Table Of ContentLecture Notes ni
lortnoC dna
noitamrofnI Sciences
detidE yb .V.A nanhsirkalaB dna amohT.M
45
.M Arat6
raeniL Stochastic Systems
htiw Constant Coefficients
A hcaorppA
Statistical
galreV-regnirpS
Berlin Heidelberg New kroY 1982
Series Editors
A.V. Balakrishnan • M. Thoma
Advisory Board
L D. Davisson • A. G. .J MacFarlane • H. Kwakernaak
.J L Massey • Ya. Z. Tsypkin • A. .J Viterbi
Author
.M Arat6
Budapest 9111
ir&vr6heF u 921
yragnuH
ISBN 3-540-12090-4 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-12090-4 Springer-Verlag NewYork Heidelberg Berlin
Library of Congress Cataloging in Publication Data
Arat6, M. ,)s~yt~M( 193|-
Linear stochastic systems with constant coefficients.
(Lecture notes in control and information sciences ; 45)
.:yhpargoilbiB .p
Includes index.
.1 Stochastic differential equations. .2 Stochastic systems.
.I Title. .II Series
QA274.23.A72 1982 519.2 82-19490
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© Springer-Verlag Berlin Heidelberg 1982
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PREFACE
Exactly twenty years ago I finished my "candidate" dissertation
at the Moscow State University on statistical problems of
multidimensional Gaussian Markovian stationary processes. The problem
to find the exact distributions of estimators was posed me by A. N.
Kolmogorov in 1959. In those years we also examined the Chandler
wobble of the Earth,s rotation. The calculations which we made on an
M-4 computer now can be carried out without troubles on a calculator
or a personal computer.
The current two decades have witnessed an exponential growth of
literature on statistics of stochastic processes. A large number of
theoretical models appeared, but is seems that there appears to be an
ever-widening gap between theory and applications in the area of
statistical inference of stochastic processes. The aim of this small
book is to attempt to reduce this gap by directing the interest of
future researchers to the application aspects of stochastic processes
on one side, and to prove that there doesnot exist separately time
series analysis (classical statistical treatment) of discrete
processes and dynamical treatment of continuous time processes, on
the other hand.
Many of the results presented here will be appearing in book form
for the first time. This is a research book written for specialists in
the common area of applications of statistics and mathematical
statistics. The graduate level students should find the book useful.
The topics in the book have been divided into three parts. The
first part (Chapter )I discusses some applications which can convince
the reader in the useful conception of Ito's integral in that form
which was developed in the fifties, using the Wiener process instead
of "white noise" process. AS a "surprising" novelty we discuss the
exact solution of the estimation problem with additive noise. It
turns out that in the constant coefficient case the Riccati equation
has an explicite solution. The second part (Chapter )2 discusses the
so called elementary Gaussian, i.e., stationary and Markov processes
in the discrete and continuous time case as the solutions of
difference and differential equations, respectively. The connection
between spectral theory and stochastic equations is also shown. The
many new developments, with their tremendous range from fundamental
theory to specific applications made it difficult to confine
VI
ourselves to an elementary treatment every where. The third part
(Chapters 3 and )4 contain the statistical investigations of linear
stochastic systems, on the basis of continuous time processes as it
was proposed by A. N. Kolmogorov in the late forties.
The needed mathematical background is given in a very short
Appendix. Those who require a more complete treatment and further
generalizatons from the mathematical point of view may be referred to
the book of Liptser, Shlryaev or Basawa, Pr. Rao. There I agree with
the famous intention of M. Bartlett: "It would, however, be a pity if
applied mathematicians or statisticians were put off from using some
of the mathematical and statistical techniques available because they
did not feel able to absorb all the more pure mathematical theory. As
a statistician I find it at times rather exasperating when the
mathematics of stochastic processes tends to become so abstract; time
spent in wrestling with it can hardly be spared unless, as of course
mathematics is best fitted to do, it deepens one's perception of the
overall theoretical picture in the probabilistic and statistical
sense."
