Spatiotemporal random fields: theory and applications
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Amsterdam ; Oxford ; Cambridge
Elsevier
[2017]
|
| Ausgabe: | Second edition |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Vorauflage unter dem Titel: "Random field models in earth sciences" Literaturverzeichnis Seite 643-652 |
| Beschreibung: | xvii, 677 Seiten Diagramme |
| ISBN: | 9780128030127 0128030127 |
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| 240 | 1 | 0 | |a Random field models in earth sciences |
| 245 | 1 | 0 | |a Spatiotemporal random fields |b theory and applications |c George Christakos (Department of Geography, San Diego State University, San Diego, California, USA; Institute of Islands and Coastal Ecosystems, Ocean College, Zhejiang University, Zhoushan, Zhejiang, China) |
| 250 | |a Second edition | ||
| 264 | 1 | |a Amsterdam ; Oxford ; Cambridge |b Elsevier |c [2017] | |
| 264 | 4 | |c © 2017 | |
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| 336 | |b txt |2 rdacontent | ||
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Datensatz im Suchindex
| _version_ | 1804177046844211200 |
|---|---|
| adam_text | Contents
Preface xv
CHAPTER I Space, Time, Space-Time, Randomness, and Probability 1
1 Introduction 1
2 Space—Time Continuum and Kolmogorov Probability Space 4
2 1 Space—Time Arguments: Points, Lags, Separations, and Metrics 4
2 2 Transformations and Invariance in Space—Time 17
2 3 Space—Time Interpretations 23
2 4 Functions of Space—Time Arguments 26
3 Random Variables in Space—Time 31
3 1 Kolmogorov’s Probability Theory 31
3 2 Useful Inequalities 35
3 3 Convergence of Random Variable Sequences 37
CHAPTER II Spatiotemporal Random Fields 39
1 Introduction 40
1 1 The Space—Time Component 41
1 2 The Randomness Component 42
2 Characterization of Scalar Spatiotemporal Random Fields 42
2 1 Probabilistic Structure 43
2 2 The Characteristic Function 50
2 3 Spatiotemporal Variability Functions: Complete (or Full) and Partial 51
2 4 Analysis in the Spectral Domain 56
2 5 Data-Independent Spatiotemporal Variability Function 57
2 6 Some Noticeable Special Cases of the Spatiotemporal Random
Field Theory 59
2 7 Space—Time Separability 60
3 Physical Insight Behind the Random Field Concept 61
3 1 Random Field Realizations 61
3 2 Probable Versus Actual 63
3 3 Probability and the Observation Effect 64
3 4 Self-consistency and Physical Fidelity 65
4 Geometry of Spatiotemporal Random Fields 69
5 Vector Spatiotemporal Random Fields 70
6 Complex Spatiotemporal Random Fields 73
7 Classifications of the Spatiotemporal Random Field Model 73
7 1 First Classification: Discrete Versus Continuous Arguments 74
7 2 Second Classification: Scalar Versus Vector Random Fields and Arguments 74
vii
viii CONTENTS
7 3 Third Classification: Probability Law Shapes 74
7 4 Fourth Classification: Space—Time Variability 75
7 5 Fifth Classification: Spatiotemporal Random Field Memory Versus
Independence 77
8 Closing Comments 78
8 1 The Methodological Importance of Space—Time 78
82A Conceptual Meeting Point for Modelers and Experimentalists 80
8 3 There Is No Assumptionless Modeling 81
CHAPTER III Space-Time Metrics 83
1 Basic Notions 83
1 1 Formal and Physical Aspects of Space—Time Metric Determination 84
1 2 Space—Time Metric Forms 87
1 3 Derived Space—Time Metrics 92
1 4 Space—Time Metric Differentials 93
1 5 Specifying Space—Time Relationships in the Covariance Function 95
2 Covariance Differential Formulas 100
3 Space—Time Metric Determination From Physical Considerations 106
4 Examples 108
5 Concerning the Zeta Coefficients 117
6 Closing Comments 118
CHAPTER IV Space-Time Correlation Theory 121
1 Focusing on Space—Time Variability Functions 121
1 1 Basics of Space—Time Correlation Theory 122
1 2 Physical Investigations Based on Space—Time Correlation Theory 123
2 Space—Time Variability Functions in Terms of Scalar Space—Time Statistics 124
2 1 Locality: One-Point Space—Time Variability Functions 125
2 2 Nonlocality: Omnidirectional Two-Point Space—Time Variability
Functions 127
2 3 Nonlocality: Direction-Specific Space—Time Variability Functions 133
2 4 Physical Considerations and Assumptions of Space—Time Variability
Functions 133
2 5 Formal and Physical Covariance Permissibility 137
3 Basic Properties of Covariance Functions 139
4 Cross—Space—Time Variability Functions 141
5 Correlation of Gaussian and Related Spatiotemporal Random Fields 144
