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Monte Carlo [Hardcover]
George Fishman (Author)

Editorial ReviewsProduct DescriptionThis volume presents a comprehensive first course in the Monte Carlo method which will be suitable for graduate and undergraduate students in the mathematical sciences and engineering, principally in operations research, statistics, mathematics, and computer science. The reader is assumed to have a sound understanding of calculus, introductory matrix analysis, probability, and intermediate statistics, but otherwise the book is self-contained. As well as a thorough exploration of the important concepts of the Monte Carlo method, the volume includes over 90 algorithms which allow the reader to move rapidly from the concepts to putting them into practice. The book also contains numerous exercises, many of them hands-on implementations of selected algorithms to demonstrate the application of these ideas in realistic settings. Software, freely available via ftp and portable across computing platforms, provides programs for pseudorandom number generation and statistical sample path data analysis. The software is suitable for use with the exercises as well as for more general applications. For professional mathematical scientists and engineers this book provides a ready reference to the Monte Carlo method, especially to implementatable algorithms for performing sampling experiments on a computer and for analyzing their results.

Product Details

• Hardcover: 728 pages
• Publisher: Springer; Corrected edition (April 25, 1996)
• Language: English
• ISBN-10: 038794527X
• ISBN-13: 978-0387945279
• Product Dimensions: 9.4 x 7.3 x 1.6 inches
• Shipping Weight: 2.9 pounds

This review is from: Monte Carlo (Hardcover)
This book gives a rigorous introduction to the main ideas behind the Monte Carlo technique and also gives many concrete examples to illustrate the important concepts. The applications of Monte Carlo are immense, and cover a wide range of fields, including finance, physics, chemistry, computational biology, geology, computational radiology, and network engineering.
The author begins, naturally, with a discussion of how to compute the volume of a high-dimensional body using standard (deterministic) methods and then shows the advantages of using Monte Carlo to find the volume. He does a fine job of assessing the errors in the volume calculation, being careful to distinguish between convergence with probability one and convergence in probability. He also explains the need for specifying confidence levels, and not just the epsilon error term, to determine the smallest sample size that guarantees an error no larger than the specified epsilon. He also gives an interesting application to network reliability in this chapter. The probability that two network nodes are connected is reduced to a volume computation which is then estimated using Monte Carlo sampling. This is an excellent example of how Monte Carlo can be used to arrive at an accurate estimate of an intractable problem. The author gives another example of this later in the chapter wherein he uses Monte Carlo methods in combinatorial probability. Also included in the chapter are some useful hands-on exercises for the reader. The sometimes tricky procedure of generating samples from a variety of distributions is the subject of the next chapter. The author is careful throughout the chapter to distinguish between results that are exact from a formal standpoint and those that are implementable in practice. In addition, a thorough discussion of the error introduced by using pseudorandom numbers in place of sequences of uniform deviates is given. This is the only book I know of that discusses this issue with the clarity it does. The author treats the inverse transform, composition, acceptance-rejection, ratio-of-uniforms, and exact-approximation in great detail. Here again a very useful set of hands-on exercises is given at the end of the chapter. Increasing the efficiency of Monte Carlo sampling is the subject of the next chapter, wherein importance sampling, control variates, stratified sampling, correlated sampling, and conditional Monte Carlo are discussed in detail. These methods are usually called variance reduction techniques, but the author gives an interesting argument about why this characterization is not really accurate. The author brings in conditional sampling, or Markov chain sampling, in the next chapter. It is this approach that has made Monte Carlo such a widely used techniques in science, finance, and network queueing problems. He gives a rather quick overview of the necessary background in Markov chains, and then moves on to discuss neutron transport. It was the intractable nature of the Boltzmann transport equation that gave Monte Carlo its first real application back in the 1940's. The much-used Metropolis method is discussed later, with close attention paid to the details. This is a section that should be read by anyone interested in Monte Carlo techniques. This is followed by a detailed discussion of Markov random fields, Gibbs sampling, and simulated annealing, all of these being heavily used in applications. And also, no book on Monte Carlo could be complete without a discussion of the three-dimensional Ising model, which is in here. The next chapter concentrates on sampling design and statistical inference, wherein the author discusses how choices of the initial (nonequilibrium) distribution, the number of steps, the number of replications, and the simulation time affect the computational and statistical efficiency of the Monte Carlo simulation. He explains very effectively these issues and also the difficulties involved with path-dependence. Network modelers will appreciate his example of routing algorithm performance. The last chapter treats in great detail procedures for generating pseudorandom numbers. The standard methods for doing this are covered, along with spectral tests and performance issues. More exotic pseudorandom generators using nonlinear recursion are also discussed, but the proofs for these are omittted. This excellent book belongs on the shelves of anyone interested in Monte Carlo techniques. The price is reasonable considering how much time it would take to collect all the results in the book from the literature. It deserves a highest recommendation.

