Spatiotemporal Random Fields: Theory and Applications, Second Edition, provides readers with a new and updated edition of the text that explores the application of spatiotemporal random field models to problems in ocean, earth, and atmospheric sciences, spatiotemporal statistics, and geostatistics, among others.
The new edition features considerable detail of spatiotemporal random field theory, including ordinary and generalized models, as well as space-time homostationary, isostationary and hetrogeneous approaches. Presenting new theoretical and applied results, with particular emphasis on space-time determination and interpretation, spatiotemporal analysis and modeling, random field geometry, random functionals, probability law, and covariance construction techniques, this book highlights the key role of space-time metrics, the physical interpretation of stochastic differential equations, higher-order space-time variability functions, the validity of major theoretical assumptions in real-world practice (covariance positive-definiteness, metric-adequacy etc.), and the emergence of interdisciplinary phenomena in conditions of multi-sourced real-world uncertainty.
1. 1. Introduction - p. 1 2. 2. Space–Time Continuum and Kolmogorov Probability Space - p. 4 3. 3. Random Variables in Space–Time - p. 31 4. Chapter II. Spatiotemporal Random Fields - p. 39 5. 1. Introduction - p. 40 6. 2. Characterization of Scalar Spatiotemporal Random Fields - p. 42 7. 3. Physical Insight Behind the Random Field Concept - p. 61 8. 4. Geometry of Spatiotemporal Random Fields - p. 69 9. 5. Vector Spatiotemporal Random Fields - p. 70 10. 6. Complex Spatiotemporal Random Fields - p. 73 11. 7. Classifications of the Spatiotemporal Random Field Model - p. 73 12. 8. Closing Comments - p. 78 13. Chapter III. Space–Time Metrics - p. 83 14. 1. Basic Notions - p. 83 15. 2. Covariance Differential Formulas - p. 100 16. 3. Space–Time Metric Determination From Physical Considerations - p. 106 17. 4. Examples - p. 108 18. 5. Concerning the Zeta Coefficients - p. 117 19. 6. Closing Comments - p. 118 20. Chapter IV. Space–Time Correlation Theory - p. 121 21. 1. Focusing on Space–Time Variability Functions - p. 121 22. 2. Space–Time Variability Functions in Terms of Scalar Space–Time Statistics - p. 124 23. 3. Basic Properties of Covariance Functions - p. 139 24. 4. Cross–Space–Time Variability Functions - p. 141 25. 5. Correlation of Gaussian and Related Spatiotemporal Random Fields - p. 144 26. 6. Correlation Theory of Complex Spatiotemporal Random Fields - p. 146 27. Chapter V. Transformations of Spatiotemporal Random Fields - p. 155 28. 1. Introduction - p. 155 29. 2. Fourier Transformation - p. 157 30. 3. Space Transformation - p. 168 31. 4. The Traveling Transformation - p. 180 32. 5. Closing Comments - p. 201 33. Chapter VI. Geometrical Properties of Spatiotemporal Random Fields - p. 203 34. 1. Introduction - p. 203 35. 2. Stochastic Convergence - p. 204 36. 3. Stochastic Continuity - p. 208 37. 4. Stochastic Differentiation - p. 216 38. 5. The Central Limit Theorem - p. 233 39. 6. Stochastic Integration - p. 234 40. Chapter VII. Auxiliary Hypotheses of Spatiotemporal Variation - p. 239 41. 1. Introduction - p. 239 42. 2. Space–Time Homostationarity - p. 250 43. 3. Spectral Representations of Space–Time Homostationarity - p. 260 44. 4. The Geometry of Space–Time Homostationarity - p. 276 45. 5. Spectral Moments and Linear Random Field Transformations - p. 297 46. Chapter VIII. Isostationary Scalar Spatiotemporal Random Fields - p. 303 47. 1. Introduction - p. 303 48. 2. Relationships Between Covariance Derivatives and Space–Time Isostationarity - p. 314 49. 3. Higher-Order Spatiotemporal Variogram and Structure Functions - p. 319 50. 4. Separable Classes of Space–Time Isostationary Covariance Models - p. 320 51. 5. A Survey of Space–Time Covariance Models - p. 324 52. 6. Scales of Spatiotemporal Dependence and the Uncertainty Principle - p. 329 53. 7. On the Ergodicity Hypotheses of Spatiotemporal Random Fields - p. 336 54. Chapter IX. Vector and Multivariate Random Fields - p. 347 55. 1. Introduction - p. 347 56. 2. Homostationary and Homostationarily Connected Cross–Spatiotemporal Variability Functions and Cross–Spectral Density Functions - p. 349 57. 3. Some Special Cases of Covariance Functions - p. 356 58. 