Information Geometry

by Arwini, Khadiga, A.; Dodson, Christopher T. J.

• ISBN13: 9783540693918
• ISBN10: 3540693912
• Format: Paperback
• Copyright: 2008-11-01
• Publisher: Springer Verlag

Summary

"This volume will be useful to practising scientists and students working in the application of statistical models to real materials or to processes with perturbations of a Poisson process, a uniform process, or a state of independence for a bivariate process. We use information geometry to provide a common differential geometric framework for a wide range of illustrative applications including amino acid sequence spacings in protein chains, cryptology studies, clustering of communications and galaxies, cosmological voids, coupled spatial statistics in stochastic fibre networks and stochastic porous media, quantum chaology. Introduction sections are provided to mathematical statistics, differential geometry and the information geometry of spaces of probability density functions."--BOOK JACKET.

Table of Contents

Preface	p. V
Mathematical Statistics and Information Theory	p. 1
Probability Functions for Discrete Variables	p. 2
Bernoulli Distribution	p. 3
Binomial Distribution	p. 3
Poisson Distribution	p. 4
Probability Density Functions for Continuous Variables	p. 6
Uniform Distribution	p. 8
Exponential Distribution	p. 8
Gaussian, or Normal Distribution	p. 9
Joint Probability Density Functions	p. 9
Bivariate Gaussian Distributions	p. 10
Information Theory	p. 11
Gamma Distribution	p. 13
Introduction to Riemannian Geometry	p. 19
Manifolds	p. 20
Tangent Spaces	p. 20
Tensors and Forms	p. 22
Riemannian Metric	p. 25
Connections	p. 26
Autoparallel and Geodesic Curves	p. 29
Universal Connections and Curvature	p. 29
Information Geometry	p. 31
Fisher Information Metric	p. 32
Exponential Family of Probability Density Functions	p. 33
Statistical a-Connections	p. 34
Affine Immersions	p. 35
Weibull Distributions: Not of Exponential Type	p. 36
Gamma 2-Manifold G	p. 37
Gamma a-Connection	p. 38
Gamma a-Curvatures	p. 39
Gamma Manifold Geodesics	p. 40
Mutually Dual Foliations	p. 42
Gamma Affine Immersion	p. 42
Log-Gamma 2-Manifold L	p. 42
Log-Gamma Random Walks	p. 45
Gaussian 2-Manifold	p. 45
Gaussian Natural Coordinates	p. 47
Gaussian Information Metric	p. 47
Gaussian Mutually Dual Foliations	p. 48
Gaussian Affine Immersions	p. 48
Gaussian a-Geometry	p. 49
Gaussian a-Connection	p. 49
Gaussian a-Curvatures	p. 50
Gaussian Mutually Dual Foliations	p. 50
Gaussian Submanifolds	p. 51
Central Mean Submanifold	p. 51
Unit Variance Submanifold	p. 52
Unit Coefficient of Variation Submanifold	p. 52
Gaussian Affine Immersions	p. 52
Log-Gaussian Manifold	p. 53
Information Geometry of Bivariate Families	p. 55
McKay Bivariate Gamma 3-Manifold M	p. 55
McKay Manifold Geometry in Natural Coordinates	p. 58
McKay Densities Have Exponential Type	p. 59
McKay Information Metric	p. 59
McKay a-Geometry	p. 60
McKay a-Connection	p. 60
McKay a-Curvatures	p. 61
McKay Mutually Dual Foliations	p. 64
McKay Submanifolds	p. 65
Submanifold M[subscript 1]	p. 65
Submanifold M[subscript 2]	p. 68
Submanifold M[subscript 3]	p. 69
McKay Bivariate Log-Gamma Manifold M	p. 71
Generalized McKay 5-Manifold	p. 72
Bivariate 3-Parameter Gamma Densities	p. 72
Generalized McKay Information Metric	p. 73
