Numerical Modelling of Random Processes and Fields: Algorithms and Applications [Reprint 2018 ed.] 9783110941999, 9783110460544

Citation preview

Numerical Modelling of Random Processes and Fields: Algorithms and Applications

Numerical Modelling of Random Processes and Fields:

Algorithms and Applications

V.A. Ogorodnikov and S.M. Prigarin

III MSPIII

Utrecht, The Netherlands, 1996

VSP BV P.O. Box 346 3700 AH Zeist The Netherlands

© VSP BV 1996 First published in 1996 ISBN 90-6764-199-5

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

CIP-DATA KONINKLIJKE BIBLIOTHEEK, DEN HAAG Ogorodnikov, Vasily A. Numerical modelling of random processes and fields: algorithms and applications / Vasily A. Ogorodnikov, Serge M. Prigarin. - Utrecht: VSP. - Fig., tab. With ref. ISBN 90-6764-199-5 bound NUGI811 Subject headings: stochastic processes / numerical methods / statistical modelling.

Printed in The Netherlands by ICG Printing b.v., Dordrecht.

Preface

Computer-aided modelling is one of the most effective means which allow one to get to the root of a natural phenomenon and to predict consequences of the mail's impact on the environment. Along with deterministic models stochastic models are finding ever increasing use. Recently, general methods of numerical modelling of random processes are being effectively developed and the area of application is rapidly expanded. As a matter of fact, the totality of results obtained in the area of numerical modelling of random functions can be distinguished as an independent division of statistical modelling. The book deals with the development and investigation of numerical methods for simulation of random processes and fields. The first two chapters describe scalar and vector-valued Gaussian models. Conventional autoregressive and moving-average schemes and their vector modifications axe indicated. Great attention is given to spectral models. In addition, several special methods axe presented. The third chapter is dealt with non-Gaussian models. The method of nonlinear transformation of Gaussian functions is studied in some detail as well as models of random fields related to stationary point flows, models based on nonlinear stochastic differential equations, etc. The issues of compatibility of marginal distributions and covariances of random processes are considered here, too. The fourth chapter of the monograph is dealt with issues of convergence of approximate models of random fields. The last two chapters deal with stochastic modelling applications. New application areas such as simulation of meteorological processes and fields, sea surface undulation and stochastic structure of clouds are presented here. Moreover, the book includes three appendixes, in which the algorithms for modelling of random numbers, some data from the random processes theory and the connections between discrete and continuous stochastic models are discussed in brief. The book is intended for experts of statistical modelling as well as specialists of application problems of Monte Caxlo methods. Chapters 1, 6 and Sections 3.4 - 3.7 are written by V.A.Ogorodnikov. Chapters 2, 4, 5, Sections 3.1 - 3.3, 3.8 and Appendixes are written by S.M.Prigarin. We would be glad to receive the readers' references about our book at the address: Computing Center, pr. Lavrentieva, 6, Novosibirsk, 630090, Russia (e-mail: [email protected]). We wish to express our gratitude to Prof. G.A.Mikhailov for moral support and stimulating ideas. We also wish to especially express our gratitude to the colleagues at the Department of Statistical Modelling at the Novosibirsk

IV

Preface

Computing Center for the close cooperation and fruitful discussions. We are grateful to Prof. B.A.Kaxgin under whose auspices Chapter 5 was written. Finally, we wish to acknowledge the help received from E.L.Seomochkina and L.A.Ulyankina who aided in the preparation of the final manuscript. The work was supported by the Russian Basic Research Foundation (Grants 93-01-00500, 93-05-14590, 95-01-00939) and by Grant No. JI7100 from the International Science Foundation and Russian Government. Academytown, Novosibirsk, Russia August, 1995

Vasily A. Ogorodnikov Serge M. Prigarin

To the reader. Each section has its own system of numbering formulas, lemmas and theorems. In references to other sections and chapters we resort to double and triple numbering. For example, (1.2.S) refers us to formula (S) from Section 1.2; Theorem 4-5 denotes Theorem 5 from the fourth section of the current chapter.

Contents

Preface

1.

Statistical simulation of discrete Gaussian processes and fields with a given correlation structure

1

1.1. 1.2.

1

1.3. 1.4. 1.5. 1.6. 1.7. 1.8.

2.

3.

Ill

Method of conditional expectations Simulation of the Gaussian vectors with correlation matrix of stationary type Regularization of the algorithm Simulation of autoregressive processes with a desired correlation structure Simulation of stationary Gaussian vector sequences and discrete spatial fields with a given correlation structure Simulation of vector autoregressive sequences Linear transformation method Special algorithms for simulation of homogeneous isotropic discrete Gaussian fields

4 6 10 16 25 28 36

Spectral models of Gaussian random fields

39

2.1. 2.2.

39 47

Construction of spectral models Spectral models of homogeneous vector fields

Numerical models of non-Gaussian processes and fields 3.1. 3.2. 3.3.

Consistency conditions of marginal distributions and covariance function Method of inverse distribution functions Models based on stochastic differential equations

59

59 65 71

VI

Contents

3.4. 3.5. 3.6. 3.7. 3.8.

"Repetition" method for simulation of random vectors and processes Method for simulation of random processes and fields on point flows Randomized models of non-Gaussian discrete processes Combined models of non-Gaussian process and fields Numerical models of certain classes of non-Gaussian homogeneous processes and fields

79 81 85 86 91

4. Convergence of numerical models of random fields in Monte Carlo method

4.1. 4.2. 4.3. 4.4. 4.5.

