Stochastic models, information theory, and Lie groups: 1 Classical Results and Geometric Methods
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| Main Author: | |
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| Format: | Book |
| Language: | English |
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Boston [u.a.]
Birkhäuser
2009
|
| Series: | Applied and Numerical Harmonic Analysis
Applied and numerical harmonic analysis |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXI, 380 S. Ill., graph. Darst. |
| ISBN: | 9780817648022 |
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MARC
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| 245 | 1 | 0 | |a Stochastic models, information theory, and Lie groups |n 1 |p Classical Results and Geometric Methods |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2009 | |
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| 490 | 0 | |a Applied and numerical harmonic analysis | |
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| _version_ | 1804141591341826048 |
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| adam_text | Titel: Bd. 1. Stochastic models, information theory, and lie groups. Classical results and geometric method
Autor: Chirikjian, Gregory S.
Jahr: 2009
Contents
ANHA Series Preface.................................................. vii
Preface................................................................ ix
1 Introduction....................................................... 1
1.1 What this Book is About......................................... 1
1.2 Different Meanings of Equality.................................... 4
1.2.1 Denning Equalities........................................ 4
1.2.2 Equality in the Sense of Zero Mean-Squared Error............. 5
1.2.3 Big-O Notation........................................... 6
1.2.4 A Philosophical View of Equality and Inequality .............. 7
1.3 Other Useful Shortcuts........................................... 8
1.3.1 Simplifying Notation....................................... 8
1.3.2 Nice, Weil-Behaved, Non-Pathological Functions............... 9
1.4 Modern Mathematical Notation and Terminology.................... 10
1.4.1 What is Modern Mathematics?.............................. 10
1.4.2 Stating Mathematical Results............................... 11
1.4.3 Sets and Mappings........................................ 11
1.4.4 Transformation Groups .................................... 17
1.4.5 Understanding Commutative Diagrams....................... 17
1.5 Transport Phenomena and Probability Flow........................ 19
1.5.1 Continuum Mechanics ..................................... 20
1.5.2 Heat Flow and Entropy.................................... 25
1.6 Organization of this Book........................................ 27
1.7 Chapter Summary............................................... 28
1.8 Exercises....................................................... 28
References.......................................................... 29
2 Gaussian Distributions and the Heat Equation..................... 31
2.1 The Gaussian Distribution on the Real Line........................ 31
2.1.1 Defining Parameters....................................... 31
2.1.2 The Maximum Entropy Property............................ 34
2.1.3 The Convolution of Gaussians............................... 36
2.1.4 The Fourier Transform of the Gaussian Distribution........... 37
2.1.5 Diffusion Equations........................................ 38
2.1.6 Stirling s Formula......................................... 39
xvi Contents
2.2 The Multivariate Gaussian Distribution............................ 40
2.2.1 Conditional and Marginal Densities.......................... 40
2.2.2 Multi-Dimensional Integrals Involving Gaussians.............. 42
2.3 The Volume of Spheres and Balls in R ............................ 43
2.4 Clipped Gaussian Distributions ................................... 45
2.4.1 One-Dimensional Clipped Gaussian Distributions.............. 45
2.4.2 Multi-Dimensional Clipped Gaussian Distributions ............ 46
2.5 Folded, or Wrapped, Gaussians.................................... 47
2.6 The Heat Equation.............................................. 48
2.6.1 The One-Dimensional Case................................. 48
2.6.2 The Multi-Dimensional Case................................ 50
2.6.3 The Heat Equation on the Unit Circle....................... 51
2.7 Gaussians and Multi-Dimensional Diffusions........................ 51
2.7.1 The Constant Diffusion Case................................ 51
2.7.2 The Time-Varying Case.................................... 52
2.8 Symmetry Analysis of Evolution Equations......................... 53
