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E-raamat: Independent Random Sampling Methods

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  • Formaat: EPUB+DRM
  • Sari: Statistics and Computing
  • Ilmumisaeg: 31-Mar-2018
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319726342
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Independent Random Sampling MethodsSuurem pilt
  • Formaat: EPUB+DRM
  • Sari: Statistics and Computing
  • Ilmumisaeg: 31-Mar-2018
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319726342
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This book systematically addresses the design and analysis of efficient techniques for independent random sampling. Both general-purpose approaches, which can be used to generate samples from arbitrary probability distributions, and tailored techniques, designed to efficiently address common real-world practical problems, are introduced and discussed in detail. In turn, the monograph presents fundamental results and methodologies in the field, elaborating and developing them into the latest techniques. The theory and methods are illustrated with a varied collection of examples, which are discussed in detail in the text and supplemented with ready-to-run computer code.

The main problem addressed in the book is how to generate independent random samples from an arbitrary probability distribution with the weakest possible constraints or assumptions in a form suitable for practical implementation. The authors review the fundamental results and methods in the field, address the latest methods, and emphasize the links and interplay between ostensibly diverse techniques.


Arvustused

The book contains more than 300 references and has more than 50 coloured figures for better understanding. The book can be recommended to all readers, who are interested in this field, e.g. engineers working in signal theory or statisticians interested in computational methods or scientists working in the fields of biology, quantitative finance or physics, where complex models that demand Monte Carlo computations are needed. (Ludwig Paditz, zbMATH 1414.62007, 2019)

