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Spatiotemporal Random Fields

Spatiotemporal Random FieldsTheory and Applications

Copyright

Dedication

Contents

Preface

I - SPACE, TIME, SPACE–TIME, RANDOMNESS, AND PROBABILITY

1. INTRODUCTION

2. SPACE–TIME CONTINUUM AND KOLMOGOROV PROBABILITY SPACE

2.1 SPACE–TIME ARGUMENTS: POINTS, LAGS, SEPARATIONS, AND METRICS

2.2 TRANSFORMATIONS AND INVARIANCE IN SPACE–TIME

2.3 SPACE–TIME INTERPRETATIONS

2.4 FUNCTIONS OF SPACE–TIME ARGUMENTS

3. RANDOM VARIABLES IN SPACE–TIME

3.1 KOLMOGOROV'S PROBABILITY THEORY

3.2 USEFUL INEQUALITIES

3.3 CONVERGENCE OF RANDOM VARIABLE SEQUENCES

II - SPATIOTEMPORAL RANDOM FIELDS

1. INTRODUCTION

1.1 THE SPACE–TIME COMPONENT

1.2 THE RANDOMNESS COMPONENT

2. CHARACTERIZATION OF SCALAR SPATIOTEMPORAL RANDOM FIELDS

2.1 PROBABILISTIC STRUCTURE

2.2 THE CHARACTERISTIC FUNCTION

2.3 SPATIOTEMPORAL VARIABILITY FUNCTIONS: COMPLETE (OR FULL) AND PARTIAL

2.4 ANALYSIS IN THE SPECTRAL DOMAIN

2.5 DATA-INDEPENDENT SPATIOTEMPORAL VARIABILITY FUNCTION

2.6 SOME NOTICEABLE SPECIAL CASES OF THE SPATIOTEMPORAL RANDOM FIELD THEORY

2.7 SPACE–TIME SEPARABILITY

3. PHYSICAL INSIGHT BEHIND THE RANDOM FIELD CONCEPT

3.1 RANDOM FIELD REALIZATIONS

3.2 PROBABLE VERSUS ACTUAL

3.3 PROBABILITY AND THE OBSERVATION EFFECT

3.4 SELF-CONSISTENCY AND PHYSICAL FIDELITY

4. GEOMETRY OF SPATIOTEMPORAL RANDOM FIELDS

5. VECTOR SPATIOTEMPORAL RANDOM FIELDS

6. COMPLEX SPATIOTEMPORAL RANDOM FIELDS

7. CLASSIFICATIONS OF THE SPATIOTEMPORAL RANDOM FIELD MODEL

7.1 FIRST CLASSIFICATION: DISCRETE VERSUS CONTINUOUS ARGUMENTS

7.2 SECOND CLASSIFICATION: SCALAR VERSUS VECTOR RANDOM FIELDS AND ARGUMENTS

7.3 THIRD CLASSIFICATION: PROBABILITY LAW SHAPES

7.4 FOURTH CLASSIFICATION: SPACE–TIME VARIABILITY

7.5 FIFTH CLASSIFICATION: SPATIOTEMPORAL RANDOM FIELD MEMORY VERSUS INDEPENDENCE

8. CLOSING COMMENTS

8.1 THE METHODOLOGICAL IMPORTANCE OF SPACE–TIME

8.2 A CONCEPTUAL MEETING POINT FOR MODELERS AND EXPERIMENTALISTS

8.3 THERE IS NO ASSUMPTIONLESS MODELING

III - SPACE–TIME METRICS

1. BASIC NOTIONS

1.1 FORMAL AND PHYSICAL ASPECTS OF SPACE–TIME METRIC DETERMINATION

1.2 SPACE–TIME METRIC FORMS

1.3 DERIVED SPACE–TIME METRICS

1.4 SPACE–TIME METRIC DIFFERENTIALS

1.5 SPECIFYING SPACE–TIME RELATIONSHIPS IN THE COVARIANCE FUNCTION

2. COVARIANCE DIFFERENTIAL FORMULAS