5 1 General Considerations 144
5 2 Gaussian Properties 144
CONTENTS ix
6 Correlation Theory of Complex Spatiotemporal Random Fields 146
6 1 Basic Notions 146
6 2 Other Types of Complex Covariance Functions 148
6 3 Gaussian Complex Spatiotemporal Random Fields 151
6 4 Complex-Valued Versus Real-Valued Covariance Functions
of Space—Time Homostationary Random Fields 152
6 5 Some Methodological Considerations 153
CHAPTER V Transformations of Spatiotemporal Random Fields 155
1 Introduction 155
2 Fourier Transformation 157
2 1 Characteristic Functions 157
2 2 Harmonizable Random Fields and Covariance Functions 158
2 3 Transfer Function and Evolutionary Mean Power 165
2 4 Fourier Transform of Vector Spatiotemporal Random Fields 167
3 Space Transformation 168
3 1 Basic Notions 168
3 2 Space Transformation of Spatiotemporal Random Fields 173
3 3 Space Transformation for Spatiotemporal Variability Functions 175
3 4 Space Transformation in the Simulation of Spatiotemporal
Random Fields 177
3 5 Space Transformation in the Solution of Stochastic Partial Differential
Equation 180
4 The Traveling Transformation 180
4 1 Basic Notions 181
4 2 Determination of the Traveling Vector 186
4 3 Traveling Transformation in Spatiotemporal Random Field Estimation:
The Space—Time Projection Technique 195
5 Closing Comments 201
CHAPTER VI Geometrical Properties of Spatiotemporal Random Fields 203
1 Introduction 203
2 Stochastic Convergence 204
3 Stochastic Continuity 208
3 1 Basic Types of Stochastic Continuity 209
3 2 Equivalence, Modification, and Separability 212
4 Stochastic Differentiation 216
4 1 Basic Notation and Definitions 217
4 2 Covariances of Random Field Derivatives 223
4 3 Mean Squarely Differentiability Conditions 227
4 4 Almost Surely Differentiability Conditions 231
X
CONTENTS
5 The Central Limit Theorem 233
6 Stochastic Integration 234
CHAPTER VII Auxiliary Hypotheses of Spatiotemporal Variation 239
1 Introduction 239
1 1 Hypothesis 1: Homostationarity 241
1 2 Hypothesis 2: Isostationarity 243
1 3 Hypothesis 3: Heterogeneity 245
1 4 Hypothesis 4: Ergodicity 246
1 5 Hypothesis 5: Separability 248
1 6 Hypothesis 6: Symmetry 249
1 7 Hypothesis 7: Locational Divergence 249
2 Space—Time Homostationarity 250
2 1 Omnidirectional Spatiotemporal Variability Functions 251
2 2 Direction-Specific Spatiotemporal Covariance Function 255
2 3 Anisotropic Features 255
2 4 Spatiotemporal Variogram and Structure Functions: Omnidirectional
and Direction Specific 257
3 Spectral Representations of Space—Time Homostationarity 260
3 1 Spectral Functions of Space—Time Homostationary Random
Fields 261
3 2 Properties of the Spectral Density Function 266
3 3 Partial Spectral Representations 268
3 4 More on Dispersion Relations 272
3 5 Separability Aspects 273
4 The Geometry of Space—Time Homostationarity 276
4 1 Differentiation Formulas: Physical and Spectral Domains 276
4 2 Stochastic Continuity and Differentiability 285
4 3 Spatiotemporal Random Field Integrability 296
5 Spectral Moments and Linear Random Field Transformations 297
CHAPTER VIII Isostationary Scalar Spatiotemporal Random Fields 303
1 Introduction 303
1 1 Basic Considerations 303
1 2 Power-Law Correlations 309
1 3 Physical Considerations of Variogram Functions 313
2 Relationships Between Covariance Derivatives and Space—Time
Isostationarity 314
3 Higher-Order Spatiotemporal Variogram and Structure Functions 319
4 Separable Classes of Space—Time Isostationary Covariance Models 320
5 A Survey of Space—Time Covariance Models 324
CONTENTS xi
6 Scales of Spatiotemporal Dependence and the Uncertainty Principle 329
6 1 Scales for Spatiotemporal Random Fields With Restricted Space—Time
Variability 330
6 2 Relationships Between Physical and Spectral Domains: The Uncertainty
Principle 334
7 On the Ergodicity Hypotheses of Spatiotemporal Random Fields 336
CHAPTER IX Vector and Multivariate Random Fields 347
1 Introduction 347
2 Homostationary and Homostationarily Connected Cross—Spatiotemporal
Variability Functions and Cross—Spectral Density Functions 349
2 1 Basic Notions and Interpretations 350
2 2 Geometry of Vector Spatiotemporal Random Fields 355
3 Some Special Cases of Covariance Functions 356
4 Solenoidal and Potential Vector Spatiotemporal Random Fields 362
5 Partial Cross-Covariance and Cross-Spectral Functions 365
6 Higher-Order Cross—Spatiotemporal Variability Functions 366
