Statistical computing in its broadest sense is an ever-growing field far too extensive to be covered in a single text. The current book has a far more manageable scope, notwithstanding its title. Its focus is on the use of Monte Carlo methods to simulate random systems and explore statistical models.
The author proposes to investigate three different kinds of questions about systems with random components. The first is to understand the normal or nominal behavior of the system. The second is to quantify the variability and estimate the magnitude of random fluctuations. The third is to explore exceptional behavior and attempt to determine the probability of specific untypical events.
The book begins with a strong section on pseudo-random number generation. This includes a discussion of methods of generating uniformly distributed pseudo-random numbers, and of using those to generate random variables with other desired distributions. This is a critical piece for getting reliable Monte Carlo simulation results. (I have run such simulations with more than a hundred random variables to generate hundreds of realizations of a physical process. A faulty pseudo-random number generator in a situation like this can introduce subtle but unreal correlations between random variables and cast doubt on all the simulation results.)
Having described how to get the pseudo-random numbers need for statistical simulations, the author proceeds to discuss how appropriate correlations can be introduced between them to get, for example, a desired covariance matrix. With all the tools in place, we move on to the heart of the book: Monte Carlo methods and Markov Chain Monte Carlo methods. The material here is pretty standard but the treatment is a little unusual. Books that focus on simulation generally explain what to do with a minimum of mathematical background. Mathematical treatments usually don’t include algorithms and code. This book has a mix of both.
Beyond the various Monte Carlo techniques, the author also has a short section on methods that can be used when no useful statistical model is available. These include Approximate Bayesian Computation (ABC) and resampling methods (primarily the Bootstrap). Finally, continuous time models get a chapter of their own that includes a discussion of Brownian motion and a bit about stochastic differential equations.
While there are a variety of examples — dealing, for example, with image analysis and the Ising model — there are not enough. The chapters on Monte Carlo techniques would benefit from at least a couple of extended examples in real applications. Several more small-scale examples would also be very desirable.
It is not clear where this book might fit in the curriculum. The author aimed it at an undergraduate audience and has limited the mathematical background accordingly, but the topics are rather specialized. This could be a supplementary source for a course in simulation and modeling, especially where Monte Carlo techniques are included.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.