3. SPACE–TIME METRIC DETERMINATION FROM PHYSICAL CONSIDERATIONS

4. EXAMPLES

5. CONCERNING THE ZETA COEFFICIENTS

6. CLOSING COMMENTS

IV - SPACE–TIME CORRELATION THEORY

1. FOCUSING ON SPACE–TIME VARIABILITY FUNCTIONS

1.1 BASICS OF SPACE–TIME CORRELATION THEORY

1.2 PHYSICAL INVESTIGATIONS BASED ON SPACE–TIME CORRELATION THEORY

2. SPACE–TIME VARIABILITY FUNCTIONS IN TERMS OF SCALAR SPACE–TIME STATISTICS

2.1 LOCALITY: ONE-POINT SPACE–TIME VARIABILITY FUNCTIONS

2.2 NONLOCALITY: OMNIDIRECTIONAL TWO-POINT SPACE–TIME VARIABILITY FUNCTIONS

2.3 NONLOCALITY: DIRECTION-SPECIFIC SPACE–TIME VARIABILITY FUNCTIONS

2.4 PHYSICAL CONSIDERATIONS AND ASSUMPTIONS OF SPACE–TIME VARIABILITY FUNCTIONS

2.5 FORMAL AND PHYSICAL COVARIANCE PERMISSIBILITY

3. BASIC PROPERTIES OF COVARIANCE FUNCTIONS

4. CROSS–SPACE–TIME VARIABILITY FUNCTIONS

5. CORRELATION OF GAUSSIAN AND RELATED SPATIOTEMPORAL RANDOM FIELDS

5.1 GENERAL CONSIDERATIONS

5.2 GAUSSIAN PROPERTIES

6. CORRELATION THEORY OF COMPLEX SPATIOTEMPORAL RANDOM FIELDS

6.1 BASIC NOTIONS

6.2 OTHER TYPES OF COMPLEX COVARIANCE FUNCTIONS

6.3 GAUSSIAN COMPLEX SPATIOTEMPORAL RANDOM FIELDS

6.4 COMPLEX-VALUED VERSUS REAL-VALUED COVARIANCE FUNCTIONS OF SPACE–TIME HOMOSTATIONARY RANDOM FIELDS

6.5 SOME METHODOLOGICAL CONSIDERATIONS

V - TRANSFORMATIONS OF SPATIOTEMPORAL RANDOM FIELDS

1. INTRODUCTION

2. FOURIER TRANSFORMATION

2.1 CHARACTERISTIC FUNCTIONS

2.2 HARMONIZABLE RANDOM FIELDS AND COVARIANCE FUNCTIONS

2.3 TRANSFER FUNCTION AND EVOLUTIONARY MEAN POWER

2.4 FOURIER TRANSFORM OF VECTOR SPATIOTEMPORAL RANDOM FIELDS

3. SPACE TRANSFORMATION

3.1 BASIC NOTIONS

3.2 SPACE TRANSFORMATION OF SPATIOTEMPORAL RANDOM FIELDS

3.3 SPACE TRANSFORMATION FOR SPATIOTEMPORAL VARIABILITY FUNCTIONS

3.4 SPACE TRANSFORMATION IN THE SIMULATION OF SPATIOTEMPORAL RANDOM FIELDS

3.5 SPACE TRANSFORMATION IN THE SOLUTION OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATION

4. THE TRAVELING TRANSFORMATION

4.1 BASIC NOTIONS

4.2 DETERMINATION OF THE TRAVELING VECTOR

4.3 TRAVELING TRANSFORMATION IN SPATIOTEMPORAL RANDOM FIELD ESTIMATION: THE SPACE–TIME PROJECTION TECHNIQUE

5. CLOSING COMMENTS

VI - GEOMETRICAL PROPERTIES OF SPATIOTEMPORAL RANDOM FIELDS

1. INTRODUCTION

2. STOCHASTIC CONVERGENCE

3. STOCHASTIC CONTINUITY

3.1 BASIC TYPES OF STOCHASTIC CONTINUITY

3.2 EQUIVALENCE, MODIFICATION, AND SEPARABILITY44TO BE DISTINGUISHED FROM COVARIANCE SEPARABILITY.

