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THEORY AND APPLICATIONS OF MONTE CARLO SIMULATIONS Edited by Victor (Wai Kin) Chan Theory and Applications of Monte Carlo Simulations http://dx.doi.org/10.5772/45892 Edited by Victor (Wai Kin) Chan Contributors Dragica Vasileska, Shaikh Ahmed, Mihail Nedjalkov, Rita Khanna, Mahdi Sadeghi, Pooneh Saidi, Claudio Tenreiro, Elshemey, Subhadip Raychaudhuri, Krasimir Kolev, Natalia D. Nikolova, Daniela Toneva-Zheynova, Kiril Tenekedjiev, Vladimir Elokhin, Wai Kin (Victor) Chan, Charles Malmborg, Masaaki Kijima, Ianik Plante, Paulo Guimarães Couto, Jailton Damasceno, Sérgio Pinheiro Oliveira Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2013 InTech All chapters are Open Access distributed under the Creative Commons Attribution 3.0 license, which allows users to download, copy and build upon published articles even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. Notice Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published chapters. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Iva Simcic Technical Editor InTech DTP team Cover InTech Design team First published March, 2013 Printed in Croatia A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechopen.com Theory and Applications of Monte Carlo Simulations, Edited by Victor (Wai Kin) Chan p. cm. ISBN 978-953-51-1012-5 free online editions of InTech Books and Journals can be found at www.intechopen.com Contents Preface VII Chapter 1 Monte Carlo Statistical Tests for Identity of Theoretical and Empirical Distributions of Experimental Data 1 Natalia D. Nikolova, Daniela Toneva-Zheynova, Krasimir Kolev and Kiril Tenekedjiev Chapter 2 Monte Carlo Simulations Applied to Uncertainty in Measurement 27 Paulo Roberto Guimarães Couto, Jailton Carreteiro Damasceno and Sérgio Pinheiro de Oliveira Chapter 3 Fractional Brownian Motions in Financial Models and Their Monte Carlo Simulation 53 Masaaki Kijima and Chun Ming Tam Chapter 4 Monte-Carlo-Based Robust Procedure for Dynamic Line Layout Problems 87 Wai Kin (Victor) Chan and Charles J. Malmborg Chapter 5 Comparative Study of Various Self-Consistent Event Biasing Schemes for Monte Carlo Simulations of Nanoscale MOSFETs 109 Shaikh Ahmed, Mihail Nedjalkov and Dragica Vasileska Chapter 6 Atomistic Monte Carlo Simulations on the Formation of Carbonaceous Mesophase in Large Ensembles of Polyaromatic Hydrocarbons 135 R. Khanna, A. M. Waters and V. Sahajwalla Chapter 7 Variance Reduction of Monte Carlo Simulation in Nuclear Engineering Field 153 Pooneh Saidi, Mahdi Sadeghi and Claudio Tenreiro Chapter 8 Stochastic Models of Physicochemical Processes in Catalytic Reactions - Self-Oscillations and Chemical Waves in CO Oxidation Reaction 173 Vladimir I. Elokhin Chapter 9 Monte-Carlo Simulation of Particle Diffusion in Various Geometries and Application to Chemistry and Biology 193 Ianik Plante and Francis A. Cucinotta Chapter 10 Kinetic Monte Carlo Simulation in Biophysics and Systems Biology 227 Subhadip Raychaudhuri Chapter 11 Detection of Breast Cancer Lumps Using Scattered X-Ray Profiles: A Monte Carlo Simulation Study 261 Wael M. Elshemey ContentsVI Preface The objective of this book is to introduce recent advances and state-of-the-art applications of Monte Carlo Simulation (MCS) in various fields. MCS is a class of statistical methods for performance analysis and decision making based on taking random samples from underly‐ ing systems or problems to draw inferences or estimations. Let us make an analogy by using the structure of an umbrella to define and exemplify the position of this book within the fields of science and engineering. Imagine that one can place MCS at the centerpoint of an umbrella and define the tip of each spoke as one engineering or science discipline: this book lays out the various applications of MCS with a goal of sparking innovative exercises of MCS across fields. Despite the excitement that MCS spurs, MCS is not impeccable due to criticisms about leak‐ ing a rigorous theoretical foundation—if the umbrella analogy is made again, then one can say that “this umbrella” is only half-way open. This book attempts to open “this umbrella” a bit more by showing evidence of recent advances in MCS. To get a glimpse at this book, Chapter 1 deals with an important question in experimental studies: how to fit a theoretical distribution to a set of experimental data. In many cases, dependence within datasets invalidates standard approaches. Chapter 1 describes an MCS procedure for fitting distributions to datasets and testing goodness-of-fit in terms of statisti‐ cal significance. This MCS procedure is applied in charactering fibrin structure. MCS is a potential alternative to traditional methods for measuring uncertainty. Chapter 2 exemplifies such a potential in the domain of metrology. This chapter