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关键词：Applications Application Monte Carlo Algorithms Algorithm computer research science course matrix

Contents
Preface ................................................................ v
Selected Notation ..................................................... xvii

I Introduction ........................................................ 1
1.1 About This Book .................................................. 4
1.2 Available Software ................................................. 9
1.3 What This Book Does Not Contain ................................... 11
1.4 Conventions ....................................................... ll
References ............................................................ 12

2 Estimating Volume and Count ....................................... 13
2.1 Volume ........................................................... 15
2.2 Error and Sample Size Considerations ................................. 19
2.3 Confidence Intervals ................................................ 27
2.4 Exploiting Regional Bounds ......................................... 34
2.5 Relative Error .................................................... 51
2.6 Network Reliability ............................................... 65
2.7 Multivariable Integration .......................................... 64
2.8 Exploiting Function Bounds ....................................... 70
2.9 Exploiting Parameter Bounds ...................................... 72
2.10 Restricting the Sampling Region .................................... 75
2.11 Reducing the Sampling Dimension .................................. 77
2.12 Counting Problems ............................................... 78
2.13 Sensitivity Analysis ............................................... 82
2.14 Simultaneous Confidence Intervals .................................. 103
2.15 Ratio Estimation ................................................. 108
2.16 Sequential Estimation ............................................. 117
Appendix ............................................................. 122
Exercises ............................................................ 127
References ........................................................... 143

3 Generating Samples ................................................ 145
3.1 Independence and Dependence ..................................... 147
3.2 Inverse Transform Method ........................................ 149
3.3 Cutpoint Method ................................................ 156
3.4 Composition Method ............................................. 160
3.5 Alias Method .................................................... 165
3.6 Acceptance-Rejection Method ...................................... 171
3.7 Ratio-of-Uniforms Method ......................................... 179
3.8 Exact-Approximation Method ..................................... 185
3.9 Algorithms for Selected Distributions ................................ 187
3.10 Normal Distribution .............................................. 189
3.12 Lognormal Distribution ........................................... 192
3.13 Cauchy Distribution .............................................. 192
3.14 Gamma Distribution .............................................. 193
3.15 Beta Distribution ................................................. 200
3.16 Student's t Distribution ............................................ 207
3.17 Snedecor's F Distribution .......................................... 208
3.18 Revisiting the Ratio-of-Uniforms Method ............................ 209
3.19 Poisson Distribution .............................................. 211
3.20 Binomial Distribution ............................................. 215
3.21 Hypergeometric Distribution ....................................... 218
3.22 Geometric Distribution ............................................ 221
3.23 Negative Binomial Distribution ..................................... 222
3.24 Multivariate Normal Distribution ................................... 223
3.25 Multinomial Distribution .......................................... 224
3.26 Order Statistics ................................................... 226
3.27 Sampling Without Replacement and Permutations ..................... 230
3.28 Points in and on a Simplex ......................................... 232
3.29 Points in and on a Hyperellipsoid ................................... 234
3.30 Bernoulli Trials .................................................. 235
3.31 Sampling from a Changing Probability Table ......................... 236
3.32 Random Spanning Trees ........................................... 241
Exercises ............................................................ 242
References ........................................................... 251


4 Increasing Efficiency ................................................ 255
4.1 Importance Sampling ............................................. 257
4.2 Control Variates ................................................. 277
4.3 Stratified Sampling ............................................... 296
4.4 Inducing Correlation .............................................. 311
4.5 Conditonal Monte Carlo .......................................... 321
Exercises ............................................................ 325
References ........................................................... 332