4. STOCHASTIC DIFFERENTIATION

4.1 BASIC NOTATION AND DEFINITIONS

4.2 COVARIANCES OF RANDOM FIELD DERIVATIVES

4.3 MEAN SQUARELY DIFFERENTIABILITY CONDITIONS

4.4 ALMOST SURELY DIFFERENTIABILITY CONDITIONS

5. THE CENTRAL LIMIT THEOREM

6. STOCHASTIC INTEGRATION

VII - AUXILIARY HYPOTHESES OF SPATIOTEMPORAL VARIATION

1. INTRODUCTION

1.1 HYPOTHESIS 1: HOMOSTATIONARITY

1.2 HYPOTHESIS 2: ISOSTATIONARITY

1.3 HYPOTHESIS 3: HETEROGENEITY

1.4 HYPOTHESIS 4: ERGODICITY

1.5 HYPOTHESIS 5: SEPARABILITY

1.6 HYPOTHESIS 6: SYMMETRY

1.7 HYPOTHESIS 7: LOCATIONAL DIVERGENCE

2. SPACE–TIME HOMOSTATIONARITY

2.1 OMNIDIRECTIONAL SPATIOTEMPORAL VARIABILITY FUNCTIONS

2.2 DIRECTION-SPECIFIC SPATIOTEMPORAL COVARIANCE FUNCTION

2.3 ANISOTROPIC FEATURES

2.4 SPATIOTEMPORAL VARIOGRAM AND STRUCTURE FUNCTIONS: OMNIDIRECTIONAL AND DIRECTION SPECIFIC

3. SPECTRAL REPRESENTATIONS OF SPACE–TIME HOMOSTATIONARITY

3.1 SPECTRAL FUNCTIONS OF SPACE–TIME HOMOSTATIONARY RANDOM FIELDS

3.2 PROPERTIES OF THE SPECTRAL DENSITY FUNCTION

3.3 PARTIAL SPECTRAL REPRESENTATIONS

3.4 MORE ON DISPERSION RELATIONS

3.5 SEPARABILITY ASPECTS

4. THE GEOMETRY OF SPACE–TIME HOMOSTATIONARITY

4.1 DIFFERENTIATION FORMULAS: PHYSICAL AND SPECTRAL DOMAINS

4.2 STOCHASTIC CONTINUITY AND DIFFERENTIABILITY

4.3 SPATIOTEMPORAL RANDOM FIELD INTEGRABILITY

5. SPECTRAL MOMENTS AND LINEAR RANDOM FIELD TRANSFORMATIONS

VIII - ISOSTATIONARY SCALAR SPATIOTEMPORAL RANDOM FIELDS

1. INTRODUCTION

1.1 BASIC CONSIDERATIONS

1.2 POWER-LAW CORRELATIONS

1.3 PHYSICAL CONSIDERATIONS OF VARIOGRAM FUNCTIONS

2. RELATIONSHIPS BETWEEN COVARIANCE DERIVATIVES AND SPACE–TIME ISOSTATIONARITY

3. HIGHER-ORDER SPATIOTEMPORAL VARIOGRAM AND STRUCTURE FUNCTIONS

4. SEPARABLE CLASSES OF SPACE–TIME ISOSTATIONARY COVARIANCE MODELS

5. A SURVEY OF SPACE–TIME COVARIANCE MODELS

6. SCALES OF SPATIOTEMPORAL DEPENDENCE AND THE UNCERTAINTY PRINCIPLE

6.1 SCALES FOR SPATIOTEMPORAL RANDOM FIELDS WITH RESTRICTED SPACE–TIME VARIABILITY

6.2 RELATIONSHIPS BETWEEN PHYSICAL AND SPECTRAL DOMAINS: THE UNCERTAINTY PRINCIPLE

7. ON THE ERGODICITY HYPOTHESES OF SPATIOTEMPORAL RANDOM FIELDS

IX - VECTOR AND MULTIVARIATE RANDOM FIELDS

1. INTRODUCTION

2. HOMOSTATIONARY AND HOMOSTATIONARILY CONNECTED CROSS–SPATIOTEMPORAL VARIABILITY FUNCTIONS AND CROSS–SPECTRAL DENSITY FUNCTIONS