4. Solenoidal and Potential Vector Spatiotemporal Random Fields - p. 362 59. 5. Partial Cross-Covariance and Cross-Spectral Functions - p. 365 60. 6. Higher-Order Cross–Spatiotemporal Variability Functions - p. 366 61. 7. Isostationary Vector Spatiotemporal Random Fields - p. 369 62. 8. Effective Distances and Periods - p. 381 63. Chapter X. Special Classes of Spatiotemporal Random Fields - p. 383 64. 1. Introduction - p. 383 65. 2. Frozen Spatiotemporal Random Fields and Taylor's Hypothesis - p. 384 66. 3. Plane-Wave Spatiotemporal Random Fields - p. 400 67. 4. Lognormal Spatiotemporal Random Fields - p. 402 68. 5. Spherical Spatiotemporal Random Fields - p. 402 69. 6. Lagrangian Spatiotemporal Random Fields - p. 407 70. Chapter XI. Construction of Spatiotemporal Probability Laws - p. 409 71. Chapter XII. Spatiotemporal Random Functionals - p. 433 72. Chapter XIII. Generalized Spatiotemporal Random Fields - p. 455 73. 1. Basic Notions - p. 455 74. 2. Spatiotemporal Random Fields of Orders ν/μ - p. 468 75. 3. The Correlation Structure of Spatiotemporal Random Field-ν/μ - p. 477 76. 4. Discrete Linear Representations of Spatiotemporal Random Fields - p. 490 77. Chapter XIV. Physical Considerations - p. 501 78. Chapter XV. Permissibility in Space–Time - p. 521 79. Chapter XVI. Construction of Spatiotemporal Covariance Models - p. 543 80. 1. Introduction - p. 543 81. 2. Probability Density Function–Based and Related Techniques - p. 545 82. 3. Delta and Related Techniques - p. 552 83. 4. Space Transformation Technique - p. 557 84. 5. Physical Equation Techniques - p. 560 85. 6. Closed-Form Techniques - p. 572 86. 7. Integral Representation Techniques - p. 580 87. 8. Space–Time Separation Techniques - p. 582 88. 9. Dynamic Formation Technique - p. 586 89. 10. Entropic Technique - p. 587 90. 11. Attribute and Argument Transformation Techniques - p. 588 91. 12. Cross-Covariance Model Construction Techniques - p. 590 92. 13. Revisiting the Role of Physical Constraints - p. 593 93. 14. Closing Comments - p. 594 94. Exercises - p. 597 95. References - p. 643 96. Appendix. Useful Mathematical Quantities, Functions, and Formulas - p. 653 97. Index - p. 665
Spatiotemporal Random Fields: Theory and Applications, Second Edition, provides readers with a new and updated edition of the text that explores the application of spatiotemporal random field models to problems in ocean, earth, and atmospheric sciences, spatiotemporal statistics, and geostatistics, among others.
The new edition features considerable detail of spatiotemporal random field theory, including ordinary and generalized models, as well as space-time homostationary, isostationary and hetrogeneous approaches. Presenting new theoretical and applied results, with particular emphasis on space-time determination and interpretation, spatiotemporal analysis and modeling, random field geometry, random functionals, probability law, and covariance construction techniques, this book highlights the key role of space-time metrics, the physical interpretation of stochastic differential equations, higher-order space-time variability functions, the validity of major theoretical assumptions in real-world practice (covariance positive-definiteness, metric-adequacy etc.), and the emergence of interdisciplinary phenomena in conditions of multi-sourced real-world uncertainty.
1. 1. Introduction - p. 1 2. 2. Space–Time Continuum and Kolmogorov Probability Space - p. 4 3. 3. Random Variables in Space–Time - p. 31 4. Chapter II. Spatiotemporal Random Fields - p. 39 5. 1. Introduction - p. 40 6. 2. Characterization of Scalar Spatiotemporal Random Fields - p. 42 7. 3. Physical Insight Behind the Random Field Concept - p. 61 8. 4. Geometry of Spatiotemporal Random Fields - p. 69 9. 5. Vector Spatiotemporal Random Fields - p. 70 10. 6. Complex Spatiotemporal Random Fields - p. 73 11. 7. Classifications of the Spatiotemporal Random Field Model - p. 73 12. 8. Closing Comments - p. 78 13. Chapter III. Space–Time Metrics - p. 83 14. 1. Basic Notions - p. 83 15. 2. Covariance Differential Formulas - p. 100 16. 3. Space–Time Metric Determination From Physical Considerations - p. 106 17. 4. Examples - p. 108 18. 5. Concerning the Zeta Coefficients - p. 117 19. 6. Closing Comments - p. 118 20. Chapter IV. Space–Time Correlation Theory - p. 121 21. 