2.8.1 Symmetries in Parameters.................................. 54
2.8.2 Infinitesimal Symmetry Operators of the Heat Equation........ 54
2.8.3 Non-Linear Transformations of Coordinates................... 57
2.9 Chapter Summary............................................... 58
2.10 Exercises....................................................... 58
References.......................................................... 60
3 Probability and Information Theory............................... 63
3.1 Probability Theory in Euclidean Space............................. 64
3.1.1 Basic Definitions and Properties of Probability Density Functions 64
3.1.2 Change of Variables ....................................... 65
3.1.3 Marginalization, Conditioning, and Convolution............... 65
3.1.4 Mean and Covariance...................................... 66
3.1.5 Parametric Distributions................................... 67
3.2 Conditional Expectation.......................................... 68
3.2.1 Jensen s Inequality and Conditional Expectation.............. 71
3.2.2 Convolution and Conditional Expectation.................... 72
3.3 Information Theory.............................................. 73
3.3.1 Entropy of Conditional and Marginal Density Functions........ 74
3.3.2 Entropy and Gaussian Distributions......................... 75
3.3.3 Mutual Information........................................ 76
3.3.4 Information-Theoretic Measures of Divergence................ 76
3.3.5 Fisher Information ........................................ 77
3.3.6 Information and Convolution ............................... 78
3.3.7 Shift and Scaling Properties................................ 80
3.4 Parameter Estimation............................................ 81
3.4.1 Unbiased Estimators....................................... 82
3.4.2 The Cramer-Rao Bound ................................... 82
3.4.3 Demonstration with Gaussian Distributions................... 84
3.4.4 The de Bruijn Identity..................................... 84
3.4.5 The Entropy Power Inequality.............................. 85
3.4.6 Entropy of a Weighted Sum of Disjoint PDFs................. 87
3.4.7 Change of Coordinates..................................... 88
3.4.8 Computation of Entropy via Discretization................... 89
Contents xvii
3.5 The Classical Central Limit Theorem.............................. 91
3.5.1 The Central Limit Theorem (Fourier Version)................. 91
3.5.2 The Central Limit Theorem (RMSD Error Version)............ 92
3.5.3 The Central Limit Theorem (Information-Theoretic Version).... 94
3.5.4 Limitations of the Central Limit Theorem.................... 94
3.6 An Alternative Measure of Dispersion.............................. 95
3.7 Chapter Summary............................................... 95
3.8 Exercises....................................................... 96
References.......................................................... 97
Stochastic Differential Equations .................................. 101
4.1 Motivating Examples ............................................ 101
4.1.1 The Discrete Random Walker............................... 102
4.1.2 Continuous-Time Brownian Motion in Continuous Space....... 103
4.2 Stationary and Non-Stationary Random Processes................... 105
4.2.1 Weak and Strong Stationarity............................... 105
4.2.2 Non-Stationary Processes................................... 106
4.3 Gaussian and Markov Processes................................... 106
4.4 Wiener Processes and Stochastic Differential Equations............... 108
4.4.1 An Informal Introduction................................... 108
4.4.2 Abstracted Definitions..................................... Ill
4.5 The Ito Stochastic Calculus....................................... 112
4.5.1 Ito Stochastic Differential Equations in Rd.................... 114
4.5.2 Numerical Approximations................................. 115
4.5.3 Mathematical Properties of the Ito Integral................... 116
4.5.4 Evaluating Expectations is Convenient for Ito Equations....... 118
4.5.5 Ito s Rule................................................ 119
4.5.6 The Fokker-Planck Equation (Ito Version).................... 120
4.6 The Stratonovich Stochastic Calculus.............................. 121
4.7 Multi-Dimensional Ornstein-Uhlenbeck Processes.................... 123
4.7.1 Steady-State Conditions.................................... 124
4.7.2 Steady-State Solution...................................... 126
4.7.3 Detailed Balance and the Onsager Relations.................. 128
4.8 SDEs and Fokker-Planck Equations Under Coordinate Changes....... 130
4.8.1 Brownian Motion in the Plane.............................. 130
