.03922240 0.03910705 0.03898959 0.03887479 0.03876099 0.03864817 0.03853632 0.03842543 0.03831548 0.03820662 0.03809852 0.03799132 0.03788502 0.03777960 0.03767548 0.03757177

0.00200401 0.00198934 0.00197482 0.00196057 0.00194651 0.00193266 0.00191903 0.00190558 0.00189223 0.00187909 0.00186617 0.00185342 0.00184083 0.00182844 0.00181620 0.00180411 0.00179217 0.00178040 0.00176880 0.00175733 0.00174601 0.00173483 0.00172379 0.00171296 0.00170219 0.00169155 0.00168106 0.00167069 0.00166043 0.00165034 0.00164032 0.00163044 0.00162065 0.00161080 0.00160151 0.00159209 0.00158277 0.00157343 0.00156447 0.00155547 0.00154658 0.00153776 0.00152907 0.00152047 0.00151197 0.00150356 0.00149519 0.00148696

Shin-yi Hsu and J. David Mason 47

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188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235

0.03746940 0.03736737 0.03726527 0.03716505 0.03706545 0.03696665 0.03686863 0.03677138 0.03667476 0.03657914 0.03648414 0.03638987 0.03629632 0.03620348 0.03611135 0.03601991 0.03592915 0.03583907 0.03574947 0.03565894 0.03557105 0.03548518 0.03539836 0.03531217 0.03522660 0.03514170 0.03505730 0.03497356 0.03489041 0.03480785 0.03472586 0.03464444 0.03456359 0.03448329 0.03440355 0.03432436 0.03424570 0.03416758 0.03409038 0.03401330 0.03393635 0.03386037 0.03378483 0.03370979 0.03363525 0.03356119 0.03348762 0.03341452

0.00147876 0.00147071 0.00146287 0.00145497 0.00144740 0.00143947 0.00143183 0.00142432 0.00141681 0.00140940 0.00140208 0.00139484 0.00138767 0.00138057 0.00137355 0.00136659 0.00135971 0.00135289 0.00134682 0.00133977 0.00133313 0.00132637 0.00131989 0.00131341 0.00130707 0.00130073 0.00129452 0.00128834 0.00128221 0.00127615 0.00127013 0.00126418 0.00125828 0.00125243 0.00124664 0.00124090 0.00123522 0.00122959 0.00122396 0.00121842 0.00121299 0.00120760 0.00120212 0.00119678 0.00119200 0.00118625 0.00118105 0.00117591

236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283

0.03334190 0.03326976 0.03319807 0.03312684 0.03305606 0.03298573 0.03291584 0.03284639 0.03277738 0.03270879 0.03264063 0.03257289 0.03250557 0.03243866 0.03237226 0.03230605 0.03224035 0.03217505 0.03211010 0.03204558 0.03198145 0.03191770 0.03185432 0.03179132 0.03172868 0.03166641 0.03160451 0.03154295 0.03148176 0.03142092 0.03136075 0.03130028 0.03124048 0.03118102 0.03112195 0.03106315 0.03100469 0.03094658 0.03088876 0.03083127 0.03077409 0.03071723 0.03066068 0.03060444 0.03054850 0.03049287 0.03043754 0.03038251

0.00117080 0.00116573 0.00116071 0.00115573 0.00115079 0.00114589 0.00114103 0.00113620 0.00113142 0.00112667 0.00112198 0.00111732 0.00111270 0.00110811 0.00110356 0.00109907 0.00109461 0.00109018 0.00108577 0.00108141 0.00107708 0.00107276 0.00106851 0.00106428 0.00106009 0.00105593 0.00105181 0.00104770 0.00104366 0.00103962 0.00103562 0.00103162 0.00102768 0.00102377 0.00101989 0.00101603 0.00101221 0.00100841 0.00100464 0.00100090 0.00099718 0.00099350 0.00098983 0.00098618 0.00098257 0.00097899 0.00097544 0.00097191

IGU Commission

48

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284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331

0.03032777 0.03027332 0.03021926 0.03016540 0.03011181 0.03005851 0.03000550 0.02995275 0.02990029 0.02984809 0.02979617 0.02974452 0.02969313 0.02964200 0.02959114 0.02954051 0.02949016 0.02943990 0.02939022 0.02934063 0.02929132 0.02924217 0.02919332 0.02914463 0.02909616 0.02904803 0.02900014 0.02895246 0.02890504 0.02885786 0.02881089 0.02876415 0.02871764 0.02867134 0.02862527 0.02857943 0.02853379 0.02848838 0.02844321 0.02839824 0.02835345 0.02830889 0.02826453 0.02822038 0.02817643 0.02813300 0.02803946 0.02804612

0.00096841 0.00096498 0.00096147 0.00095804 0.00095463 0.00095125 0.00094789 0.00094455 0.00094124 0.00093813 0.00093470 0.00093146 0.00092824 0.00092505 0.00092187 0.00091872 0.00091559 0.00091247 0.00090938 0.00090631 0.00090325 0.00090022 0.00089721 0.00089422 0.00089126 0.00088831 0.00088538 0.00088247 0.00087959 0.00087671 0.00087385 0.00087102 0.00086820 0.00086540 0.00086262 0.00035985 0.00085712 0.00085439 0.00085165 0.00084898 0.00084630 0.00084363 0.00084099 0.00083836 0.00083575 0.00083312 0.00083054 0.00082792

332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379

0.02800298 0.02796003 0.02791728 0.02787473 0.02783245 0.02779028 0.02774830 0.02770651 0.02766491 0.02762349 0.02758225 0.02754120 0.02750033 0.02745931 0.02741880 0.02737847 0.02733831 0.02729832 0.02725851 0.02721841 0.02717889 0.02714013 0.02710100 0.02706204 0.02702325 0.02698462 0.02694612 0.02690782 0.02686968 0.02683171 0.02679389 0.02675623 0.02671873 0.02668138 0.02664379 0.02660715 0.02657027 0.02653354 0.02649703 0.02646053 0.02642425 0.02638811 0.02635213 0.02631629 0.02628060 0.02624505 0.02620964 0.02617437

0.00082542 0.00082287 0.00082035 0.00081784 0.00081535 0.00081248 0.00081042 0.00080797 0.00080654 0.00080313 0.00080073 0.00079834 0.00079598 0.00079294 0.00079131 0.00078898 0.00078667 0.00078439 0.00078211 0.00077988 0.00077761 0.00077527 0.00077304 0.00077031 0.00076860 0.00076640 0.00076421 0.00076203 0.00075987 0.00075772 0.00075558 0.00075346 0.00075134 0.00074924 0.00074719 0.00074507 0.00074300 0.00074093 0.00073988 0.00073687 0.00073484 0.00073283 0.00073083 0.00072884 0.00072686 0.00072489 0.00072293 0.00072098

N

p

a

N

K

a

380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427

0.02613924 0.02610426 0.02606941 0.02603471 0.02600003 0.02596559 0.02593129 0.02589713 0.02586310 0.02582665 0.02579286 0.02575920 0.02572567 0.02569227 0.02565899 0.02562585 0.02559558 0.02556324 0.02553051 0.02549790 0.02546541 0.02543305 0.02540302 0.02536822 0.02533669 0.02530481 0.02527306 0.02524142 0.02520989 0.02517852 0.02514666 0.02511549 0.02508443 0.02505348 0.02502265 0.02499193 0.02496132 0.02493082 0.02490044 0.02487016 0.02484000 0.02480994 0.02477999 0.02475012 0.02472045 0.02469082 0.02466130 0.02463188

0.00071905 0.00071712 0.00071520 0.00071330 0.00071142 0.00070954 0.00070766 0.00070579 0.00070394 0.00070233 0.00070049 0.00069867 0.00069685 0.00069505 0.00069320 0.00069146 0.00068942 0.00068760 0.00068584 0.00068408 0.00068234 0.00068060 0.00067867 0.00067719 0.00067545 0.00067375 0.00067205 0.00067036 0.00066869 0.00066701 0.00066541 0.00066375 0.00066213 0.00066049 0.00065886 0.00065708 0.00065563 0.00065403 0.00065243 0.00065085 0.00064927 0.00064770 0.00064643 0.00064457 0.00064302 0.00064146 0.00063992 0.00063839

428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475

0.02460257 0.02457336 0.02454425 0.02451525 0.02448635 0.02445755 0.02442885 0.02440025 0.02437175 0.02434335 0.02431505 0.02428684 0.02425874 0.02423073 0.02420281 0.02417500 0.02414727 0.02411965 0.02409211 0.02406439 0.02403704 0.02400978 0.02398262 0.02395555 0.02392856 0.02390167 0.02387487 0.02384816 0.02382153 0.02379500 0.02376886 0.02374250 0.02371623 0.02369004 0.02366394 0.02363793 0.02361200 0.02357737 0.02355147 0.02353473 0.02350914 0.02348363 0.02345820 0.02343286 0.02340760 0.02338241 0.02335731 0.02333217

0.00063687 0.00063536 0.00063385 0.00063235 0.00063086 0.00062937 0.00062789 0.00062642 0.00062496 0.00062350 0.00062205 0.00062060 0.00061916 0.00061773 0.00061631 0.00061489 0.00061348 0.00061207 0.00061067 0.00060930 0.00060792 0.00060654 0.00060516 0.00060380 0.00060243 0.00060108 0.00059973 0.00059839 0.00059705 0.00059572 0.00059437 0.00059305 0.00059173 0.00059042 0.00058912 0.00058782 0.00058653 0.00058601 0.00058473 0.00058269 0.00058142 0.00058016 0.00057890 0.00057775 0.00057639 0.00057514 0.00057391 0.00057269

IGU Commission

50

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476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523

0.02330723 0.02328237 0.02325758 0.02323288 0.02320838 0.02318383 0.02315936 0.02313497 0.02311065 0.02308641 0.02306224 0.02303815 0.02301413 0.02299019 0.02296620 0.02294241 0.02291869 0.02289504 0.02287147 0.02284796 0.02282453 0.02280118 0.02277789 0.02275467 0.02273153 0.02270845 0.02268544 0.02266251 0.02263964 0.02261684 0.02259411 0.02257145 0.02254885 0.02252633 0.02250387 0.02248160 0.02245928 0.02243702 0.02241483 0.02239270 0.02237064 0.02234864 0.02232671 0.02230484 0.02228304 0.02226130 0.02223945 0.02221783

0.00057147 0.00057025 0.00056904 0.00056783 0.00048693 0.00048598 0.00048503 0.00048409 0.00048315 0.00048221 0.00048128 0.00048035 0.00047942 0.00047850 0.00047760 0.00047668 0.00047577 0.00047486 0.00047396 0.00047306 0.00047216 0.00047127 0.00047038 0.00046949 0.00046861 0.00046773 0.00046685 0.00046598 0.00046511 0.00046424 0.00046338 0.00046252 0.00046166 0.00046081 0.00045995 0.00045909 0.00045825 0.00045741 0.00045657 0.00045573 0.00045490 0.00045407 0.00045325 0.00045242 0.00045160 0.00045079 0.00044999 0.00044918

524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571

0.02219628 0.02217478 0.02215335 0.02213198 0.02211068 0.02208943 0.02206824 0.02204712 0.02202605 0.02200504 0.02198410 0.02196321 0.02194238 0.02192162 0.02190091 0.02188025 0.02185974 0.02183921 0.02181873 0.02179831 0.02177794 0.02175764 0.02173738 0.02171719 0.02170822 0.02168825 0.02165694 0.02163697 0.02161705 0.02159718 0.02157737 0.02155762 0.02153792 0.02151827 0.02149868 0.02147914 0.02145965 0.02144021 0.02142076 0.02140143 0.02138215 0.02136292 0.02134374 0.02132462 0.02130555 0.02128653 0.02126762 0.02124870

0.00044837 0.00044644 0.00044563 0.00044482 0.00044517 0.00044438 0.00044359 0.00044280 0.00044202 0.00044124 0.00044045 0.00043968 0.00043890 0.00043813 0.00043736 0.00043659 0.00043582 0.00043445 0.00043431 0.00043355 0.00043280 0.00043205 0.00043130 0.00043056 0.00042879 0.00042804 0.00042834 0.00042761 0.00042688 0.00042615 0.00042542 0.00042470 0.00042398 0.00042326 0.00042254 0.00042183 0.00042111 0.00042041 0.00041971 0.00041900 0.00041830 0.00041760 0.00041690 0.00041621 0.00041552 0.00041482 0.00041413 0.00041344

Shin-yi Hsu and J. David Mason

51

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572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619

0.02122983 0.02121101 0.02119222 0.02117350 0.02115483 0.02113620 0.02111763 0.02109910 0.02108063 0.02106225 0.02104386 0.02102553 0.02100724 0.02098901 0.02097081 0.02095267 0.02093457 0.02091652 0.02089852 0.02088056 0.02086264 0.02084478 0.02082696 0.02080918 0.02079145 0.02077376 0.02075612 0.02073858 0.02072103 0.02070352 0.02068594 0.02066864 0.02065126 0.02063393 0.02061664 0.02059939 0.02058214 0.02056498 0.02054786 0.02053079 0.02051376 0.02049677 0.02047982 0.02046291 0.02044605 0.02042922 0.02041244 0.02039570

0.00041276 0.00041208 0.00041140 0.00041073 0.00041005 0.00040938 0.00040871 0.00040804 0.00040738 0.00040671 0.00040605 0.00040501 0.00040473 0.00040408 0.00040343 0.00040278 0.00040213 0.00040148 0.00040084 0.00040020 0.00039956 0.00039892 0.00039928 0.00039765 0.00039702 0.00039639 0.00039576 0.00039513 0.00039451 0.00039389 0.00039327 0.00039265 0.00039203 0.00039142 0.00039081 0.00039020 0.00038959 0.00038899 0.00038838 0.00038778 0.00038718 0.00038658 0.00038599 0.00038539 0.00038480 0.00038421 0.00038362 0.00038303

620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667

0.02037900 0.02036234 0.02034572 0.02032914 0.02031260 0.02029601 0.02027955 0.02026313 0.02024675 0.02023042 0.02021411 0.02019785 0.02018163 0.02016545 0.02014930 0.02013320 0.02011713 0.02010110 0.02008511 0.02006916 0.02005324 0.02003736 0.02002152 0.02000572 0.01998995 0.01997422 0.01993853 0.01994288 0.01992726 0.01991168 0.01989613 0.01988062 0.01986515 0.01984971 0.01983431 0.01981894 0.01980361 0.01978831 0.01977305 0.01975783 0.01974264 0.01972748 0.01971262 0.01969776 0.01968271 0.01966769 0.01965271 0.01963776

0.00038245 0.00038186 0.00038128 0.00038070 0.00038012 0.00037955 0.00037898 0.00037840 0.00037783 0.00037726 0.00037670 0.00037621 0.00037557 0.00037500 0.00037444 0.00037388 0.00037332 0.00037277 0.00037221 0.00037166 0.00037111 0.00037056 0.00037001 0.00036947 0.00036892 0.00036838 0.00036784 0.00036730 0.00036676 0.00036623 0.00036569 0.00036516 0.00036463 0.00036410 0.00036357 0.00036304 0.00036251 0.00036199 0.00036147 0.00036095 0.00036043 0.00035991 0.00035937 0.00035884 0.00035833 0.00035781 0.00035730 0.00035679

IGU Commission

52

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668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715

0.01962284 0.01960796 0.01959263 0.01957782 0.01956303 0.01954829 0.01953357 0.01931889 0.01950424 0.01948963 0.01947504 0.01946049 0.01944597 0.01943149 0.01941703 0.01940261 0.01938822 0.01937386 0.01935954 0.01934524 0.01933101 0.01931675 0.01930255 0.01928838 0.01927424 0.01926013 0.01924605 0.01923201 0.01921799 0.01920401 0.01919005 0.01917613 0.01916223 0.01914837 0.01913454 0.01912073 0.01910696 0.01909321 0.01907949 0.01906581 0.01905215 0.01903852 0.01902492 0.01901135 0.01899781 0.01898430 0.01897082 0.01895736

0.00035629 0.00035578 0.00035532 0.00035481 0.00035431 0.00035381 0.00035330 0.00035280 0.00035231 0.00035181 0.00035133 0.00035084 0.00035035 0.00034986 0.00034937 0.00034889 0.00034840 0.00034792 0.00034744 0.00034696 0.00034648 0.00034600 0.00034552 0.00034505 0.00034457 0.00034410 0.00034363 0.00034316 0.00034269 0.00034222 0.00034175 0.00034129 0.00034083 0.00034036 0.00033990 0.00033944 0.00033898 0.00033853 0.00033807 0.00033761 0.00033716 0.00033671 0.00033626 0.00033581 0.00033536 0.00033491 0.00033446 0.00033402

716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763

0.01894393 0.01893054 0.01891717 0.01890382 0.01889051 0.01887722 0.01886397 0.01885074 0.01883753 0.01882382 0.01881067 0.01879756 0.01878541 0.01877235 0.01875905 0.01874604 0.01873305 0.01871998 0.01870705 0.01869410 0.01868123 0.01866841 0.01865563 0.01864283 0.01863006 0.01861731 0.01860459 0.01859190 0.01857923 0.01856659 0.01855397 0.01854138 0.01852881 0.01851627 0.01850376 0.01849127 0.01847880 0.01846636 0.01845395 0.01844156 0.01842919 0.01841685 0.01840454 0.01839225 0.01837998 0.01836774 0.01835552 0.01834332

0.00033357 0.00033313 0.00033269 0.00033226 0.00033181 0.00033137 0.00033093 0.00033050 0.00033006 0.00032967 0.00032924 0.00032881 0.00032830 0.00032788 0.00032747 0.00032704 0.00032662 0.00032620 0.00032578 0.00032536 0.00032496 0.00032453 0.00032411 0.00032367 0.00032326 0.00032284 0.00032243 0.00032201 0.00032160 0.00032119 0.00032078 0.00032037 0.00031996 0.00031955 0.00031696 0.00031655 0.00031834 0.00031794 0.00031753 0.00031713 0.00031673 0.00031633 0.00031593 0.00031554 0.00031514 0.00031475 0.00031435 0.00031396

Shin-yi Hsu and J. David Mason 53

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a

764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811

0.01833115 0.01831900 0.01830688 0.01829478 0.01828271 0.01827066 0.01825863 0.01824663 0.01823465 0.01822270 0.01821076 0.01819885 0.01818697 0.01817510 0.01816326 0.01815145 0.01813965 0.01812788 0.01811614 0.01810441 0.01809271 0.01808103 0.01806937 0.01805773 0.01804612 0.01803453 0.01802296 0.01801142 0.01799984 0.01798834 0.01797686 0.01796540 0.01795397 0.01794255 0.01793116 0.01791984 0.01790849 0.01789716 0.01788585 0.01787456 0.01786330 0.01785206 0.01784083 0.01782963 0.01781845 0.01780729 0.01779562 0.01778450

0.00031357 0.00031318 0.00031279 0.00031240 0.00031201 0.00031162 0.00031124 0.00031085 0.00031047 0.00031008 0.00030970 0.00030932 0.00030894 0.00030856 0.00030818 0.00030781 0.00030743 0.00030705 0.00030668 0.00030630 0.00030593 0.00030556 0.00030519 0.00030482 0.00030445 0.00030408 0.00030371 0.00030335 0.00030298 0.00030262 0.00030226 0.00030189 0.00030153 0.00030117 0.00030081 0.00030045 0.00030009 0.00029973 0.00029937 0.00029902 0.00029866 0.00029831 0.00029795 0.00029760 0.00029725 0.00029685 0.00029654 0.00029619

812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850

0.01777341 0.01776233 0.01775128 0.01774025 0.01772924 0.01771824 0.01770775 0.01769679 0.01768586 0.01767495 0.01766405 0.01765318 0.01764233 0.01763150 0.01762068 0.01760989 0.01759911 0.01758836 0.01757762 0.01756691 0.01755621 0.01754553 0.01753488 0.01752424 0.01751362 0.01750302 0.01749244 0.01748188 0.01747134 0.01746082 0.01745031 0.01743983 0.01742936 0.01741891 0.01740848 0.01739807 0.01738768 0.01737731 0.01736695

0.00029584 0.00029554 0.00029519 0.00029485 0.00029451 0.00029416 0.00029378 0.00029344 0.00029310 0.00029276 0.00029242 0.00029208 0.00029174 0.00029140 0.00029106 0.00029073 0.00029039 0.00029006 0.00028972 0.00028939 0.00028906 0.00028872 0.00028839 0.00028806 0.00023773 0.00028740 0.00028708 0.00028675 0.00028642 0.00028610 0.00028577 0.00028545 0.00028513 0.00028480 0.00028448 0.00028416 0.00028384 0.00028352 0.00028320

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References CLARK, P. J., and F. C. EVANS (1954). "Distance to Nearest Neighbor as a Measure of Spatial Relationship in Population." Ecology, 35:445-53. DACEY, M. F. (1964). "Two Dimensional Random Point Patterns: A Review and an Interpretation." Regional Science Association, Papers and Proceedings, 13:41-45. DACEY, M. F. (1965). "Numerical Measure of Random Sets." Technical Report No. 5, Geographical Information Systems Project. Department of Geography, Northwestern University. DACEY, M. J., and T. TUNG (1962). "The Identification of Randomness in Point Patterns." Journal of Regional Science, 4:83-96. GRANEY, R. W. (1969). "Tests of Concentration and Identification of Mixed Samples." Unpublished dissertation in the Department of Statistics, University of Georgia, Athens. HERTZ, P. (1907). "Uber den geigerseitigen durchschnittlichen Abstand von Punkten, die mit bekannter mittlerer Dichte im Raume angeordnet sind." Mathematische Annalen, 67:387-98. Hsu, S. (1969). "On the Use of the Nearest Neighbor Technique in Geographic Studies." Paper presented at the meeting of the Southeast Division of the Association of American Geographers, Tallahassee, Fl. KENDALL, M. G. (1963). Geometrical Probability. New York: Hafner. KENDALL, M. G., and A. STUART (1969). The Advanced Theory of Statistics. New York: Hafner (3rd. ed). KING, L. J. (1969). Statistical Analysis in Geography. Englewood Cliffs, N.J.: Prentice-Hall. MILLER, R. L., and J. S. KAHN (1962). Statistical Analysis in the Geological Science. New York: Wiley. MOORE, P. J. (1954). "Spacing in Plant Population." Ecology, 35 :222-27. PARZEN, E. (1962). Stochastic Process. San Francisco: Holden-Day. SKELLAM, J. G. (1952). "Studies in Statistical Ecology: I. Spatial Pattern." Biometrika, 39:346-462.

Parametrization of the Gamma Probability Density Function for Examining and Modeling Space Use BARRY S. WELLAR (University of Kansas)

and GERALD J. LACAVA (Georgia State University)

The potential for using finite families of gamma (F) probability density functions (p.d.f.'s), and the pointwise maximum curve (PMC) of the families, in the description, analysis, planning, and prediction of space use is explored. The paper proceeds as follows. First, the three-parameter (a, ß, s) I' p.d.f. is compared and contrasted with two-parameter (ß, s) p.d.f.'s such as the normal and negative exponential (a special F). Second, a mathematical model of the F p.d.f. is developed.' Third, the characteristics of, and decision rules for, determining the a, ß, and s parameters of the model are presented. The paper is then concluded by discussions of the F density as a tool for describing, analyzing, planning, and predicting such space uses and activities as traffic flow, networks, market areas, and diffusion of phenomena. Although an occasional reminder is presented in the later applications sections of the paper, we do not confine ourselves to using the F curves solely in a probabilistic context. By way of illustration, if it is apparent that insights are to be gained by reference to curve fitting in a given situation, then the notion of probability is subjugated. As inspections of the diagrams reveal, only minor modifications are required to include the word probable in the axes labels without jeopardizing the generality associated with the situations studied. I. This model is an extension of one previously developed by Wellar and LaCava (1972). 55

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Comparing and Contrasting r and Two-Parameter Probability Density Functions Parameters common to all p.d.f.'s are the location parameter, ß, and the scale parameter, s. In addition to the ß and s parameters, F has a third parameter, namely an intensity or shape parameter, a.2 The suggestive relationship between p.d.f. parameters and moments requires that we briefly report on the r p.d.f. moments. Although we do not deal specifically with moments in this paper, and are primarily interested in other bases for comparing and contrasting two- and three-parameter p.d.f.'s, the following discussion does contribute toward making distinctions between such p.d.f.'s (or, more simply, densities). The mean or first moment for a r density with parameters a, ß, and s is computed as follows:

EX_ rx —

(x — ß)°e- cx-eu=

r(a)sa

~ e

—

dx,

r(a)s°

9

—

+ß

13)2e - (x-8)13

r(a + 1)sa + 1

(x — ß)a-'e -cz a~r= dx, r(a)s°

dx dx + ß

~ r (x

J

— ß)'- Ie-i:-au= dx. r(a)s°

Note that both integrands are now r density functions. Therefore, both integrals are equal to 1; and

ß. The above technique is a standard procedure for dealing with r densities. The property ar[a] = r[a + 1] is exploited whenever the integrand is a density except for a discrepancy in the exponent on [X — ß] in the numerator and the argument of the F function and the exponent on s in the denominator. The second noncentral moment is

EX=as+

EX 2 = a[a + l]s2 + 2ß[as] + ß 2 . The variance, or second central moment, is computed by the formula VarX = E[X 2 ] — [EX] 2 ,

and is found to be Var X = as2 .