1. Focusing on Space–Time Variability Functions - p. 121 22. 2. Space–Time Variability Functions in Terms of Scalar Space–Time Statistics - p. 124 23. 3. Basic Properties of Covariance Functions - p. 139 24. 4. Cross–Space–Time Variability Functions - p. 141 25. 5. Correlation of Gaussian and Related Spatiotemporal Random Fields - p. 144 26. 6. Correlation Theory of Complex Spatiotemporal Random Fields - p. 146 27. Chapter V. Transformations of Spatiotemporal Random Fields - p. 155 28. 1. Introduction - p. 155 29. 2. Fourier Transformation - p. 157 30. 3. Space Transformation - p. 168 31. 4. The Traveling Transformation - p. 180 32. 5. Closing Comments - p. 201 33. Chapter VI. Geometrical Properties of Spatiotemporal Random Fields - p. 203 34. 1. Introduction - p. 203 35. 2. Stochastic Convergence - p. 204 36. 3. Stochastic Continuity - p. 208 37. 4. Stochastic Differentiation - p. 216 38. 5. The Central Limit Theorem - p. 233 39. 6. Stochastic Integration - p. 234 40. Chapter VII. Auxiliary Hypotheses of Spatiotemporal Variation - p. 239 41. 1. Introduction - p. 239 42. 2. Space–Time Homostationarity - p. 250 43. 3. Spectral Representations of Space–Time Homostationarity - p. 260 44. 4. The Geometry of Space–Time Homostationarity - p. 276 45. 5. Spectral Moments and Linear Random Field Transformations - p. 297 46. Chapter VIII. Isostationary Scalar Spatiotemporal Random Fields - p. 303 47. 1. Introduction - p. 303 48. 2. Relationships Between Covariance Derivatives and Space–Time Isostationarity - p. 314 49. 3. Higher-Order Spatiotemporal Variogram and Structure Functions - p. 319 50. 4. Separable Classes of Space–Time Isostationary Covariance Models - p. 320 51. 5. A Survey of Space–Time Covariance Models - p. 324 52. 6. Scales of Spatiotemporal Dependence and the Uncertainty Principle - p. 329 53. 7. On the Ergodicity Hypotheses of Spatiotemporal Random Fields - p. 336 54. Chapter IX. Vector and Multivariate Random Fields - p. 347 55. 1. Introduction - p. 347 56. 2. Homostationary and Homostationarily Connected Cross–Spatiotemporal Variability Functions and Cross–Spectral Density Functions - p. 349 57. 3. Some Special Cases of Covariance Functions - p. 356 58. 4. Solenoidal and Potential Vector Spatiotemporal Random Fields - p. 362 59. 5. Partial Cross-Covariance and Cross-Spectral Functions - p. 365 60. 6. Higher-Order Cross–Spatiotemporal Variability Functions - p. 366 61. 7. Isostationary Vector Spatiotemporal Random Fields - p. 369 62. 8. Effective Distances and Periods - p. 381 63. Chapter X. Special Classes of Spatiotemporal Random Fields - p. 383 64. 1. Introduction - p. 383 65. 2. Frozen Spatiotemporal Random Fields and Taylor's Hypothesis - p. 384 66. 3. Plane-Wave Spatiotemporal Random Fields - p. 400 67. 4. Lognormal Spatiotemporal Random Fields - p. 402 68. 5. Spherical Spatiotemporal Random Fields - p. 402 69. 6. Lagrangian Spatiotemporal Random Fields - p. 407 70. Chapter XI. Construction of Spatiotemporal Probability Laws - p. 409 71. Chapter XII. Spatiotemporal Random Functionals - p. 433 72. Chapter XIII. Generalized Spatiotemporal Random Fields - p. 455 73. 1. Basic Notions - p. 455 74. 2. Spatiotemporal Random Fields of Orders ν/μ - p. 468 75. 3. The Correlation Structure of Spatiotemporal Random Field-ν/μ - p. 477 76. 4. Discrete Linear Representations of Spatiotemporal Random Fields - p. 490 77. Chapter XIV. Physical Considerations - p. 501 78. Chapter XV. Permissibility in Space–Time - p. 521 79. Chapter XVI. Construction of Spatiotemporal Covariance Models - p. 543 80. 1. Introduction - p. 543 81. 2. Probability Density Function–Based and Related Techniques - p. 545 82. 3. Delta and Related Techniques - p. 552 83. 4. Space Transformation Technique - p. 557 84. 5. Physical Equation Techniques - p. 560 85. 6. Closed-Form Techniques - p. 572 86. 7. Integral Representation Techniques - p. 580 87. 8. Space–Time Separation Techniques - p. 582 88. 9. Dynamic Formation Technique - p. 586 89. 10. Entropic Technique - p. 587 90. 11. Attribute and Argument Transformation Techniques - p. 588 91. 12. Cross-Covariance Model Construction Techniques - p. 590 92. 13. Revisiting the Role of Physical Constraints - p. 593 93. 14. Closing Comments - p. 594 94. Exercises - p. 597 95. References - p. 643 96. Appendix. Useful Mathematical Quantities, Functions, and Formulas - p. 653 97. Index - p. 665