7 Isostationary Vector Spatiotemporal Random Fields 369
7 1 Direct (Lag-Based) Space—Time Isostationarity 369
7 2 Composite Lag-Field—Based Space—Time Isostationarity 372
7 3 Links With Solenoidal and Potential Spatiotemporal
Random Fields 378
8 Effective Distances and Periods 381
CHAPTER X Special Classes of Spatiotemporal Random Fields 383
1 Introduction 383
2 Frozen Spatiotemporal Random Fields and Taylor’s Hypothesis 384
2 1 Basic Notions 385
2 2 Spectral Domain Analysis 389
2 3 Differential Equation Representations 391
2 4 Extensions of the Frozen Random Field Model 395
2 5 Integrals of Frozen Spatiotemporal Random Fields 399
2 6 Vector Frozen Spatiotemporal Random Fields 399
3 Plane-Wave Spatiotemporal Random Fields 400
4 Lognormal Spatiotemporal Random Fields 402
5 Spherical Spatiotemporal Random Fields 402
6 Lagrangian Spatiotemporal Random Fields 407
CHAPTER XI Construction of Spatiotemporal Probability Laws 409
1 Introduction 409
2 Direct Probability Density Model Construction Techniques 411
2 1 The Independency Techniques 412
xii CONTENTS
2 2 The Spherical Symmetry Technique 412
2 3 The Transformation Technique 413
3 Factora-Based Probability Density Model Construction Techniques 414
4 Copula-Based Probability Density Model Construction Techniques 418
5 Stochastic Differential Equation—Based Probability Density Model
Construction Techniques 421
5 1 The Transformation of Variables Approach 422
5 2 The Characteristic Function Approach 425
5 3 The Functional Approach 426
6 Bayesian Maximum Entropy—Based Multivariate Probability Density
Model Construction Techniques 428
7 Methodological and Technical Comments 431
CHAPTER XII Spatiotemporal Random Functionals 433
1 Continuous Linear Random Functionals in the Space—Time Domain 433
1 1 Basic Notions 433
1 2 Generalized Fourier Transform 436
1 3 Space—Time Characteristic Functionals 439
1 4 Functional Derivatives 441
2 Gaussian Functionals 447
CHAPTER XIII Generalized Spatiotemporal Random Fields 455
1 Basic Notions 455
1 1 The Notion of Generalized Spatiotemporal Random Field 456
1 2 Generalized Spatiotemporal Random Field Properties and Physical
Significance 461
1 3 Homostationary Generalized Spatiotemporal Random Fields 464
2 Spatiotemporal Random Fields of Orders v/n 468
2 1 Departure From Space—Time Homostationarity 468
2 2 Space—Time Detrending 470
2 3 Ordinary Spatiotemporal Random Field-iV/u Representations of the
Generalized Spatiotemporal Random Field-vl/i 474
2 4 Determination of the Operator Qv/^ and Its Physical Significance 475
3 The Correlation Structure of Spatiotemporal Random Field-vlfx 477
3 1 Space—Time Functional Statistics 477
3 2 Generalized Spatiotemporal Covariance Functions 479
3 3 Generalized Spectral Representations and Permissibility of Generalized
Covariances 481
3 4 Generalized Covariance Function Models 484
CONTENTS xiii
4 Discrete Linear Representations of Spatiotemporal Random Fields 490
4 1 Space—Time Random Increments 490
4 2 Space—Time Variogram Analysis 495
CHAPTER XIV Physical Considerations 501
1 Spatiotemporal Variation and Laws of Change 501
2 Empirical Algebraic Equations 504
3 Physical Differential Equations 506
4 Links Between Stochastic Partial Differential Equation and Generalized
Random Fields 512
4 1 Links in Terms of the Random Functional 513
4 2 Links in Terms of the Detrending Operator 515
5 Physical Constraints in the Form of Integral Relationships, Domain
Restrictions, and Dispersion Equations 518
CHAPTER XV Permissibility in Space-Time 521
1 Concerning Permissibility 521
2 Bochnerian Analysis 522
2 1 Main Results 523
2 2 Limitations of Bochnerian Analysis 525
3 Metric Dependence 528
4 Formal and Physical Permissibility Conditions for Covariance Functions 529
4 1 Permissibility Conditions for Space—Time Homostationary Covariance
Functions 530
4 2 Permissibility Conditions for Space—Time Isostationary Covariance
Functions 532
4 3 Permissibility Conditions for Generalized Spatiotemporal Covariance
Functions 535
4 4 Permissibility Conditions for Spatiotemporal Covariance Matrices 537
5 More Consequences of Permissibility 540
CHAPTER XVI Construction of Spatiotemporal Covariance Models 543
1 Introduction 543
2 Probability Density Function—Based and Related Techniques 545
2 1 Linking Directly Covariance Models and Probability Density
Functions 545
2 2 Using Polynomial-Exponential Functions 548
2 3 Using Spectral Functions 550