In this book our main purpose is to investigate the most simple
dynamical stochastic models, the linear stochastic differentlal
(difference) systems with constant coefficients. At the first moment
it seems that such processes are not more interesting, we know
everything about them and there are manymore sophisticated models
which have been studied. From the statistical point of view this is
not the case as we have many unsolved problems till now. The
elementary processes, if all their components are observable, have
the advantage that a set of sufficient statistics exists, which is not
the case when we have an elementary process with additive noise and
this latter case will be studied here in detail.
I believe that the basic premise in model building is that
complicated systems, and all real systems are, as a rule, complicated,
do not always need complicated models. It is advisable to fit
relatively simple models to the given data and to increase the
complexity of the model only if the simpler model is not satisfactory.
Of course models with a degree of complexity beyond a certain level
often perform poorly and only in this case we shall use more
complicated models.
In constructing models for the given data our goal is to
understand the process and summarize the entire available set of
observations. We say that it is not enough that a model be consistent
V
with the numerical observations, we want the model to be the
"simplest." A model that has too many parameters and variables is
considered unsatisfactory.
The initial basis of this work is my dissertation written in
Moscow and the lectures that I gave at the Budapest University L.
EStv~s in the statistics of stochastic processes. In 1974 with A.
Benozdr, A. Kr/unli, and J. Pergel we wrote a preprint in this matter
and planned to write it up in book form, but then we learned that the
book of Liptser and Shiryaev came out and the idea was dropped. In
1981A. Kolmogorov encouraged me to summarize some results, which are
not well known and widespread, in this formal manner. I tried to give
everywhere exact results, as they are available only in a few cases
in statistics, and not asymptotic results, but the connection between
them is discussed. Later in the second volume I plan to return to the
computer programs, data analysis and exercises and to more computer
applications in this field.
The "decimal system" of numbering the chapters, sections and
subsections has been used. Equations have been numbered separately
for each section. The bibliography is cited in [ ] if it is a book
and with the author and reference year in case of a paper (e.g.,
Bartlett (1951)).
I am convinced that even in the investigation of linear stochastic
systems there are many gaps and even more in this book, a number of
methods and results are not included, which may have been successful
in practice, and this is my own responsibility.
ACKNOWLEDGMENTS
I would like to than kfirst of all my teacher Andrej Nikolaevic
Kolmogorov from whom I learned stochastic processes and the
statistical approach. I thank him not only for the explanation of an
entirely new field of research but for all the continuous help and
encouragement he has provided.
I am very grateful to my Hungarian friends, A. Benczdr, A. Kramli,
and J. Pergel with whom I discussed many of the problems treated in
this book for a couple of years and with whom we wrote a first version
in 1974.
I would llke to thank my friends in the USSR, Ju. Rozanov, A.
Shiryaev, A. Novikov and many others for their help, when I was at the
IV
Moscow State University and Steklov Mathematical Institute,and for
encouraging me to undertake this book.
I am pleased to acknowledge the help of Ju. Prokhorov and A.
Balakrishnan for giving me the possibility to write this book at the
Probability Department of Steklov Mathematical Institute and the
System Science Departments University of California, Los Angeles,
respectively, and A. Bagchl for many discussions.
I would llke to thank Mrs.Loetitia Loberman and Ms. Ginger Nystrom
for their careful typing from a marginally legible manuscript, and
Miss Andrea Bajusz for typing the final version of this book.