From the mean, we see the translation adjustment made possible by the location parameter, ß. This flexibility is available in all p.d.f.'s. The expression for the variance, however, indicates that while the scale parameter, s, affects the shape of the curve it does not do so completely, as is the case with two-parameter families. 2. Although discussed in more detail below, the parameters are briefly characterized as follows: ß locates the maximum of the curve on the horizontal axis, s is a measure of the dispersion of the curve, and a determines the shape of the curve.

Barry S. Weltar and Gerald J. LaCava 57

Figure 1 Two I Members (Curves) With Same Scale (s = 1) and Same Location of Maximum Values (X = 2) but with Different Maximum Values Y 1.1

9

7

5

a =2 3

a =3 1 2

3

4

5

6

To return to our thrust of primary interest, and using the brief reference to moments as context, the significance of the a parameter can be explained by considering the mathematical relationships between the variables and the densities which they generate. Three properties of the p.d.f.'s are important to our modeling: (1) maximum value, M, of a curve; (2) shape of a curve as measured by the dispersion; and (3) location of the maximum value. In the two-parameter (ß and s) families, the location parameter, ß, provides (3). Properties or characteristics (1) and (2) are determined by the scale parameter s. The model user is limited to controlling only one of (1) and (2), and the second property is dictated. In the three-parameter case, ß again gives (3). Now, a and s can be used to provide (1) and (2). To summarize, for the two-parameter families two of the characteristics ((I), (2), (3)) can be realized, but no control can be exercised over the third. For the three-parameter family, two can be realized and there is still a parameter available for controlling the remaining characteristic. Figure 1 contains two V densities which are identical in characteristics (2) and (3), but have different maximum values. This could not be accomplished with two members of a two-parameter family.

7 X

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The relationship between the F and two-parameter p.d.f.'s is further explicated by referring to specific densities, and Figures 2 and 3. In Figure 2 a family of t functions is presented. The negative exponential, a two-parameter p.d.f., which is commonly referred to as a distance decay curve, is included as a special member of the F family in Figure 2a, when a = 1. The negative exponential differs from the other curves in that it is monotonic, and the a parameter, being fixed at a = 1, has no bearing on the shape of the curve. In Figure 2b a magnified indicator function (MIF) is incorporated into the family of F's but not discussed, since it is examined in detail in Section 3. It is included here to illustrate diagrammatically how the pointwise maximum curve may be adjusted for those cases where a 4 1 (no negative exponential member in the family of F's), and it is desired that the left slope of the gamma closest to the origin not decrease to zero.3 In Figures 3a and 3b a bell-shaped two-parameter p.d.f. and a F p.d.f. are represented by families of curves associated with a hierarchy of foci or centers, such as those established by the principles of Central Place Theory. At a very general level, the curves could be interpreted as reflecting the role of distance in effecting change in market areas, influence, interaction, etc., and the pointwise maximum curves (PMC's) as being two-dimensional mappings. A distinguishing characteristic of the F which is noted at this point and discussed more fully in later sections concerns the nature of the curves associated with the foci in Figures 3a and 3b. Each bell-shaped curve in 3a increases up to its focus and decreases at the same rate to the right of the focus. However, each F curve in 3b increases to its maximum at the focus, and then decreases to the right of its focus at a rate which is smaller in magnitude than that operating to the left of its focus.' Development of the Mathematical Model The F family is modified in this discussion by the addition of a magnified indicator function, and some of the F's are truncated at x = ß so that all are bounded.5 These modifications are included to achieve rigor and improved goodness-of-fit. (Properties of F densities are presented in Appendix A.) 3. If T's are used to describe traffic congestion, with the CBD as the origin, then a magnified indicator function would serve to illustrate that congestion does not disappear in the CBD. Obviously, the T property of each curve integrating to I, i.e., "one" is destroyed for that curve to which a magnified indicator function is attached. Inclusion of the MIF, therefore, depends not only upon the nature of the phenomenon under study, and research design, but upon the significance of preserving the integrity of the curve so that it integrates to I. 4. As the location of the maximum shifts to the right, and if the r member starts at the origin, then the difference in rate of change will diminish to the point where the r will take on the shape of a flattened bell-shaped curve, such as the one for C4 in Figure 3a. 5. This is done in the case of a's = I, because the changing ß's and s's could cause the value of the curve to be arbitrarily large.

Figure 2 (a) Negative Exponential Truncated (b) Family of Gammas, No Negative Exponential, Magnified Indicator Function (MIF) Included Y

(A)

Y

.9

(B)

.9

.7

mif

5 /

pmc

/

....-

3

pmc i •

1i\i• \ /\ \ \

4

2

y\

1

5

6

7X

2

0

3

4

5

7X

Figure 3 (a) Family of Curves Representing a Bell Shaped Two Parameter P.D.F. (b) Family of Curves Representing a Three Parameter P.D.F. (A) Y

(B)

Y

pmc mif pmc

/

is

Cl

C2 C3

C4

X

C2

C4

X

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Define, for a > 1, {(x r..13 .3(X) —

— ß)°-1e—(x—BUs x

f(a)s° 0,

OSx< ß.

Fora = 1, ' r1.,z.s(x) =

1 e-(x- pits s

0_ 1, i = 1, ..., K, with at least one s; = 1. Let x1 = s,(a, — 1) + ß, = min {s,(a1 — 1) + fl1 l

(x1) >

Define Gay a ~ ,,(x) = G1,0, x,)(x) where — 1)a'- 1e-(a' -1) G = max I a (x) — (a1 , (a, )s and 1i0,X1)(x) is the indicator function, which is 1 on [0,x1) and 0 elsewhere. Definition. is called an urban (or regional or other geographic area) family of T's if and only if" has the form = {G„",,,,} tr„,,„,,„0,.„,,,,,e JK, for some positive integer K).' JK is called the index set for the urban family. Definition. A ray gamma, r' , is a function expressible as the pointwise maximum of an urban family of r's; that is,

r.(x) = max{F(x):Fe , an urban family of I"s}.s In order to determine the I'„ of the family of I's, it is necessary to first prescribe the urban family. This requires enunciation of the decision rules for selecting the triples comprising the index set. Before that can be done, however, it is necessary 6. The peak nearest the origin is labeled 1. 7. An urban family would be applicable for the study of urban space use, such as intraurban traffic flows and market areas. A regional or other family of r"s would be applicable for the study of interurban flows and market relationships. 8. The term ray is used in a conventional mathematical sense, that is, as a set consisting of a point P on a straight line and all points of the line on one side of P. In a related study Tobler (1969) used the term line spectrum.

Barry S. Wellar and Gerald J. LaCava 61

to elaborate on the role of parameters. (See also, Eagleman (1968), Greenwood and Durand (1960), Niedercorn (1971), and Voorhees (1968).) Description of the F Function Parameters As noted above the index-set triples consist of the location, scale, and intensity parameters. The location parameter, ß, simply translates the F horizontally. Once the maximum value and scale have been determined, ß can be used to shift the curve so that the maximum occurs at the desired point on the horizontal axis. The intensity parameter, a, together with s, yields the shape of the curve. Further, the maximum value of the F is a function of a and s. Upon fixing s, the maximum can be controlled by means of a. That is, the maximum value of ra.es(X) is M= (a — 1)°- e-ca -n

r(a)s and this equation is used to determine the intensity parameter. The scale parameter, s, is a measure of the dispersion of the curve—the smaller values of s yielding steeper, more compact curves. The dispersions of two F's, with ß's and a's held constant, are compared by computing the ratio of the twoscale parameters. Thus, the F with the steepest slope in the family has s = 1 and the others have s satisfying s > 1. For all curves in the family, a shift of distance s in a positive direction from the maximum results in the same increase in the areas under the curves.' The magnitude of this increase, say E, is determined using s= 1. Decision Rules for Selection of Index-Set Triples A new ordered triple of parameters results for each peak in the data (e.g., when graphed in two-space). If K is the number of F's in the family, then K — 1 is the number of times the scale changes. The following decision rules are used to determine K, a, ß, and s: (1) K is equal to the number of peaks as one moves from, say, C, to C (or from the CBD through other centers to the urban fringe, or any other comparable type of hierarchy). (2)One of the si's, say s,, is set equal to 1. Let Mr be the maximum value of the gamma. Then, (a M = r— (a.) 9. This is evident upon inspection of the integral equation s, 1 x'- '1 I y°, _ e adx= f jaa! I) r(a) s s a J 1 r(a) — using the change of variabley =

xis. The integrand on the letthand side is the r density for 13 = 0.

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For a given M,, a, can be determined10 (see Appendix B, Part 1). B, is then chosen so that the F attains its maximum at x = (a, — 1)+ ß,. (3) To determine the s;, it is first necessary to set s. Because of the nature of the si and since s, = 1, s satisfies the equation e=

°' x a.- l e- x

dx.

a, - 1 I'(a.) Hence, for a given a„ a can be found using a statistical table of the I" distribution for s = 1. s; can now be determined as mentioned above, that is, s; is the distance to the right of the maximum such that an increase of a is achieved. (4) Choose a; to satisfy (a, — I)a' 'e-lar-tl = r(a) , where M; is the maximum value of the peak. (5)Choose ß; such that s;(a; — 1) + ß. = x0, where x0 is the point at which the maximum occurs. Determination of the triples in index set JK is primarily aimed at use of the PMC, or I'*. In the event that only one F density function is to be used, that is, when the effect of only one center or focus is modeled, the same decision rules can be employed. In that case, however, s, the scale parameter, may be dictated by the metric of, for example, the existing highway network, population concentration, etc. However, a and ß can be determined using the above decision rules. In addition to considering rules associated with the index-set triples, it is useful at this point to briefly consider some of the intuitive aspects of the PMC. The PMC, as a r , is not particularly significant in modeling the effect of only one focus or center. t t We can, of course, think of this case as a "degenerate" F',, with K = 1. It seems, however, that building-block developments are most appropriately associated with the synergism arising from combining two or more members of the I family. That is, the PMC is particularly valuable for modeling the effect of several centers on each other and on the surrounding region. t z Such modeling is considered more useful to a variety of decision-makers and hence decision rules are described in terms of developing ray F.'s for K > 1. 10. The expression on the righthand side can be evaluated for an assortment of values of;, preferably on a digital computer. The results can be arranged in a table allowing the user to find an a, for each M, (Appendix B, Part 2). 11. This is an exercise in fitting a single curve, which ignores the interrelationships (mathematical and interpretational) between curves representing the effect of two or more centers on each other. 12. In these cases, scale becomes relative and one of the scale parameters can be set equal to 1.

Market Comma nd

Figure 4 Depiction of Market Command Along a Ray of Finite Family of Densities

mif

, I

/

AI/

1 1~

1/ I-

** c ` pm •

;~~ , i~ Cl

-.1..-- C2

1~l

i

\

\

~\ ~~~ ~~--~~

~

-- C3

I

`~

C4

ierarchy of Center

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The F Density as Descriptor of Space Use In this section several diagrams and Table 1 are used to examine the F density as a descriptor of space use. In Table 1, four activities or phenomena are offered for consideration in terms of yielding the kinds of distributions that can be described by F densities. Interpretations of distributions for all the selected activities are quite similar and hence, since we wish to examine several phenomena in the different phases of inquiry, market command and diffusion along a ray are selected for discussion. Table 1 1' as Descriptor of Space Use Activity or Phenomenon to Be Described

State of Space Use

1. Intensity of space use: population/unit distance along a ray frontage foot cost/unit distance along ray number of services/unit distance along ray number of customers/unit distance along ray 2. Rate of adoption of an innovation along a ray 3. Command of market along a ray newspaper readers range of goods and services

Descriptor 1. Pointwise Maximum Curve (PMC)

Hypothesized and/or Observed

and/or

2.t Family and/or 3. F Family Member

4. Traffic flow along a ray

In examining Figure 4, which depicts market command along a ray, individual F's can represent inter- or intracity relationships. It is suggested, along the lines of Central Place Theory, that according to size, bundle of goods and services offered, etc., centers have varying amounts of command along the ray. In exploring the T density as a tool for mapping how phenomena or innovations diffuse along a ray, related additional interpretational perspectives can be generated. For example, members of the family could depict diffusion of a phenomenon along a ray, but at different times; or, the members could depict diffusion of different phenomena among a population along a ray. And, conversely, the members could also depict diffusion of the same phenomena among different populations. For the most part, goods and services could replace phenomena or innovations, thereby further extending the range of the F to describe market areas (along a ray, which could be, for example, a twenty-mile-wide band along 1-70 from St. Louis to Denver, or along 401 from Toronto to Windsor).

Barry S. Weltar and Gerald J. LaCava 65

The I Density as a Tool for Analyzing Space Use

Much research in the marketing, traffic flow, land-use, and diffusion fields can be related quite readily to our discussion of the F as descriptor of space use. However, little interpretive work has been done to date that could be used directly to explain why members of the F family take on the shapes that they do as representations of certain spatially distributed phenomena or activities. Since market studies have a long tradition in geographic research, they are examined in this section as we attempt to illustrate how the I density can be used to analyze space use. As shown in Figure 4 a finite family of is is used to depict (probable) quantities of goods and services sold or market area (two-space) controlled by a hierarchy of centers.' 3 A magnified indicator function is incorporated to keep the first member from going to zero, and as such serves to represent realistically the (probable) command of the major center, such as the CBD, in the CBD. And, as might be expected as a general rule, given a ceteris paribus environment with distance being the only variable, the power of the CBD as a marketplace declines with distance. The second center to the right of the CBD and the others as well are regarded as being of major significance in this section, however, due to the different shapes of the curves on each side of the respective maximum values. It is hypothesized that the shape of the curve for C2 varies as it does due to the variation in competition provided by Cl (the CBD) to the left and C; (i > 3) to the right. Related hypotheses can be erected for all the remaining centers, and are diagrammatically illustrated by the shape of the curves in Figure 4. The position that change is solely a function of distance may be somewhat unrealistic, as might our not distinguishing between air and road distance, or among travel time, cost of vehicle operation, effort, and other factors. We are attempting to incorporate some or all of these factors in an ongoing research project which is briefly discussed in the following section on planning of space use. As far as these few comments on analysis are concerned, however, we are hopeful that others will be induced to pursue theoretical or applied investigations in this area by our simplified explanation of why market area command is as we hypothesize and illustrate by means of Figure 4. The F Density as a Tool for Planning Space Use As a general rule planning is most effective if one has successfully analyzed at least some of the many relationships between the demand for and supply of space.

r.

13. As noted above, since the could be n units wide it seems appropriate to refer to the area of the ray, even though the ray is a line in its strict sense and as such is only one-dimensional.

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In this section attention is focused on transportation planning, and a considerable number of variations of distance as a change agent are introduced. t a The inherent complexity of a transportation system (facilities and services, trip generation, flows, and trip-end distributions) can be illustrated by considering just one of the components which influences or shapes traffic patterns or flows, that is, the highway network. Network characteristics which influence travel patterns or flow are topology or structure, capacity (per link or segment, or route), and density of the facility or system. Each of the network characteristics may be involved in varying degrees and combinations in terms of their impact on trip generation, flow, and distribution. For example, Figure 5 depicts a general relationship between probable return to investment per kind of facility and decreasing traffic flow (or trip generation) away from some center or focus.15 The vertical axis could be made more specific by referring to (probable) time savings, vehicle miles traveled, congestion reduction, etc. instead of (probable) return to investment. t 6 The horizontal axis on the other hand could represent similar or different topologies, capacities, or densities in similar and different kinds of space use. Hence, the axes could be made to approximate very closely a variety of realities. Due to the fact that it is more difficult to attach meaning to the curves than it is to draw them or generate them mathematically, a brief, suggested explanation of why the intensity might change for the members of Figure 5 is offered. The setting for the network is one in which vehicle density decreases away from the center, which may be shifted or relocated. Hence, traffic flow or trip generation builds up or increases along the ray in the direction of the center (destination)." For the facility labeled local, rate of return increases with increased traffic flow, to a point. Then, due to congestion resulting from the limited capacity, return to investment declines to zero (or, conceivably, becomes negative), that is, traffic comes to a 14. The materials which follow were first examined in detail in a graduate transportation seminar in the Department of Geography, University of Kansas, where three projects involving the State of Kansas, the Kansas City-Topeka Corridor, and the Douglas County-City of Lawrence highway networks were designed to test the working hypothesis advanced in this paper. (Our outcome of the work is the development of a dissertation topic by T. O. Graff which will involve the density in an investigation of recreation space use.) To some extent the efforts of Fisher and Boukidis (1963) and Levinson and Roberts (1965) were used in developing the underlying conceptual frames and the hypothesis to be tested in the projects.

r

15. Although the origin is labeled Network center, it probably could be labeled population, economic, and other terms which are synonymous with focal points of activity, without much if any loss of generality. 16. For further, related discussions see Beckman (1967), Ellis and Milstein (1967), Haight (1963), Smeed (1961), Wardrop (1961), Werner (1968), and Wohl (1968). 17. As noted earlier, depending upon the situation at hand the words probable, hypothetical, or actual could be read into the text or illustrations with only minor or possibly no loss of generality with respect to the phenomena being discussed or depicted.

Barry S. Weltar and Gerald J. LaCava 67

Figure 5 Relationship Between Rate of Return to Investment and Traffic Flow for a Ray of Highway Facilities

Rate of Return To Investm

Y

pmc

/

F eeway Expressway /

/~

./ ...0"----

J_

Network Canto of Focus

First Order Arterial

econd Order Arterial ~~~, — J~ eighborhood >h."~ \ \ 1

1

1—~

Traffic Flow o Trip Generation Decreasing Away from Center X

halt. The next orders of system facilities experience similar series of events, but at higher orders of magnitude. One hypothesis to account for the rate of return decreasing at an increasing rate for higher-order facilities is that due to the accelerated accumulation of vehicles on, say, the freeway, traffic grinds to a halt more quickly and at higher orders of magnitude (e.g., vehicles per unit distance) once a critical point is passed than for the other lower-order facilities. We have not yet isolated the set of assumptions with which we plan to investigate this problem, so what we are proposing for consideration at this time is what might be termed a working hypothesis. One suggested use of the F density in transportation planning is suggested by the pointwise maximum curve, PMC. As seen by Figure 5 the PMC combines various members to produce a (probabilistically) highest rate of return surface in two space. Let us suppose that the urban setting receives a shock, such as construction of a new plant, subdivision, or shopping center.1e Let us further suppose that the highway planner wishes to effect change in the PMC as a whole, or in part. One question to be resolved is for which segments of which facility, or 18. Or conversely, the trip generation capabilities of areas could change, leading to variations in the rate of return for different facilities. See Wohl (1968).

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combination of facilities, should resources be expended.19 On the basis of the preceding discussion he has the option of working with topology, capacity, and/or density, each of which may influence in varying degrees the spatial and temporal aspects of traffic generation flow and distribution. Although topology, capacity, and density are highly suggestive in terms of available alternatives, several are listed to relate planning to the mathematical model. In order to effect a change in flow in a developed or developing network, or in return to investment, the planner could do so by doing any of the following separately, or in combination: (1) add or delete links; (2) add or delete lanes; (3) make lanes or links reversible; (4) make lanes or links unidirectional or bidirectional; (5) add, delete, synchronize, or desynchronize traffic lights. As might be anticipated, these factors when used with topology, capacity, and density of the network or parts of the network comprise the parameters of the f's which are the basis for planning. Hence, if the planner wishes, for example, that (1) a certain range be associated with each center or land-use activity, or (2) the network be designed in accordance with an accessibility hierarchy, or (3) an upper limit on congestion be established for each center, he has at his disposal some combination of the five factors (and there are probably others) to use in parametrizing the family of f's to achieve his objectives. It seems likely that this approach would be most useful for planning purposes (in the short run at least) as a heuristic device. This is attributed primarily to such factors as the potentially numerous calculations involved, problems associated with solving differential equations, and the magnitude of output that the technique can generate. It is our expectation that at least some of these problems can be resolved by recourse to a hybrid digital-analog computer configuration such as the one currently under development and testing at the University of Kansas.20 As far as implementation is concerned, however, the degree to which that is attained depends very much upon advances in planning methodology and in the articulation of transportation policy in general. For the most part this will involve development of multidimensional, multivariable paradigms for conceptualizing networks in an urban (and/or regional) systems context. We are hopeful that this study not only contributes to that end, but that it will evolve to the point where it can be used in postulating, and appreciating the meaning of, different space-use and transportation planning alternatives. 19. Although the PMC does not integrate to 1 as long as it is made up of more than one curve, it is, of course, possible to integrate each segment to yield good approximations of the area under the PMC. For purposes of determining planning strategies, sensitivity analysis could be performed whereby return/cost ratios for different facilities are incorporated into the PMC; this would yield a basis for a comprehensive network (or subnetwork) development program in which levels of expenditure are directly related to levels of performance. 20. This research is being directed by Professor Robert E. Nunley, Department of Geography, University of Kansas.

Barry S. Weltar and Gerald J. LaCava 69

The F Density as a Predictor of Space Use The model developed earlier in the paper can be made a predictor in much the same way that the gravity and other allocation and interaction models can be so transformed, that is, by changing the values of variables, parameters, constants, etc. Hence, although the F model may be more mathematically sophisticated than the gravity or regression-type models, for example, it can be readily transformed into a predictive model. As with the other models, however, unless of course they are recursive in nature, hypothesized, or real, change is exogenous to the model and must be introduced by the user. One application of the r density as a predictor of space use would be in the delimitation of market areas (other topics are suggested by Table 1 and subsequent discussions of I in the description, analysis, and planning of space use). Bearing in mind that the location, scale, and intensity parameters all have a bearing on the PMC, real or hypothesized changes could be incorporated in one or several members of the family and the PMC would change accordingly; or, the predicted impact of a change in one member on another in terms of market command could also be calculated by varying any parameter or combination of parameters as desired. Just as there have been considerable testing and calibration of other forecasting models, so must there be testing of the applicability of the r density. The potential applications suggested in preceding sections show a wide range of topics remaining to be investigated. Conclusion This paper discusses the gamma probability density function as a tool for describing, analyzing, planning, and predicting space use. Our objective was to develop a mathematical model based on the F density whereby decision rules for the parameters could be incorporated into attempts to model space use. In addition, a number of possible research areas were suggested and discussed for each of the descriptive, analytic, planning, and predictive components of F-based models of space use. Although we do not deal explicitly with notions of process in this paper, the implicitly process-oriented nature of the illustrations of examples provides a basis for extending work in that direction. Acknowledgments The authors are indebted to Mr. F. L. Speir, Department of Geography, University of Kansas, for the drawings and to Mrs. Sandra Beldt, Department of Quantitative Methods, Georgia State University, for the calculations in Appendix B, Part 2.

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Appendix A Properties of r Probability Density Functions 1. The maximum of ra,ß,,(x) is attained at x = s(a — 1) + fl. (x — ß)a-1 a -(x-fills

ra.s.,(x) =

(a)sa

Therefore, e—(x— eus r,,s,s(x) =r(å~a (a — 1)(x — ß)a-2 (x — ß)a-2 e-(x-BNs r

— ß)a -1 e—(x—eusl, x > fl

]

La— 1

r(a)s'

—1(x

-- (x — fl) , x > s

ß.

J

Now,

(x — ßr2 e -(x -sus r(a)sa

> 0, for x > P.

So, r;,a,,(x) = 0 if and only if, a — 1 - -s(x —

ß)= 0, or x= s(a — 1)+ ß.

(a —

2. The maximum value is

p =(s(a ra.s.s(s(a — 1) + ß)

r(a)s —

(a)S2

(s(a — (a —

1) + ß — ß)a -1 —cs(a-1)+s—e1/s 1))a-1 e-542- 1113

r(cc)s' e-- (a - 1)

l)a-1

r(«)s

3. For I < a < 2, ra,a,,(x) hasa point of inflection at x = s(a — 1) + fl + s a — 1. For a > 2, r",,(x) has two points of inflection which occur at x =s(a — 1)+ß± s e-(x-eus(s(a — i) + ß — x) r."4,3(x) = r(a)s.+1 [(a — 2)(x — ß)a-3 —S (x — ß)a-2 e-(x-s)/s(s(a — 1) + fl — x) — (x — ß)a-2 e-(x-vusl , x >

JJ

fl,

— ß\a-3e-(x-s/ [(cc — 2)(s(a — 1) + ß — x) r(a)sa + x—ß — (s(a— 1)+ß—x)—(x—ß) ,x>ß,

(x

— (x—

e-a-eus ) a+1 Iß)(ccs

[s(a-2)(a— 1)-2(a— 1)(x—ß)+(x

sß)2J

, x> ß.

