Multiscale Modeling: A Bayesian Perspective (Springer Series in Statistics)

a Wide Variety Of Processes Occur On Multiple Scales, Either Naturally Or As A Consequence Of Measurement. This Book Contains Methodology For The Analysis Of Data That Arise From Such Multiscale Processes. The Book 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. The Bayesian Approach Also Facilitates The Use Of Knowledge From Prior Experience Or Data, And These Methods Can Handle Different Amounts Of Prior Knowledge At Different Scales, As Often Occurs In Practice. the Book Is Aimed At Statisticians, Applied Mathematicians, And Engineers Working On Problems Dealing With Multiscale Processes In Time And/or Space, Such As In Engineering, Finance, And Environmetrics. The Book Will Also Be Of Interest To Those Working On Multiscale Computation Research. The Main Prerequisites Are Knowledge Of Bayesian Statistics And Basic Markov Chain Monte Carlo Methods. A Number Of Real-world Examples Are Thoroughly Analyzed In Order To Demonstrate The Methods And To Assist The Readers In Applying These Methods To Their Own Work. To Further Assist Readers, The Authors Are Making Source Code (for R) Available For Many Of The Basic Methods Discussed Herein. 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 "A wide variety of processes occur on multiple scales, either naturally or as a consequence of measurement. This book contains methodology for the analysis of data that arise from such multiscale processes. The book 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. The Bayesian approach also facilitates the use of knowledge from prior experience or data, and these methods can handle different amounts of prior knowledge at different scales, as often occurs in practice." "The book is aimed at statisticians, applied mathematicians, and engineers working on problems dealing with multiscale processes in time and/or space, such as in engineering, finance, and environmetrics. The book will also be of interest to those working on multiscale computation research. The main prerequisites are knowledge of Bayesian statistics and basic Markov chain Monte Carlo methods. A number of real-world examples are thoroughly analyzed in order to demonstrate the methods and to assist the readers in applying these methods in their own work. To further assist readers, the authors are making source code (for R) available for many of the basic methods discussed herein."--Jacket 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.