3 Delta and Related Techniques 552
3 1 Basic Decomposition 552
xiv CONTENTS
3 2 Normalized Angular Spectrum Decomposition 554
3 3 Normalized Frequency Spectrum (or Coherency Function)
Decomposition 555
4 Space Transformation Technique 557
5 Physical Equation Techniques 560
5 1 Covariance Construction From Stochastic Partial Differential
Equation Representations 560
5 2 Covariance Construction From Algebraic Empirical Relationships 570
6 Closed-Form Techniques 572
7 Integral Representation Techniques 580
8 Space—Time Separation Techniques 582
9 Dynamic Formation Technique 586
10 Entropie Technique 587
11 Attribute and Argument Transformation Techniques 588
11 1 Attribute Transformation 588
11 2 Argument Transformation 589
12 Cross-Covariance Model Construction Techniques 590
13 Revisiting the Role of Physical Constraints 593
14 Closing Comments 594
Exercises 597
References 643
Appendix 653
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| id | DE-604.BV044038265 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:41:51Z |
| institution | BVB |
| isbn | 9780128030127 0128030127 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029445363 |
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| owner_facet | DE-11 DE-83 |
| physical | xvii, 677 Seiten Diagramme |
| publishDate | 2017 |
| publishDateSearch | 2017 |
| publishDateSort | 2017 |
| publisher | Elsevier |
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| spelling | Christakos, George 1956- Verfasser (DE-588)123140331 aut Random field models in earth sciences Spatiotemporal random fields theory and applications George Christakos (Department of Geography, San Diego State University, San Diego, California, USA; Institute of Islands and Coastal Ecosystems, Ocean College, Zhejiang University, Zhoushan, Zhejiang, China) Second edition Amsterdam ; Oxford ; Cambridge Elsevier [2017] © 2017 xvii, 677 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Vorauflage unter dem Titel: "Random field models in earth sciences" Literaturverzeichnis Seite 643-652 Geowissenschaften (DE-588)4020288-4 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Zufälliges Feld (DE-588)4191094-1 gnd rswk-swf Geowissenschaften (DE-588)4020288-4 s Zufälliges Feld (DE-588)4191094-1 s Mathematisches Modell (DE-588)4114528-8 s DE-604 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029445363&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Christakos, George 1956- Spatiotemporal random fields theory and applications Geowissenschaften (DE-588)4020288-4 gnd Mathematisches Modell (DE-588)4114528-8 gnd Zufälliges Feld (DE-588)4191094-1 gnd |
| subject_GND | (DE-588)4020288-4 (DE-588)4114528-8 (DE-588)4191094-1 |
| title | Spatiotemporal random fields theory and applications |
| title_alt | Random field models in earth sciences |
| title_auth | Spatiotemporal random fields theory and applications |
| title_exact_search | Spatiotemporal random fields theory and applications |
| title_full | Spatiotemporal random fields theory and applications George Christakos (Department of Geography, San Diego State University, San Diego, California, USA; Institute of Islands and Coastal Ecosystems, Ocean College, Zhejiang University, Zhoushan, Zhejiang, China) |
| title_fullStr | Spatiotemporal random fields theory and applications George Christakos (Department of Geography, San Diego State University, San Diego, California, USA; Institute of Islands and Coastal Ecosystems, Ocean College, Zhejiang University, Zhoushan, Zhejiang, China) |
| title_full_unstemmed | Spatiotemporal random fields theory and applications George Christakos (Department of Geography, San Diego State University, San Diego, California, USA; Institute of Islands and Coastal Ecosystems, Ocean College, Zhejiang University, Zhoushan, Zhejiang, China) |
| title_short | Spatiotemporal random fields |
| title_sort | spatiotemporal random fields theory and applications |
| title_sub | theory and applications |
| topic | Geowissenschaften (DE-588)4020288-4 gnd Mathematisches Modell (DE-588)4114528-8 gnd Zufälliges Feld (DE-588)4191094-1 gnd |
| topic_facet | Geowissenschaften Mathematisches Modell Zufälliges Feld |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029445363&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT christakosgeorge randomfieldmodelsinearthsciences AT christakosgeorge spatiotemporalrandomfieldstheoryandapplications |