Barry S. Wellar and Gerald J. LaCava 71

Now, (x — fir -2 e-(x-ß)/s > 0, r (a).a+

for x > P.

Therefore, i å;0,,(x) = 0 only if (x—ß)2 2(a— 1)(x—ß)+ s(a — 2)(a— 1)=0. s

By the quadratic formula, 2(a— 1)f J4(a— 1)2 — 4s(a — 2)(a — 1)/s x— ß= 2/s =s(a — 1)±sJ(a— 1)(a— 1 —(a- 2)), =s(a — 1)± s a— 1. Therefore, x = s(a — 1) + ß ± s a — I . x = s(a — 1) + ß + s a — 1 is a point of inflection of r,,0,,(x) whenever a > 1. However, x = s(a — 1) + ß — s a — 1 is a point of inflection of ra,0,,(x) only if s(a— 1)+ ß — s a— 1 > ß or s(a —1)> s a-1. This holds for a — 1 >a — 1, or a-1>1, or a > 2. 4. Fora > 1,0 < y < s(a — 1), ra.a.=(s(a— 1)+ ß +y)—ra.e.,(s(a — 1)+ ß —y)>0; that is, is rise more quickly than they drop. Note that the inequality actually holds for 0 0. Now,

7

Y

xo — ➢

x' e- xis dx > 0 and can be integrated by parts. r(a)sa

-

xa-1

xa + Y xa - 1 e s/,

J

xa

-Y

r(aka

=

e-431x0+ Y

X4/1- Y

rws.-1 xa -Y xo+Y x.-1 e-x/,

xa-,

= r(a)sa-1

I xa- ➢

J

xo

(a — 1)

JY

e—x/3

r(a)sa-1

e-4. dx. Ra — Ds.-1

p x

x0 —

X"- 2

a

-2

Next, xo„ + ➢ xa-2 e- xi' J r(a — 1)sa dx > 0, since the integrand is 11_1.0,3(x).

X0-Y

dx'

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Therefore, e-1 e —s/sl so+s — (x0 + y)a-1 e- Ise +Y)/s (x0 — y)a -1 e-0,0- v)1 0 < r(a)sa -1 a1 r(a)s r(a)s°SO-Y

= sra.0.3(x0 + y) — sra.0.3(xo — y). Since s z 1 > 0, ra,0,3(xo + y) > ra,0,,(x0 — y), completing the proof. 5. Fora > — 1, s > — 1, max ra,ß,,(x) < 1. max ra p s(x) = r p AS (CC — 1) + ß) •

•

(a — I )a-1

(*- 1) r(a)s Let xo = s(a — 1) + ß. We show that, fora z 1, s z 1, ra,ø,s(xo) < r1 ,ß,1 (ß) = 1. Since

(a — l)a-1 e-la- 1) r(ak is a decreasing function of s, it attains its maximum with respect to

sats= I. Therefore,

(a — 1)a

-1 Co- 1)

r(a)s

It suffices to show that (a — 1)a-1 e-ta -1►

r ,(a)

(a

(a — ir-1 e-4a-1) 5

r(a)

e (a-1) — 1)a -1 r(a)

•

is maximized by a = 1.

= max ra.0.1(x) z ra.13.1(a + ß), as-1 e-a

r(a)

as e-a

— r(a + 1) = max ra+ l.ß.l (x)• It remains to be shown that, for 1 < a 5 2, 1)a -1 T(a) _ (a-

a -la- u

r(a)

is maximized by a = 1; or equivalently, that T(a) is a nonincreasing function of a for 1 S a < 2. We show that T'(a) S 0. Now, ela- 1 )(toga - 1) - 1 )

T(a) =

r(a)

Barry S. Weltar and Gerald J. LaCava 73

Therefore, T'(a) —

r(a)(a — Ir-' e-'°-' )log(a —

1)

r2(a)

r'(a)(a — 1)°-' e °- ') r2 (a)

r(a)(a — Ir-' e-(°-' ) [j'log(a — 1) —

1-3(0 -1 e- c°-'► Now, r(a)(a — 1)° > 0. r3(«) Fora z 1.47,

I'(å

> 0, log(a — 1) < O.

Therefore, log(a — 1) — For 1 S a < 1.47,

r(a)

T()

a)

< 0.

> — .6, and log(a — 1) < — .75.

Therefore,

!og(a-1)— r'(a) < r(a)

—.15.

Therefore, for 1 5 a 5 2

!og(a — 1) —

r(a)
0 the propensity to locate in a subarea increases as x increases. It decreases as x decreases when c < 0. If the number of objects in a subarea is kept constant, then the probability of locating there increases as n increases when c < 0. It decreases with n, however, when c > 0 and is a constant function over all n when c = 0. In fact, when c = 0, as has been pointed out, no diffusion process

Arthur Getis 85

takes place at all and formula (1) reduces to a binomial distribution. Therefore, the initial attributes of each subarea and a bias c representing the process is all that is needed to predict the x in j*. There are a number of difficulties with such a relatively simple scheme. First, the diffusion takes place within subareas and not between them. One must assume that subarea boundaries are not breached. In essence the location process described is really the sum of J processes, each independent of the other. Second, no coordinate position is given to the location of an object. The only spatial information about an object is that it is found within the)" subarea, but there is no knowledge of its place within the subarea. Another drawback, related to this, is that analyses of patterns must be in terms of frequency distributions. The number of objects per subarea rather than their spacing must be the subject for study. This limits detailed analyses of spatial patterns. These problems are not insurmountable, but it is apparent that much caution must attend any study using the Polya model as it has thus far been developed. In particular, the sample areas must conform to the districts for which data are available. In most cases, these districts will be of irregular size and have various potentialities for growth and change. The differences from district to district may be a function of their differing sizes or the number of objects contained within them. In addition, there may be limitations on the number of objects that can possibly be contained in them. These differences from district to district are in addition to the potential myriad of nonsize related characteristics. If the b/r ratio and the c parameter are unlikely to express these differences, then, of course, the Polya model as developed is inappropriate. The first problem, that dealing with impervious boundaries, may be alleviated to some extent in the following way. Suppose it is suspected that the propensity for diffusion is a function of the location of a subarea relative to other subareas. Consider, too, that a certain set of subareas contiguous to the subarea representing the center of diffusion will be most affected by the central subarea. If it is known or hypothesized what effect the position of one subarea has on another, then in the assignment of b/r and c values the spatial relationships can be built into the model. This, however, does not imply that diffusion takes place between subareas. It merely means that the process of diffusion is affected within subareas as a function of the spatial position of the subarea relative to other subareas. The problem of object location will be less troublesome if the subareas are small. If it is possible to use small subareas, the number of possible locations for an object is small and the generated patterns, though still indeterminate representations, will be more concise than otherwise. In any case, analysis must proceed using grouped data and it may be wasteful to attempt to use sample subareas of very small size. Esthetically, there may be justification for proceeding with small areas, but the effect of size limitations on probabilities (attributes) may become more difficult to handle.

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The model developed so far does not consider the distribution of the x objects among the subareas of the region. Most likely, it would be more useful to have a model which summarizes the many subarea diffusion processes. In this situation we would assume that each j1b subarea has its own attribute measure, p, at the start of the process. If it is assumed that c is constant throughout the region, we have 1 P,(x; p, n, y) = -J-PP(x; p, n, y) J j= 1

(9)

where PJ represents the probability of obtaining a subarea with x objects contained within it. PJ(x; p, n, y) is the same as (4); the subscript implies that there is a distribution of x objects in n trials for each j' subarea. By dividing by J the sum of the Pi's for any x gives the proportion of j subareas containing x objects. PJ (x; p, n, y) is also a Polya distribution with moments J

ntPi J=1

111 =

(10)

J

142 = npq(1 + ny)/(1 + y).

(11)

The expected number of objects in a subarea is E(xx) J

With equation (9) it is possible to weight each j differently, thereby relieving the parameters p and y of the potential burden of representing so many different subarea characteristics. This may be done by increasing J to represent parts of subareas such that any!' area is equal to all others in some specified attribute. The p value would be reformulated to allow for this adjustment. Now, if the values of the parameters are to be estimated from empirical experience, it may be useful to determine these values from the sample data. Suppose each subarea has a set of n trials associated with it and the probability of a success (drawing a black bead) varies from subarea to subarea. Two parameters can describe the Polya distribution for all subareas. These are u = b/c and w = r/c. These can be estimated (Skellam, 1948) by using factorial moments—about the origin (z(9) is the g' factorial moment). Set R9 =

— (n—g+ 1)(g+f1— 1) µ'(g- 1) g+Q+ —1 µ'(g)

for g= 1,2,3

(12)

where n represents the number of trials for each subarea and Q and ra are estimated from sample data. Then u=

R1R2 — (n — 1)R1 (n — 1)R1 — nR2

(13)

Arthur Getis 87

w

n R—1.

(14)

In this model knowledge of the way in which the p's are distributed among the subareas is useful for a number of reasons. More will be said about this later. Suffice it to say now that the p's may be distributed in such a way that it will be more convenient to use another probability model to generate theoretical distributions. For the moment, however, it seems worthwhile to show how the model developed this far may be applied. Our first example will be a straightforward demonstration using hypothetical data while the second example, based on data gathered from Wisconsin, more nearly illustrates how modifications must be made in the model in order to better approximate real world situations. An Example: The Hypothetical Diffusion of Color Television Sets As an example of the use of a Polya distribution let us suppose that our interest is in the evolution of the pattern of residences where color televisions are owned. One must assume that the diffusion does not cross subarea boundaries. This is a somewhat unrealistic approach, as has been mentioned before, but a distance effect can be built into the analysis. The sample areas may be census tracts. The propensity for an object (say, 100 color sets) to be placed in a tract is assumed to be a function of family income and distance from a leadership tract. Thep value of each of the j census tracts can be made to depend on the proportion of all families above some critical income level. The tracts can be weighted according to the number of families contained within them. Of course, one would apply a separate p value to a portion of a tract if its special income characteristics were known. The diffusion parameter, y, is assumed to be a constant (c = 1) for the entire region. If it is judged that too much realism is lost by the assumption of a constant, then the PJ formulation must be abandoned. Also, let us suppose that by drawing ten times from the urn for the region as a whole, we will have moved from time period to to 11. A frequency distribution of objects within tracts can now be generated according to the Polya scheme. An analysis of results might include: (1) comparisons between the theoretical and observed number of tracts having x objects at time t i ; (2) partitioning of tracts according to various characteristics and studying the significance of x values; (3) studies of the rates of change within certain census tracts, between groups of tracts or for the entire region. We will not enter into a discussion of the shortcomings of this approach. Much has been said previously about the character of the Polya distribution. Also, no mention will be made here of the numerous ways in which the analysis might be modified to include more realistic assumptions. Table 1 below contains the relevant information.

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Table 1 Some Characteristics of Hypothetical Census Tracts Tract Number

No. of families (F)

Proportion of families haying yearly income >$12,000

Distance from Tract I (in miles: center to center)

1 2 3 4 5 6 7 8 9 10 11 12

1000 500 2000 1000 500 1000 1000 2000 3000 2000 1000 1000

.80 .80 .20 .10 .10 .15 .80 .50 .60 .15 .25 .10

0.3 0.2 0.4 0.5 1.5 4.0 2.5 1.5 6.0 5.0 6.0

Attribute assignment (p)

.8 .8 .2 .1 .1 .1 .2 .2 .4 .0 .0 .0

The attribute assignment was made using the relationship pi = 1; /d1 for all j having d > _ 1.0 where 1i is the proportion of families in tract j having yearly income of more than $12,000 and di is the distance in miles from the center of tract j to the center of the leadership tract, designated as tract 1. di is assumed to be one for all tract distances less than one mile. The probability distributions for tracts having p values of .8, .4, .2, .1, and 0 are given in Table 2. Table 2 Probability Distributions for Tracts Having p Values of .8, .4, .2, .1, and 0 x

.8

.4

0 1 2 3 4 5 6 7 8 9 10

.001 .002 .010 .016 .028 .056 .084 .120 .161 .250 .286

.066 .110 .135 .144 .140 .126 .105 .080 .054 .030 .011

P (x; p, 10, y) .2 .286 .250 .161 .120 .084 .056 .028 .016 .010 .002 .001

.1 .474 .263 .139 .070 .032 .014 .005 .002

.0 1.000

The value .120 at x = 7 in the column headed .8 means that for tracts 1 and 2 the probability of receiving seven objects during the period to and t 1 is .120 if there were ten drawings each from urns representing the tracts.

Arthur Getis 89

Since it is the entire region that is being described by the Polya model, we develop the distribution of objects within tracts by formula (9). Here it is assumed that ten drawings are made for the region as a whole. Each tract is weighted according to its proportion of the region's families. Thus, we have w1 = J

'

(15)

EF

1= 1

where w.; is the weight for tract j and F is the number of families in region j. Formula (9) now becomes

E

1

J J=1

w,•• P,(x ;p, n,Y)•

(16)

The probability distribution in this case is x

P

0 1 2 3 4 5 6 7 8 9 10

.425 .140 .097 .077 .060 .049 .037 .031 .028 .029 .029

and the mean is n Epjw1 = 2.26. 1= 1

This value is interpreted as the number of objects within the region. These are allocated to the tracts by np1 w1 for each j. In our example we have the distribution given in Table 3. Each object represents 100 color television sets. This distribution could influence the selection ofp values if further study is required. Another Example: The Diffusion of Propane Tanks in Grant County, Wisconsin, 1952-62 In this example an attempt is made to retain the essence of the Polya model while modifying it so that we may deal with a particular problem, i.e., predicting the

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Table 3 Allocation of Color Television Sets among Hypothetical Census Tracts Using Polya Model Tract

w

Objects

Color TV sets

1 2 3 4 5 6 7 8 9 10 11 12

.0625 .0312 .1250 .0625 .0312 .0625 .0625 .1250 .1875 .1250 .0625 .0625

.500 .250 .250 .062 .031 .062 .125 .250 .750 .000 .000 .000

50 25 25 6 3 6 12 25 75 0 0 0

1962 distribution of 1,108 propane tanks among the thirty-three townships of Grant County, Wisconsin. The data were gathered by Brown (1963) for his Markov approach to a similar problem. In our approach we specify ap and a y value for each township. These values are dependent on existing conditions in time period t. Thep refers to a population factor (propensity for purchasing a tank) and y refers to the number of tanks extant in the townships (reinforcement effect). popi and ,y Pi = , pop., E i-1

—

J

tanks in j at time t

(17)

E tanks in j at time t i=1

The weighting factor is P; + ,yi J

t

= (1952, 1953, 1954, ... , 1962).

(18)

E P; + ;=1 E ,Yi

;=1

Our purpose is to predict the 1962 distribution of the 1,108 propane tanks among the townships by using data from earlier time periods. With this set of data one may proceed by stages, i.e., year to year or longer periods. Here we will report on two experiments. In the first we take 1952 data and attempt to predict the 1962 distribution (see Table 4), and second we take 1957 data and attempt to predict the 1962 distribution. In the first case, less than 1 percent of the total number of tanks together with the population factor are used to forecast the 1962 pattern. In the second case about 16 percent of the data is used, and it is clear that here fairly good values are obtained. Figure 4 gives some idea of the difficulty involved in making accurate predictions. Each of the four townships shown had different variations in their growth rates and each is representative of several other townships.

Arthur Getis 91

Table 4 Observed and Expected Number of Propane Tanks for 1962 in Townships of Grant County, Wisconsin, Using a Population and Reinforcement Effect Model Township

Beetown Bloomington Boscobel Cassville Castle Rock Clifton Ellenboro Fennimore Glen Haven Harrison Hazel Green Hickory Grove Jamestown Liberty Lima Little Grant Marion Millville Mt. Hope Mt. Ida Muscoda N. Lancaster Paris Patch Grove Platteville Potosi Smelser S. Lancaster Waterloo Waterstown Wingville Woodman Wyalusing Total

Population 1960

956 1326 3030 1796 440 978 559 2467 758 557 2021 497 1490 643 678 391 414 163 590 615 1317 553 1362 618 6957 1927 2458 4776 636 642 1001 339 760 43715

Actual propane tanks 1952

1

2

1 3

1

8

Predicted propane tanks 1962

Actual propane tanks 1962

Predicted propane tanks 1962

12 17 108 23 6 12 7 170 9 7 25 6 19 8 9 5 5 2 7 8 17 7 17 8 158 24 240 60 8 78 13 4 9

5 51 117 80 1 9 8 79 12 5 71 2 9 10 12 7 2 6 26 3 39 8 32 24 153 49 77 125 3 15 15 6 47

10 71 124 44 5 10 10 90 11 6 74 5 16 10 36 4 4 2 20 10 32 6 25 24 173 42 72 100 7 21 21 4 18

1108

1108

1108

Actual propane tanks 1957

16 26 7

1 18 1 15

1 8

4 l 5 3 5 28 6 13 14 4 3 3 182

There are no essential differences between this approach and many of the simulation experiments performed in recent years. In this context, however, we associate our method to a particular type of probability model, one which we claim may be used to represent processes responsible for spatial patterns.

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Figure 4

160 150 140 130

PROPAN E

TAN KS-O BS E RVED

120

I I0 Platteville I0 0 90 80 70 60 50 40 30 20

IO 0 1952

1954

1956

1958

1960

1962

Arthur Getis 93

Relation to the Negative Binomial Distribution Skellam (1948) shows that theoretical distributions based on the Polya distribution and the negative binomial distribution are not much different. This is convenient for two reasons. First, it is much easier to derive theoretical values using the negative binomial and, second, the negative binomial is a limiting distribution of the Polya model. Let us first consider the relationship between the two distributions in an expository fashion. Although Skellam (1948) did not call his distribution the Polya, nor did he speak of urns, he was able to show that if the probability of a success varied from one set of trials to another according to a distribution with no severe limitations imposed on its fluctuations, the result will be a Polya distribution. For our purposes, this simply means that if p varies from subarea to subarea in any of a wide variety of ways, the Polya model will suffice. The implication of this is that the Polya model can be considered a compound distribution, i.e., a distribution which describes expectations which vary from subarea to subarea. This is in contrast to a single Poisson process where the expected number of objects in a subarea does not vary. If one assumes that the distribution curve of the expected number of objects in a subarea can take on a wide range of shapes, say in relation top above, then the resulting distribution describing objects in subareas can be negative binomial. This has been proved by many writers using the very general Pearson type III distribution (unimodal or J-shaped) representing the distribution of Å where A = np. Going at this in another direction it has been shown that when all subareas have similarp's, but when one assumes a large number of trials (n -► co) and subareas (J -• co) with c > 0, and a constant denoted by n/(J - 1), the Polya distribution has as its limit the negative binomial distribution (Bosch, 1963, pp. 210-11). Rewrite (3) as P(x; n, u, v) =

r(n + 1) r(u + x)r(v + n — x)r(u + v) — x + 1) r(u)r(v)r(u + v + n) r(x + 1)1-(n

(19)

and let n, v -> co. If u is fixed and v - co we can replace r(u + v) by vuF(v). In like manner when n -> co we have r'(u + v + n) - (v + n)°r(v + n). Therefore (19) becomes v°nX r(u + x) r(x + 1)r(u) (v + n)" +X

(20)

which when written (I, + x — 1"( v y( n 1X n+v n+v x

(21)

is the clearly recognizable form of the negative binomial distribution. Therefore, it seems that in many circumstances, if one can hypothesize a process described by

94

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the Polya urn model, then the advantages of using the negative binomial model may be exploited. Negative Binomial Models The negative binomial distribution itself is said to arise from any of a number of processes irrespective of its link to the Polya model. Anscombe (1950, pp. 360-61) lists four negative binomial processes, three of which have possibilities for representation of spatial patterns. Briefly, these are: (1) Randomly distributed colonies.... Groups of objects scattered randomly, the number of objects in the groups are distributed independently in a logarithmic distribution and the total count has a negative binomial distribution. Spatially envisage groups of varying size scattered about without any discernible pattern. The number of objects per unit area will be distributed in a negative binomial fashion. (2) Immigration-birth-death process.... Each object has a constant birth and death rate and there is a constant rate of immigration which leads to a negative binomial distribution for the population size. Spatially we might view an area at different time periods. The number of objects in that area at different time periods has the negative binomial distribution. (3) Heterogeneous Poisson sampling.... This is the compound Poisson distribution mentioned earlier. The mean of the Poisson varies randomly from area to area. The negative binomial distribution arises for the number of objects per area if the mean has a Type III distribution. Since in each case a negative binomial model is the result, it is impossible, after observing the existence of a negative binomial distribution, to determine which process was responsible for it. There is a substantial literature which has been developed recently attesting to the wide applicability of the negative binomial distribution as a descriptive device. Most researchers are fully aware of the pitfalls in working backwards toward the process instead of away from the process as we have been suggesting here. Let us look more closely at the relationship between the two distributions. The negative binomial distribution depends on two parameters : the mean m and an exponent k. The probability of observing a value x is P(x; k, m) =

rn k+x— 1 k k x im +k k+m

(22)

The Exponent k In terms of the Polya distribution, given the necessary conditions for its negative binomial limit, the value k corresponds to the ratio between the number of red or black beads and the c parameter (the replacement value). In other words, k is the propensity for replacement. If k is to be estimated from the raw data, it represents

Arthur Getis 95

a measure of aggregation characteristic of the entire study area. Hypotheses based on the value of k are most helpful in that low values of k imply great clustering, and high values little clustering. This is evident from the formula for the variance z

V =m+

(23)

k

Pielou (1969, p. 94) points out that k does not change from time period to time period when a population either increases or decreases in size owing to random births or deaths. If tests on sample data fail to bear this out, there is good reason to believe that the birth or death process is most likely density dependent, provided we can be certain there is no migration from subarea to subarea. In order to estimate k from sample data the criterion of at least 90 percent efficiency is suggested since there are a number of ways that the estimation can be accomplished. Probably the safest, most reliable method is that of maximum likelihood (Bliss and Fisher, 1953A, 1953B). There are, however, simpler procedures which in many cases compare favorably with the maximum likelihood method (Bliss and Fisher, 1953 A, B; Anscombe, 1950, pp. 369-72). In any case, the value of k obtained from these simpler methods can be used as the initial estimate of k for the maximum likelihood approach. Roughly speaking, use the first method described below when k > 1 and the second method when k < 1. If k is very large, say when (k + m)(k + 2)/m > 15, then the maximum likelihood method should be used. Method 1: mo0 k V

(24)

— mo

where V and mo are the variance and the mean of the sample data. mo is the best estimate of m. Method 2: Equate the observed proportion of subareas having no objects to the expected proportion. Then choose k by successive approximations to satisfy no

(k — J+ kmolk )

(25)

where no is the observed proportion of zero subareas and J is the total number of subareas. In the maximum likelihood method an estimate of k called k' is used as a first approximation of k and in a series of steps a maximum likelihood k is obtained. This is achieved by finding a zi for a k', which equals zero. The subscript refers to the step number in the estimation process. zi =

E( +s

/ ) N In l 1 + I x

(26)

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Ax is the accumulated frequencies up to x (but not including the frequency associated with x) starting with the highest x value; x is the index number referring to objects in a subarea; N is the total number of objects in the region and x is the mean number of objects in a subarea. Usually k; is calculated from (24). If zl is positive, then make k'. > lei ; otherwise make k'2 < k;. Continue to vary k; (usually by interpolation) until z; is close to or equal to zero. In Table 5 are shown the k values for each of eleven distributions (one for each year) of the propane tanks discussed earlier. Several trends are evident—decreasing k values for the period 1953-56 and increasing values from 1956. The decreasing k values indicate a clustering or intensification while an increasing k implies that a diffusion process is taking place. Note, however, that there is a leveling off toward the end of the study period signifying an end to the diffusion. Table 5 Negative Binomial k Values for Distribution of Propane Tanks in Grant County, Wisconsin, for Each Year 1952-62 Maximum Likelihood k

1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962

Anscombe Method I k

.23 .47 .36 .32 .28 .40 .51 .61 .68 .72 .74

Negative Binomial Relationship to the Logarithmic Series Continuing with the negative binomial distribution, it is interesting to note that when k 0, i.e., when there is great clustering the probability value for x = 0 tends to infinity. However, upon excluding this term as being, in general, unobservable, one obtains the logarithmic distribution. Thus the logarithmic distribution is a limiting form of a truncated negative binomial distribution (Pielou, 1969, pp. 206-08; Anscombe, 1950, pp. 375-79). This can be useful when clusters are to be accounted for without regard for clusterless areas. In this case, the smallest cluster of objects is size 1. More important is the fact that when there is a strong tendency for clustering the value of k --> 0 as has already been shown. Equating the negative binomial distribution to unity after eliminating the prob-

Arthur Getis 97

ability associated with x = 0 and allowing k —> 0 yields the logarithmic distribution (Pielou, 1969, pp. 206-07). This can be shown by first taking the negative binomial distribution of (22), setting r = m/(k + m) and s = k/(k + m) and finding P when x = 0

P(0; k, s) = sk.