2.1 BASIC NOTIONS AND INTERPRETATIONS

2.2 GEOMETRY OF VECTOR SPATIOTEMPORAL RANDOM FIELDS

3. SOME SPECIAL CASES OF COVARIANCE FUNCTIONS

4. SOLENOIDAL AND POTENTIAL VECTOR SPATIOTEMPORAL RANDOM FIELDS

5. PARTIAL CROSS-COVARIANCE AND CROSS-SPECTRAL FUNCTIONS

6. HIGHER-ORDER CROSS–SPATIOTEMPORAL VARIABILITY FUNCTIONS

7. ISOSTATIONARY VECTOR SPATIOTEMPORAL RANDOM FIELDS

7.1 DIRECT (LAG-BASED) SPACE–TIME ISOSTATIONARITY

7.2 COMPOSITE LAG-FIELD–BASED SPACE–TIME ISOSTATIONARITY

7.3 LINKS WITH SOLENOIDAL AND POTENTIAL SPATIOTEMPORAL RANDOM FIELDS

8. EFFECTIVE DISTANCES AND PERIODS

X - SPECIAL CLASSES OF SPATIOTEMPORAL RANDOM FIELDS

1. INTRODUCTION

2. FROZEN SPATIOTEMPORAL RANDOM FIELDS AND TAYLOR'S HYPOTHESIS

2.1 BASIC NOTIONS

2.2 SPECTRAL DOMAIN ANALYSIS

2.3 DIFFERENTIAL EQUATION REPRESENTATIONS

2.4 EXTENSIONS OF THE FROZEN RANDOM FIELD MODEL

2.4.1 Spectral Domain Decomposition

2.4.2 Nonstationary Representation

2.4.3 Elliptic Representation

2.4.4 Random Velocity

2.4.5 Heterogeneous Representation

2.5 INTEGRALS OF FROZEN SPATIOTEMPORAL RANDOM FIELDS

2.6 VECTOR FROZEN SPATIOTEMPORAL RANDOM FIELDS

3. PLANE-WAVE SPATIOTEMPORAL RANDOM FIELDS

4. LOGNORMAL SPATIOTEMPORAL RANDOM FIELDS

5. SPHERICAL SPATIOTEMPORAL RANDOM FIELDS

6. LAGRANGIAN SPATIOTEMPORAL RANDOM FIELDS

XI - CONSTRUCTION OF SPATIOTEMPORAL PROBABILITY LAWS

1. INTRODUCTION

2. DIRECT PROBABILITY DENSITY MODEL CONSTRUCTION TECHNIQUES

2.1 THE INDEPENDENCY TECHNIQUES

2.2 THE SPHERICAL SYMMETRY TECHNIQUE

2.3 THE TRANSFORMATION TECHNIQUE

3. FACTORA-BASED PROBABILITY DENSITY MODEL CONSTRUCTION TECHNIQUES

4. COPULA-BASED PROBABILITY DENSITY MODEL CONSTRUCTION TECHNIQUES

5. STOCHASTIC DIFFERENTIAL EQUATION–BASED PROBABILITY DENSITY MODEL CONSTRUCTION TECHNIQUES

5.1 THE TRANSFORMATION OF VARIABLES APPROACH

5.2 THE CHARACTERISTIC FUNCTION APPROACH

5.3 THE FUNCTIONAL APPROACH

6. BAYESIAN MAXIMUM ENTROPY–BASED MULTIVARIATE PROBABILITY DENSITY MODEL CONSTRUCTION TECHNIQUES

7. METHODOLOGICAL AND TECHNICAL COMMENTS

XII - SPATIOTEMPORAL RANDOM FUNCTIONALS

1. CONTINUOUS LINEAR RANDOM FUNCTIONALS IN THE SPACE–TIME DOMAIN

1.1 BASIC NOTIONS

1.2 GENERALIZED FOURIER TRANSFORM

1.3 SPACE–TIME CHARACTERISTIC FUNCTIONALS

1.4 FUNCTIONAL DERIVATIVES

2. GAUSSIAN FUNCTIONALS

XIII . GENERALIZED SPATIOTEMPORAL RANDOM FIELDS

1. BASIC NOTIONS

1.1 THE NOTION OF GENERALIZED SPATIOTEMPORAL RANDOM FIELD

1.2 GENERALIZED SPATIOTEMPORAL RANDOM FIELD PROPERTIES AND PHYSICAL SIGNIFICANCE

1.3 HOMOSTATIONARY GENERALIZED SPATIOTEMPORAL RANDOM FIELDS

2. SPATIOTEMPORAL RANDOM FIELDS OF ORDERS ν/μ

2.1 DEPARTURE FROM SPACE–TIME HOMOSTATIONARITY

2.2 SPACE–TIME DETRENDING

2.3 ORDINARY SPATIOTEMPORAL RANDOM FIELD-ν/μ REPRESENTATIONS OF THE GENERALIZED SPATIOTEMPORAL RANDOM FIELD-ν/μ

2.4 DETERMINATION OF THE OPERATOR Qν/μ AND ITS PHYSICAL SIGNIFICANCE

3. THE CORRELATION STRUCTURE OF SPATIOTEMPORAL RANDOM FIELD-ν/μ

3.1 SPACE–TIME FUNCTIONAL STATISTICS

3.2 GENERALIZED SPATIOTEMPORAL COVARIANCE FUNCTIONS

3.3 GENERALIZED SPECTRAL REPRESENTATIONS AND PERMISSIBILITY OF GENERALIZED COVARIANCES