4.8.2 General Conversion Rules.................................. 134
4.8.3 Coordinate Changes and Fokker-Planck Equations ............ 134
4.9 Chapter Summary............................................... 136
4.10 Exercises....................................................... 136
References.......................................................... 138
Geometry of Curves and Surfaces.................................. 141
5.1 An Introduction to Geometry Through Robotic Manipulator Kinematics 142
5.1.1 Forward (or Direct) Kinematics............................. 142
5.1.2 Reverse (or Inverse) Kinematics............................. 144
5.2 A Case Study in Medical Imaging................................. 147
5.2.1 A Parametric Approach................................----- 148
5.2.2 An Implicit Approach...................................... 151
5.3 Differential Geometry of Curves................................... 154
5.3.1 Local Theory of Curves.................................... 155
xviii Contents
5.3-2 Global Theory of Curves................................... 157
5.4 Differential Geometry of Surfaces in R3 ............................ 159
5.4.1 The First and Second Fundamental Forms.................... 161
5.4.2 Curvature................................................ 163
5.4.3 The Sphere............................................... 166
5.4.4 The Ellipsoid of Revolution................................. 167
5.4.5 The Torus................................................ 169
5.4.6 The Gauss-Bonnet Theorem and Related Inequalities..........170
5.5 Tubes.......................................................... 171
5.5.1 Offset Curves in R2........................................ 171
5.5.2 Parallel Fibers, Ribbons, and Tubes of Curves in R3........... 172
5.5.3 Tubes of Surfaces in R3.................................... 175
5.6 The Euler Characteristic: From One Dimension to N Dimensions...... 176
5.6-1 The Euler Characteristic of Zero-, One-, and Three-Dimensional
Bodies................................................... 176
5.6-2 Relationship Between the Euler Characteristic of a Body and
Its Boundary............................................. 178
5.6.3 The Euler Characteristic of Cartesian Products of Objects...... 179
5.7 Implicit Surfaces, Level Set Methods, and Curvature Flows........... 179
5.7.1 Implicit Surfaces.......................................... 179
5.7.2 Integration on Implicitly Defined Surfaces and Curves in R3 .... 181
5.7.3 Integral Theorems for Implicit Surfaces ...................... 184
5.7.4 Level Sets and Curvature Flows............................. 184
5.8 Chapter Summary............................................... 186
5.9 Exercises....................................................... 186
References.......................................................... 190
6 Differential Forms................................................. 193
6.1 An Informal Introduction to Differential Forms on R ................ 194
6.1.1 Definitions and Properties of n-Forms and Exterior Derivatives.. 194
6.1.2 Exterior Derivatives of (n - 1)-Forrns on Mn for n = 2,3........ 197
6.1.3 Products of Differential Forms.............................. 199
6.1.4 Concise Notation for Differential Forms and Exterior Derivatives 200
6.2 Permutations................................................... 201
6.2.1 Examples of Permutations and Their Products................ 201
6.2.2 The Sign of a Permutation .................................202
6.2.3 Multi-Dimensional Version of the Levi-Civita Symbol.......... 202
6.3 The Hodge Star Operator........................................ 204
6.4 Tensor Products and Dual Vectors................................. 206
6.5 Exterior Products............................................... 207
6.5.1 A Concrete Introduction to Exterior Products ................ 207
6.5.2 Abstract Definition of the Exterior Product of Two Vectors.....208
6.5.3 The Exterior Product of Several Vectors......................209
6.5.4 Computing with Exterior Products.......................... 210
6.5.5 The Exterior Product of Two Exterior Products............... 210
6.5.6 The Inner Product of Two Exterior Products................. 211
6.5.7 The Dual of an Exterior Product............................ 211
6.6 Invariant Description of Vector Fields.............................. 211
6.7 Push-Forwards and Pull-Backs in Rn............................... 213
6.7.1 General Theory...........................................213
Contents xix
6.7.2 Example Calculations...................................... 214
6.8 Generalizing Integral Theorems from Vector Calculus................ 221
6.8.1 Integration of Differential Forms ............................ 221