Introduction Weak convergence of probability measures and random functions Conditions of weak convergence in spaces C and Cp Convergence of spectral models of the Gaussian homogeneous fields Convergence of one class of non-Gaussian models Remark on allowance for bias of estimates constructed by approximate models Appendix to Chapter 4. Some applications of the Jane-Marcus central limit theorem in statistical simulation

96 96 97 98 106 113 115 118

5. Simulation of random fields in stochastic problems of the atmosphere - ocean optics 5.1.

5.2.

Numerical modelling of stochastic structure of cumulus clouds for investigation of the solar radiation transfer in the atmosphere Simulation of the undulated sea surface and study of its optical properties by Monte Carlo method

123

123 134

6. Hydrometeorological applications of statistical simulation methods 6.1

Influence of uncertainty in initial data on forecasting accuracy

144

145

Contents

6.2. 6.3. 6.4. 6.5. 6.6. 6.7.

On accuracy of temperature vertical profiles expansion into a series by eigenvectors of sampling covariance matrix Investigation of some features of excursions of air temperature time series Probabilistic models of dry and rainy days time series Probability properties of precipitation amount Approximation of empirical probability distribution of daily rainy precipitation sums Probabilistic model of non-stationary vector sequences in applications to some joint time series and spatial fields of different weather elements

Appendix 1. Synopsis of the theory of stochastic processes

VII

149 153 159 171 175

181

195

Appendix 2. On correspondence between discrete and continuous linear homogeneous stochastic models

210

Appendix 3. Coding of multiplicative generators of pseudorandom numbers

225

References

231

1. Statistical simulation of discrete Gaussian processes and fields with a given correlation structure

1.1. Method of conditional expectations In the general case, n-dimensional Gaussian vector £ with an arbitrary covariance matrix R

R =

fii ri2

r12 r22

-fin

n

r In 7"2n •••

(1)

fnnj

and zero mean can be obtained by the linear transformation

(2)

£ = An, where y>\, (fi2,..., m. The correlation function is uniquely defined by equation (5) for arbitrary regression coefficients provided that all the roots of equation (2) lie inside the unit circle of the complex plane. Now, we consider an inverse problem to find out whether it is possible to construct the m-th-order autoregressive process such that its correlation function ph coincide with a given correlation function rj, at initial points. We show that this problem has a unique solution [92]. Thus, keeping in mind that the correlation function is even, we obtain from (5): hpo H hpi -\ hpm-l

h bmpm-1 = plf 1- bmpm-2 = P2, H

(6)

1- bmpo = Pm,

where po = 1. Since pn are uniquely defined by the parameters b\,... ,bm, in (6) we have identities. Now, in equation (6), we replace po,.--,pm with the initial values ro = l , r i , . . . ,rm of the given correlation function, and consider the obtained relations as a system of equations for 6[m] = (bj,... ,bm)T Rmb[m] - rm, where rm =

(ri,..,rm)T,

(7)

1. Simulation of dicrete Gaussian processes and fields

12

1

l rm-2

Lrm_i

1

Equation (7) is uniquely solvable as Rm is a positive definite matrix (because 1, r i , . . . , rm are initial values of a positive definite correlation function r/,). We show that thus determined parameters (provided the matrix i ? m + i is positive definite) can be considered as coefficients of an autoregressive difference equation of type (1), in other words, the corresponding characteristic equation (2) has only roots lying inside the unit circle of the complex plane. Since the autoregressive equation has only one stationary solution of type (3) and only one correlation function satisfying equation (5), then the initial values of this correlation function coincide with l , r i , . . . , r m due to equation (7). All consequent values are obtained using (5). The process described by equation (1) will transfer a stationary state after rather a large number of steps (beginning with arbitrary initial values of £ 0 we obtain 0 < A ( S S T ) < 1. In particular, 0 < max A ( S S T ) < 1.

(8)

Obviously, A(S) - A ( S T ) = A ( B ) = A ( B T ) . Assume, that A ( B ) is a real eigenvalue of the matrix B, and

and y>(n) = ( n on the two-dimensional integral lattice, where m, n = . . . — 1,0,1, can be represented in the form of an infinite moving-average

,1. Simulation of dicrete Gaussian processes and fields

32

£m,n

oo ^ 1 GijVm+itn+ji

=

i,j= —oo

oo ^ ]

a

(7)

ij
0 simultaneously for all m and n.

1.7. Linear transformation method

33

1.7.3. T h e use of partial differential equations In some cases the linear integral transformation

£(x) = J A(x, \)(*),

K M ) l e = 0 =V>W,

where ip(t) and ip(t) are independent of one another random processes of the white noise type with the intensities 4 abd c^,, correspondingly. For any 6 > 0 spectral densities and, also, cross-spectral densities may be written in the form:

/««(w;0)

=

fvv(u>) 9) = e'

[4cos 2(Va6

u0) + £ c\sm 2(y/ab

c% cos 2(Vâb

u>6) H— c 2 s i n 2 ( V a 6 u>0)

/ « » ( w ; 0 ) = - / „ « ( « ; 0) = Ô

e

uO)

(11)

— 2u> 8

The normalized auto-correlation functions of the processes u(9, r), v{6, r ) and cross-correlation functions, corresponding to the spectral densities (11) can be presented as

Puu(r-, 6) = e"^ 8* Pw(t-,9)

[Au(6) +

2 =_ e e-r /se [AV(B) +

Bu{6)chf3r],

(12)

Bv{6)ch^r],

(13)

/»««(r; 0) = Pvu(T; 0) = e ~ T 2 / * e c u v ( 0 ) s h p T , where ¡3 = V a t / 2 , and A„(