3.4 GENERALIZED COVARIANCE FUNCTION MODELS

4. DISCRETE LINEAR REPRESENTATIONS OF SPATIOTEMPORAL RANDOM FIELDS

4.1 SPACE–TIME RANDOM INCREMENTS

4.2 SPACE–TIME VARIOGRAM ANALYSIS

XIV - PHYSICAL CONSIDERATIONS

1. SPATIOTEMPORAL VARIATION AND LAWS OF CHANGE

2. EMPIRICAL ALGEBRAIC EQUATIONS

3. PHYSICAL DIFFERENTIAL EQUATIONS

4. LINKS BETWEEN STOCHASTIC PARTIAL DIFFERENTIAL EQUATION AND GENERALIZED RANDOM FIELDS

4.1 LINKS IN TERMS OF THE RANDOM FUNCTIONAL

4.2 LINKS IN TERMS OF THE DETRENDING OPERATOR

5. PHYSICAL CONSTRAINTS IN THE FORM OF INTEGRAL RELATIONSHIPS, DOMAIN RESTRICTIONS, AND DISPERSION EQUATIONS

XV - PERMISSIBILITY IN SPACE–TIME

1. CONCERNING PERMISSIBILITY

2. BOCHNERIAN ANALYSIS

2.1 MAIN RESULTS

2.2 LIMITATIONS OF BOCHNERIAN ANALYSIS

3. METRIC DEPENDENCE

4. FORMAL AND PHYSICAL PERMISSIBILITY CONDITIONS FOR COVARIANCE FUNCTIONS

4.1 PERMISSIBILITY CONDITIONS FOR SPACE–TIME HOMOSTATIONARY COVARIANCE FUNCTIONS

4.2 PERMISSIBILITY CONDITIONS FOR SPACE–TIME ISOSTATIONARY COVARIANCE FUNCTIONS

4.3 PERMISSIBILITY CONDITIONS FOR GENERALIZED SPATIOTEMPORAL COVARIANCE FUNCTIONS

4.4 PERMISSIBILITY CONDITIONS FOR SPATIOTEMPORAL COVARIANCE MATRICES

5. MORE CONSEQUENCES OF PERMISSIBILITY

XVI - CONSTRUCTION OF SPATIOTEMPORAL COVARIANCE MODELS

1. INTRODUCTION

2. PROBABILITY DENSITY FUNCTION–BASED AND RELATED TECHNIQUES

2.1 LINKING DIRECTLY COVARIANCE MODELS AND PROBABILITY DENSITY FUNCTIONS

2.2 USING POLYNOMIAL-EXPONENTIAL FUNCTIONS

2.3 USING SPECTRAL FUNCTIONS

3. DELTA AND RELATED TECHNIQUES

3.1 BASIC DECOMPOSITION

3.2 NORMALIZED ANGULAR SPECTRUM DECOMPOSITION

3.3 NORMALIZED FREQUENCY SPECTRUM (OR COHERENCY FUNCTION) DECOMPOSITION

4. SPACE TRANSFORMATION TECHNIQUE

5. PHYSICAL EQUATION TECHNIQUES

5.1 COVARIANCE CONSTRUCTION FROM STOCHASTIC PARTIAL DIFFERENTIAL EQUATION REPRESENTATIONS

5.2 COVARIANCE CONSTRUCTION FROM ALGEBRAIC EMPIRICAL RELATIONSHIPS

6. CLOSED-FORM TECHNIQUES

7. INTEGRAL REPRESENTATION TECHNIQUES

8. SPACE–TIME SEPARATION TECHNIQUES

9. DYNAMIC FORMATION TECHNIQUE

10. ENTROPIC TECHNIQUE

11. ATTRIBUTE AND ARGUMENT TRANSFORMATION TECHNIQUES

11.1 ATTRIBUTE TRANSFORMATION

11.2 ARGUMENT TRANSFORMATION

12. CROSS-COVARIANCE MODEL CONSTRUCTION TECHNIQUES

13. REVISITING THE ROLE OF PHYSICAL CONSTRAINTS

14. CLOSING COMMENTS

USEFUL MATHEMATICAL QUANTITIES, FUNCTIONS, AND FORMULAS

1. FACTORIALS—MULTIFACTORIALS

2. GAMMA FUNCTIONS

3. BESSEL FUNCTIONS

4. GEGENBAUER POLYNOMIALS

5. THE N-DIMENSIONAL SPHERE AND ITS SURFACE AREA1

6. COORDINATE SYSTEMS IN N-DIMENSIONS

7. DELTA FUNCTIONS

8. SPACE–TIME VECTORS AND MULTIINDEXES (CONVENTIONS AND FORMULAS)

9. USEFUL INTEGRALS

10. MISCELLANEOUS

11. SPECIAL CASES—EXAMPLES

EXERCISES

CHAPTER I

CHAPTER II

CHAPTER III

CHAPTER IV

CHAPTER V

CHAPTER VI

CHAPTER VII

CHAPTER VIII

CHAPTER IX

CHAPTER X

CHAPTER XI

CHAPTER XII

CHAPTER XIII

CHAPTER XIV

CHAPTER XV

CHAPTER XVI

REFERENCES

FURTHER READING

Index

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K

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