Freund Bivariate Exponential 4-Manifold F	p. 74
Freund Fisher Metric	p. 75
Freund Natural Coordinates	p. 76
Freund a-Geometry	p. 77
Freund a-Connection	p. 77
Freund a-Curvatures	p. 78
Freund Foliations	p. 80
Freund Submanifolds	p. 81
Independence Submanifold F[subscript 1]	p. 81
Submanifold F[subscript 2]	p. 82
Submanifold F[subscript 3]	p. 83
Submanifold F[subscript 4]	p. 84
Freund Affine Immersion	p. 87
Freund Bivariate Log-Exponential Manifold	p. 87
Bivariate Gaussian 5-Manifold N	p. 88
Bivariate Gaussian Fisher Information Metric	p. 89
Bivariate Gaussian Natural Coordinates	p. 90
Bivariate Gaussian a-Geometry	p. 91
a-Connection	p. 91
a-Curvatures	p. 94
Bivariate Gaussian Foliations	p. 98
Bivariate Gaussian Submanifolds	p. 99
Independence Submanifold N[subscript 1]	p. 99
Identical Marginals Submanifold N[subscript 2]	p. 101
Central Mean Submanifold N[subscript 3]	p. 103
Affine Immersion	p. 105
Bivariate Log-Gaussian Manifold	p. 106
Neighbourhoods of Poisson Randomness, Independence, and Uniformity	p. 109
Gamma Manifold G and Neighbourhoods of Randomness	p. 110
Log-Gamma Manifold L and Neighbourhoods of Uniformity	p. 111
Freund Manifold F and Neighbourhoods of Independence	p. 112
Freund Submanifold F[subscript 2]	p. 113
Neighbourhoods of Independence for Gaussians	p. 114
Cosmological Voids and Galactic Clustering	p. 119
Spatial Stochastic Processes	p. 120
Galactic Cluster Spatial Processes	p. 121
Cosmological Voids	p. 125
Modelling Statistics of Cosmological Void Sizes	p. 126
Coupling Galaxy Clustering and Void Sizes	p. 130
Representation of Cosmic Evolution	p. 132
Amino Acid Clustering With A.J. Doig	p. 139
Spacings of Amino Acids	p. 139
Poisson Spaced Sequences	p. 141
Non-Poisson Sequences as Gamma Processes	p. 142
Local Geodesic Distance Approximations	p. 145
Results	p. 148
Why Would Amino Acids Cluster?	p. 151
Cryptographic Attacks and Signal Clustering	p. 153
Cryptographic Attacks	p. 153
Information Geometry of the Log-gamma Manifold	p. 154
Distinguishing Nearby Unimodular Distributions	p. 155
Difference From a Uniform Distribution	p. 157
Gamma Distribution Neighbourhoods of Randomness	p. 157
Stochastic Fibre Networks With W.W. Sampson	p. 161
Random Fibre Networks	p. 161
Random Networks of Rectangular Fibres	p. 164
Log-Gamma Information Geometry for Fibre Clustering	p. 168
Bivariate Gamma Distributions for Anisotropy	p. 169
Independent Polygon Sides	p. 171
Multiplanar Networks	p. 175
Correlated Polygon Sides	p. 179
McKay Bivariate Gamma Distribution	p. 182
McKay Information Geometry	p. 184
McKay Information Entropy	p. 188
Simulation Results	p. 191
Stochastic Porous Media and Hydrology With J. Scharcanski and S. Felipussi	p. 195
Hydrological Modelling	p. 195
Univariate Gamma Distributions and Randomness	p. 196
Mckay Bivariate Gamma 3-Manifold	p. 196
Distance Approximations in the McKay Manifold	p. 198
Modelling Stochastic Porous Media	p. 200
Adaptive Tomographic Image Segmentation	p. 201
Mathematical Morphology Concepts	p. 203
Adaptive Image Segmentation and Representation	p. 209
Soil Tomographic Data	p. 214
Quantum Chaology	p. 223
Introduction	p. 223
Eigenvalues of Random Matrices	p. 226
Deviations	p. 229
References	p. 235
Index	p. 247
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