(27)

Now define

P'(x; k, s) — P(x, k, s) 1 — P(0; k, s)

(28)

This gives

P'(x; k, r, s)

x) skrxx rx I (k) sk

(29)

Collecting all k terms together and calling them C (a constant) we will then be in a position to express the limiting form of P'(x; k, r, s) as k —> 0.

P'(x; k, r, s) — C

1'(k + x) x r x!

(30)

where

C = sk/F(k)(1 — sk). Then

lim P'(x;k,r,$)= 0r(x)rx = 0 x r for = 1,2,...

k-0

x!

x

(31)

and 0 = 1im, k -•o

C=

-1

In(1 — r).

(32)

Thus, we have the logarithmic distribution Ø P(x;0,r)= x (x= 1, 2,...)

(33)

where Ø

=

—

1/In s.

In many instances when n is large, distributions display the logarithmic series characteristics. If one can hypothesize the nature of the expected distribution by way of the Polya model, it seems that reasonable explanations for such phenomena as rank size or species abundance are forthcoming.

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Method of Estimation for the Logarithmic Series Distribution

The value r may be estimated from the likelihood equation (Patil, 1962, p. 69). x = P(F)

(34)

where z is the mean of a random sample of size N and u = Ør/1 — r, which is the first moment of (33). In the table provided by Patil (1962, p. 70), find r by observing the position of z in the body of the table. Then find logio e e _ —loge (1 — r) and substitute into (33). Weighting Subareas The ability to weight each j subarea in a meaningful way depends to a large extent on knowledge of the spatial processes which are operating in each and all j. There are schemes which might be developed which can provide a framework for assigning weights. One such scheme was briefly explained in our earlier example of the diffusion of color television sets. Another simple weight might derive from the density of the points in each j relative to the entire region's density. This is a type of location quotient which, in general, is designed to show relative concentrations of objects of a specified type in a subarea. Any of a number of location quotients may be used depending, of course, on the nature of the objects. Taking another view, one might presume that the structure of weights conforms to some measure of the potential interaction between a subarea and all other subareas. This gravity approach can be formulated as X

wr j

=E du

(35)

where xj represents the number of objects in the f' subarea, or it could represent some other attribute of the subareas, and du is some measure of distance between subarea i and j. Another approach might be to consider not only the amount of potential interaction between any two subareas, but also the length of common boundary between the subareas. Cliff and Ord (1969, p. 33) suggest a weighting system of the form wo = dd a [gr(!)] ø

(36)

where q,(j) is the proportion of the perimeter of subarea i (not counting study area boundaries) which is in contact with subarea), and a and ß are parameters which regulate the effect of distance and common boundaries, respectively. The authors point out that the choice of a functional form for w;j must lie with the investigator. In all of these schemes it is suggested that E w;j or E w; is standardized by setting it equal to one.

Arthur Getis 99

Contiguity At a particular stage in the analysis it may be helpful to measure the degree of contiguity between subareas. By contiguity we mean the spatial autocorrelation, i.e., the degree of similarity among subareas which have some spatial relationship (such as nearness). The need for a measure is a result of a desire to evaluate the evolving pattern for evidence of diffusion or agglomeration. Cliff and Ord (1969, p. 32) provide a correlation coefficient which measures the degree of positive or negative spatial autocorrelation. This is n n E E zi zpv, i=1

r=

j=1

i#J

~E E wiJ i*j i=1 n

(37)

n

Zz

i=1 j=1

where n is the size of the sample (if J is small, n would most likely be equal to J), zi and zj are the deviations from the mean of the sample values (the number of objects in subareas) for the ith and jth subareas, and wii are the weights as defined in (36). If tests are needed, sample values must be assumed normal. Unfortunately, the r value does not have a bounded fixed range between —1 and + 1, but Cliff and Ord (1969, pp. 37-39) provide a method for ascertaining the limits of r for any set of zt's and wo's. In conclusion, it seems that many of the possible spatial processes outlined at the beginning of the paper can be realistically modeled by Polya-type urn formulations.

References ANSCOMBE, F.J. (1950). "Sampling Theory of the Negative Binomial and Logarithmic Series Distributions." Biometrika, 37:358-82. BLISS, C. I., and R. A. FISHER (1953A). "Fitting the Negative Binomial Distribution to Biological Data." Biometrics, 9:176-87. Buss, C. I., and R. A. Fisi-ma (1955B). "Note on the Efficient Fitting of the Negative Binomial." Biometrics, 9:188-200. Boscl-t, A.

J. (1963). "The Polya Distribution." Statistica Neerlandica, 17 :201-13.

L. A. (1963). "The Diffusion of Innovation: A Markov Chain-Type Approach." Discussion Paper No. 3, Department of Geography, Northwestern University, Evanston, Ill. BROWN,

CLIFF, A. D., and J. K. ORD (1969). "The Problem of Spatial Autocorrelation." In A. J. Scott (ed.), Studies in Regional Science. London: Pion, pp. 25-55.

F. (1969). "A Hypergeometric Family of Discrete Probability Distributions: Properties and Applications to Location Models." Geographical Analysis, 1: 296-316.

DACEY, M.

F., and G. POLYA (1923). "Uber die Statistik verketteter Vorgänge." Zeitschriftfür angewandte Mathematik und Mechanik, 3:279-89.

EGGENBERGER,

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HARVEY, D.

(1969). Explanation in Geography. New York: St. Martin's Press.

PATIL, G.

P. (1962). "Some Methods of Estimation for the Logarithmic Series Distribution." Biometrics, 18:68-75.

PØU, E. C. (1969). An Introduction to Mathematical Ecology. New York: Wiley-lnterscience. SKELLAM, J. G. (1948). "A Probability Distribution Derived from Binomial Distribution by Regarding the Probability of Success as Variable between the Sets of Trials." Journal of the Royal Statistical Society, Series B, 10:257-61.

Use of Two-stage Least Squares to Solve Simultaneous Equation Systems in Geography DAVID R. MEYER (University of Massachusetts, Amherst)

1

The concept of system has a long history in geographic thought. However, this history of systems thinking has not been highly formalized nor has it occupied the center of geographic thought (Harvey, 1969, p. 467). By the 1960s, as Harvey (1969, p. 468) has noted, geographers were being urged to think explicitly in systems terms. Some early exhortations included statements by Blaut (1962), Chorley (1962), Ackerman (1963), and Berry (1964). Yet in spite of the present popularity of systems terminology in geographic research, systems concepts and systems analysis have not been operationalized to any great extent. One reason for this situation is that theory, which provides a basis for making evaluative judgments concerning the nature of the system, is weakly developed in geography (Harvey, 1969, p. 469). Given the present development of theory in geography, it is doubtful if systems concepts and analysis will be useful in providing substantial insight into geographic problems. Most use of systems ideas will probably continue to be in terms of general organizing frameworks in which the main theme is that most things are interrelated rather than the more important theme that some things are more related than other things. The task for geography would seem to be to identify and assess the structure of these significant interdependencies among geographic phenomena. Only then will we be in a position to make beneficial use of systems theory and systems analysis at a level above the trivial. As an example, interdependencies among some components which comprise the residential character of a city will be identified. It will be proposed that methods 101

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for solving simultaneous equation systems, in particular two-stage least squares, provide one technique for assessing the structure of geographic interdependencies. This technique will be discussed and its use exemplified. Before discussing the example of residence in the city, some clarification of terminology is necessary. The term interdependent is used to refer to mutually or reciprocally dependent relations. Thus if two phenomena are interdependent each stands in a dependent relation to the other. By the exclusion of single unidirectional relations (A affects B) from the discussion, no implication is intended that these are less important than interdependent relations. Rather the focus is on interdependent relations since these are the relations which are most weakly developed in geography and which are necessary future building blocks for geographical systems theory. Residence in the City: An Example of Geographic Interdependencies The residential character of a city results from a complex interaction of many decision-makers such as tax assessors, home builders, real estate dealers, households, and planning officials and various environmental components such as highways, housing, parks, and industrial sites. From this complexity two components, housing and households, are chosen to illustrate geographic interdependencies in the residential character of a city. Further simplification is introduced by assuming that households engaged in residential choice decisions are primarily interested in housing in residential areas not household characteristics. This is not totally unrealistic if it is assumed that for given characteristics of housing there are enough residential areas to choose from so that the household can find compatible neighbors. Even with these simplifications households and housing exist in a complex web of interdependencies. Important characteristics of households which are relevant to their demand for housing include stage in life cycle, socioeconomic status, and ethnicity. When choosing a residential location, households base their choice on such housing criteria as age, quality, size, value, rent, and type—single-family versus multi-family (Brown and Moore, 1970; Butler et al., 1969; Rossi, 1955; and Simmons, 1968). That is, residential location decisions are made based on the spatial distribution of housing opportunities in the city so that the spatial distribution of households (Y) which results is a function of the spatial distribution of housing (X), or Y =f(X)•

Within residential areas households change in character. Children grow up and leave home. Income may rise or decline. Occupational status may go up or down. The housing ages, structural alterations are made, and maintenance expenditures increase or decrease. The result is that housing, measured by such characteristics as quality, size, value, and rent, adjusts to the characteristics of households. Thus

David R. Meyer 103

the spatial distribution of housing (X) is a function of the spatial distribution of households (Y), or X = g(Y). Hence the residential character of our simplified city of households (Y) and housing (X) is determined by two simultaneous relations : Y = f(X)

and X = g(Y). This is a simple simultaneous equation system. A more complex model of the residential character of a city would of course include other relations which taken together would comprise a system of simultaneous equations. Such equation systems cannot be estimated by ordinary least squares; other techniques must be used. Two-stage least squares, a single equation method of estimation, is one useful technique. Before discussing the use of this technique, it is helpful to clarify estimation problems in simultaneous equation systems. Estimation Problems in Simultaneous Equations Consider the following simple model of the residential character of a city at one point in time: Q = f(Y) Y = g(Q,R) where for each residential area, Q is average quality of housing, Y is average household income, and R is average size of housing. The variable R is exogenous and variables Q and Y are endogenous to the model. As is evident from the model, both relations are needed to determine the values of the two endogenous variables. In brief the model states that housing quality in a residential area depends on household income. At the same time the type of households attracted to a residential area defined by income level depends on the housing characteristics of the residential area, namely quality and size. A statistical formulation of the model is Q1 = a1 + a2 Y + cis

Y ==ß1+

(la)

ß2Q1 + ß3R,+

E21

(lb)

where i is a particular residential area, a's and ß's are parameters, and E's are random disturbances. The properties of the c's given i = 1, 2, ..., n residential areas are assumed to be E(E1;) = 0; E(E21) =

0

E(Ei1) = ai; E(41) = Qi

gene u) = 0; E(E21E2j) = 0, 1 J.

(2a) (2b)

(2c)

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Furthermore, R is assumed to be independent of each of the E's. That is E(RtEp) — 0; E(R4E2t) — 0.

(3)

These assumptions are characteristic of the normal linear regression model. A simplifying assumption is added that the disturbances are uncorrelated across equations. Thus (4)

E(E11E21) = 0.

If this assumption were not valid, then the parameter estimates would not be asymptotically efficient. In this case methods in which all equations of a system are estimated simultaneously, such as three-stage least squares, are more appropriate (Kmenta, 1971, p. 573). Given the above assumptions, we can derive the statistical properties of the resulting estimates. Equations (I) are the structural form of the model. They can be solved for the endogenous variables to yield the reduced form of the model: Cat+«2ßi) + ( a2ß3 )RI+ (a2E2i +E1i) 1— \ 1— a2ß2 1 11—azfz \ \ / ßl + ßzi \{ (ß211 +E2i y ( ß3 —_ I 1\1 — ß22 1 + I — '+ — ßzaz 1

Qt =

(5)

Both endogenous variables are expressed as functions of the exogenous variable R;. The covariances of endogenous variables and disturbances can be derived using the reduced form equations. For example, EQieli —

—

al + a2l\ ß 1 1 — a2 ß2

a2 1

1 — a2ß2

EEtt +

1

a2 3 ER1E,, + ß — azßz

+ 1 — azßz

a2EE1 if 2i

E€11€11

(6a)

0 0,

because from the assumptions concerning the E's: EE11 = 0, ER; E1, = 0, Ec11c2i = 0, and &1;e i = aj. In similar fashion it can be shown that EQ,E2t =

EYtEtt = EY,•E2t =

az alz 0 0 1 — a2ß2 ßz I— ßzaz a

a2 z 1 — ß2a2

(6b)

1z 0 0

(6c)

0.

(6d)

0

Hence both Q. and Y, which are used as explanatory variables in the statistical model, are correlated with the disturbance terms. Therefore a direct application of ordinary least squares to equations (1) will not yield consistent estimates of the parameters a and fl (Johnston, 1972, p. 343).

David R. Meyer 105

It is possible to derive consistent estimates of the parameters of the model in equations (1). Rewriting the reduced form of the model (equation 5), we have Qi=1111+ 1112Ri+ vli

(7a)

Y•=1121 + 1122Ri + v21

(7b)

where

_

a l + azf1

1—az~z '1112

azß3

1

—a z~z ' vli

=

a2 2, +

Eli

1 — a zÅz

and 1121

YI

+ N2a l

= 1 — O2a 2

N3

, 11 zz = 1

— Q 2a2

vz~

ß2E1, + t2i I

— 32a2

By making the same assumptions as in equations (2-4), it can be shown that the ordinary least-squares estimators of the reduced form coefficients in equations (7) are unbiased. These reduced form estimators can then be used to derive consistent estimates of the structural model parameters (Goldberger, 1964, pp. 291-92). This can be done if it is possible to express the structural model parameters in terms of the reduced form parameters (Kmenta, 1971, p. 539). This is the problem of identification, which is very important in the analysis of simultaneous equation systems. Identification In identification an important issue is the relation between the unrestricted and the restricted version of the model. The reduced form of the previous model (equations 7) represents the unrestricted version of the model while the structural model (equations 1) represents the restricted version of the model. If the parameters of a structural equation can be uniquely estimated from the parameters of the reduced-form model, then the structural equation is said to be exactly identified or just identified. Alternatively, if there exists more than one way of estimating the parameters of a structural equation then the structural equation is said to be overidentified. Finally, a structural equation is said to be underident fled if there is no way of estimating the parameters of the structural equation from the reduced form model (Fisher, 1966, p. 31). An exactly identified or overidentified equation is said to be identified. Conversely, an equation is not identified if it is underidentified. The problem of identification can be illustrated using the above residential model (equations 1). The question of identification concerns the possibility of solving the structural parameters (equations 1) in terms of the reduced-form parameters (equations 7). The structural model is + a2 Y + t1;

(la)

gq Yi = Y1 + 1 2Q1 + 1333R, + E2i

(lb)

Qi

=

al /

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and the reduced-form model is Q1= 71 11 + 7112R1 + v1i

(7a)

Y! = 7121 + i22Ri + v21.

(7b)

Substituting for Qi and Y,• in equations (1) from equations (7) we get 7111

+ 7112R1 +

71 21

+ n22R1 + V21 = ßl + ß2(n11 + i12/2r + V11) + ß3R1 + E2,.

Vli = 11 + 12(7121 + 71 22R1 + v21) + E11

(la') (lb')

The disturbances can be ignored because in identification we are only concerned with the relation of the structural-model parameters to the reduced-form parameters. Rewriting equations (1') we get 7111 + il2Rr = (11

+ 127121)

+ 12rz22R1

7121 + 7122R1 = (Y1 + ß27111) + (ß27112 + ß3)R1•

(la") (lb")

According to the theoretical model, the equations hold for all i. Therefore both sides of each equation are equivalent in form. Equating equivalent elements of la", 7111 = 11 + 1127121; 7112 = 127122

so that we have two equations in two unknowns. Therefore, 11 =7111

71 12 71 12 — 7121 - ;112= , 7122

7122

that is, the a's are uniquely determined. Therefore structural model (la) is exactly identified. The same procedure applied to equation (lb") yields 7121 = P1 + ß271l1; 7122 = ß27112 + ß3.

There are three unknowns but only two equations; therefore there is no solution for the ß's. Hence structural equation (I b) is underidentified. General methods for determining the identification status of any given structural equation have been developed. Since the discussions are rather elaborate, only a brief sketch following Kmenta's (1971, pp. 541-46) discussion will be given here. For further discussions the reader should refer to Fisher (1966, pp. 32-64), Goldberger (1964, pp. 313-18), Johnston (1972, pp. 356-65), and Theil (1971, pp. 489-93). We begin with the general structural form of a simultaneous equation system with G equations: ß11Y1n

+ ßß12y2n + ...•+ PIGYGn +

YI

lxln + Yl2x2,, + • .. + 71&XKn = U1,,

ß2GYGn+Y21X1n+Y22x2n+• . •+Y22XXn= 142n ß21Yln+ß22Y2n+•• -• +

(8)

• GGYGn + Yc1x1n + YG2X2n + "' + YGKXKn = Ucn• ßG1Y1n + ßc2Y2n + • •• +ß

They's are endogenous variables, the x's are predetermined variables, and the u's are stochastic disturbances. The n = 1, 2, ..., N represents the observation whether it be a time period, urban place, or residential area. Although there are

David R. Meyer 107

G endogenous variables and K predetermined variables in this simultaneous system, in any given equation some of the ß's and y's will be zero. That is, some variables in the equation system do not appear in a given equation. In addition, one of the y's is taken as the dependent variable in each equation. Hence ß = l for this y. Also, if a constant term appears in an equation then one of the x's equals unity for all observations. With G = number of endogenous variables in equation system, K = number of predetermined variables in equation system, we define G° = number of endogenous variables in gi° equation,

G°° = G — G° = number of endogenous variables excluded from gib equation, Kb = number of predetermined variables in g'h equation, Kbb = K — K° = number of predetermined variables excluded from g`h equation. Using these definitions we can state the order condition for identification: Kbb > G° — 1.

(9)

Exclusion restrictions, which consist of specifying that certain variables do not appear in a given structural equation, are commonly but not solely used in model building. In this case the order condition for identification (9) can be interpreted to mean that the number of predetermined variables excluded from an equation must be greater than or equal to the number of endogenous variables included in the equation minus one. Although the order condition is necessary for identification it is not sufficient. A necessary and sufficient condition for identification is that rank(xabb) = G° — 1

(10)

where nabb is a submatrix of the reduced form coefficients with dimensions G° x K". Equation (10) is called the rank condition for identification. However, in practice the rank condition generally cannot be used to determine the identification status of a structural equation because it requires knowledge of the reduced-form coefficients which must be estimated assuming identifiability. Therefore reliance must be placed in general on the order condition to determine the identification status of a structural equation. With the qualification that the following are necessary but not sufficient conditions for identification, some general rules are Kbb > G° — 1: overidentification; Kbb = G° — 1: exact or just identification; and Kbb < G° — 1: underidentification.

If the rank condition is satisfied, then all three rules are unchanged. However, if the rank condition is not satisfied, then a structural equation is underidentified. In practice, if the order condition is satisfied then the rank condition will almost always be satisfied.

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Since structural coefficients in underidentified equations cannot be estimated, these equations are of no further interest for our purposes here. Rather the question is how can parameters of simultaneous equations which are identified (i.e., exact or overidentified) be consistently estimated? One useful method is the two-stage leastsquares estimation technique which has the virtue that it can be applied to both exact and overidentified equations whereas another commonly used technique, indirect least squares can only be applied to exactly identified equations (Goldberger, 1964, p. 327). Two-stage Least Squares

The two-stage least-squares method is designed to estimate structural equations one at a time and is therefore known as a single-equation method. Other such methods include (k)-class estimators and limited-information (least-variance ratio) estimators. If it is suspected that disturbances are correlated across equations, then system methods of estimation including three-stage least squares and full-information maximum likelihood are also available. For discussions of these single and system methods of estimation the reader is referred to Goldberger (1964), Johnston (1972), Kmenta (1971), and Theil (1971). Recall that one problem in a system of simultaneous equations is to derive consistent estimates of the structural model parameters. This arises because endogenous variables appear as predictors of another endogenous variable (dependent variable in a particular equation) and these endogenous variables are usually correlated with the disturbance terms in the structural model. The two-stage leastsquares method involves purging the values of the endogenous variables appearing as explanatory variables of their correlation with the error in each of the structural equations. Thus consistent estimates of parameters can be derived. To illustrate two-stage least squares, take a particular structural equation from the system of simultaneous equations and write it as y=Y,ß+X,y+ u where y: n x 1 vector of observations on dependent variable, Y, : n x (G° — 1) matrix of observations on other endogenous variables included in the equation, ß: (G° — 1) x 1 vector of structural coefficients of variables in Y1, Xl : n x Kb matrix of observations on predetermined variables included in equation, y: Kb x 1 vector of coefficients of variables in X1, and u: n x 1 vector of disturbances in the equation. The matrix Y, can be partitioned as Yi

= [Y2Y3' • • •, yc°]

where each of they's is an n x 1 vector.

(11)

David R. Meyer 109

The first stage consists of regressing each of the y's on all predetermined variables in the complete model : y2 =Xn2 + v2 y3 =Xir3 +v3 yGa = XirG

(12)

+ VGA .

These are the reduced-form equations. Each X is an n x K matrix of all predetermined variables, each it is a K x 1 vector of reduced-form coefficients, and each v is an n x 1 vector of reduced-form disturbances. The values of the y's computed from (12) then form a new matrix I1. In the second-stage values of ? replace values of Y1 in (11). These values are linear functions of the exogenous variables. Thus the second stage consists of estimating the structural equation y= p1 ß + X,y+u.

(13)

The estimates derived from (13) are consistent. If (11) is exactly or just identified it can be shown that the estimates derived from two-stage least squares are the same as those derived from indirect least squares (Goldberger, 1964, p. 334). As an illustration of two-stage least squares the following model will be examined, Q =f(Y, A)

(14a)

Y = g(Q, R, S)

(14b)

where the observations refer to residential areas and Q: housing quality, Y: household income, A: housing age, R: housing size (number of rooms), S: incidence of single-family dwellings.