1 Introduction 1(26)
1.1 The Monte Carlo Method: A Brief History
1(3)
1.2 The Need for Monte Carlo
4(5)
1.2.1 Numerical Integration
4(2)
1.2.2 Importance Sampling
6(2)
1.2.3 Quasi-Monte Carlo
8(1)
1.2.4 Inverse Monte Carlo
8(1)
1.3 Random Number Generation
9(2)
1.3.1 Random, Pseudo-Random, Quasi-Random
9(2)
1.4 Pseudo-Random Number Generators
11(5)
1.4.1 Nonlinear Recursions
11(1)
1.4.2 Chaotic Pseudo-Random Number Generators
12(2)
1.4.3 The Middle-Square Generator
14(1)
1.4.4 Linear Congruential Generators
14(2)
1.5 Random Sampling Methods
16(3)
1.5.1 Direct Methods
17(1)
1.5.2 Accept/Reject Methods
17(1)
1.5.3 Markov Chain Monte Carlo (MCMC)
18(1)
1.5.4 Importance Sampling
18(1)
1.5.5 Hybrid Techniques
18(1)
1.6 Goal and Organization of This Book
19(2)
1.6.1 Motivation and Goals
19(1)
1.6.2 Organization of the Book
20(1)
References
21(6)
2 Direct Methods 27(38)
2.1 Introduction
27(1)
2.2 Notation
28(1)
2.2.1 Vectors, Points, and Intervals
28(1)
2.2.2 Random Variables, Distributions, and Densities
29(1)
2.2.3 Sets
29(1)
2.3 Transformations of Random Variables
29(12)
2.3.1 One-to-One Transformations
30(3)
2.3.2 Many-to-One Transformations
33(4)
2.3.3 Deconvolution Method
37(1)
2.3.4 Discrete Mixtures
38(1)
2.3.5 Continuous Mixtures: Marginalization
39(1)
2.3.6 Order Statistics
39(2)
2.4 Universal Direct Methods
41(12)
2.4.1 Inversion Method
41(4)
2.4.2 Vertical Density Representation (VDR)
45(4)
2.4.3 The Fundamental Theorem of Simulation
49(1)
2.4.4 Inverse-of-Density Method
50(3)
2.5 Tailored Techniques
53(3)
2.5.1 Recursive Methods
53(2)
2.5.2 Convex Densities
55(1)
2.6 Examples
56(4)
2.6.1 Multiplication of Independent Uniform Random Variates
56(2)
2.6.2 Sum of Independent Uniform Random Variates
58(1)
2.6.3 Polynomial Densities with Non-negative Coefficients
59(1)
2.6.4 Polynomial Densities with One or More Negative Constants
60(1)
2.7 Summary
60(2)
References
62(3)
3 Accept-Reject Methods 65(50)
3.1 Introduction
65(1)
3.2 Rejection Sampling
66(8)
3.2.1 Acceptance Rate
69(1)
3.2.2 Distribution of the Rejected Samples
69(1)
3.2.3 Distribution of the Accepted and Rejected Samples with Generic L > 0
70(1)
3.2.4 Different Application Scenarios
70(1)
3.2.5 Butcher's Version of the Rejection Sampler
71(1)
3.2.6 Vaduva's Modification of the Butcher's Method
72(1)
3.2.7 Lux's Extension
73(1)
3.3 Computational Cost
74(6)
3.3.1 Further Considerations About the Acceptance Rate
75(3)
3.3.2 Squeezing
78(1)
3.3.3 Sibuya's Modified Rejection Method
79(1)
3.4 Band Rejection Method
80(6)
3.4.1 Preliminaries
80(2)
3.4.2 Generalized Band Rejection Algorithm
82(3)
3.4.3 Payne-Dagpunar's Band Rejection
85(1)
3.5 Acceptance-Complement Method
86(3)
3.6 RS with Stepwise Proposals
89(5)
3.6.1 Strip Methods
90(2)
3.6.2 Inversion-Rejection Method
92(2)
3.7 Transformed Rejection Method
94(5)
3.7.1 Transformed Rejection and IoD Method
97(2)
3.8 Examples
99(4)
3.8.1 RS for Generating Order Statistics
99(1)
3.8.2 Mixtures with Negative Coefficients
100(2)
3.8.3 Pdfs Expressed as Sequences of Functions
102(1)
3.9 Monte Carlo Estimation via RS
103(7)
3.9.1 Recycling Rejected Samples
105(1)
3.9.2 RS with a Generic Constant L > 0
106(4)
3.10 Summary
110(1)
References
111(4)
4 Adaptive Rejection Sampling Methods 115(44)
4.1 Introduction
115(1)
4.2 Generic Structure of an Adaptive Rejection Sampler
116(3)
4.2.1 Proposal Densities
117(1)
4.2.2 Generic Adaptive Algorithm
117(2)
4.3 Constructions of the Proposal Densities
119(28)
4.3.1 Standard Adaptive Rejection Sampling
119(6)
4.3.2 Derivative-Free Constructions for Log-Concave pdfs
125(2)
4.3.3 Concave-Convex ARS
127(1)
4.3.4 Transformed Density Rejection
128(4)
4.3.5 Extensions of TDR
132(3)
4.3.6 Generalized Adaptive Rejection Sampling
135(12)
4.4 Performance and Computational Cost of the ARS Schemes
147(1)
4.4.1 Acceptance Rate
147(1)
4.4.2 Probability of Adding a New Support Point
148(1)
4.5 Variants of the Adaptive Structure in the ARS Scheme
148(5)
4.5.1 Deterministic Test for Adding New Support Points
149(2)
4.5.2 An Adaptive Rejection Sampler with Fixed Number of Support Points
151(2)
4.6 Combining ARS and MCMC
153(2)
4.6.1 Adaptive Rejection Metropolis Sampling
153(1)
4.6.2 A Procedure to Build Proposal pdfs for the ARMS Algorithm
154(1)
4.7 Summary
155(1)
References
156(3)
5 Ratio of Uniforms 159(38)
5.1 Introduction
159(2)
5.1.1 A Remark on Inverse Densities
160(1)
5.2 Standard Ratio of Uniforms Method
161(6)
5.2.1 Some Basic Considerations
163(1)
5.2.2 Examples
164(3)
5.3 Envelope Polygons and Adaptive RoU
167(2)
5.4 Generalized Ratio of Uniforms Method
169(2)
5.5 Properties of Generalized RoU Samplers
171(5)
5.5.1 Boundary of Ag
171(1)
5.5.2 How to Guarantee that Ag is Bounded
171(3)
5.5.3 Power Functions
174(2)
5.6 Connections Between GRoU and Other Classical Techniques
176(7)