5 Random Tours ..................................................... 335
5.1 Markov Processes ................................................ 338
5.2 Random Walk ................................................... 343
5.3 Markov Time .................................................... 347
5.4 Score Processes .................................................. 347
5.5 Neutron Transport ............................................... 347
5.6 Buffer Exceedance on a Production Line ............................. 353
5.7 Fredholm Equations of the Second Kind ............................. 358
5.8 Catastrophic Failure .............................................. 363
5.9 First Passage Time ............................................... 368
5.10 Random Stopping Time ........................................... 371
5.11 Generating Random Points from a Target Distribution ................. 376
5.12 Generating a Uniformly Distributed Point on a Finite Set .............. 378
5.13 Generating All Coordinates in a Bounded Region on Each Step ......... 380
5.14 Metropolis Method ............................................... 384
5.15 Sampling Coordinates One at a Time ................................ 388
5.16 Markov Random Fields ........................................... 396
5.17 Gibbs Sampling .................................................. 402
5.18 Simulated Annealing .............................................. 403
5.19 Bayesian Posterior Distributions .................................... 406
5.20 Edge Effects ..................................................... 411
5.21 Time to Stationarity .............................................. 413
5.22 Spectral Structure ................................................ 415
5.23 Bounds on Error ................................................. 418
5.24 Varying Initial Conditions ......................................... 423
5.25 Random Walk on a Hypercube ..................................... 424
5.26 Conductance .................................................... 425
5.27 More About a Random Walk on a Hypercube ........................ 430
5.28 An Alternative Error Bound for Stationarity .......................... 431
5.29 Sampling from a Hyperrectangular Grid ............................. 433
5.30 Sampling from a Hyperrectangle .................................... 435
5.31 Product Estimator ................................................ 437
5.32 Estimating the Volume of a Convex Body ............................ 442
5.33 Estimating the Permanent ......................................... 452
5.34 Coupling ........................................................ 459
5.35 Strong Markov Time ............................................. 464
5.36 Strong Stationary Dual Process ..................................... 467
5.37 Thresholds ...................................................... 472
Exercises ............................................................ 475
References ........................................................... 488


6 Designing and Analyzing Sample Paths ............................. 493
6.1 Problem Context ................................................. 495
6.2 A First Approach to Computing Confidence Intervals .................. 500
6.3 Warm-Up Analysis ............................................... 505
6.4 Choosing a "Good" Initial State or a "Good" n o ...................... 515
6.5 Strictly Stationary Stochastic Processes .............................. 520
6.6 Optimal Choice of Sample Path Length t and Number of Replications n .. 525
6.7 Estimating Required Sample Path Length ............................ 531
6.8 Characterizing Convergence ........................................ 532
6.9 An Alternative View of var Xt ....................................... 535
6.10 Batch Means Method .............................................. 540
6.11 Batch Means Analysis Programs .................................... 565
6.12 Regenerative Processes ............................................ 572
6.13 electing an Optimal Acceptance Scheme for Metropolis Sampling ....... 578
Exercises ............................................................ 580
References ........................................................... 585

7 Generating Pseudorandom Numbers ................................ 587
7.1 Linear Recurrence Generators ...................................... 589
7.2 Prime Modulus Generators ......................................... 592
7.3 Generators with M = 2*Beta (Beta >= 3) ............................. 598
7.4 Mixed Congruential Generators ..................................... 601
7.5 Implementation and Portability ..................................... 602
7.6 Apparent Randomness ............................................. 607
7.7 Spectral Test ..................................................... 611
7.8 Minimal Number of Parallel Hyperplanes ............................ 617
7.9 Distance Between Points ........................................... 620
7.10 Discrepancy ...................................................... 621
7.11 Beyer Quotient ................................................... 627
7.12 Empirical Assessments ............................................ 628
7.13 Combining Linear Congruential Generators .......................... 634
7.14 j-Step Linear Recurrence .......................................... 645
7.15 Feedback Shift Register Generators ................................. 649
7.16 Generalized Feedback Shift Register Generators ...................... 658
7.17 Nonlinear Generators ............................................. 670
Appendix ............................................................. 675
Exercises ............................................................ 676
References ........................................................... 679


Author Index .......................................................... 683
Subject Index ......................................................... 689

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