Q and Y are endogenous variables and A, R, and S are predetermined variables. Equation (14a) states that housing quality in residential areas is a function of household income and age of housing. The higher the income of households the greater the demand for good quality housing. Since deterioration accompanies the aging process, it is suggested that the older the housing the lower the housing quality. However, characteristics of households (e.g., income) in residential areas would also appear to be a function of the characteristics of housing. This is so because for any household searching for a dwelling, the existing supply of housing distributed among residential areas must be taken as given. Therefore the level of income of households attracted to a particular residential area will depend upon the characteristics of housing in the residential area. Thus equation (14b) states that household income is a function of housing quality and size and the incidence of single-family dwellings. All three variables are expected to be positively related

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to income. Formulating model (14) as a statistical model we have Q; _ al + a2Y + a3A; + E1; Y = ß1 + ß2Q, + ß3R1 + ßaSS + Eat

(15a) (15b)

where i is a particular residential area, a's and ß's are parameters, and E's are random disturbances. The model will be estimated using a sample of forty-two census tracts, defined as residential areas, in Atlanta, Georgia, in 1960 (U.S. Bureau of the Census, 1962). The forty-two tracts are conceived as a sample of all possible delimitations of tracts in Atlanta in 1960. Definitions of the census variables are: proportion of sound housing with all plumbing facilities (Q), median family income (Y), proportion of housing built before 1940 (A), median number of rooms (R), and proportion of single-family dwellings (S). Since the variables chosen apply only to the Black population, the model refers specifically to Black residential areas. Before equations (15) can be estimated, their identification status needs to be checked. In (15a) the predetermined variables R; and Si have been excluded which implies K" = 2 and both endogenous variables are included so that G° = 2. Therefore, K6b >

Ga

-1

which means the order condition (equation 9) is satisfied and by the general rules the equation is overidentified. In equation (15b) the predetermined variable A. has been excluded which implies K6b = 1 and both endogenous variables are included so that G° = 2. Therefore, K bb

=G°

—

1

which means the order condition (equation 9) is satisfied and by the general rules the equation is exactly or just identified. In the first stage the two endogenous variables, housing quality (Q) and family income (Y), are each regressed in turn on all predetermined variables in the model: housing age (A), housing size (R), and single-family dwellings (S). This is the reduced form of the model. We have = —0.294 + 0.325R; — 0.520S; — 0.2I8A,, R2 = .66 (.285) (.060) (.120) (.161) =212.6+709.3R,+515.151 -82.5A1,R 2 —.65 (788.8) (167.4) (332.3) (445.1) where the standard errors are enclosed in parentheses. For the second stage the computed values Q, and p are substituted for the respective predictor variables Q. and Y,, in equations (15) giving

Q,= 0.118+0.000181,-0.312A; + w,; (.380)

(.00009)

(.217)

(16a)

David R. Meyer 111

Y; = 324.1 + 378.70; + 586.4R; + 712.0S; + w21. (16b) (1322.9) (2014.8) (773.2) (1068.7) These two-stage least-squares estimates (16) can be compared with the ordinary least-squares estimates of the equivalent equation. The latter estimates are Q. = 0.386 + 0.00011 Y; — 0.432A, + z11, R2 = .40 (17a) (.254) (.00006) (.175) Y,. = 385.3 + 476.4Q; + 549.4R, + 761.3S; + z2i, R2 = .66 (17b) (478.5) (431.9) (205.0) (392.1) Comparing equations (16a) and (I7a), it is evident that the two-stage leastsquares estimates (16a) differ somewhat from the ordinary least-squares estimates. The coefficient of Y• in (16a) is about 60 percent larger than the coefficient of Y in (17a) while the coefficient of A; in (16a) is about 30 percent smaller (in absolute value) than the coefficient of A; in (17a). Thus the two-stage least-squares equation reveals that in the prediction of housing quality (Q;), income (Y,-) is a substantially more important predictor while housing age (As) is a somewhat less important predictor relative to the same variables in the ordinary least-squares equation. On the other hand, the coefficients of all the two-stage estimates in (16b) are not substantially different from the corresponding ordinary estimates in (17b). Hence two-stage estimates do not seem to improve upon the ordinary estimates in the prediction of Y,• while there is relatively significant improvement in the prediction of Q;. Although the coefficients of (16b) are not substantially different from the coefficients in (17b), the standard errors of coefficients in (16b) are very large relative to the coefficients and from two to five times larger than the standard errors in (17b). Since a high degree of multicollinearity leads to large standard errors, this condition may be present in the model. An examination of the simple correlations reveals that the correlation between Q and R is high (.691) to begin with. Thus when Q is included in the second stage (16b) with both R and S, the correlation between Q and a linear combination of R and Swill be even greater than 0.691. Although two-stage estimates have the virtue of consistency over ordinary estimates in simultaneous equations, the problem of multicollinearity in twostage least squares is potentially as serious as in ordinary least squares. In fact it is likely that the multicollinearity problem might be more serious in the case of two-stage least squares because endogenous variables from the reduced form are included in the second stage with some of the same predetermined variables used to derive values of the endogenous variables in the reduced form. Conclusions As geographers develop more complete geographic models which take into account significant interdependencies, simultaneous equation methods such as two-stage least squares will be found increasingly useful. However, benefits will

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accrue to geography through the use of such methods only to the extent to which they are linked to sound theoretical models. The proper use of the techniques is relatively straightforward. However, the development of a sound simultaneous equation model to which the methods are applied is of a significantly higher order of difficulty. The temptation may be to create simultaneous equation models ad nauseam. Yet if such models are to be truly useful and not just statistical exercises, they must be based on good theory, a condition equally as applicable to the "simpler" single equation model. Acknowledgments

This research was supported in part by a Faculty Research Grant from the University of Massachusetts, Amherst. Special thanks are due to Professor James Kindahl, Department of Economics, University of Massachusetts, for his comments on the manuscript. He is, of course, not responsible for any errors that remain. References E. A. (1963). "Where Is a Research Frontier?" Annals of the Association of American Geographers, 53:429-40.

ACKERMAN,

BERRY, B.

J. L. (1964). "Approaches to Regional Analysis: A Synthesis." Annals of the Association of American Geographers, 54:2-11.

BLAtrr, J. M. (1962). "Object and Relationship." Professional Geographer, 14:1-7. L. A., and E. G. MooRE (1970). "The Intra-Urban Migration Process: A Perspective." Geografiska Annaler, 52:1-13.

BROWN,

BUTLER, E. W., S. F. CHAPIN, JR., G. C. HEMMENS, E. J. KAISER, M. A. STEGMAN, and S. F. Weiss (1969). Moving Behavior and Residential Choice: A National Survey. National Cooperative Highway Research Program Report 81, Washington, D.C.: Highway Research Board.

R. J. (1962). "Geomorphology and General Systems Theory." U.S. Geological Survey Professional Paper 500-B.

CHORLEY,

FISHER,

F. M. (1966). The Identification Problem in Econometrics. New York: McGraw-Hill.

GOLDBERGER, HARVEY,

A. S. (1964). Econometric Theory. New York: Wiley.

D. (1969). Explanation in Geography. London: Edward Arnold.

JOHNSTON, J. (1972). Econometric Methods (2nd ed.). New York: McGraw-Hill. KMENTA, J. (1971). Elements of Econometrics. New York: Macmillan. Rossi, P. H. (1955). Why Families Move. Glencoe, 111.: The Free Press. SIMMONS, J.

W. (1968). "Changing Residence in the City: A Review of Intraurban Mobility." Geographical Review, 58:622-51.

THEIL,

H. (1971). Principles of Econometrics. New York : Wiley.

U.S. BUREAU OF Ø CENSUS (1962). U.S. Censuses of Population and Housing: 1960. Census Tracts. Final Report PHC (1)-8, Washington, D.C.: U.S. Government Printing Office.

Optimum Urban Population Densities EMILIO CASETTI (Ohio State University)

This paper discusses the derivation of urban population densities that maximize an aggregate locational welfare function. The welfare function is defined as the unweighted sum of household utility functions, and may be justified as an interpretation of the "ethical norms" of the community. The paper shows that, within the context discussed, the optimum urban population densities generated by a class of utility functions depend only upon the households' preference ordering, and not upon the scaling of their preferences. Derivation of Optimum Densities An ideal setting is considered, characterized by an homogeneous circular urban area of radius h, with all jobs concentrated in a central point called the CBD, and inhabited by P identical one-person households. The households prefer to reside close to their place of work and dislike crowding, so that their locational preferences are made of a combination of centrality preferences and preferences for noncongested sites. Indicate by u(s, D) the combinations of distance from the CBD and of population density D yielding to a household a. utility level of u where åu/ås < 0, åu/ D < 0, and 02u/3D2 < 0. The locational welfare of an aggregate of households is defined as the sum of the locational utilities of the households in the aggregate. Specifically, call I the locational welfare function of the P households in the system. It is easy to show that in the context considered any population density surface that maximizes I has to be symmetric on the CBD, so that all points at the same distance s from the CBD are characterized by the same population density. This means that any optimum surface should be expressible as a function D(s) of s. In order to prove this point let us suppose that a population of p(s) households has to be distributed along a circle of radius s in such a manner that the aggregate locational welfare p(s)u is a maximum. 52u/aD2 < 0 implies that an increase in D produces a more than proportional increase in congestion disutility. Therefore minimum disutility, and consequently, maximum locational welfare must involve an even distribution of population 113

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along the circle. Therefore any population distribution that maximizes the locational welfare of the households in the system is symmetric on the CBD and is expressible as some function D(s) of s. Indicate by b(s) the function specifying the population densities at distances from the CBD from zero to h that make the sum of the locational utilities of the P households a maximum. The problem of deriving b may be formulated in the following manner. The relationship between population p(s) and population density D(s) at distance s from the CBD is p(s) = 2irsD(s). The total urban population is h

f

h

p(s) ds = f 27rsD(s) ds = P.

0

0

The locational welfare of the persons residing at distance s from the CBD is u(s, D)p(s). Therefore the aggregate locational welfare I corresponding to the population distribution identified by the function D(s), for 0 < s < h is h

I

= f

21rsu(s, D)D(s) ds.

Hence, the optimum density distance relationship b(s) is such that h

I =

f

2rrsub ds = maximum

and h

J = f 2rrsb ds = P.

Determining b constitutes a degenerate isoperimetric problem in the calculus of variations. Because of its degeneracy (absence of dD/ds under integral signs), the determination of sufficient conditions capable of insuring that extremum functions correspond to a maximum is somewhat atypical and here is arrived at from first principles. The necessary conditions for an extremum could be obtained by a straightforward application of the Euler equation. Here however it is convenient to derive them explicitly since the determination of the sufficient conditions for a maximum is an extension of the mathematical procedures involved. Define F(s, D) and G(s, D) as follows F = 271suD

(1)

G = 2ntsD.

(2)

Let no and ql be two arbitrary functions of s with continuous second derivatives and bounded, defined over 0 < s < h. Let co and cI be arbitrary constants. Let D(s) bean extremum function, and D(s) _ D(s) + £0110(s) + E,,i i(s) a variation of this function. Assume that b exists. In order to obtain it, maximize

Emilio Casetti 115

5

140, EI) = J F(s, D + Eono + E1n1)d 0

subject to the condition that h

G(s, D + Eono + E1n1)ds = P 0 and require that the E0, E1 yielding the constrained maximum be equal to zero. To this effect the Lagrangian function L is formed

J(EO, El) = f

d(J — P) where A is a Lagrangian multiplier. The first-order conditions for an extremum are L=I —

al, =å Et

—2 åJ—o

i=0, 1

(3)

(4)

aL åA = —(.1 — P) = O.

(5)

By differentiation under the integral sign it is obtained that

al dE

h OF

0, I = f aD n,~ t=

al

h ac

eE T = f aD nt

1

(6)

= 0' I

(7)

(4), (6), and (7) imply that

h aE = f (åD — ap nrd' = o

r = 0, 1.

(8)

By a slight extension of the fundamental lemma of the calculus of variations the integral to the right-hand side of (8) equals zero for any arbitrary function if and only if

OF 8D —

~

aG = 0.

(9)

'8

Since it is required that the E's yielding a constrained maximum be equal to zero,

and since

OF OF OD D=„_a — eD ac OGl 3D ,4a„a o = fib from (9) we have that the necessary condition for an extremum is

OF _ 2 OG_ 0 ob ab

(10)

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If equation (10) defines implicitly an extremum function D containing A as one of its parameters, and if for some values of Å D satisfies the integral constraint (5), under which conditions is this function the required maximum function? Sufficient conditions for the extremum to be a maximum may be derived as follows : Eo, El yield a maximum for I(EO, El) subject to the condition that J(E0, El) = P if the following holds 0 B=

aJ

al

ago

åE1

al

02L

82L

alo

aEp

aepaE1

aJ

aZL

a2L

aE1

.3E,'X,

8E1

> 0.

(11)

By expanding the bordered determinant B, and after a few manipulations we obtain h

h

h

B = 2A0 A1 f Hrlot l ds — Ai f

H7öds —

0

0

Ag f Hni ds > 0

(12)

0

where

i =0,1

A,= I and a2F

z

. D2.

H=

aD2 For any given duplet of arbitrary ?I's the A's above are constants, and may be moved under the integral signs in (12). That is to say h

h

h

B = f 2H(Aonl)(A111o)ds — f H(A0o)2 ds — f H(A0 41)2ds > O. 0

0

(13)

0

By collecting the three expressions in the left-hand side of (13) under one integral sign and factoring, the following is obtained h

B = — f H(Aog1 — A1 , o)2 ds > 0.

(14)

0

Inequality (14) will hold if H < 0 for 0 5 s S h, which proves that s

2

H=

(15)

— aD2 < is a sufficient condition for a maximum of J(E0, El) subject to the condition that J(E0, El) = P. Since the optimum I occurs for Eo — El = 0, in correspondence to which D = D, (15) may be rewritten as a2F

aD2

a2G

Emilio Casetti 117

Hence, if equation (10) defines implicitly an extremum function D(s, A) that for some 2 satisfies the integral constraint J = P, a sufficient condition for D to be the required maximum is that it satisfies inequality (16). Using the definitions of Fand G in equations (1) and (2) in conjunction with (10) and (16), we obtain that in the problem under consideration the necessary condition for an extremum is DA D -2+u=0

(17)

while a sufficient condition for an extremum function D(s, A) defined implicitly by (17) and consistent with the integral constraint to be the required function is that it satisfies inequality (18) z (18) + D åD2 0. Role of Transport Costs Transportation costs increase at a decreasing rate with respect to distance. Hence, satisfaction of low transportation costs (K) decrease at a decreasing rate. In Figure 2 it can be noticed that K'(t) < 0 and K"(t) > 0. Now let us look at the function q (amount of living space) with respect to t (distance from the center of the city). By the Chain Rule for derivatives (5) q'(t) = q'(K) • K'(t). Both terms on the right are negative, so q'(t) is positive. For the second derivative q"(t) = q"(K) • K'(t) + q'(K)K"(t).

(6)

An analysis of the terms on the right will show q"(t) to be negative. The function q(t) is graphed in Figure 3, where q increases at a decreasing rate with respect to t. By definition, density is the inverse of q, namely density rises as living space falls and density falls as living space rises. Density = cq-1, where c is a constant. This yields Density(t) < 0, Density"(t) > 0, and the negative exponential graph in Figure 4. This gives us the model of urban density as defined by Clark (1951) and presented elsewhere by Papageorgiou (1971).

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Figure 2

Figure 3

q

t

The above conclusions are not in accord with the empirical function developed by Sherratt (1960), which is that Density = Doe-°`Z, based on the normal distribution with mean at the city center. The shape of the curve indicated by this equation is indicated in Figure 5. It should be noted that Density"(t) < 0 in some instances. Nor, by extension, are they in accord with Dacey's (1968) work which is also based on the normal distribution. Similarly, these conclusions are not in

Kingsley E. Haynes and Milton I. Rube 125

Figure 4

Figure S

0

C a)

in

t

accord with the population density functions of Newling (1969) which is that Density = Daeb`-"2 where Density"(t) > 0 can occur on an interval of t when b > 0 (Figure 6). The implicit conflict between land-use theory as presented by Alonso and the empirical findings of Newling is brought out again via the bid rent curves of Alonso. These bid rent curves result from the indifference curves discussed above.

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Figure 6

c

0

>

0

t

If, as according to Newling, greater space in terms of lower density exists in the central portion of the city, the bid rent curves of Alonso would be approximately vertical, not extending beyond the area of low central density. According to Alonso, rent is bid for land away from the center only because of the utility of the greater space available. If greater space as well as lower transportation costs existed in the center, the city would be in a market disequilibrium (i.e., everyone would rush to the center of the city). One conclusion, of course, is that these models are not based upon sound premises. Empirical evidence consistently shows that in most cases population Figure 7 (B)

Kingsley E. Haynes and Milton I. Rube 127

densities are lower in some central ring around the city center than farther out from the center. We could suggest functions that would yield the graphs depicted in Figures 7A, B, but these would leave us with functions having discontinuous derivatives. On the other hand, we could measure distance from the ring of maximum density and this may, in fact, be an adequate adjustment. The main question is, however, should we discard the studies involving the models of Sherratt and Newling or is it not obvious that they have given us evidence to question the model of residential location as outlined above? An Environmental Approach Alonso (1964, p. 11) states "The model ... may be preferable for its economy of assumptions." The statement is made in reference to arguments for the exclusion of environmental factors. It is precisely this lack of realism that has led Kain (1971) to suggest this approach of extending simplified urban models is bankrupt and that urban simulation offers the only hope for the future. A simple alternative, however, is to increase the flexibility of our urban theory by incorporating an environmental approach. With site selection being only a trade-off of space versus transportation costs, there is no doubt that the center should have the highest density and, as we have seen, Clark's density conditions follow. What factors leave portions of the city center partially vacant at lower density than areas farther out? In terms of residential density it is clear that as a city increases in scale &larger and larger proportion of its central space is occupied by nonresidential activities. However, it is suggested here that a similar centrifical role is played by a set of environmental factors such as age of the neighborhood, health and sanitation problems, race, high crime rate, lack of proper school and recreation facilities, and a host of others. These are factors which are excluded in the present models of urban land use. They are not considered for simplicity's sake. The question then is, should we discard our empirical findings concerning population density and reject plausible models of population density functions because we are logically tied to a theory which contradicts our empiricism and models? We are thus left with strong motivation to modify the residential location model to include some of the environmental factors omitted. A plausible hypothesis is that much of the environmental effect is due to deterioration of the structures and neighborhoods, and the growing nature of metropolises. The in-migrant poor do not occupy the new structures which are built to relieve the increased demand for housing, but move to the center of the city while more of the center of the city is vacated by higher income groups. This filtering process makes the center part of the city as poor or poorer than before the expansion of the metropolis. Focusing our environmental approach on urban decay, we introduce a new variable, r, which denotes the positive utility of living in a quality neighborhood.

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Figure 8

Figure 9

i 1

t

Since we are only attempting to provide a theoretical framework within which the empirical distribution of urban population can be explained, we have left "quality" intentionally vague. Clearly it is a general environmental variable that would include racial, socioeconomic, physical, and location elements that attract urban residents. If we make r a function of distance from the center of the city, the

Kingsley E. Haynes and Milton I. Rube 129

Figure 10

a

t

utility function for the urban resident becomes u = u[q(t), K(t), r(t)].

(7)

So as to keep r(t) consistent with q(t), let large r represent a low degree of decay. This decay function is closely related to the concept of neighborhood. There is a relatively constant desirability for an area within a given radius of the city center. As one leaves the so-called "core," there is a large change to higher desirability. This leads to something very similar to a step function. When this ecological function is operationalized in terms of urban decay or age of residential capital stocks, rapid growth is likely to lead to a single sharp rise as is indicated in Figure 8. Slow growth should lead to a step function of the kind suggested in Figure 9. Let us concentrate on the fast growth situation. Approximating that function with a smooth curve (which should better represent reality), we obtain an S curve (Figure 10) with r'(t) > 0 r"(t) > 0, for t < a; r"(t) < 0, for t > a.

By the implicit function theorem we can assume q is a function of K and r. (8) q = q[K(t), r(t)]• For constant K assume a standard indifference curve between q and r (Figure 11).

Thus dq_ äq dK aq dr dt öK dt + år • dt d 2q ö2q dK aq d2K a2q dr aq d 2r dt 2 = aK 2 dt + OK dt 2 + ar2 dt + ar • di'

(9) (10)

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Figure 11

We are primarily interested in density as a function of t. But since density = cqthe respective derivatives with respect to t will simply be of opposite sign as those of q. Instead of analyzing the nature of dq/dt and d 2q/dt 2 directly, it will be illuminating to consider the functions q[K(t)] (disregarding r) and q[(t)] (disregarding K). In this model q depends on K and r, therefore q[K(t)] and q[r(t)] do not exist as independent functions. However, the first terms on the right of (9) and (10) are the derivatives of q[K(t)] (disregarding r) and the latter terms are the derivatives of q[r(t)]. In essence, then, the behavior of q(t) is a combination of the Figure 12 (A)

(B)

a tn

0

0

Kingsley E. Haynes and Milton I. Rube 131

Figure 13

hypothetical functions of q[K(t)] and q[r(t)]. The nature and derivatives of q[K(t)] were discussed in the section on the Alonso model. Let us now look at q[r(t)]. Disregarding K we have q[r(t)]' = q'(r) • r'(t) < 0.

(11)

Therefore density' (t) > 0 [q(r(t))]" =

q„(r) . r' (t) + q' (r) • r'' (t)

(12)

positive for t > a, positive or negative for t < a. Therefore, with Density"(t) < 0, t > a, and Density(t) positive or negative t < a, the curves are of the form suggested in Figures 12A and B.-Taking into account both K and r and referring back to (9) and (10) we find dq/dt is either positive or negative depending on which has greater effect, K or r. For t up to a point slightly past the transitional zone between core and noncore we can surmise that r might have enough effect to make dq/dt < 0 and hence d (Density)/dt > 0, which accounts for the Newling model with b > 0. No definitive statement can be made as regards the sign of the second derivative and this would allow for the Sherratt model or the Newling model with b 5 0.

A Classification of Density Functions A more precise analysis of r(t) may allow for a three-part classification of density functions. For a new city r(t) would be close to horizontal. All the transportation conditions of Wingo would be of primary importance, and we would expect a density function consistently decreasing at a decreasing rate (Figure 13). This is similar to the model suggested by Clark (1951).

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For a city with a steady and continuous growth we might expect a moderate effect of r and d2q/dt 2 changing from negative to positive in some interval, while dq/dt remains positive over the entire domain. Translating this density we would obtain the situation depicted in Figure 14. This is similar to the model suggested by Sherratt (1960). Figure 14

t

For a city which experienced extremely rapid growth at a particular time in its history, we might expect r(t) to be very pronounced in its effect resulting in a density of the type illustrated in Figure 15. This is similar to the model suggested by Newling (1969). The typology presented above is similar to Newling's classification based on age of the city. This classification, however, is based on environmental quality together with the effect of rapid versus slow growth. Since spatial growth (expansion) and the transportation age of a city are not independent, both classifications have a certain interrelatedness. Implications of the Environmental Viewpoint It should be noted that this model ignores some of the restrictive assumptions of spatial and economic equilibrium which are typical of existing theories of urban location. Kain (1971, p. 28) has pointed out that equilibrium models provide no information about the process of residential adjustment or the effects of the time path of such adjustments. The failure to consider explicitly these adjustment

Kingsley E. Haynes and Milton L Rube 133

mechanisms may be only a general weakness in economic theory but it is a critical fault in urban analysis since existing urban housing has a powerful effect both on the type of new investments and on their location. This usually means that new housing construction will occur on vacant land at the urban periphery unless there are compelling locational advantages to the contrary. The consequence is that the Figure 15

t

spatial distribution of housing and hence residential population will be dependent on the timing of development. Furthermore, the prices that determine demand at any point in time are market prices that reflect on the environment or ecology of residential characteristics that are external to the piece of property or housing unit under consideration. Unfortunately, until recently the lack of a theoretical framework, the shortage of empirical evidence, and the priorities of model builders has prevented attempts to develop an environmentally oriented model of land use. Parenthetically, it may now be possible to suggest a complex of price gradients for residential land with different environmental characteristics. Important empirical evidence for this suggestion has been outlined by Staszheim (1971). It should be clear that no explicit density distribution is inherent in the utility function derived in this paper. The purpose was to establish a generalized utility function and land-use theory compatible with the density findings of Newling and others, which, although empirically sound, were not in accord with classical landuse theory. In a similar manner to the presentation above an alternative but

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explicit density function could be derived from the utility function developed here following Casetti (1971). This would give equilibrium land values and yet allow for environmental considerations, though assumptions regarding identical income and preferences for households would be required.3 The latter is not an assumption all analysts are willing to make (Pines, 1972). There are other problems with the present model. One of the fundamental questions of urban land use is why do the poor occupy the more expensive land in the center of the city while the rich occupy the cheaper land in the suburb? The land-use model of Alonso explains this phenomenon in terms of the poor outbidding the others in the center of the city because low incomes make transportation costs more important. The poor consequently give up space for low transportation costs. At first glance, we see the environmental factors as causes for lower land cost in a certain ring close to the center. The question posed above still remains, however, since it is generally the rule that costs are indeed higher toward the center for any interval of t, although Yeates (1965) questions this in his study of Chicago land values. Another traditional answer is derived from the ecological view presented by Hawley (1950). That is, land in the city core is not valued as highly for residential property as it is for potential business property (caused by speculation in the hoped-for expansion of the business core). This is based on the assumption of continued city growth. This assumption, however, may no longer apply in light of the 1970 census which showed very low growth rates for cities with over two million people. A further environmental explanation may be the factor of racial exclusion. The core of most American cities is primarily a black ghetto for there is no free market for Negro housing (Grodzin, 1962, pp. 85-123). There are low-income suburbs but many poor blacks are excluded from them. The core of the city remains expensive and occupied by blacks because they have no other place to live. The supply of housing available to black people is limited; their numbers and hence their demand is large and increasing. The price structure of the ghetto is then autonomous of the structure outside the ghetto. In this sense, the ghetto can be viewed as the equivalent of a zoned area, and in terms of the operations of the urban land 3. With r defined as a function of distance t, we could allow for the derivation of equilibrium land values based on the following: u(z, q, t) = Maximum pf+P(s)q+kt= y where i is Casetti's rent function and z and q are the equilibrium values for z and q. The modification necessary would be in Casetti's example of u(z, q, t) = z°gbe-". A different functional relationship between u and z, q and t would be necessarily based on the specific function assigned to u(r(t)). Any attempt to establish the specific functional relationships for u(r(t)) and u(z, q, t) can be left for further investigation based on empirical evidence.