shows that MCS can overcome the limitations of traditional methods and work well on a wide range of applica‐ tions. MCS has been extensively used in the area of finance. Chapter 3 presents various sto‐ chastic models for simulating fractional Brownian motion. Both exact and approximate methods are discussed. For unfamiliar readers, this chapter can be a good introduction to these stochastic models and their simulation using MCS. MCS has been a popular approach in optimization. Chapter 4 presents an MCS procedure to solving dynamic line layout prob‐ lems. The line layout problem is a facility design problem. It concerns with how to optimally allocate space to a set of work centers within a facility such that the total intra traffic flow among the work centers is minimized. This problem is a difficult optimization problem. This chapter presents a simple MCS approach to solve this problem efficiently. MCS has been one major performance analysis approach in semiconductor manufacturing. Chapter 5 deals with improving the MCS technique used for Nanoscale MOSFETs. It intro‐ duces three event biasing techniques and demonstrates how they can improve statistical es‐ timations and facilitate the computation of characteristics of these devices. Chapter 6 describes the use of MCS in the ensembles of polyaromatic hydrocarbons. It also provides an introduction to MCS and its performance in the field of materials. Chapter 7 discusses variance reduction techniques for MCS in nuclear engineering. Variance reduction techni‐ ques are frequently used in various studies to improve estimation accuracy and computa‐ tional efficiency. This chapter first highlights estimation errors and accuracy issues, and then introduces the use of variance reduction techniques in mitigating these problems. Chapter 8 presents experimental results and the use of MCS in the formation of self-oscillations and chemical waves in CO oxidation reaction. Chapter 9 introduces the sampling of the Green’s function and describes how to apply it to one, two, and three dimensional problems in parti‐ cle diffusion. Two applications are presented: the simulation of ligands molecules near a plane membrane and the simulation of partially diffusion-controlled chemical reactions. Simulation results and future applications are also discussed. Chapter 10 reviews the appli‐ cations of MCS in biophysics and biology with a focus on kinetic MCS. A comprehensive list of references for the applications of MCS in biophysics and biology is also provided. Chap‐ ter 11 demonstrates how MCS can improve healthcare practices. It describes the use of MCS in helping to detect breast cancer lumps without excision. This book unifies knowledge of MCS from aforementioned diverse fields to make a coher‐ ent text to facilitate research and new applications of MCS. Having a background in industrial engineering and operations research, I found it useful to see the different usages of MCS in other fields. Methods and techniques that other research‐ ers used to apply MCS in their fields shed light on my research on optimization and also provide me with new insights and ideas about how to better utilize MCS in my field. In‐ deed, with the increasing complexity of nowadays systems, borrowing ideas from other fields has become one means to breaking through obstacles and making great discoveries. A researcher with his/her eyes open in related knowledge happening in other fields is more likely to succeed than one who does not. I hope that this book can help shape our understanding of MCS and spark new ideas for novel and better usages of MCS. As an editor, I would like to thank all contributing authors of this book. Their work is a valuable contribution to Monte Carlo Simulation research and applications. I am also grate‐ ful to InTech for their support in editing this book, in particular, Ms. Iva Simcic and Ms. Ana Nikolic for their publishing and editorial assistance. Victor (Wai Kin) Chan, Ph.D. Associate Professor Department of Industrial and Systems Engineering Rensselaer Polytechnic Institute Troy, NY USA PrefaceVIII Chapter 1 Monte Carlo Statistical Tests for Identity of Theoretical and Empirical Distributions of Experimental Data Natalia D. Nikolova, Daniela Toneva-Zheynova, Krasimir Kolev and Kiril Tenekedjiev Additional information is available at the end of the chapter http://dx.doi.org/10.5772/53049 1. Introduction Often experimental work requires analysis of many datasets derived in a similar way. For each dataset it is possible to find a specific theoretical distribution that describes best the sam‐ ple. A basic assumption in this type of work is that if the mechanism (experiment) to generate the samples is the same, then the distribution type that describes the datasets will also be the same [1]. In that case, the difference between the sets