Preface

Monte Carlo integration is an innovative tool used in wide-ranging applications in science, mathematics, engineering, medicine, and economics. On a deeper level, it has become the go-to tool in physics, chemistry, biology, astronomy, digital signal processing, machine learning, control systems, robotics, medical, and economic applications. The MathWorks® has provided MATLAB® and Simulink® software proven to be invaluable in this pursuit. MATLAB® and Simulink® provide the necessary computational and simulation capability to address the plethora of applications. Numerous Monte Carlo integration and numerical integration examples based on this software are intended to be complementary allowing further insight into complex computations.

Preceding Monte Carlo integration numerical integration accomplished numerical analysis involving multiple methods to evaluate a definite integral. Both computational methods with known, exact expressions and simulation methods for unknown or unavailable expressions are used. MATLAB® and Simulink® tools are invoked to obtain accurate results of the integration. The extension to Monte Carlo integration is a natural step to where the underlying computations are based on random variables.

In this book, the reader may choose to follow the traditional approach that proceeds from introductory material to increasingly more complicated explanations. Alternatively, the presentation often allows the reader to scan the available examples and proceed directly to an example of interest while diminishing the use of prior information. By having the MATLAB® and Simulink® programs adjoining the computations, a close-coupling permits quick back-and-forth study of the program code and the computations. Where necessary, mathematical explanations are included for clarification.

This book focuses on Monte Carlo integration methods where an introductory aspect is pursued in terms of traditional, i.e., deterministic methods. Use of Monte Carlo integration via simulation provides an estimate of the actual integral based on random sampling. The latter case is a tribute to the recent advances in computational techniques via computer simulation involving random processes. The principal attention of the presented examples focuses on MATLAB® where extensive Simulink® examples are provided subsequently.

An important goal in providing the detailed examples presented here is to expand the reader's knowledge and ability to develop their own programs. As a consequence, an interactive approach in use of the provided examples is recommended. The reader can realize maximum benefit by adopting the suggested actions listed below:

• Replicate specific examples to ensure an understanding of the computations.
• Modify the inputs of the selected example for exploring the consequences of the ensuing changes.
• Replace the function in the integration routine with a user-defined alternative and execute the modified example.

Scope

This book introduces the reader to a wide selection of Monte Carlo-based numerical integration techniques. The selection of the method to use is dependent on the users' application likely involving random number generation and expectations of functions of random variables. MATLAB® is the naturally chosen computational technique for providing accurate, efficient solutions of often complicated, intractable problems. As a complementary tool, Simulink® examples, involving Monte Carlo-based numerical integration, is included to explore unlimited topics in diverse technologies.

It is noted that the MATLAB® and Simulink® programs presented here assumes that the reader has rudimentary knowledge of MATLAB® and Simulink®. The MathWorks® has excellent tutorials on both of these topics; see https://www.MathWorks.com

Organization of the Book

Chapters 1-13 encompass numerical integration methods where MathWorks® programs in MATLAB® can perform integration of any well-defined function. In Chapters 4-13, Monte Carlo integration via MATLAB® encompasses many computational and simulation-based methods whose function is intractable, complicated, or undefined. These methods include accept-rejection sampling, importance sampling, Metropolis-Hastings, Gibbs sampling, slice sampling, Hamiltonian Monte Carlo sampling, and sequential Monte Carlo, also known as particle filtering. This last subject delves into nonlinear, non-Gaussian models and is more indicative of realistic cases.

In Chapter 14, Monte Carlo Integration via Simulink® addresses numerous examples focusing on science, engineering, astronomy/cosmology, neural networks, robotics, Kalman filtering, and sequential Monte Carlo or particle filtering.

Chapters 2-13 contain detailed MATLAB® examples and the accompanying code. Chapter 14 contains multiple Simulink® models concluding in particle filtering.

Chapters 2-14 contain suggested problems where solutions appear on a website.

Chapter 1

This chapter outlines the framework for performing numerical integration using Monte Carlo methods. Integration using computer-based discrete methods replaces continuous integration where convergence is assured via the Law of Large Numbers (LLN). Following a historical introduction to numerical integration, the focus quickly advances to modern Monte Carlo integration methods where intractable integrals can be evaluated using pseudo-random sampling. In the subsequent chapters, numerous Monte Carlo integration methods are presented concluding with sequential Monte Carlo or particle filters where the general application involves nonlinear, non-Gaussian computations.

Chapter 2

Numerical integration methods for deterministic or nonrandom applications have been a mainstay for the evaluation of definite integrals prior to the availability of laptop computers possessing extensive memory and fast computational capability. Prior advances in computational techniques involving Legendre-Gauss quadrature methods that remove the restriction of equally spaced intervals in the integrand have been successful in evaluating a myriad of complicated integrals. Currently modern laptop computers with extensive software capability propelled the advance of numerical integration methods where MATLAB®'s program integral is a premier tool in this regard. A comparison of MATLAB®'s integral program with standard techniques such as rectangular, trapezoidal, and Simpson's method establishes a baseline for the study of Monte Carlo integration methods.

Chapter 3

This chapter represents a compendium of examples using MATLAB® integral as well as MATLAB® programs trapz and quad. Program trapz computes the integral of a vector via the trapezoidal method where for matrices the computation produces a row vector with the integral over each column. Program quad uses recursive adaptive Simpson quadrature to determine the integral of scalar-valued function with an absolute error tolerance. Examples include power spectrum estimation of a linear system, blackbody radiation emittance and integration of the probability density for Cauchy and standard normal distributions. This preliminary background establishes the starting point where Monte Carlo integration can be recognized as the pinnacle of computational methods.

Chapter 4

Random number generation is the backbone of Monte Carlo integration where multiple samples of the integrand are needed. Chebyshev's inequality bounds the integration error. For a sufficiently large sample size, the estimate of the integral is normally distributed. Using discrete samples of the integrand, the sample mean provides an estimate of the integral with a sample variance that decreases with the number of samples. When the integration involves a random variable with a known probability density, the Monte Carlo sample mean provides an estimate of the integral using randomly generated samples.

Chapter 5

Monte Carlo integration is instrumental in integrating complicated or undefined functions. Using computer-based computations with random sampling, two principal binary methods applied for arbitrary functions are hit-or-miss and accept-reject, also referred to as rejection sampling. The function to be integrated is assumed to have a maximum value within the integration interval. In the Monte Carlo algorithm, randomly generated samples are dropped into the area of interest and the probability that the samples lie inside the region of interest are estimated thereby allowing an estimate of the integral to be obtained. This method is applicable to multiple integrals involving three-dimensional objects such as a sphere.

Chapter 6

Monte Carlo methods are frequently used in the integration of normal distributions. In this chapter, MATLAB® program integral, a sampled version of the integral and normally distributed samples generated from MATLAB® routine randn are used in conjunction with a MATLAB® program ksdensity for estimating the probability density function. An important computation with normal distributions involves the estimation of the tail of the distribution. Investigating multiple methods leads to a comparison of the various techniques in terms of the sample size and variance of the estimate.

Chapter 7

This chapter presents...