5.6.1 Extended Inverse-of-Density Method
176(4)
5.6.2 GRoU Sampling and the Transformed Rejection Method
180(3)
5.6.3 Role of the Constant CA
183(1)
5.7 How Does GRoU Work for Generic pdfs?
183(6)
5.7.1 IoD for Arbitrary pdfs
184(1)
5.7.2 GRoU for pdfs with a Single Mode at x = 0
185(1)
5.7.3 GRoU for pdfs with a Single Mode at x not = to 0
186(1)
5.7.4 GRoU for Arbitrary pdfs
187(1)
5.7.5 Summary
188(1)
5.8 Rectangular Region Ag
189(2)
5.8.1 Yet Another Connection Between IoD and GRoU
190(1)
5.9 Relaxing Assumptions: GRoU with Decreasing g(u)
191(2)
5.9.1 General Expression of a r.v. Transformation
192(1)
5.10 Another View of GRoU
193(1)
5.11 Summary
194(3)
References
195(2)
6 Independent Sampling for Multivariate Densities 197(52)
6.1 Introduction
197(1)
6.2 Notation
198(1)
6.3 Generic Procedures
199(8)
6.3.1 Chain Rule Decomposition
199(1)
6.3.2 Dependence Generation
200(3)
6.3.3 Rejection Sampling
203(2)
6.3.4 RoU for Multivariate Densities
205(2)
6.4 Elliptically Contoured Distributions
207(3)
6.4.1 Polar Methods
208(2)
6.5 Vertical Density Representation
210(2)
6.5.1 Inverse-of-Density Method
212(1)
6.6 Uniform Distributions in Dimension n
212(4)
6.6.1 Points Uniformly Distributed in a Simplex
213(1)
6.6.2 Sampling Uniformly Within a Hypersphere
214(2)
6.6.3 Points Uniformly Distributed Within a Hyperellipsoid
216(1)
6.7 Transformations of a Random Variable
216(8)
6.7.1 Many-to-Few Transformations (m > n)
217(1)
6.7.2 Few-to-Many Transformations: Singular Distributions (m < n)
218(3)
6.7.3 Sampling a Uniform Distribution on a Differentiable Manifold
221(3)
6.8 Sampling Techniques for Specific Distributions
224(8)
6.8.1 Multivariate Gaussian Distribution
224(1)
6.8.2 Multivariate Student's t-Distribution
225(1)
6.8.3 Wishart Distribution
225(1)
6.8.4 Inverse Wishart Distribution
226(1)
6.8.5 Multivariate Gamma Samples
227(1)
6.8.6 Dirichlet Distribution
227(1)
6.8.7 Cook-Johnson's Family
228(1)
6.8.8 Some Relevant Bivariate Distributions
229(3)
6.9 Generation of Stochastic Processes
232(13)
6.9.1 Markov Jump Processes
232(2)
6.9.2 Gaussian Processes
234(3)
6.9.3 Wiener Processes
237(1)
6.9.4 Brownian Motion
238(2)
6.9.5 Poisson Processes
240(3)
6.9.6 Dirichlet Processes: "Rich Get Richer"
243(2)
6.10 Summary
245(1)
References
246(3)
7 Asymptotically Independent Samplers 249(18)
7.1 Introduction
249(2)
7.2 Metropolis-Hastings (MH) Methods
251(3)
7.2.1 The Algorithm
251(1)
7.2.2 Invariant Distribution of the MH Algorithm
252(1)
7.2.3 Acceptance Rate in MH-Type Methods
253(1)
7.3 Independent Generalized MH Methods with Multiple Candidates
254(3)
7.3.1 Independent Multiple Try Metropolis Algorithms
254(2)
7.3.2 Ensemble MCMC Method
256(1)
7.4 Independent Doubly Adaptive Rejection Metropolis Sampling
257(8)
7.4.1 Adaptive Rejection Sampling (ARS)
258(1)
7.4.2 Adaptive Rejection Metropolis Sampling
259(1)
7.4.3 Structural Limitations of ARMS
260(1)
7.4.4 IA2RMS Algorithm
261(2)
7.4.5 Convergence of the Chain and Computational Cost
263(1)
7.4.6 Examples of Proposal Constructions for IA2RMS
263(2)
7.5 Summary
265(1)
References
265(2)
8 Summary and Outlook 267(4)
References
270(1)
A Acronyms and Abbreviations 271(2)
B Notation 273(2)
B.1 Vectors, Points, and Intervals
273(1)
B.2 Random Variables, Distributions and Densities
273(1)
B.3 Sets
274(1)
B.4 Summary of Main Notation
274(1)
C Jones' RoU Generalization 275(4)
C.1 Possible Choices of t(v, u)
277(1)
References
278(1)
D Polar Transformation 279
Luca Martino is currently a research fellow at the University of Valencia, Spain, after having held positions at the Carlos III University of Madrid, Spain, the University of Helsinki, Finland and the University of São Paulo, Brazil. His research interests are in the fields of statistical signal processing and computational statistics, especially in connection with Bayesian analysis and Monte Carlo approximation methods.

David Luengo is an Associate Professor at the Technical University of Madrid, Spain. His research interests are in the broad fields of statistical signal processing and machine learning, especially Bayesian learning and inference, Gaussian processes, Monte Carlo algorithms, sparse signal processing and Bayesian non-parametrics. Dr. Luengo has co-authored over 70 research papers, which were published in international journals and conference volumes.

Joaquín Míguez is an Associate Professor at the Carlos III University of Madrid, Spain. His interests are in the fields of applied probability, computational statistics, dynamical systems and the theory and applications of the Monte Carlo methods. Having published extensively and lectured internationally on his research, he was a co-recipient of the IEEE Signal Processing Magazine Best Paper Award in 2007.
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