Multiscale modeling. A Bayesian perspective

✍ Scribed by Ferreira M., Lee H.

• Publisher: Springer
• Year: 2007
• Tongue: English
• Leaves: 257
• Series: Springer Series in Statistics
• Category: Library

✦ Synopsis

This highly useful book contains methodology for the analysis of data that arise from multiscale processes. It brings together a number of recent developments and makes them accessible to a wider audience. Taking a Bayesian approach allows for full accounting of uncertainty, and also addresses the delicate issue of uncertainty at multiple scales. These methods can handle different amounts of prior knowledge at different scales, as often occurs in practice.

✦ Table of Contents

Preface......Page 7
Contents......Page 9
Part I: Introduction......Page 13
1 Introduction......Page 15
2.1 Markov Random Fields......Page 19
2.2 Gaussian Processes......Page 22
3 Illustrative Example......Page 29
Part II: Convolutions and Wavelets......Page 33
Convolution and Wavelet Overview......Page 35
4.1 Convolutions......Page 37
4.2 Multiscale Convolutions......Page 45
5 Wavelet Methods......Page 51
5.1 Background......Page 52
5.2 Continuous Wavelet Transform......Page 53
5.3 Scaling Function......Page 54
5.4 Discrete Wavelets and the Discrete Wavelet Transform......Page 55
5.5 Bayesian Nonparametric Regression with Wavelets......Page 58
5.6 Other Statistical Applications of Bayesian Wavelet Analysis......Page 64
Part III: Explicit Multiscale Models......Page 67
6 Overview of Explicit Multiscale Models......Page 69
6.2 Classi.cation and Regression Trees......Page 71
7 Gaussian Multiscale Models on Trees......Page 75
7.1 The Model......Page 77
7.2 Covariance Structure......Page 78
7.3 Estimation When θ Is Known......Page 81
7.4 Estimation When θ Is Unknown......Page 88
8 Hidden Markov Models on Trees......Page 91
8.1 HMMs in 1-D......Page 92
8.2 HMMs on Trees......Page 93
8.3 Estimation When θ Is Known......Page 94
8.5 Application: Image Classi.cation......Page 96
9 Mass-Balanced Multiscale Models on Trees......Page 99
9.1 Introduction......Page 100
9.2 Gaussian Case......Page 101
9.3 Poisson Case......Page 105
10 Multiscale Random Fields......Page 109
10.1 Two-Level Model......Page 111
10.2 Model with Several Levels......Page 114
10.3 The Multiscale Model as an Application of Je.rey’s Rule......Page 118
10.4 Didactic Example: Three-Level Model......Page 119
10.5 Posterior Simulation......Page 121
10.6 A Simulated Example......Page 123
11.1 Introduction......Page 125
11.2 Model Construction......Page 126
11.3 Properties of Hidden Resolution Models......Page 131
11.4 Incorporating Periodicities......Page 142
11.5 Inference and Prediction......Page 144
11.6 Examples......Page 148
12 Change of Support Models......Page 157
12.1 Point-Level Spatial Processes......Page 158
12.2 Inferring Intermediate-Level Processes......Page 161
12.3 Block-Level Spatial Processes......Page 163
Part IV: Implicit Multiscale Models......Page 165
13 Implicit Computationally Linked Model Overview......Page 167
13.1 Simulated Annealing......Page 169
13.2 Simulated Tempering......Page 170
13.3 Simulated Sintering......Page 173
13.4 Multigrid Methods......Page 175
14.1 Metropolis Coupling......Page 177
14.2 Multiscale Metropolis Coupling......Page 179
14.3 Sequential Parallel Tempering......Page 187
14.4 Extensions......Page 188
15.1 The Basics of Genetic Algorithms......Page 191
15.2 Multiscale Genetic Algorithms......Page 194
15.3 Multiscale Genetic Algorithm-Style MCMC......Page 197
15.4 Example......Page 203
Part V: Case Studies......Page 205
16.1 Introduction......Page 207
16.2 Multiscale Modeling......Page 210
16.3 Implicit Multiscale Methods......Page 221
17 Single Photon Emission Computed Tomography Example......Page 225
17.1 Metropolis Coupling......Page 229
17.2 Genetic Algorithms......Page 231
18 Conclusions......Page 235
References......Page 237
Index......Page 255