6.8.2 The Inner Product of Forms................................ 222
6.8.3 Green s Theorem for a Square Region in R2 .................. 223
6.8.4 Stokes Theorem for a Cube in R3........................... 223
6.8.5 The Divergence Theorem for a Cube in R3 ................... 224
6.8.6 Detailed Examples ........................................ 224
6.8.7 Closed Forms and Diffusion Equations....................... 226
6.9 Differential Forms and Coordinate Changes......................... 227
6.10 Chapter Summary............................................... 228
6.11 Exercises....................................................... 229
References.......................................................... 232
Polytopes and Manifolds........................................... 233
7.1 Properties and Operations on Convex Polytopes in M.n............... 234
7.1.1 Computing the Volume and Surface Area of Polyhedra......... 235
7.1.2 Properties of Minkowski Sums .............................. 235
7.1.3 Convolution of Bodies...................................... 237
7.2 Examples of Manifolds........................................... 238
7.3 Embedded Manifolds, Part I: Using Vector Calculus ................. 246
7.3.1 The Inner Product of Vector Fields on Manifolds Embedded in R™246
7.3.2 An Example: A Hyper-Spherical Cap........................ 247
7.3.3 Computing Normals Extrinsically Without the Cross Product... 250
7.3.4 The Divergence Theorem in Coordinates..................... 253
7.3.5 Integration by Parts on an Embedded Manifold............... 254
7.3.6 Curvature................................................ 257
7.4 Covariant vs. Contravariant....................................... 258
7.4.1 Tensors.................................................. 258
7.4.2 Derivatives and Differentials................................ 259
7.5 Embedded Manifolds, Part II: Using Differential Forms............... 261
7.5.1 Push-Forwards and Pull-Backs (Revisited).................... 261
7.5.2 Expressing Pull-Backs of Forms in Coordinates................ 262
7.5.3 Volume Element of an Embedded Manifold................... 263
7.5.4 Conversion to Vector Notation.............................. 264
7.5.5 General Properties of Differential Forms on Embedded Manifolds 265
7.6 Intrinsic Description of Riemannian Manifolds ...................... 266
7.6.1 Computing Tangent Vectors and Boundary Normals in Local
Coordinates .............................................. 269
7.6.2 Stokes Theorem for Manifolds.............................. 270
7.6.3 The Gauss-Bonnet-Chern Theorem ......................... 277
7.7 Fiber Bundles and Connections ................................... 278
7.7.1 Fiber Bundles............................................. 279
7.7.2 Connections.............................................. 281
7.8 The Heat Equation on a Riemannian Manifold...................... 283
7.9 Chapter Summary....................................•.......... 283
7.10 Exercises....................................................... 284
References.......................................................... 286
Contents
Stochastic Processes on Manifolds................................. 289
8.1 The Fokker-Planck Equation for an Ito SDE on a Manifold: A
Parametric Approach............................................ 289
8.2 Ito Stochastic Differential Equations on an Embedded Manifold: An
Implicit Approach............................................... 291
8.2.1 The General Ito Case...................................... 291
8.2.2 Bilinear Ito Equations that Evolve on a Quadratic Hyper-Surface
in R .................................................... 292
8.3 Stratonovich SDEs and Fokker-Planck Equations on Manifolds........293
8.3.1 Stratonovich SDEs on Manifolds: Parametric Approach........293
8.3.2 Stratonovich SDEs on Manifolds: Implicit Approach........... 294
8.4 Entropy and Fokker-Planck Equations on Manifolds................. 295
8.5 Examples....................................................... 296
8.5.1 Stochastic Motion on the Unit Circle ........................ 296
8.5.2 The Unit Sphere in R3: Parametric Formulation............... 297
8.5.3 SDEs on Spheres and Rotations: Extrinsic Formulation......... 299
8.5.4 The SDE and Fokker-Planck Equation for the Kinematic Cart .. 299
8.6 Solution Techniques ............................................. 300
8.6.1 Finite Difference and Finite Elements........................ 300
8.6.2 Non-Parametric Density Estimation ......................... 301
8.6.3 Separation of Variables: Diffusion on SE(2) as a Case Study .... 301
8.7 Chapter Summary............................................... 308
8.8 Exercises....................................................... 309