Kingsley E. Haynes and Milton I. Rube 135

market the two systems (ghetto and nonghetto) can be regarded as separate and independent and any marginal substitution between the uses may be assumed to be negligible. As a result, the rent bid curve for urban land in large cities cannot be represented by a single curve. Acknowledgments The authors would like to express their thanks to Dr. Arthur Getis for his considerable substantive and editorial suggestions, many of which have been incorporated. Similarly, comments by Drs. Casetti and Papageorgiou have been very helpful. This paper is a revised version of Rutgers University, Department of Geography, Discussion Paper No. 3, "An Ecological Consideration of Urban Density and Land Use" (June, 1971). Any errors are the sole responsibility of the authors. References ALONSO, W.

(1964). Location and Land Use. Cambridge: Harvard University Press.

CASETTI, E. (1971). "Equilibrium Land Values and Population Densities in an Urban Setting." Economic Geography, 47:16-20. CLARK, C. (1951).

"Urban Population Densities." Journal of the Royal Statistical Society, Series A, 114:490-96. DACEY, M. F. (1968). "A Model for the Areal Distribution of Population in a City with Multiple Population Centers." Tijdschrift voor Economische en Sociale Geografie, 59:232-36. GRODZIN,

M., ed. (1962). American Race Relations Today. Glencoe, Ill.: The Free Press.

HAWLEY, A.

H. (1950). Human Ecology. New York: Ronald Press.

J. F. (1971). "The N.B.E.R. Urban Simulation Model and Urban Economics in the 1970's." Reports on Research Underway in Urban and Regional Studies, Fifty-First Annual Report of the National Bureau of Economic Research, pp. 21-31. KAIN,

B. (1969). "The Spatial Variation of Urban Population Densities." Geographic Review, 59:242-52.

NEWLING,

PAPAGEORGIOU, G. J. (1971). "A Theoretical Evaluation of the Existing Population Density Gradient Functions." Economic Geography, 47:21-26. PINES, C. (1972). "The Equilibrium Utility Level and City Size: A Comment." Economic Geography, 48:439-43.

SHERRArr, G. (1960). "A Model for General Urban Growth." In C. W. Churchman and M. Veerhulst (eds.), Management Sciences and Techniques. New York: Pergamon Press, pp. 147-59. STASZHEIM, M. R. (1971). "The Demand for Residential Housing Services, Housing Markets, and Metropolitan Development." In N.B.E.R. Urban Simulation Model, II, ch. 4. WINGO,

L., JR. (1961). Transportation and Urban Land. Washington, D.C.: Resources for the

Future. YEATES, M.

H. (1965). "Some Factors Affecting the Spatial Distribution of Chicago Land Values." Economic Geography, 41:57-70.

Deducing Psychic Transport Costs from the Transportation Problem MARK D. MENCHIK (University of Wisconsin, Madison)

This paper is an attempt to define and calculate "psychic transport costs" by deduction from observed spatial behavior. The approach taken here is to assume that observed behavior is rational, given subjective psychic transport costs, and is guided by the linear programming Transportation Problem model of behavior. Under these assumptions, the task is to "turn the Transportation Problem around" by defining psychic transport costs as those that make observed behavior "optimal" and thus are implicit in "subjectively optimal" behavior.' A lengthy introduction will supply some of the background for this effort. Many of our theories and models of spatial behavior assume the rational minimization of transport costs, rational total cost minimization, or profit or utility maximization, where transport costs are one of the important considerations of the location problem. Examples include the works of Alonso (1965), Lösch (1954), von Thünen (Hall, 1965), Weber (1929), and the literature on interregional linear programming (Stevens (1961)). Viewing such theoretical constructs as positive theories (that is, as theories of actual human behavior rather than as normative guides to efficient action), we may at times find less than perfect correspondence between these theories and the real world. One response to this state of empirical affairs is to reject rationality and optimization as realistic theoretical bases for explaining human behavior. Another approach, though, is to assert that actual spatial behavior is guided by influences other than simple and objective distance or transport costs. Thus, for example, shipping decisions may be based on other than the simple transport costs 1. This idea for the definition and measurement of psychic transport costs was suggested to me by Benjamin H. Stevens, in a conversation that took place some time ago. 136

Mark D. Menchik 137

for each route and transport mode. They may be based on the time taken, risks of spoilage or breakage, the reliability of shipping for a given route/mode combination, the shipper's familiarity with the route and mode, limited information about alternatives, and so on. With all these considerations, many quite difficult to define, much less measure, actual behavior may be optimal with regard to these considerations and information. The notion being developed here is Herbert Simon's concept of subjective or intended rationality, "behavior that is rational, given the perceptual and evaluation premises of the subject" as opposed to objective rationality, "behavior that is rational as viewed by the experimenter" (Simon, 1957, p. 278; see also p. 200 and chs. 14-15). Given this approach to explaining the limited correspondence between objective distance measures and human spatial behavior, behavioral geographers have defined and, in some cases, measured various notions of psychic distance in the context of discussing space preferences and mental maps. Examples include the work of Gould (1969), Rushton (1969), and Stea (1969). Measurements of psychic distance have often been made directly by means of experimental or questionnaire procedures, rather than indirectly through the actual spatial behavior to be explained. One criticism of such approaches has been that the accuracy and validity of the psychic distance measurements depend heavily on the accuracy, design, and phrasing of the measurement techniques used. Thus, omission of some of the causes of the divergence between objective and subjective transport costs (as listed in the preceding paragraph) may cause inaccurate measurement of the latter. In sum, the direct measurement of psychic distances may be criticized for unrealism and inaccuracy. This paper takes another approach to the measurement of psychic distances or transport costs, because here such quantities are measured, indeed defined, from actual spatial behavior. Given observed behavior plus a theory of that behavior based on transport costs and other behavioral influences and constraints, psychic transport costs may be deduced from observed behavior, in the context of that theory. In other words, psychic transport costs, when substituted into that theory of spatial behavior, are those that make observed behavior optimal. The approach taken here may be called realistic in the sense that psychic transport costs, as defined and measured, have a strong logical basis in the observable behavior of interest. The approach further allows explicit use of a well-defined theory or model of spatial behavior. Moreover, this definition and measurement of psychic transport costs does not require identification and measurement of the causes of the divergence of subjective from objective transport costs. Anything that influences psychic transport costs will, by definition, influence observed behavior, and such an influence will appear in the calculated psychic transport costs. Notice that here psychic transport cost is taken as a unidimensional numerical measure of the friction of space that may have none of the properties of distance, even of psychic distance.

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The spatial behavior considered in this paper will be flows between a finite number of spatially discrete origins and destinations. Such flows may consist of goods to be shipped, persons traveling, or information being communicated. The behavioral model used is that of the linear programming Transportation Problem (Dorfman, Samuelson, and Solow, 1958; Hadley, 1962). The Transportation Problem was chosen because of its simplicity, spatial generality, and considerable theoretical importance. Behaviorally, the Transportation Problem is simple because shipping patterns are determined only by shipping cost minimization, as constrained by the maximum capacity of each origin, and minimum requirements of each destination. Mathematically, the Transportation Problem represents a simple and important subclass of linear programming problems that can be solved by simpler and more efficient algorithms than those used for general linear programs. The considerable spatial generality of the Transportation Problem comes from the fact that the unit transport cost from a given origin to a given destination can be any quantity. This allows highly complex and perhaps even realistic transport cost structures, where the direction along a route influences shipping costs, and a roundabout route may be the cheapest one. By their very definition, it would appear that psychic transport costs might have no simple correspondence with Euclidean (map) distance or objective transport costs so the spatial generality of the Transportation Problem seems very useful as a basis for calculating psychic transport costs. Indeed, the use of a model assuming a uniform transport plane might initially severely restrict our psychic transport costs. Additionally, interest in the Transportation Problem on the part of location theorists stems from the interpretation of variables in the dual program (discussed later) as representing location rents or relative locational advantages of the various origins and destinations. Finally, quite a different interpretation can be given to this paper's efforts. Assuming that observed flows are guided by the Transportation Problem, and with only this information, what can this observed behavior tell us about the underlying transport costs? In other words, given this approach, what spatial information is contained in the observed flows? Here, information is used in a substantive spatial sense (rather than in terms of formal information theory) to indicate the locational understanding that can be gained from observed behavior. The Transportation Problem: Notation and Duality Properties To set up the general Transportation Problem, assume there are m origins (e.g., factory sites) of the flows considered with the i subscript indicating origin and where i = 1, 2, ... , m; m = 1, 2 .... There are n destinations (e.g., cities) for these flows, denoted by the j subscript, where j = 1, 2, ... , n; n = 1, 2 .... Let c;; be the transport cost per unit flow between i and j and x;1 the (nonnegative) flow level

Mark D. Menchik 139

between i and j, measured in items shipped, persons, trips, or whatever units correspond to the behavior considered. Origin i has capacity ki; the total number of units shipped out of i may not exceed ki. Destination j has requirement ri; the total number of units shipped into j may not be less than rj. The cost of shipping xi, units between i and j is c jxv. This quantity, summed over i and j, forms the objective function. With this information we may form the direct (or primal) transportation problem as

E E ctj xu

Minimize z =

1=11=1

Subject to: n

E xu 5 kt j=1 t

(i

= 1, ... , m) , n)

xuZrj(1 -1,

xu > 0 (i = 1, ...,m;j= 1,...,n).

(I)

Simply, the problem is to find that nonnegative level of the xu's that minimizes total transport costs (the objective function) while not exceeding origin capacity constraints but satisfying the minimum destination requirements. We additionally assume, as is common, that m

n

E kt= j=1 E rj t =1

(2)

or that total capacity in the system equals total requirements. With this added condition any set of xe's satisfying the constraints and nonnegativity restrictions of (1) must satisfy the capacity and requirement constraints exactly. In other words, there is no unused capacity in the system, nor are there shipments more than satisfying minimum requirements. Thus, the constraints of (1) may be phrased as equalities, rather than inequalities. Furthermore, any one constraint of (1) may be omitted, without changing the problem's solution. Suppose the nth requirement constraint is dropped. Then any set of nonnegative xe's satisfying the m capacity constraints and the first n — 1 requirement constraints will, by (2), also satisfy the omitted requirement. Assuming (2), the program (1) may be rewritten as the Primal Transportation Problem: m n

Minimize z =

E E cuxu i=1 J=1

Subject to: n

E xu = kt j=1 E —xu J.

(i

= 1, ... , m)

= — ri (j=1,...,n-1)

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0 (where

(i= 1,.. ,m;j= 1,...,n)

E k; = E rJ I.

t=1

(3)

J=1

The first n — 1 requirement constraints have been multiplied by minus one for later algebraic convenience.' Associated with the Primal Transportation Problem (3) is the Dual Transportation Problem: n-1

Maximize Z =

E k;u, — J=1 E riv, !=1

Subject to: u; —vJ 0.

(11)

The left-hand side may be defined as the cost of shipping one unit indirectly by way of i and j where these correspond to included routes in that shipment pattern, that is, x”, x,j, and x;, are all strictly positive. We may further interpret the left-hand side as the sum of costs of shipping a unit from /to], then shipping it backwards from destination] to origin i by reducing the direct flow from i to j thus reducing total transport cost by c,j, then shipping that unit from i to J. If no pair i and j exist in that solution with the positive flows required by (11), we can always find a more indirect route possessing the requisite positive flows. The optimality criterion (11) is easily deduced from the form of the dual constraints and the primal-dual relationship (5). To prove that Solution III always forms a set of psychic transport costs, we will show that expression (11) holds as an equality for all I, J regardless of whether xi, is zero or strictly positive. Substituting the formula for the costs (10) find (I +

j) —(i+ j) +(i+J)=(I+

J)

establishing that any feasible pattern of x;j's is optimal for the c,j's of Solution 1II.

Q.E.D.

Mark D. Menchik 143

Figure 1 An Example of the Nonuniqueness of Psychic Transport Costs; (a) Optimal Flow Matrix, Destination 1

2

Capacity:

a

1

2

3

b

4

0

4

5

2

Origin :

Requirements:

(b) Alternative Psychic Transport Cost Matrices Consistent with the Diagrammed Flows 1

2

a

2

2

b

5

5

CI =

4

1

4

1

CII =

2

3

3

4

3

5

CIII=

C1 =

5 C2 =

1

6

1

3

5

3

4

1

6

4

C4 =

C3 =

1

6

7

4

3

1

C5 =

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Let us give a numerical example of alternative sets of psychic transport costs for a given problem. Figure la indicates the capacity and requirement constraints and the observed flows, assumed optimal. Observe that xbZ = 0. This corresponds to a basic solution. For linear programs a basic solution is one in which the number of variables positive is at most equal to the number of constraints. These solutions are important for it is well known that if an optimum exists, at least one optimum corresponds to a basic solution. Figure lb presents solutions of forms I, II, and III for that problem (CI, C11, C1 ) as well as five other sets of psychic transport costs, C1 through C5. That these latter numbers are, in fact, psychic transport costs may be confirmed by reference to the Direct/Indirect Shipping Cost Optimality criterion of (11). Clearly the eight solutions of Figure 1 (and there are many more) indicate the high degree of nonuniqueness present in the psychic transport costs for this simple problem. One would hope that some useful properties are preserved among all the solution sets, but this seems not to be the case. Relative orderings of origindestination pairs by psychic closeness are not preserved. The pair b2 is the only one with zero flow, perhaps indicating that these points are farther apart than the others. And in C1, cb is the largest, but in C5 it is the smallest indicating that for that solution b and 2 are the closest in a psychic transport cost sense. Similarly, orderings of closeness of all origins to a given destination, or all destinations from a given origin, vary across the sets of psychic transport costs. A frightening consequence of Proposition 1 and Figure 1 is that a set of psychic transport costs need have little to do with the original data, especially the magnitude of flows between different ij pairs. Proposition 1 indicates this by proving the existence of solutions to all Transportation Problems that are the same whatever the x's. Here is one set of solutions, at least, having nothing to do with the observed flows. Similarly, a wide range of psychic distances is suggested by one set of observed flows in Figure 1. Clearly, such a property is highly undesirable. The following section will discuss the reason for this state of affairs. At this point, it is possible to state an intuitive reason for the demonstrated undesirable properties of our psychic transport costs. These costs correspond in a mathematical sense to the objective function of the Primal Transportation Problem, and not the constraints. Less formally, our reasoning about turning the Transportation Problem around to get psychic transport costs is also based on the objective function and not the constraints. That is why Solutions I and II may contradict unreflected intuition: these solutions allow the constraint set to adjust to observed behavior, rather than using the objective function for this purpose since the objective function is invariant across all feasible flow sets. Problems of this sort may be said to be constraint determined rather than strongly influenced by the objective function. Indeed the very generality of the Transportation Problem comes from the possibility of constraint determination. As an example, given two origins a and b, the latter with a much greater capacity than the former, and a

Mark D. Menchik 145

high requirement destination 1 much closer to a than to b, an optimal solution may nevertheless have b shipping to 1 since a cannot satisfy all of l's demand. The same reasoning suggests why solutions always exist, or why no observed behavior is internally inconsistent, in the psychic transport cost sense. With a flat enough objective function (i.e., one like those of Solutions I and II) the problem becomes constraint determined. The data include the ki's and rf's and so these constraints will clearly be consistent with the observed x's. Thus, the possibility of constraint determination in the Transportation Problem, while adding to its generality, makes the Transportation Problem a difficult model for the computation of psychic transport costs based only on the objective function and may also contradict our objective-function—based intuition. The Algebraic Basis for the Computation of Psychic Transport Costs This section examines the information at our disposal in computing psychic transport costs, in order to explain the nonuniqueness of the c is and why these quantities seem to have so little connection with the observed flows. The mathematical structure of the Transportation Problem is simply that of the primal and dual programs and the primal-dual relationships above. However the only relationships (equations or inequalities) useful in calculating the c*'s are those containing these quantities, and somehow influenced by the problem's data: x's, ki's, and ri's. The possibly informative relationships are thus limited to one setting the primal and dual objective functions equal at optimum, and the mn dual constraints. The former is established by primal-dual relationship (6) and, encouragingly, contains all the x's and cut's. The latter set does not contain any x:'s but we know (by primal-dual relationship (5)) that the sign (but not the magnitude of x*) tells us about the dual constraint corresponding to that in pair. If x* > 0 then that constraint holds as an equality. However, if it is zero, we know nothing further about the constraint since the converse of (5) is false. All we can say in that case is that the constraint holds as a nonstrict inequality, that is, the right-hand side may be equal to or (strictly) greater than the left-hand side. Thus, the only information we have about the magnitude (not just the sign) of the flows is the equation of primal and dual objective functions n-1

— r= 1

E rlvv = i

J= 1

m n

cux7.

(12)

=1 J= 1

We shall reproduce a standard result (Dantzig, 1963, p. 301) indicating that (12) does not help us, and so the computation of psychic transport costs is based only on the signs of the x's 4 Notice that the dual constraints that hold as equalities, and (12), are all linear equations in the unknowns cu, ui, and u given the data x, k;, and ri. We will show that (12) is linearly dependent on the constraint set, or that the equation may be expressed as a weighted sum (a linear combination) of 4. 1 am indebted to Lalita Sen for supplying the reference to this result.

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those constraints holding as equalities. Let us multiply each of the dual constraints by the corresponding x j, a known constant. Where x j = 0, those nonstrict inequality constraints will drop out. Where xi > 0, we obtain — vjxj = c;jxv (where vn = 0). Summing across all i and j find n

m n

m

m

n

E u; j=1 L X— E Uj E X~ _ E E c;jx. 1.1 j=1 1=1 1=1 j=1 Inserting the primal capacity and requirement constraints, find n-1

m

n

E u ki — j=1 E vjrj = 1=1j=1 E CIA i=1 which is just (12). Since (12) is linearly dependent on the other equations at our disposal we know from the theory of linear equation systems (Hadley, 1961) that it provides redundant information, and thus may be dropped from the equation system without changing that system's solution. We can now attack the question of the nonuniqueness of the cg's. At optimum, among the system of linear equalities and inequalities that is the dual constraint set, there are mn + m + n — 1 unknowns: mn co's, m u;'s and n — 1 v1's. For the whole set a maximum of all mn constraints can hold as linear equalities, when all x j's are positive. In this situation (the best possible for our purposes) there are m + n — 1 more unknowns than equations, suggesting a high degree of nonuniqueness in the solutions. All the equations have zero constant terms, so are homogeneous, and the constraint equations are linearly independent of each other, so the equation system is said to be of full rank. Under these circumstances it is easily shown (Hadley, 1961, pp. 173-77) that the solution subspace is an (m + n — 1)-dimensional hyperplane passing through the origin of an (mn + m + n — 1)-dimensional hyperspace with axes corresponding to each of the unknowns above. When not all possible flows are positive, the constraint set has inequalities replacing some of the equations. Reasoning only from the equations, we find the dimensionality of the solution subspace to be even greater, and so is the degree of nonuniqueness in the calculation of the c's. Clearly, one may deal with the indeterminacy in the problem by adding additional equations, or other information limiting the sets of possible psychic transport costs. A simple addition comes immediately to mind: the psychic transport costs are in a large sense arbitrary, so they can be conveniently scaled. One way to do this is to choose the larger of m and n. Suppose this is m, then we require Ecj=1 (i =1,...,m) j=1

simply that the psychic transport costs to all the destinations for one origin sum to unity, and this holds for all origins. This adds m linear nonhomogeneous equa-

Mark D. MenØ

147

lions to the problem, reducing the dimensionality of the solution subspace, but the remaining solution subspace is still highly nonunique. Geometric Properties of Psychic Transport Cost Problems This section takes an alternative, geometric, approach to understanding the properties of psychic transport costs for the Transportation Problem. Using psychic transport costs as the measure of distance between pairs of points, it is useful to try to gain additional information by mapping or diagramming these points, the problem's origins and destinations. We can thus find if identifiable geometric patterns correspond to either psychic transport costs or Transportation Problems of specified characteristics. The geometric approach may be useful because at times our main motivation in inferring psychic transport costs from actual behavior is to find the perceived spatial arrangements of the points in an intuitive geometric sense, rather than finding relative distances among pairs of points. Furthermore, a geometric approach lends concreteness to a highly abstract problem that, thus far, seems devoid of simple substantive conclusions. However, it is clear that, given a set of points and psychic transport costs, there may be no simple, unique, or even internally consistent way of mapping these points. This is because the extreme spatial generality of the Transportation Problem, mentioned in the introduction, goes beyond the limits of ordinary or Euclidean geometry. Defining the distance between an origin and destination as the psychic transport cost from the former to the latter may be misleading if we have no guarantee that the distance in the reverse direction is defined, much less the same as in the forward direction. To deal with the extreme geometric generality of the Transportation Problem, without losing the precision of definition we need, we will proceed slowly and formally, adding additional requirements to convert the problem into one we can manipulate using standard mathematical procedures. These procedures establish a precise correspondence between our origins, destinations, and psychic transport costs, and the location of points in a familiar Euclidean space. We will be using the apparatus of distance geometry or metric topology (see Berge, 1963, or, more specifically Blumenthal, 1970) which deals with algebraic structures on which a notion of distance is defined, the geometric properties of such structures, and distance-preserving transformations. To start, let us repeat an important, and standard Definition ofa semimetric space (Blumenthal, 1970, p. 7). A semimetric space S is (i) a nonempty collection of points p in S, where (ii) for all ordered pairs of points p and q in S there is defined a semimetric function d(p, q) with the following properties: (iii) (positive definiteness) for all p and q in S d(p, q) z 0; (iv) (identity) d(p, q) = 0 if and only if p = q, that is they form the same point in that semimetric space; and (v) (symmetry) d(p, q) = d(q, p) for all pairs of points in S. (13)

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A semimetric space is a generalization of the more common and restrictive metric space, itself a generalization of the familiar Euclidean space. In a metric space it is additionally required that d(p, q) satisfy the Triangle Inequality. This property states that for p, q, and r any three points in the metric space we must have d(p, q) < d(q, r) + d(p, r). For a Euclidean space, of course, the metric has the familiar form. A semimetric space is an exceedingly general topological structure, whose very few requirements on the semimetric (or distance function) lead to bizarre and counter-intuitive distance properties. For examples of general semimetric and metric spaces and their properties, see Blumenthal (1970). However even a semimetric space imposes additional requirements over the still more general geometry implicit in the Transportation Problem. We need a semimetric space, though, in order to apply the machinery of distance geometry to the task at hand. To obtain such a space the origins and destinations of a Transportation Problem must be converted into the points of the space, and what is more difficult, the psychic transport costs must be converted into a semimetric on all pairs of points in the space. That task may be assisted by an alternative characterization of a semimetric in a space containing a discrete collection of points. The semimetric may be completely defined by the elements of a square matrix whose rows and columns are identified with the points of the discrete space. If the i`h and jth rows and columns of the matrix correspond to points p and q, then d(p, q) is simply the `h element of the matrix. Then property (v) above requires this matrix to be symmetric; property (iv) requires it to have zeroes on the main diagonal and perhaps elsewhere, when points are indistinct; and property (iii) is satisfied by having all the remaining elements nonnegative. To obtain a semimetric we thus must first require that all c is be nonnegative. This loses no generality for we can always add the same constant to all psychic transport costs, and it is well known (Dantzig, 1963, p. 305) that this addition of a constant does not change the optimal solution to any Transportation Problem. Now label the points of the Transportation Problem so that the origins correspond to the first m rows and columns of the matrix in Figure 2, and the destinations correspond to the last n rows and columns of that matrix. That matrix is partitioned into four rectangular submatrices or blocks. The upper off-diagonal block, m by n in dimension, simply contains the c*'s of the problem, adjusted to nonnegativity. Let the lower off-diagonal block be the transpose of that submatrix; in other words we are requiring that the distance from j to i be the same as from i to j or that cl = cJ. The elements of the main diagonal corresponding to the distance from a point to itself are naturally set equal to zero. The only remaining elements are the off-diagonal ones in the m by m and n by n blocks corresponding to the transport costs within the origin and destination sets. These may be set at any convenient and consistent set of nonnegative numbers—actual values are easily found, as is shown below.