CONTENTS
Chapter I
Case Studies, Problems and Their Statistical Investigation I
1.1. Introductionary Remarks 1
1.2. The Brownian Motion 6
1.3. The Torsion Pendulum and Electrical Circuits 13
1.4. The Chandler Wobble 27
1.4.1. The Rotation of the Earth 27
1.4.2. The Mathematical Description and Statistical
Investigation 33
1.5. System Descrition 44
1.6. Measurement Analysis in Computer Systems 52
1.6.1. Measurement of Performance 52
1.6.2. Round off Errors in Solutions of Ordinary
DifferentialEquations 60
1.7. Sunspot Activity 70
1.8. Kalman Filtering with Explicite Solutions (Signal Plus
Noise Case) 79
Chapter 2
Elementary Gaussian Processes 95
2.1. Processes with Discrete Time 95
2.1.1. Main Theorems 95
2.1.2. Structure of Degenerate and Deterministic Processes 100
2.1.3. Spectral Representation of Processes, Autoregressive
and Moving Average Type Processes 103
2.2. Processes with Continuous Time 118
2.2.1. Main Theorems 118
2.2.2. Stationary Oaussian Processes with Rational Spectral
Density Functions 127
2.3. Density Functions and Sufficient Statistics 137
2.3.1. The Discrete Time Case 137
2.3.2. Some Auxiliary Theorems 144
2.3.3. The Radon- Nikodym Derivatives with Respect
to the Wiener Measure 151
2.3.4. Unobservable Components 159
IIIV
Chapter 3
The Maximum Likelihood Estimators and their Distributions
in the One Dimensional Case 169
3.1. The Basic Principles of Statistical Estimation Theory 169
3.2. The Unknown Mean 176
3.3. The Unknown 177
3.4. Two Unknown Parameters 185
3.5. The Discrete Time Case 193
3.5.1. Single Parameters 193
3.5.2. Distribution of the Derivatives of Likelihood
Function 198
3.5.3. Asymptotic Distribution of Maximum Likelihood
Estimates 207
3.5.4. Results Obtained for Discrete Analogues of the
Continuous-tlme Case 210
3.6. The Moments of Estimators and Asymptotic Theory 214
3.6.1. Sequential Estimation 218
Chapter 4
Multi-Dimensional Processes 221
4.1. The Complex Process 221
4.2. Construction of Confidence Intervals for the Parameter A 225
4.3. Estimation of the Period 236
4.4. The Uknown Mean 239
4.4.1. The Complex Process 239
4.4.2. Linear Regression 240
4.4.3. Correct Estimates 242
4.4.4. Pitman,s Estimates 245
4.4.5. Admissible Estimates 247
4.4.6. Minimax Weigths in Trend Detection 250
4.5. Real Roots and Other Special Cases 253
4.6. Multi-Dimensional Case, Asymptotic Theory 258
Appendix A
Linear Differential Equations with Constant Coefficients 263
I. Preliminary Definitions and Notations, Matrices 263
XI
.2 Linear Systems with Constant Coefficients 268
Appendix B
Probability Bakcground 272
.~ Gaussian Systems 272
.2 Some Basic Concepts in Probability Theory 280
General Bibliography
Books 288
References 294
Authors' index 308
CHAPTER 1
C_ASE STUDIES~ PROBLEM ~ AND THEIR STATISTICAL INVESTIGATION
1.1 Introductionary Remarks
The statistical theory of stochastic processes may be regarded as the
main tool to find the connection between mathematical investigations
of stochastic processes, on one side, and such applications as stoch-
astic control, optimization, filtering, information processes and
communication networks, on the other side. The statistical examination
of linear dynamical systems with constant coefficients has its beginning
in the forties. Both theoretical results and concrete practical appli-
cations have shown an accelerated progress in the last twenty years.
Our book may seem in some sense old fashioned, as we are using mostly
the classical terminology of mathematical statistics and try to get the
results in the framework of this theory. Although we do not deny the
influence of communication theory, nonlinear filtering or information
processing, a systematic discussion will be given only in estimation
theory of these processes, what is called identification in engineering
practice, and most of the examples discussed will be of scientific
type than technological. The classical theory of mathematical statistics
in discrete time stochastic processes (time series) under this pressure
of applications became closely connected with the investigations of
continuous time processes. And it gives considerable success as the
concrete examples illustrate this statement. The first order auto-
regressive process ~(n), which fulfils the stochastic difference equa-
tion
(i.i.i) ~(n) =~(n-l) + e(n),
where ~(n) is a Gaussian white noise, having two unknown parameters
,9 ~ = D2~(n), has been investigated for a long time. There were many
attempts to find the distribution of the estimators of the unknown
parameter .9 It turned out (see Ch. 3. Section 3, Theorem i) that exact
distribution we get only in the continuous time case, when ~(t) is the
solution of the stochastic differential equation
(I.i.i') d~(t) = - l~(t)dt + dw(t),
where w(t) is a Wiener process, = 9 ~l.&t
