Theory and Applications of Monte Carlo Simulations

Victor (Wai Kin) Chan (Ed.) InTech, 364 pp. ISBN: 978-953-51-1012-5

The book ‘Theory and Applications of Monte Carlo Simulations’ is an excellent reference for researchers interested in exploring the full potential of Monte Carlo Simulations (MCS).

The book focuses on the latest advances and applications of (MCS). It presents eleven chapters that show to which extent sampling techniques can be exploited to solve complex problems or analyze complex systems in a variety of engineering and science topics. The book explores most of the issues and topics in MCS including goodness of fit, uncertainty evaluation, variance reduction, optimization, and statistical estimation. All these concepts are extensively discussed and examples of solutions are given in order to help both researchers and practitioners. Perhaps the most valuable parts of the book are the examples of ground- breaking applications of MCS in financial systems modeling and estimation of transition behavior of organic molecules, but every single chapter teaches at least one new application or explains a useful concept about MCS. The aim of this book is to combine knowledge of MCS from diverse fields to facilitate research and new applications of MCS. Any creative researcher will be able to take advantage of the presented techniques and can find plenty of applications in their scientific field.

Monte Carlo Simulations

Monte Carlo simulations are an extensive class of computational algorithms that rely on repeated random sampling to obtain numerical results. One runs simulations a large number of times over in order to obtain the distribution of an unknown probabilistic entity. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to obtain a closed-form expression, or infeasible to apply a deterministic algorithm, as it also happens in medical and socioeconomic issues. Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration and generation of draws from a probability distribution.

MCS are particularly useful in modeling phenomena with significant uncertainty in inputs such as the calculation of risk in business and evaluation of multidimensional definite integrals with complicated boundary conditions. In applications such as oil exploration problems, Monte Carlo–based predictions of failure, cost overruns and schedule overruns are routinely better than human intuition or alternative methods.

The modern version of the Monte Carlo method was first applied in the late 1940s by Stanislaw Ulam, while he was working on nuclear weapons projects at the Los Alamos National Laboratory. MCS tend to follow this pattern: First a domain of possible inputs is defined, then inputs are randomly generated from a probability distribution over the domain.

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A deterministic computation on the inputs is performed and finally the results are aggregated

There is no consensus about the definition of MCS. Sawilowsky establishes a classification of three different concepts: Simulation, a Monte Carlo method, and a Monte Carlo simulation: a simulation is a representation of reality, a Monte Carlo method is a technique that can be used to solve a mathematical or statistical problem, and a Monte Carlo simulation (MCS) uses repeated sampling to determine the properties of some phenomenon or behavior.

It is important to distinguish between these three concepts. An example of a simulation is to use a uniform distribution [0,1] to figure out the number of tails and heads when the number obtained is higher or lower than 0.5. A Monte Carlo method would be to toss a number of coins and compute the ratio of heads vs. tails a number of times to determine the behaviour of this variable. A MCS would be to use a large number of random uniform variables to simulate the behavior of tossing a coin. As it can be seen, these three concepts are related but they are different.

MCS sample probability distribution for each variable of the system to produce a large number of possible outcomes. The results are analyzed to get probabilities of each different outcome. The main difference with traditional ‘what if’ analysis is MCS can provide samples in the very low probability regions and it makes possible a better analysis of the phenomenon under study. Thus, MCS are particularly helpful for simulating situations with considerable uncertainty in inputs and systems with a significant number of coupled degrees of freedom.

Book insights

The book begins with a chapter that develops a platform in Matlab that allows users to test eleven different kinds of distributions in their datasets and add new types of distributions. The correct identification of the underlying distribution is a key issue prior to data analysis and comparison with previous researches with different parameters. The Kolmogorov- Smirnov test is widely used as it is conservative but it is not always the most appropriate test. As the authors show, it is always advisable to use MCS in order to correctly identify the distribution of the data with a higher likelihood than other conventional methods.

In addition to data analysis, MCS can improve the estimation of measurement uncertainties using the law of propagation of uncertainties that is based on several assumptions that are not always applicable. As chapter 2 explains the main errors are the following:  There is significant non-linearity in a portion of processes that make unacceptable Taylor series analysis.  The central limit is not always valid as data does not tend to a Normal distribution in all the cases.  Conventional methods need to compute the degrees of freedom and sometimes this calculation is arbitrary and therefore inadequate. In this way, MCS can cope with broader problems related to uncertainties in measurement and replicate the results of other approaches for simple models. Then, their usage would be highly advisable.

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Chapter 3 shows how MCS can be divided between exact and approximate schemes. Exact schemes capture the whole length of the sample size and the approximate methods truncate the covariance structure to obtain robust results. These schemes are applied to a specific example in a finance model but it could be easily applied to a number of models with time dependence. MCS allows computing long-range time dependence under different assumptions and makes it possible to get to the optimal result in terms of accuracy, robustness and computing speed in the case of big data being involved in the analysis.

The fourth chapter presents a procedure to find robust solutions to dynamic line layout problems. This approach finds solutions that achieve minimum criteria for each scenario and estimates the form of the volume distance distribution of each parameter. Nonlinear variables are not a problem for this approach and a limited number of random samples are usually enough to reach a value sufficiently near to the global optimal solution. The sort of problems that this method is capable of solving can be unmanageable with common computational tools and it can be used in a large number of scenarios.

Self-consistent event biasing schemes for MCS are also studied in the book. Three- dimensional Monte Carlo device simulator (MCDS 3-D) and the associated models/kernels such as event biasing and effective quantum potential are used in the computation of bias of subatomic particles. This method has the potential to simulate a high number of chemical, biological and social problems with a high degree of accuracy while still including the desired degree of randomness.

The accuracy, precision and relative error in MCS are also commented on chapter 7. The main variable of the analysis is accuracy, but precision should also be taken into account. In most of the occasions there is a trade-off between accuracy and precision and the researcher should choose the appropriate parameters to optimize the results. The variance reduction is a main goal to achieve better results when applying MCS.

Figure 1. Schematic diagram of the definition for accuracy and precision

Source: Theory and Applications of Monte Carlo Simulations, page 159

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In addition, random walk based kinetic MCS can describe diffusion and trafficking in biological systems. Each object, for example a molecule, can be randomly sampled and then the possible movements can be simulated according to predefined probabilities. The applications include complex systems of rules of cellular signaling, as extensively studied in chapter 10. The last chapter describes, with an example, how MCS is also used to differentiate tissues in medical tests that only show small differences between carcinomas and fibroglandular tissue, thus improving the diagnosis.

In summary, this book is an excellent reference on the new fields of application of MCS, a highly adaptable technique that allow studying multiple problems in different areas of knowledge. The theoretical and applied examples described in the book are valuable not only to researchers in the fields of the related examples but also to any researcher in biological systems, physics, engineering and social sciences. However, this book is not a guide for beginners to MCS and previous knowledge is required to follow most of its contents.

The book is published under a Creative Commons 3.0 license and is freely available at www.intechopen.com/books/theory-and-applications-of-monte-carlo-simulations

By Irene Begoña Lara Universidad Autónoma de Madrid

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