Mark D. Menchik 149

Figure 2 Matrix of Values of Semimetric for Pairs of Points m origins

n destinations

0 0 m origins •

lJ •

0 0 n destinations

• • •

0 We have put additional restrictions on the problem, but only those necessary to obtain a semimetric space. Whether these additional restrictions are worthwhile may be judged from the insights provided by this geometric analysis. We can now apply a very powerful tool of distance geometry to relate this general semimetric space, its points and distance properties, to a more familiar and much more restrictive Euclidean space. We seek to establish a congruent imbedding between the semimetric space and a Euclidean space of specified and small dimension. The congruent imbedding establishes an isometry between the two spaces, that is, an identity-preserving (i.e., one-to-one) transformation between points of the two spaces that also preserves the distances of psychic transport costs between point pairs in a given space. Thus, all of the distance geometric properties of the parent semimetric space may be precisely represented in the Euclidean space by a simple and accurate sketch. This imbedding will furthermore be irreducible in that no lower dimensional Euclidean space may be used for this purpose, so the irreducible imbedding represents the simplest Euclidean representation of the Transportation Problem. Having established the imbedding, we can thus investigate psychic transport costs by applying our intuition to the strong and simple properties of Euclidean

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spaces, unafraid of imprecise metaphoric representations. Another purpose is served. Imbeddable semi-metric spaces are less general than might be thought, for they have the restrictive distance properties of a Euclidean space. By testing for congruent imbedding we thus see if the parent semi-metric space has stronger distance geometric properties (e.g., a Euclidean structure of distances) than might have initially been assumed. Roughly, we can avoid "fooling ourselves" about the generality of our spaces. We can now present the imbedding theorem. Theorem. (Irreducible Congruent Euclidean Imbedding; Blumenthal (1970, p. 94). A semimetric space S is irreducibly congruently imbeddable in a subspace of an r-dimensional Euclidean space (E,) if and only if (i)S contains r + 1 points congruent with an (r + 1)-tuple of E„ that is not a subset of a lower dimensional Euclidean space; and (ii)each (r + 3)-tuple containing the r + 1 points above is congruently contained in a subset of Er, but not one of smaller dimension than r. Intuitively, congruent imbedding of any tuple (double, triple, quadruple, ...) of points means that the distance relations among pairs of those points must satisfy the Euclidean metric appropriate to that dimensional space. This requirement may be satisfied by establishing certain purely algebraic properties of a matrix of inter-point distances, like that in Figure 2 (Blumenthal, 1960, ch. 3). Here, however a more concrete approach is used. We will establish a diagram of the points in Euclidean space, hence a transformation between the two spaces and a coordinate system in the Euclidean space, and show that inter-point distances for this representation satisfy the Euclidean metric. As an aside, in the theorem above notice that the recurrent appearance of the phrase "... but not in a subset of lower dimensionality than Er ..." is necessary to make the imbedding irreducible. Now onto the meat of this section. Two general types of psychic transport problems are considered here. The first is a problem of arbitrary size, where for at least some of the origins and some of the destinations, all possible flows occur at a positive level. The second type of problem has two origins and destinations where, at optimum, only three of the four possible flows occur at positive levels. For the first type of problem, we shall show that the semimetric space consisting of those origins and destinations among which all flows are positive, is irreducibly congruently imbeddable in a Euclidean space of dimension one. That is, these points and their distances are representable in a finite length segment of the real line, which itself runs from plus to minus infinity. Proposition 2 (First Congruent Imbedding). (1) Suppose for a Transportation Problem there exists m* origins (2 < m* < m), forming the set of subscripts 1*, and n* destinations (2 < n* < n) forming the set

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J*, where for all ij pairs where i is in I* and j in J*, x > 0. This psychic transport cost problem is converted into a semimetric space by the procedures above. (2) Further suppose that at least one of the following conditions is satisfied for the set of psychic transport costs among the points in sets I* and J* : (i) (Distinctness of pair of origins, H, I). There exist three points, H, I and J (H and I in I*, J in J*) so that cH, c;, or (ii) (Distinctness of pair of destinations, J, K). There exist points H, J, K (H in I*, J and K in J*) so that cÅ, # eh. or (iii) (Distinctness of origin-destination pair, H, J). There exist points H and J (H in I*,J in J*) so that cH, 0 0. Figure 3 First Imbedding of Points into the Real Line

— CO
+CO G

origins

Then The semimetric space containing the origins and destination sets I* and J* is congruently imbeddable in a finite section of the real line and no lower dimensional Euclidean space. Furthermore, for each set of psychic transport costs there exists a representation (like that of Figure 3) where all the origins are on one side of all the destinations, along the real line. Proof The proof proceeds by establishing the representation and coordinate system of Figure 3, which satisfies the distance properties of the imbedding theorem. The requirement of part (2) of the statement of the theorem merely insures that at least two distinct points will exist in our representation. Two non-distinct points are congruent with the single point of Eo rather than forming a line segment in E„ so if distinctness is not established neither is irreducible imbedding in El. For simplicity assume (at least) requirement (i) to have been satisfied; also initially assume that the distination J used to phrase that requirement is the same point as below. Thus there exist two distinct origins H and I. (All the points below are in sets I* or J*.) Now choose a destination J which is the closest destination to origin I, or that ct > c , for all j in J* ; of course, several points may qualify. For that J the optimal level of the corresponding dual variable vj (where the star indicates the optimal value) is set at zero, and place J at the origin in Figure 3. This is done without loss of generality. Recall that any dual variable may be set at zero; there it was v„, here for convenience of representation, it is v,. Since xH, and 4 are positive we know that cN, — uy and c! , = of . Since the two costs must be unequal by the distinctness assumed above we can assume cH, > cf,.

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0, while H is to the right of I Thus, we can give origin I the coordinate location u! = on the real line (Figure 3) at 4 = cH, > u; z 0. This establishes the irreducible congruent imbedding between two points in the semimetric space and two distinct points on E,. Distances are preserved, as is seen from the diagram's construction. This construction suggests how the other points may be placed, and the rest of the imbedding theorem satisfied. In case (only) requirement (ii) or (iii) is satisfied, a similar representation may be constructed. To place the other points on the real line, consider two general destinations K and L, placed at uK and 4. The coordinate locations are calculated by using 4K = 4 — uK (and a similar equation for 4). By the selection of J, v; < 0 for all j in J*, so the destinations occupy points, not necessarily distinct, on the non-positive section of the real line. Similarly, locate general origins F and G at uF and 4 using 0 < cF, = uF* — 4 where 4 = 0. The origins will occupy positions on the nonnegative section of the real line. To show that this construction satisfies the second requirement of Blumenthal's congruent imbedding theorem, namely, the congruence of quadruples of points, let H and I be the specially selected distinct points, while K and L, H and G are the general points representing origins or destinations. We must show that for any quadruple of points containing H and I, the psychic transport costs among pairs of such points satisfy the E, metric. For points p and q with coordinate locations P and Q this metric is d(p, q) = IP — The requirement is easily seen to be satisfied by the consistent construction of Figure 3, placing the points on the real line. For completeness, one may see algebraically that the E, metric is satisfied by using Figure 3's coordinate representation to calculate interpoint distances for origin-destination, origin-origin, and destination-destination pairs, and comparing these E, distances to the semimetric of the psychic transport cost problem. Recall that we have let interorigin and interdestination distances be any convenient and consistent set of nonnegative quantities. Thus, for the two origins Fand G we find d(F, G) = 14 — utl > 0. If the largest u* (i in I*) and the smallest of (j in J*) are both finite, then these points lie in a finite-length line segment of the real line, and the imbedding proof is complete for that set of

c j's. However, there are many sets of c is and along with them many sets of u; 's and of 's. The representations will differ across these solution sets, given the same point sets I* and J* and the same x*'s. The whole array of points may expand or contract. Within the origin and destination sets (1* and J*) the relative positions of individual points will change. Point J may no longer be at the extreme right of J* as it is no longer closest in a psychic transport cost sense. If we change the representation slightly, the whole array may shift along the real line, or be reflected across the origin, since the origins can easily be represented on the left side of the destinations. Through all these changes, however, the appropriate dual constraints hold as equalities, since their 4's remain positive. Thus, similar construction procedures lead to some invariant geometric properties. These are the isometric representations of those origins and destinations on a finite-length section of E„ with all the origins to one side of all the destinations. The proof is complete. Q.E.D. After a lengthy proof, some discussion of the significance of the results may be in order. We have shown that, given points among which all flows are positive, the geometric structure is much stronger and simpler than may have been expected. For all the generality of the Transportation Problem, psychic transport costs, and semimetric spaces and for all the nonuniqueness of the cJ 's, we wind up with a

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simple invariant geometric pattern. This pattern however is the only useful information thus far extracted. Relative distances and positions change, the pattern remains. The collinearity of this representation suggests a simple measure of points' locational advantage or disadvantage with regard to all others in the sets 1* and J*. The difference in the locational advantages of origins H and 1 with respect to destination J is simply uH — ill. The same difference in locational advantage holds with respect to all other destinations. Now, given an optimal solution to any Transportation Problem it is well known that the dual variables may be interpreted as locational rents or locational advantages of the appropriate origins or destinations (Stevens, 1961). However, this interpretation is general, locational advantage being defined with regard to the problem as a whole. When the points correspond to an irreducible array in E2, or a more general metric or nonmetric space, specific locational advantage (i.e., with regard to specific points) may differ from general locational advantages. Figure 4 A Hypothetical Euclidean Representation of Four Points

1

For example, consider the quadrilateral of Figure 4 with origins a and b and destinations 1 and 2. Destination 1 is closer to origin a than destination 2 is, so 1 has a locational advantage over 2 with regard to origin a. However the situation is reversed with regard to origin b, since 1 is further from b than 2 is. Thus, while the specific locational advantages of 1 and 2 vary with the particular origins, the general locational advantages, expressed by the dual variables of and v?, are constant throughout. Only in the case of the collinearity of Figure 3 are specific locational advantages for a pair of destinations the same whatever the origin. Furthermore, specific and general locational advantages are the same throughout. Notice that the irreducibility of the above imbedding indicates that we have the simplest Euclidean representation preserving the original distance geometric properties. But simplest is used here only in the formal geometric sense of lowest

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dimension. In a specific problem understanding may be better served by a less economical higher dimensional representation. For example, consider as in Figure 5 a special case of the problem just discussed where there are two origins (H and I), two destinations (J and K), all flows are positive, and all four origindestination psychic transport costs are the same positive quantity, c?.. This is consistent with the First Congruent Imbedding Proposition, leading to two sets of nondistinct points H and I and J and K, separated on the real line by c*. (Figure 5a). However another consistent but higher dimensional representation is making the four points vertices of a rhombus whose side is c*. (Figure 5b). Figure 5 Alternative Representations for A Simple Problem with Origins H, I, Destination J, K; x*y„ xåa, xii+ xia Z 0; and

CÅI — CHa—CIJ— CIa — >0 (B) J

(A)

HI

C " J.K

One-Dimensional Representation

K Two-Dimensional Representation

This example indicates that while no distance geometric properties are lost in going from the E2 to the E1 representation, the appearance is quite different. The points do not have to be on a line; they could in fact form a rhombus on the plane. In specific cases the two-dimensional representation may be more useful than the one-dimensional one, since the former is more general. However the rhombus demands more information, for its angles vary even as the sides stay the same length. Without this information many different rhombuses are possible. Indeed it is useful to know that the entire distance geometric content of the problem can be represented in a simple line segment. If one believes in geometric parsimony, a higher-dimensional representation would be justified only by a specific purpose not served by the lower-dimensional one.

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The second of this section's applications of imbedding procedures is to a Transportation Problem with two origins (a and b) and two destinations (1 and 2). At optimum, one and only one flow is taken to be zero; without loss of generality we can thus require that 42 = 0, obtaining the pattern of Figure 1, and corresponding to a basic solution for this problem. The four dual constraints to this problem are: ua — v; = cal

(14a)

uå = ca2 uy — vl = C61 ur 5 c2.

(14b) (14c) (14d)

Subtracting (14c) from (14a), then (14d) from (14b), obtain Cal

— cD1 Z å2 — cb2.

(15)

We can begin to interpret this inequality in terms of the relative distances among the origin-destination pairs. Initially consider all four points to be close to one another in a psychic transport sense. This would cause all four flows to be positive. Now let us rearrange the points so that only 42 is zero. This is done by moving origin b; alternatively destination 2 could be rearranged. Intuitively origins a and b compete, based on their psychic transport costs, to serve destination 2. For a to take care of all this demand, so that 42 = 0, 42 must be large relative to a2, making the quantity (42 — 42) small. Now, moving b away from the whole cluster of a, 1, and 2 would be inconsistent with the problem, since we want b close enough to at least destination 1 to be consistent with the observed fact that 4 is positive. We can similarly view a and b competing to serve destination 1. Since 41 is positive, b cannot fare too poorly in this competition, 41 must not be too large relative to cal , and so the quantity (cal — 41) should be large. Small values of (cat — c 2) may thus be viewed as b's locational disadvantage as compared to a, with regard to 2. Large values of (cal — 41) may be interpreted as b's locational advantage (or at least, absence of a large locational disadvantage) compared to a, with regard to 1. To make the psychic transport costs consistent with observed flows we can say that the latter advantage should be made at least as great as the former disadvantage, arriving at ca2 —

cå1z :2 — C:2

which is simply (15). We can carry these notions further, obtaining geometric and algebraic simplification by adding additional information in the form of normalizing assumptions, as suggested earlier. Previous discussion considered what might be called column or origin competition: the competition of the two origins (down a column) to serve a given destination. In parallel fashion we can speak of the row or destination competition across a given row, where two destinations compete to be served by the same origin. Suppose we want to normalize to show the relative locational

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advantage of one destination over another in that competition. Then we would require c 1 + a2 = 1 and

(16a)

l + cb2 = I .

(16b)

Cp

Notice that the application of this normalization has no relative effect on comparisons of psychic transport costs among destinations for a given origin for it just scales the costs. However such normalization does influence interorigin psychic transport cost comparisons, because the invented scale of (16) is oriented to one origin at a time. Inserting (16) into (15) find cez

? a2

(17a)

and cb1

5 c;1 .

(17b)

This is easily interpreted, for these comparisons of distances (the ones induced by the psychic transport costs) are similar to the prenormalization comparisons, but simpler and stronger. Given the invented scales imposed by (16), the inequalities of (17) indicate that b must be at least as far from 2 as a is for 42 to be zero, but must nevertheless be at least as close to 1 as a is, for 41 to be positive. This is very symmetrical: b's absolute locational disadvantage as compared to a with regard to destination 2 must be balanced by b's absolute locational advantage over a with regard to destination 1. We thus repeat, with a very different approach, a general property of linear programming models of location. It is the relative locational advantage of origins, especially with regard to specific destinations, that is important. An origin that has relative locational disadvantages (compared to a competing origin) with regard to several destinations will not ship to any of them. The relative distances of (17), combined with (16), enable a simple mapping of the four points. Rather than beginning with an imbedding proof, let us begin simply, looking for a representation in E2. Without loss of generality we can locate destinations 1 and 2 on the plane as shown in Figure 6. (The case of these points' nondistinctness is discussed shortly.) First locate origin a. By (16a) this point must be located so that the sum of distances (or psychic transport costs) from a to 1 and a to 2 must equal the constant unity. But this is just the definition of an ellipse—the locus of points the sum of whose distances to two fixed points (here, l and 2) equals a constant. Thus point a can be anywhere on this ellipse. And b must be located on the same ellipse as well, by (16b). Notice here that even before we look at the information provided by the dual, the normalizing assumptions are sufficient to give us a surprisingly definite pattern of the four points.

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Figure 6 Two-Dimensional Representation of a Simple Problem with Origins a and b, Destinations 1 and 2. Origins a and b must be on the ellipse. With a at a', for example, b may be anywhere in the counterclockwise arc from a' to a'.

Adding information from the dual, it is sufficient to look at either (17a) or (17b) since, given (16), one implies the other. Thus, given any location of origin a on the ellipse it is sufficient, by (17b) that b be in some location on the ellipse that is at least as close to 1. Suppose a is located in Figure 6 at a', then b may be located anywhere on the closed arc from a' counterclockwise to a'. If a is located at a", the point on the ellipse farthest from 1, b may be located anywhere on the ellipse, including a". At the other extreme, if a is located at the point closest to l,a"', then b may be located only at that point, where (17a and b) hold as equalities. Of course the actual location of a and b in any specific problem is uniquely determined by cal and cb, (or cå2 and c z). However, we have shown the vast nonuniqueness of these quantities. For the problem at hand the only invariant algebraic properties are the expressions (16) and (17), and the only invariant geometric property is the location of a and b on an ellipse whose foci are 1 and 2, with the positions of a and b interrelated by (17). When points 1 and 2 are nondistinct, the ellipse collapses into a circle with points a and b having the same distance from 1 and 2. This is equivalent to Ca, = Ca2 = Cy, = c 2 = 1/2, for which an alternative representation is a rhombus like that of Figure 5b. The elliptical representation of Figure 6 is an appealing one, accustomed as we are to the map plane. However, it is not the simplest (i.e., lowest-dimensional) Euclidean representation, as a representation on the real line is possible. Thus, this situation parallels the first problem of this section, where (in a special case)

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representations of both Figures 5a and 5b were possible. Here the E1 representation is introduced in the context of an imbedding proof. Since this proposition is proved similarly to the preceding one, the formal proof may proceed more quickly and intuitively. Proposition 3 (Second Imbedding). Suppose we have a psychic transport problem with two origins (a and b) and two destinations (1 and 2) where, at optimum 42 = 0 but 41, 42 , 41 > 0. The c,j's are normalized by (16). Further suppose that the four points and the (adjusted) psychic transport costs are made to form a semimetric space as before. Then this semimetric space is irreducibly congruently imbeddable in a finite subspace of E1, as shown in Figure 7. In this representation origins a and b are in the closed unit line segment whose endpoints are destinations 1 and 2, with b on the side of a corresponding to point 1. Figure 7 One-Dimensional Irreducible Euclidean Congruent Imbedding of the Problem of Figure 6 2

I

I

I

Proof By (16) the sum of the distances from a to 1 and 2, as well as the sum of the distances from b to 1 and 2, must be unity. This is accomplished by defining a unit length line segment anywhere along the real line with endpoints 1 and 2. Points a and b may be anywhere in this line segment, including the endpoints. We have previously shown (expressions (14) to (17)) that inequalities (17) must hold. This is accomplished by putting b on the side of a closer to I. Points a and b may be indistinct in which case (17) holds as two equalities. It is easy to see how this representation satisfies the two requirements of the congruent imbedding theorem. First of all, points 1 and 2 are always distinct and always have the distance properties required by (16). Second, the only quadruple is that of all four points of the problem. By construction, this represents the distance relations of the problem and is not representable in Ea. Notice that for different psychic transport cost solutions to the given problem a and b shift their position on the unit line segment, but b is always to the right of, or on, a, that is, the side of 1. Minor changes in representational conventions can shift the line segment along the real line (though not change its length) or reverse it. However the geometric properties of the proposition's statement remain invariant for all psychic transport cost solutions. Q.E.D.

Two morals may be drawn: the normalization of (16) enables us to give this problem, more complex and general than the preceding one of this section, a simpler and stronger representation. The present problem is more complex since 42 = 0 causes the nonstrict inequality of (14d) to replace the equality of the corresponding dual constraint in the preceding problem. (Observe, however, that the all-flows-positive case appears as a special case of the dual of the present

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problem, (14) where (14d) holds as an equality.) We thus have one fewer equation, but the normalization of (16) adds two additional equations, with constant terms to boot. This adds additional structure. The second moral comes from the alternative representations of Figures 6 and 7. While Figure 6 may be more realistic for those accustomed to the map plane, it is based on the same problem and contains no more information than Figure 7. The greater apparent generality of the elliptical representation stems from the fact that more information is necessary to locate a and b uniquely (even given the positions of 1 and 2) than is the case in the collinear representation. In the former case we need, crudely, four pieces of information : the locations in two coordinates of both a and b, constrained however by their position on the ellipse and by (17). In the latter case we need only two, much simpler, numbers: the position of a in the unit interval, and the position of b to one side of a. Conclusions and Suggestions for Further Research The introduction posed two questions to be answered by the body of this paper. First, may this paper's procedures be used to define and compute a useful set of psychic transport costs and, second, what spatial information (that is, the arrangement of origins and destinations), implicit in observed flows, may be discerned by the methods developed here? Two approaches were developed that are roughly parallel with the two questions. The first dealt with algebraic properties of psychic transport costs, such as their existence, nonuniqueness and whether any useful properties are invariant across alternative solutions to a psychic transport problem. The second studied the geometric patterning of points induced by the psychic transport costs of certain types of Transportation Problems. At this point, it is possible to give only highly preliminary answers to the questions phrased earlier. All results are, of course, based on the use of a Transportation Problem model of behavior, where our only information is the naming of origins and destinations, observed flows, and capacities and requirements simply linked to these flows. As suggested below, a different behavioral model might give different results, as might the presence of additional information. Similarly, there might be different approaches to the same problem with different definitions of psychic transport costs, different normalization procedures, and different distance geometric treatments. For the simple problem posed here, some of the results are surely counter-intuitive and are just beginning to be understood. However, given this paper's preliminarity, certain answers are beginning to emerge. The paper's approach does not lead to the computation of spatially useful transport costs. The high generality of the Transportation Problem and the small amount of information provided lead to many more unknowns than equations (or even inequalities), and consequently a high dimension of nonuniqueness in the subspace characterizing the psychic transport cost solutions to a given problem. Thus far, no spatially useful properties of the psychic transport costs appear

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invariant across this solution subspace. Worse than the nonuniqueness is the fact that the magnitudes of the observed flows do not influence computed psychic transport costs, only whether given flows are positive or zero. Much information is thus lost. Geometrically, the picture looks much brighter even though thus far only two special types of psychic transport problems have been investigated. For a set of origins and destinations such that all flows are positive, we get the surprising result that these points and the distances among them induced by their psychic transport costs may always be represented on a straight line with Euclidean distances and with all the origins to one side of all the destinations (Figure 3). In this case the specific locational advantage of an origin with respect to each destination individually, as well as the general locational advantage for the problem as a whole, is simply indicated by that point's position on the line. In the general Transportation Problem general and specific location advantages are often quite different. While the nonuniqueness of the psychic transport costs causes the position of specific points to vary, the geometric pattern for this problem always holds. It may also be possible to give this problem a two-dimensional representation. A second psychic transport problem studied is that with two origins and two destinations, where three of the four flows are positive. This situation, of course, corresponds to the important linear programming basic solution. With the psychic transport costs normalized, a one-dimensional Euclidean representation is again possible (Figure 7) which, however, differs from the previous one in that the locational advantage of a given origin differs across the two destinations. A two-dimensional Euclidean representation is also possible. Throughout, the algebra has been made very simple by the linear equations and inequalities of the Primal and Dual Transportation Problems, and in the primaldual relationships. This advantage also occurs in more general linear programming models of location. Another methodological point refers to the use of distance geometric procedures, particularly congruent imbedding. These procedures are powerful tools for the task at hand, enabling us to understand the properties of spaces more general than the familiar Euclidean ones, or even the more general metric spaces. Given the very general geometry implicit in the psychic transport problem, we are interested in a representation that is simple and has strong distance properties (for example, a low-dimensional Euclidean representation) but that loses none of the original problem's distance geometric content. This may be done by a congruent imbedding that preserves the points' identities and the distances between pairs of points as well. If the congruent imbedding is also irreducible, the case of the one-dimensional representations, then we know we have the lowest-dimension Euclidean space capable of this isometry. The tool of congruent imbedding might

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be used more widely to help establish the geometry implicit in a set of distances. Let us discuss the spatial meaning of some of the results obtained. In a previous section the existence of certain standard or invariant solutions to all psychic transport problems was explained as due to constraint determination. While we may think of the psychic transport costs only as part of the Transportation Problem's objective function, they are also influenced by the remainder of the problem, that is, the capacity and requirement constraints. Thus, a destination close to a highcapacity origin has a different psychic transport cost than one equally close to a lower-capacity origin. This is because the psychic transport costs are closely related to the dual variables, showing the locational advantage of each origin (or destination) with respect to the distances to all the destinations (or origins) and all their requirements (or capacities) as well. This consequence of the constraints' influence on psychic transport costs seems an intuitively useful property. We might want a high-capacity origin to be closer to a destination than might otherwise be the case, because its influence is felt over a wide objective distance. Suppose we define psychic transport costs in a general structure, such as an interregional linear programming model. Then psychic transport costs might be influenced by other data appearing in the constraints, such as the comparative production costs of competing regions. A low-cost region would thus be closer than an otherwise identical high-cost one. Similarly, consider building into a linear program those real world conditions (cited in the introduction) that make actual behavior differ from that of simple optimizing models. Suppose the shipper faces a variety of transport modes and shipment routes having different reliabilities in terms of known probabilities of arrival within stated times. We can use a chance-constrained or another stochastic linear programming model (Hadley, 1964). In this case optimal behavior is based on these probabilities, as would be the psychic transport costs here. Our psychic transport costs thus might derive in part from transport reliabilities, even though we would not previously know how much of a behavioral influence is reliability, compared to the other data of the problem. Furthermore, the addition of more data in the form of additional constraints might resolve the indeterminacy in the calculation of psychic transport costs. A general advantage of mathematical programming models is that quite subtle and realistic behavioral influences are often easily phrased as constraints, even linear ones. Here we can enrich a psychic transport problem by adding, as constraints, real world complexities that make actual behavior diverge from simple cost or distance minimization. If the resulting psychic transport costs are congruently imbeddable in a Euclidean one- or two-space, we have a transformation for converting a complex situation into a uniform transport plain representation. Another advantage of adding to the constraint set is that it may thus be possible to avoid the Transportation Problem's linear dependence of the equation of the

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primal and dual objective functions on the constraint set. The consequence of this dependence is that only the signs of the observed flows, not their magnitudes, influence the computation of psychic transport costs. Another suggestion for further research is that congruent imbeddings other than irreducible Euclidean ones be experimented with. As mentioned, a higherdimensional Euclidean representation (even higher than three) though less economical, may be more intuitively insightful, as well as more useful in suggesting additional information that would reduce the indeterminacy that even the lowest-dimensional representation has. Perhaps non-Euclidean representations would be useful, particularly spaces like spherical ones, where distances are measured on the surface of a sphere of specified radius. Spherical spaces are not only geographically appealing, but have well-known mathematical properties. Other non-Euclidean metric spaces might be used (Blumenthal, 1970). Above it was suggested that extra information be added to the problem through additional constraints in the original model. Alternatively, one could add additional information by assumed relationships between psychic transport costs and observed flows, or with the optimal levels of the dual variables treated as locational rents. Outside information could also be used, such as a specification relating psychic and objective transport costs. Acknowledgments This investigation was supported by NSF Grant GI-29731. The author is indebted for the helpful comments of Nazir A. Dossani, Lalita Sen, Tony E. Smith, and Benjamin H. Stevens. None is responsible for any errors appearing in the preceding discussion. References ALONSO, W.