References.......................................................... 309
Summary.......................................................... 313
References.......................................................... 314
Review of Linear Algebra, Vector Calculus, and Systems Theory .. 315
A.I Vectors......................................................... 315
A.I.I Vector Spaces............................................. 315
A.1.2 Linear Mappings and Isomorphisms.......................... 317
A.1.3 The Scalar Product and Vector Norm........................ 317
A.I.4 The Gram-Schmidt Orthogonalization Process................ 319
A.1.5 Dual Spaces.............................................. 319
A.1.6 The Vector Product in S3.................................. 320
A.2 Matrices ....................................................... 321
A.2.1 Matrix Multiplication and the Trace......................... 322
A.2.2 The Determinant.......................................... 323
A.2.3 The Inverse of a Matrix.................................... 325
A.2.4 Pseudo-Inverses and Null Spaces............................326
A.2.5 Special Kinds of Matrices .................................. 327
A.2.6 Matrix Norms ............................................ 328
A.2.7 Matrix Inequalities........................................ 331
A.3 Eigenvalues and Eigenvectors..................................... 331
A.4 Matrix Decompositions........................................... 334
A.4.1 Jordan Blocks and the Jordan Decomposition................. 334
A.4.2 Decompositions into Products of Special Matrices............. 335
A.4.3 Decompositions into Blocks................................. 336
A.5 Matrix Perturbation Theory...................................... 337
Contents xxi
A.6 The Matrix Exponential.......................................... 338
A.7 Kronecker Products and Kronecker Sums........................... 340
A.8 Complex Numbers and Fourier Analysis............................ 342
A.9 Important Inequalities from the Theory of Linear Systems............ 344
A.10 The State-Transition Matrix and the Product Integral ............... 346
A.11 Vector Calculus................................................. 349
A.ll.l Optimization in Rn........................................ 349
A.11.2 Differential Operators in Rn................................ 351
A.11.3 Integral Theorems in R2 and R3............................. 351
A.11.4 Integration by Parts in Rn.................................. 353
A.11.5 The Chain Rule........................................... 353
A.11.6 Matrix Differential Calculus................................ 354
A.12 Exercises....................................................... 356
References.......................................................... 360
Index.................................................................. 363
|
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| id | DE-604.BV025738845 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:18:18Z |
| institution | BVB |
| isbn | 9780817648022 |
| language | English |
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| series2 | Applied and Numerical Harmonic Analysis Applied and numerical harmonic analysis |
| spelling | Chirikjian, Gregory S. Verfasser aut Stochastic models, information theory, and Lie groups 1 Classical Results and Geometric Methods Boston [u.a.] Birkhäuser 2009 XXI, 380 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied and Numerical Harmonic Analysis Applied and numerical harmonic analysis (DE-604)BV025603055 1 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019343177&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chirikjian, Gregory S. Stochastic models, information theory, and Lie groups |
| title | Stochastic models, information theory, and Lie groups |
| title_auth | Stochastic models, information theory, and Lie groups |
| title_exact_search | Stochastic models, information theory, and Lie groups |
| title_full | Stochastic models, information theory, and Lie groups 1 Classical Results and Geometric Methods |
| title_fullStr | Stochastic models, information theory, and Lie groups 1 Classical Results and Geometric Methods |
| title_full_unstemmed | Stochastic models, information theory, and Lie groups 1 Classical Results and Geometric Methods |
| title_short | Stochastic models, information theory, and Lie groups |
| title_sort | stochastic models information theory and lie groups classical results and geometric methods |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019343177&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV025603055 |
| work_keys_str_mv | AT chirikjiangregorys stochasticmodelsinformationtheoryandliegroups1 |