will be captured not through changing the type of the distribution, but through changes in its parameters. There are some advantag‐ es in finding whether a type of theoretical distribution that fits several datasets exists. At first, it improves the fit because the assumptions concerning the mechanism underlying the experi‐ ment can be verified against several datasets. Secondly, it is possible to investigate how the variation of the input parameters influences the parameters of the theoretical distribution. In some experiments it might be proven that the differences in the input conditions lead to quali‐ tative change of the fitted distributions (i.e. change of the type of the distribution). In other cases the variation of the input conditions may lead only to quantitative changes in the output (i.e. changes in the parameters of the distribution). Then it is of importance to investigate the statistical significance of the quantitative differences, i.e. to compare the statistical difference of the distribution parameters. In some cases it may not be possible to find a single type of dis‐ tribution that fits all datasets. A possible option in these cases is to construct empirical distri‐ butions according to known techniques [2], and investigate whether the differences are statistically significant. In any case, proving that the observed difference between theoretical, or between empirical distributions, are not statistically significant allows merging datasets and operating on larger amount of data, which is a prerequisite for higher precision of the statistical results. This task is similar to testing for stability in regression analysis [3]. © 2013 Nikolova et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Formulating three separate tasks, this chapter solves the problem of identifying an appropri‐ ate distribution type that fits several one-dimensional (1-D) datasets and testing the statistical significance of the observed differences in the empirical and in the fitted distributions for each pair of samples. The first task (Task 1) aims at identifying a type of 1-D theoretical distribu‐ tion that fits best the samples in several datasets by altering its parameters. The second task (Task 2) is to test the statistical significance of the difference between two empirical distribu‐ tions of a pair of 1-D datasets. The third task (Task 3) is to test the statistical significance of the difference between two fitted distributions of the same type over two arbitrary datasets. Task 2 can be performed independently of the existence of a theoretical distribution fit valid for all samples. Therefore, comparing and eventually merging pairs of samples will always be pos‐ sible. This task requires comparing two independent discontinuous (stair-case) empirical cu‐ mulative distribution functions (CDF). It is a standard problem and the approach here is based on a symmetric variant of the Kolmogorov-Smirnov test [4] called the Kuiper two-sample test, which essentially performs an estimate of the closeness of a pair of independent stair-case CDFs by finding the maximum positive and the maximum negative deviation between the two [5]. The distribution of the test statistics is known and the p value of the test can be readily estimated. Tasks 1 and 3 introduce the novel elements of this chapter. Task 1 searches for a type of the‐ oretical distribution (out of an enumerated list of distributions) which fits best multiple da‐ tasets by varying its specific parameter values. The performance of a distribution fit is assessed through four criteria, namely the Akaike Information Criterion (AIC) [6], the Baye‐ sian Information Criterion (BIC) [7], the average and the minimal p value of a distribution fit to all datasets. Since the datasets contain random measurements, the values of the parame‐ ters for each acquired fit in Task 1 are random, too. That is why it is necessary to check whether the differences are statistically significant, for each pair of datasets. If not, then both theoretical fits are identical and the samples may be merged. In Task 1 the distribution of the Kuiper statistic cannot be calculated in a closed form, because the problem is to compare an empirical distribution with its own fit and the independence is violated. A distribution of the Kuiper statistic in Task 3 cannot be estimated in close form either, because here one has to compare two analytical distributions, but not two stair-case CDFs. For that reason the dis‐ tributions of the Kuiper statistic in Tasks 1 and 3 are constructed via a Monte Carlo simula‐ tion procedures, which in Tasks 1 is based on Bootstrap [8]. The described approach is illustrated with practical applications for the characterization of the fibrin structure in natural and experimental thrombi evaluated with scanning electron microscopy (SEM). 