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HALL, P. (1965). Von Thünen's Isolated State. C. Wartenberg, trans. Oxford: Pergamon Press. LØcH, A. (1954). The Economics of Location. W. H. Woglom, trans. New Haven: Yale University Press. RUSHTON, G. (1969). "Analysis of Spatial Behavior by Revealed Space Preference." Annals of the Association of American Geographers, 59:391-400. SIMON, H.

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Science, 3:15-26. (1929). Theory of the Location of Industry. C. Friedrich, trans. Chicago: Chicago University Press.

WEBER, A.

A Model for Location of Service Facilities in a Non-western Urban Environment P. D. MAHADEV (University of Mysore)

and K. R. RAO (University of Pittsburgh)

When one thinks of city planning, one is inclined to think mainly of new buildings, superhighways, spaghetti intersections, massive community facilities such as stadia, landscaping, and city beautification. This type of planning concerns itself mainly with the physical aspects of urban environment, with artifacts that involve brick and mortar, steel and glass, asphalt and cement. However, it is becoming apparent to all that city planning involves more than these aspects of development. A human city is a place where people can live, work, play, study, and pursue a variety of cultural activities. This study attempts to define some of the services that are necessary for better community life by placing a premium on the needs of the households for accessibility to those services. This emphasis is generally in line with the ideas of Wurster (1960, 1962) about the scale of neighborhood that essentially stresses this aspect. LeCorbusier (1947) and Mumford (1938, 1956) also outline the importance of human aspects of planning while drawing up the physical plans for communities. Though most of these scholars suggest the ideal end products of planning, they do not suggest the methods of evaluation of the services and their accessibility to the households, and they refrain from suggesting any prescriptive measures to rectify the shortcomings in neighborhood development. The first stage in the planning process of a residential neighborhood is accomplished by Herbert and Stevens' model (1960). Their model helps in distributing 164

P. D. Mahadev and K. R. Rao 165

residential activity with reference to transportation costs. Further progress was made in the same direction by Lowry (1964) where some of the activities connected with the residential population are distributed with reference to the transportation and workplace locations. Ochs' model (1969) based on linear programming on the urban spatial organization took into account Lowry's model and developed it further by trying to find the minimum cost combination of housing type, its location, and traffic flow. The model presented in this paper concerns itself with the second stage of planning, i.e., the provision of services for the already existing residential clusters. In physical planning of a city, the social aspects of community needs are too often overlooked. In particular, the non-Western city planners' preoccupation with the traditional comprehensive planning methods limits the utility of plans when allocation decisions for choosing specific schemes on a spatial basis are attempted. This paper analyzes at the neighborhood level the problems of a household in terms of its daily need for access to service facilities. The scale of treatment is restricted to the level of the neighborhood planning unit, which is a neighborhood unit formed for microlevel planning purposes. Given the definition used in this paper, these neighborhood units are expected to be self-contained with respect to education, health, recreation, and retail shopping facilities. This definition of a neighborhood is presented in Solow (1969) as elementary and primary education, health centers and neighborhood clinics, recreation needs for children in terms of playlots and parks, and shopping units catering to day-to-day needs. Thus, the paper analyzes at a neighborhood level the household needs of a community for service facilities. The optimal distribution is determined through the procedures of linear programming. Although linear programming is mathematically quite complex, it can be reduced to a very simple form in the practical application of its use in this particular context. Definition of the Problem The non-Western urban planner is confronted with scant statistical evidence to describe the social and urban structure of the communities with which he works. Even in the Western world, the use of quantitative measurements and mathematical models by practicing urban planners at the community level is infrequent. The present model: (1) aims to be descriptive by delineating the existing distribution of service facilities occurring throughout an area; (2) attempts to pinpoint at a neighborhood or community level any deficiency caused by the lack of such facilities; (3) indicates a remedial measure to overcome any deficiencies by suggesting areas for the location of additional facilities; and (4) evaluates possible priorities for the location of public facilities. Thus the model is prescriptive both in making the choice and in fixing a priority in location among various neighborhoods.

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In the context of the above, the present model is designed to analyze the accessibility of the service facilities (both individually and collectively) to the households in the residential neighborhoods. For this purpose, data for each of the neighborhood units are aggregated and compared to those for the other units. This facilitates identifying the neighborhoods that are deficient in accessibility to the service facilities for improvement by way of prescriptive measures. The prescriptive measures for enhancing accessibility to the service facilities can take the form of increased capital investments, marginal improvements and changes in the land use, creation of underpasses, and provision of ambulance service to the hospitals. The Study Area The city of Mysore, with a population of about 350,000, is located in the southern part of India. It is spread over an area of 14.4 square miles and has recently been divided into twenty-eight neighborhood units for the purposes of planning. Each neighborhood unit consists of an average of 12,500 people and varies in areal size. Being a non-Western city, it is characterized by the absence of widespread use of automobiles for commuting. Most of the people walk to attend to their daily chores and to reach community facilities, and this makes distance the most important factor, and effort associated in negotiating the streets and traffic the major obstacle. The central areas in the northern part of the city are characterized by a mixture of commercial and high-density, low-income residential uses. Toward the north, on the fringes, industrial and low-income residential uses occur. The eastern part is dominated mostly by recreation areas such as a race course, zoological gardens, golf links, with a scattering of low-density and high-income residential clusters. The southern part, however, is mostly residential with a sewage farm and a burial ground on the fringes, whereas the western part is marked by an unusually high degree of concentration of educational land uses. Housing for the University of Mysore, the affiliated colleges, and research institutions borders the residential areas nearer the core. The generalized land-use pattern indicates a sectoral development superimposed by a circular pattern (Figure 1). The distribution of population also signifies a pattern associated with nonWestern cities, a population peak near the center along with a highest percentage of commercial land. The highest density of 287 persons per acre occurs in the neighborhood unit 4 located at the center of the city. The population density drops to about 150 persons per acre in the neighboring blocks, dropping further to 10 to 15 persons per acre toward the periphery.' 1. The data used in this paper form a part of the land-use and socioeconomic survey information collected for the preparation of the Comprehensive Master Plan of Mysore, and for a research project of Mahadev (1972). For details of data used in the present study refer to Table 2.

P. D. Mahadev andK. R. Rao

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More than two decades have elapsed since the transfer of power from the king to the elected representatives of the people in the state and during this period there has been an expansion of the city accompanied by locational shifts in land uses. Such expansion, superimposed on the earlier spatial organization of the service facilities, which was basically nonoptimal in character, has resulted in a complex pattern particularly in terms of accessibility of the neighborhood units. Therefore the present spatial arrangement of service facilities warrants an evaluation in view of emphasizing optimum locational considerations in the future. Formulation of the Problem and Methodology In the spatial arrangement of land uses and service facilities described above, the residential clusters in the city have unequal access to the educational, health, recreation, and shopping facilities. The suboptimal accessibility of some of the neighborhoods to the service facilities have been further accentuated by manmade and topographical barriers. Such differences in accessibility are mainly a result of differences in physical distance, the difference in time consumed to travel, and the costs incurred in reaching these facilities, indicating a positive correlation between distance, time, and cost. Hence, distance has been taken as the variable, substituting for time and cost. The other resource assimilation factor is the effort that is necessary to negotiate the barriers in reaching these facilities. This draws upon the tolerance of the households in overcoming these barriers. Hence the formulation of the problem envisages identifying the variations of accessibility from one planning area to another, and determining the deficiency in the neighborhoods. The rationale behind the accessibility, barriers, tolerances, and distances is that every housewife or any member of a household can afford to travel certain distances and expend certain amounts of tolerance to fulfill his daily chores. Beyond a point it becomes suboptimal to travel longer distances or increase the tolerance level. Thus the formulations are designed to assess the accessibility characteristics of the service facilities within the constraints of capacity of the households to expend energy to commute the distances and overcome obstacles. Let Ali (j = 1,2,3,4) be the number of trips per day (accessibility index) to reach (1) education, (2) health, (3) recreation, and (4) retail shopping facilities. In this formulation it is assumed that there is no interaction among these variables as such. Within the constraints outlined below, a high X; indicates optimal accessibility while a low X1 indicates suboptimal conditions. Thus in this formulation Xi, being the journey to the facilities, is considered to be an ACTIVITY. Let b. (i = 1,2) be the amounts of resources available (energy expended in covering the distance or tolerance and convenience spent in overcoming the barriers) for the journey to the four activities (X)). These are input CONSTRAINTS. In this formulation

P. D. Mahadev and K. R. Rao 169

b, = maximum distance to be covered in transportation (2 miles);2 b2 = maximum tolerance' and convenience available to a household which can be expended if confronted with traffic barriers and natural hurdles. (The maximum value of this measure is 1.) Let T be the overall measure of accessibility to the neighborhood community facilities. Thus, it becomes the measure of effectiveness of the total number of trips occurring in the given cluster of households in any one neighborhood. T is in effect the index of transportation overhead occurring in the given family cluster. The objective, then, is to maximize T for the given assumptions. T is a function of Xi and can be written as

T = CjXj where Cj is the price4 of T associated with a unit increase in each activity (Xj) which optimizes T. In this formulation, C3 (j = 1,2,3,4) is the price associated with unit increases in trips to educational, health, recreational, and shopping facilities. C thus refers to values associated with trips to the respective facilities. au = constant amount of resources (b1) used per unit activity (X) undertaken such that ai,X3 < bi. Now the mathematical statement of a general form of linear programming problem has emerged. It is required to: Maximize the objective function, T = ECjXj;

(1)

subject to the restrictions, aij X, < b1; and

(2)

with the nonnegativity conditions (since activities cannot have negative values), Xj < O.

(3)

As mentioned above, bj and Cj are also taken as constants. By maximizing the objective function, one has available an indication of the self-containment of the neighborhoods. Here linear programming permits the optimization of T such that the adequacy of facilities serving an area is indicated. Areas with smaller T-maximum imply a deficiency in respect to these facilities. Having decided upon the area where a deficiency is indicated, one then needs to examine the shadow prices of X. While the variables in the basis do not increase the objective function at this stage of optimality, shadow prices indicate the 2. In addition to the linear distance, money spent and time consumed can be used in the equation (Rao, 1971). 3. An obstacle rated as intolerable has an expenditure of a unit of tolerance. Moderate impediments are rated 1/2 and tolerable hindrances and obstructions are taken as 0. (This method of evaluation, as cited by Hodge (1963), is used for comparison and ranking, rather than using absolute values for desirability.) 4. From the examination of the demographic and socioeconomic content of the population in the neighborhoods 1 to 28, Cj would be the same for all the areas, though this would vary for different facilities. Adopted values of Cj are: C, = 3 (EF); C2 = 2 (HF); C3 = 1 (RF); and C4 = 4 (SF). In practice, it is suggested that these weights be determined by the community itself through a participatory process.

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amount of increase possible in the objective function resulting from a unit increase in a specific variable. These therefore indicate, for any neighborhood or set of neighborhoods whose T-maximum is low, where planners, administrators, and policy-makers should concentrate their efforts. For example, if the variables X1 and X3 appear in the basis of final iteration (at optimality) with X2 and X4 having intrinsic values, then this implies that an increase of 1 unit in X1 and X3 will not increase the T-maximum while an increase of 1 unit in X2 and X4 will. Even the amount of desirable inputs required for improving the objective function by increasing X2 and X4 can be calculated. Data The data utilized in this study form part of a wider range of data collected by the Department of Town Planning, Mysore, and by one of the authors of this paper. The data represent a 10 percent sample of the households of the city. In each street every tenth household was chosen for field investigations. The questions pertaining to this specific study included inquiries as to the distances traveled in miles to reach each of the facilities, such as education (EF), health (HF), recreation (playgrounds) (RF), and retail shopping (SF).5 One of the reasons the field investigators were assigned to fill in the questionnaires was the widespread illiteracy in many of the households and the varying concept of distance. Table 1 Data Format Neighborhoods

X1

1 2

Tolerances expended < 1 mile

Distances covered 5 2 miles X2

X3

X4

X1

X2

X3

X4

(EF) (HF) (RF) (SF)

(EF) (HF) (RF) (SF)

all

a12

a13

a1 4

az1

a22

a23

a24

all

alt

a 13

a14

az1

a22

a23

a24

28 Mode for the Neighborhood

5. Education, health, recreation, and retail shopping are used as neighborhood service facilities in this study. Education refers to primary education. Health facilities refer to the primary health facilities such as health centers and public dispensaries and clinics. Recreational facilities are the children's playlots and parks; shopping facilities are the retail shops selling a small variety of vegetables and some groceries, cigarettes, soft drinks, etc., which cater to the daily needs of the neighborhood residential clusters. These services are expected to be provided in the neighborhoods, and it is assumed that a neighborhood is a geographical area wherein social relationships and community participation can be maximized. A detailed discussion on this concept is found in Solow, et al. (1969).

P. D. Mahadev and K. R. Rao 171

Detailed maps were used to identify streets and the number of streets and intersections to be negotiated in reaching these facilities. Personal observation and a count of vehicular traffic revealed the nature and degree of obstacles in reaching these facilities. In addition to the information gathered by means of questionnaires, maps were used to measure the distances to check the answers given by the households. In a small number of cases the children skipped the neighboring schools and went to schools farther off, but the number of such cases was negligible and hence was ignored. The data is presented conceptually in Table 1. The actual data for 28 neighborhoods is presented in Table 2. The data as listed can be treated in the form of equations for use in a linear programming model. The simplex method was used to solve the equations. It should be mentioned that this model fulfills the basic assumptions of linear programming and thereby overcomes limitations of its use, which are: (1)The proportionality assumptions of the constraint functions and the objective functions are satisfied by the model suggested. Proportionality in the constraint functions suggest b, cc X; such that b; = a;; X;, where a;; is a constant amount of resource i used per unit of activity X. The proportionality assumption of the objective function requires : T cc X; such that T = CA, where C; is the price associated with a unit increase or decrease of X. As already mentioned both of these criteria are fulfilled. (2) The additivity assumption requiring X;C; = T and X;a;; = b; is likewise satisfied since the activities X; are additive with respect to the total accessibility (T) and also with regard to each resource consumption (b;). The total measure of accessibility (T) and each total resource consumption (b;) resulting from the joint performance of the activities (X;) must equal the respective sums of the quantities resulting from each individual activity. This is apparent from equations (1) to (3). (3) The divisibility assumption requires that the decision variables (X;) have significance only if they have integer values. Fractional decision variables are permissible when treated as comparative value, and in the model presented in this study this form is adopted. Thus, the decision variables (X;), even though in fractional levels, are indicative of comparative value. All of the coefficients in the linear programming model (a;;, b;, and C;) are assumed to be known constants. In reality they are neither known nor constant. Linear programming models can be based on a prediction of the future course of action, the coefficients can be based on a prediction of the future conditions. In the present case the coefficients used were based on the actual experience and conditions found in the study area. Interpretation of the Results In the analysis of the location of service facilities, the primary aim was to explain the locational aspects of the service facilities and their accessibility to the

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Table 2: The Data Distances and Obstacles in Reaching Neighborhood Facilities Density Number NHU Area of pop. of No. (acres) (persons houseper acre) holds

EF

HF

RF

SF

(X1)

(X2)

(Xi)

(X4)

Obstacles (in fractional units) of households from EF HF RF SF (X1) (X2) (X3) (X4)

0.25 0.5 0.75 0.5 1.0 0.5 0.5 0.5

Distances (in miles) of households from

1 2 3 4 5

142 248 94 64 122

85.7 80.4 168 287.6 198.7

2130 3400 2400 2620 3630

0.5 0.5 0.125 0.25 0.25

0.625 0.25 0.25 0.125 0.125

0.5 0.75 0.25 0.25 0.75

0.5 0.25 0.125 0.125 0.125

0.5 1.0 1.0 0.5 1.0

6 7 8 9 10

381 176 60 164 165

30.6 17.0 75.5 11.0 36.2

5100 550 1610 350 1080

0.25 0.5 0.125 0.75 0.125

0.5 0.75 0.125 0.75 0.25

0.25 0.125 0.125 0.75 0.125

0.75 1.0 0.125 0.75 0.125

11 12 13 14 15

304 239 258 15 259

10.5 138.0 94.1 88.0 57.5

1420 5330 3560 200 2990

1.0 0.5 0.75 1.0 0.5

0.5 0.25 0.5 1.0 0.5

0.25 1.0 0.5 1.0 0.4

1.0 0.25 1.0 1.25 0.3

16 17 18 19 20

211 350 482 316 273

195.0 5.5 21.1 91.0 160.5

7630 250 1910 5100 6540

0.25 0.25 0.25 0.25 0.25

0.25 0.25 0.25 0.4 0.25

0.4 0.25 0.5 0.25 1.0

0.2.5 0.25 0.25 0.3 0.25

21 22 23 24 25

228 228 222 904 102

50.9 64.4 62.6 4.9 66.1

1800 2700 3120 820 2130

0.25 0.25 0.25 0.5 0.5

0.4 0.4 0.25 1.0 0.5

0.4 0.25 0.25 0.5 0.5

0.3 0.3 0.25 0.75 0.5

0.75 0.25 0.25 0.5 0.25 0.5 0.5 0.25 0.5 0.5

1.0 1.0

0.5 -

1.0 1.0

26 27 28

160 284 310

10.0 49.6 8.8

330 3090 180

0.75 0.5 0.25

1.0 0.75 0.25

1.0 0.4 0.25

2.0 0.6 2.0

1.0

1.0

1.0

1.0

-

-

-

-

1.0

0.25 0.25 0.50 1.0

1.0

1.0

1.0

1.0

1.0 0.5 0.25 0.25 1.0 1.0 0.5 1.0 1.0 1.0 1.0

1.0 0.5 1.0

households in the neighborhood units. The analysis and interpretation of the results (Table 3) can be broadly organized into five sections concerning each of the four community facilities and the overall accessibility of all four taken together. The overall accessibility of these service facilities to the neighborhood units is taken up first for discussion. Overall Accessibility (T Maximum) Areas having optimum accessibility (64 units) can be examined first (Figure 2).

P. D. Mahadev and K. R. Rao 173

Table 3 The Optimum Results

"Zj - CI evaluation row: shadow

Vectors in the basis: (values of variables) Value of NHU objective Structural variables No. function: T-max Xi X2 X3 X4 SF EF HF RF

1 2 3 4 5 6 7 8 9 10 11

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

16 32 64 64 64 16 20 64 4 64 8 32 8 4 27 32 32 32 26 33 11.5 16 32 6 8 4 13.3 24

5.3 -

8

1 3 -

- - 0.5 2 2 1.3 4.0

4.0

16 8 2 2 4 -

Slack variables S3

1

1 1.25

1 1 1

16

8

2 -

1 0.75

1 1 1

6.7 8

8 8 6.5 8.2

2

2

2 4.0

4 8

3.3

-

S2

1 1 1 1 I

4 8 16 16

16 1 10 -

prices or intrinsic values

I I

0.8

1

1

1 1

Values in slack

Vectors in the row

EF

HF

RF

SF S'

1 1 1 1 5 5 1 1 1 1 5 5 1 3.6 1 1 1 0.3 1 1 1 0.3 -

3 2 6 2 2

3 11 7 7 23 2 3 15 7 4.3 5 3 7 2.3 1 7 3 2 2.2 1.7 -

- 8 - 16 - 32 - 32 - 32 5 12 - 8 - 32 5 5 4 - 16 -

4

4 6 2 2 2.1 4.6 2 2 2 3.3 2 0.6

6 2 4 3 1.2 3.0 -

1

2 1 5 20

S2

4 4 4

8 4

13 16 16 16 13.3 16

2.5

6.5 16

16

2 0.8

6 4 2.4

6.7

12.0

The neighborhoods 3, 4, 5, 8, and 10 fall into this category, and when space relationship is considered, it is found that the first three neighborhoods are near the core of the city and the last two (8 and 10) are at the fringes. Neighborhoods 3, 4, and 5 are the oldest parts of the city and their location near the center gives

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Figure 2 General Accessibility

r I

F

V

~

e.lii I

9 14 15

f

~4-

\

1

12 r`

1

\

.~

7

'`...,'' 28

2

~

Tank

3

{

.w:..-"::

—jPa lac e 7 -\

r

1

k 27

I

16

--~

19

( j 5......--.._....

f

....~

1,,,

.

21

26

Optimum

M

25 €

64+ Units

32 to 63 16 to 31 8to15

24

7710 to 7

Least Optimum

— — Neighborhood boundary

them easy access to the service facilities. Another reason for the optimum accessibility is that they have been served well for a long time because of their proximity to the king's court.

P. D. Mahadev and K. R. Rao 175

Neighborhoods 8 and 10, in spite of having fringe location, show optimality. This is mainly because of the isolated and independent development of a tuberculosis sanitarium with all the service facilities in neighborhood 8. The optimality of neighborhood 10 can be attributed to its development as a middle-class residential area by the governmental agency and private housing of higher-income groups. This developmental and economic aspect enabled the area to support economically the basic services, thereby making the neighborhood optimal in accessibility. The lowest accessibility of four units has been recorded in neighborhoods 9, 14, and 26 which are also located on the fringes but for different reasons. The low accessibility is the result of the combination of low-income groups and the existing low-density population making the location of services economically nonviable. Neighborhood 26 especially encompasses a low-income village. The other neighborhoods range from eight units to thirty-three units indicating varying degrees of optimality. Neighborhood 25, though located nearer the other populated parts of the city, can be classified as a fringe neighborhood with high population density and low income. The analysis of neighborhood 6 confirms this view. This neighborhood, containing an extremely dense (over 200 persons per acre) village pocket with nearly 10,000 inhabitants, has sixteen units of accessibility in spite of its proximity to the work centers. Fringe neighborhood 24 is planned as an industrial estate with residential development adjoining the industrial sites, but as yet the locality does not have the necessary threshold population to support the facilities, and the absence of such facilities has deterred population increase. A parallel situation exists in neighborhood 9. The above localities that are deficient in accessibility are a contrast to neighborhoods 2, 3, 4, and 20 which contain work centers abutting the residential development. These neighborhoods also happen to be near the palace with easy accessibility to the center of the city. The neighborhoods that had integrated development, such as 8, 10, and 28, are higher in the scale of optimality than the ones that had haphazard development. The general pattern that emerges from this analysis suggests that optimality fades from the core to the periphery except in cases where the neighborhoods had an independent and/or integrated development. Educational Facilities The neighborhoods on the fringe that have institutional backing and where subsidies are locally available have optimum accessibility to the educational facilities. This is illustrated in Figure 3. Since neighborhoods 6, 24, and 28 contain residential campuses of educational and industrial institutions that subsidize the educational facilities, the rating in optimality is rather high despite the low density of population. As a contrast, neighborhood 7 has low accessibility rating due to low density of population and lack of institutional support.

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Figure 3 Educational Facilities

There are, however, several areas that have schools in the neighborhood but rank lower in optimality because of high density of population (about 200 per acre), resulting in inadequacy of the existing facilities (5, 12, 13). Some of these

176

P. D. Mahadev and K. R. Rao 177

Figure 4 Health Facilities

I