2. Theoretical setup The approach considers N 1-D datasets χ i = ( x 1 i , x 2 i , , x n i i ) , for i=1,2,…,N. The data set χ i contains n i >64 sorted positive samples (0< x 1 i ≤ x 2 i ≤ ≤x n i i ) of a given random quantity under equal conditions. The datasets contain samples of the same random quantity, but under slightly different conditions. Theory and Applications of Monte Carlo Simulations2 [...]... − invCDF i (0.25) e e e 3 4 Theory and Applications of Monte Carlo Simulations The non-zero part of the empirical density PDF ei (.) is determined in the closed interval x1i − Δdi ; xnii + Δui as a histogram with bins of equal area (each bin has equal product of densi‐ ty and span of data) The number of bins bi is selected as the minimal from the Scott [9], Sturges [10] and Freedman-Diaconis [11] suggestions:... convenient evaluation of these structures with respect to their chemical stability and resistance to enzymatic degradation [16] 19 20 Theory and Applications of Monte Carlo Simulations Figure 6 Fibrin structure on the surface and in the core of thrombi A Following thrombectomy thrombi were wash‐ ed, fixed and dehydrated SEM images were taken from the surface and transverse section of the same thrombus... points of fibrin fibers Sample size (N), mean (meane in μm), median (mede in μ m), standard deviation (stde), inter-quartile range (iqre, in μm) of the empirical distributions over the 12 datasets for different thrombin concentrations (in U/ml) and buffers are presented 13 14 Theory and Applications of Monte Carlo Simulations Figure 1 SEM image of fibrin used for morphometric analysis Figure 2 Steps of. .. cited 28 Theory and Applications of Monte Carlo Simulations Measurement uncertainty is a quantitative indication of the quality of measurement results, without which they could not be compared between themselves, with specified reference values or to a standard According to the context of globalization of markets, it is necessary to adopt a universal procedure for estimating uncertainty of measurements,... 9.18e-03 2.06e-04 1.00e+00 Table 4 P-values of the statistical test for identity of stair-case distributions on pairs of datasets The values on the main diagonal are shaded The bold values are those that exceed 0.05, i.e indicate the pairs of datasets whose staircase distributions are identical 17 Theory and Applications of Monte Carlo Simulations 4.3 Task 3 – Identity of fitted distributions As concluded... [12] is a measure for the closeness of the two ‘stair‐ i1 i2 case’ empirical cumulative distribution functions CDF sce (.) and CDF sce (.): i1 i2 i2 i1 V i1,i2 = max{CDF sce ( x ) − CDF sce ( x )} + max{CDF sce ( x ) − CDF sce ( x )} x x (8) 7 8 Theory and Applications of Monte Carlo Simulations The distribution of the test statics V i1,i2 is known and the p-value of the two tail statistical test with... Resistance of Fibrin Containing Red Blood Cells Arterioscl Thromb Vasc Biol 2011; 31 2306-2313 25 26 Theory and Applications of Monte Carlo Simulations [18] Palisade Corporation Guide to Using @RISK – Risk Analysis and Simulation Add-in for Microsoft Excel, Version 4.5 USA: Palisade Corporation; (2004) [19] Geer Mountain Software Corporation Inc Stat::Fit - Version 2 USA: Geer Mountain Software Corporation... simulation cycles The table contains the maximal values for minPvalue, j and meanPvalue, j , and the minimal values for AICj and BICj across the datasets for each distribution type The bold and the italic values are the best one and the worst one achieved for a given criterion, respectively 15 Theory and Applications of Monte Carlo Simulations file: length/L0408full ; variable:t5 0.5 0 0 0.5 1 2 2.5 PDF... Fig 8) It is noteworthy that the type of distribution was not changed by the drug, only its parameters were modified This example underscores the applicability of the designed procedure for testing of statistical hypotheses in situations when subtle quantita‐ tive biological and pharmacological effects are at issue 21 22 Theory and Applications of Monte Carlo Simulations Figure 8 Changes in the fibrin... points in fibrin Panels a and b: scaling Panel c: selection of region of interest Panel d: taking a measurement Monte Carlo Statistical Tests for Identity of Theoretical and Empirical Distributions of Experimental Data http://dx.doi.org/10.5772/53049 4.1 Task 1 – Finding a common distribution fit A total of 11 types of distributions (Table 1) are tested over the datasets, and the criteria (3)(6) are . THEORY AND APPLICATIONS OF MONTE CARLO SIMULATIONS Edited by Victor (Wai Kin) Chan Theory and Applications of Monte Carlo Simulations http://dx.doi.org/10.5772/45892 Edited. Biasing Schemes for Monte Carlo Simulations of Nanoscale MOSFETs 109 Shaikh Ahmed, Mihail Nedjalkov and Dragica Vasileska Chapter 6 Atomistic Monte Carlo Simulations