This thesis is composed of three main parts. Each chapter is concerned with characterizing some properties of a random ensemble or stochastic process. The properties of interest and the methods for investigating them di er between chapters. We begin by establishing some asymptotic results regarding zeros of random harmonic mappings, a topic of much interest to mathematicians and astrophysicists alike. We introduce a new model of harmonic polynomials based on the so-called "Weyl ensemble" of random analytic polynomials. Building on the work of Li and Wei [28] we obtain precise asymptotics for the average number of zeros of this model. The primary tools used in this section are the famous Kac-Rice formula as well as classical methods in the asymptotic analysis of integrals such as the Laplace method. Continuing, we characterize several topological properties of this model of harmonic polynomials. In chapter 3 we obtain experimental results concerning the number of connected components of the orientation-reversing region as well as the geometry of the distribution of zeros. The tools used in this section are primarily Monte Carlo estimation and topological data analysis (persistent homology). Simulations in this section are performed within MATLAB with the help of a computational homology software known as Perseus. While the results in this chapter are empirical rather than formal proofs, they lead to several enticing conjectures and open problems. Finally, in chapter 4 we address an industry problem in applied mathematics and machine learning. The analysis in this chapter implements similar techniques to those used in chapter 3. We analyze data obtained by observing CAN tra c. CAN (or Control Area Network) is a network for allowing micro-controllers inside of vehicles to communicate with each other. We propose and demonstrate the e ectiveness of an algorithm for detecting malicious tra c using an approach that discovers and exploits the natural geometry of the CAN surface and its relationship to random walk Markov chains. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2018. / FAU Electronic Theses and Dissertations Collection

We consider the application of multilevel Monte Carlo methods to elliptic partial differential equations with random coefficients. Such equations arise, for example, in stochastic groundwater ow modelling. Models for random coefficients frequently used in these applications, such as log-normal random fields with exponential covariance, lack uniform coercivity and boundedness with respect to the random parameter and have only limited spatial regularity. To give a rigorous bound on the cost of the multilevel Monte Carlo estimator to reach a desired accuracy, one needs to quantify the bias of the estimator. The bias, in this case, is the spatial discretisation error in the numerical solution of the partial differential equation. This thesis is concerned with establishing bounds on this discretisation error in the practically relevant and technically demanding case of coefficients which are not uniformly coercive or bounded with respect to the random parameter. Under mild assumptions on the regularity of the coefficient, we establish new results on the regularity of the solution for a variety of model problems. The most general case is that of a coefficient which is piecewise Hölder continuous with respect to a random partitioning of the domain. The established regularity of the solution is then combined with tools from classical discretisation error analysis to provide a full convergence analysis of the bias of the multilevel estimator for finite element and finite volume spatial discretisations. Our analysis covers as quantities of interest several spatial norms of the solution, as well as point evaluations of the solution and its gradient and any continuously Fréchet differentiable functional. Lastly, we extend the idea of multilevel Monte Carlo estimators to the framework of Markov chain Monte Carlo simulations. We develop a new multilevel version of a Metropolis Hastings algorithm, and provide a full convergence analysis.

In summary, the viewpoint of random graphs indeed provides us an opportunity of improving some existing machine learning algorithms. / In this thesis, we establish three machine learning models on random graphs: Heat Diffusion Models on Random Graphs, Predictive Random Graph Ranking, and Random Graph Dependency. The heat diffusion models on random graphs lead to Graph-based Heat Diffusion Classifiers (G-HDC) and a novel ranking algorithm on Web pages called DiffusionRank. For G-HDC, a random graph is constructed on data points. The generated random graph can be considered as the representation of the underlying geometry, and the heat diffusion model on them can be considered as the approximation to the way that heat flows on a geometric structure. Experiments show that G-HDC can achieve better performance in accuracy in some benchmark datasets. For DiffusionRank, theoretically we show that it is a generalization of PageRank when the heat diffusion coefficient tends to infinity, and empirically we show that it achieves the ability of anti-manipulation. / Predictive Random Graph Ranking (PRGR) incorporates DiffusionRank. PRGR aims to solve the problem that the incomplete information about the Web structure causes inaccurate results of various ranking algorithms. The Web structure is predicted as a random graph, on which ranking algorithms are expected to be improved in accuracy. Experimental results show that the PRGR framework can improve the accuracy of the ranking algorithms such as PageRank and Common Neighbor. / Three special forms of the novel Random Graph Dependency measure on two random graphs are investigated. The first special form can improve the speed of the C4.5 algorithm, and can achieve better results on attribute selection than gamma used in Rough Set Theory. The second special form of the general random graph dependency measure generalizes the conditional entropy because it becomes equivalent to the conditional entropy when the random graphs take their special form-equivalence relations. Experiments demonstrates that the second form is an informative measure, showing its success in decision trees on small sample size problems. The third special form can help to search two parameters in G-HDC faster than the cross-validation method. / Yang, haixuan. / "August 2007." / Advisers: Irwin King; Michael R. Lyu. / Source: Dissertation Abstracts International, Volume: 69-02, Section: B, page: 1125. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 184-197). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract in English and Chinese. / School code: 1307.

by Man-lai Tang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1991. / Includes bibliographical references. / Chapter PART I ： --- ANALYSIS OF DATA WITH POLYTOMOUS VARIABLES --- p.1 / Chapter Chapter 1 --- Introduction --- p.1 / Chapter Chapter 2 --- Estimation of the Model with Incomplete Data --- p.5 / Chapter §2.1 --- The Model --- p.5 / Chapter §2.2 --- Two-stage Estimation Method --- p.7 / Chapter Chapter 3 --- Generalization to Several Populations --- p.16 / Chapter §3.1 --- The Model --- p.16 / Chapter §3.2 --- Two-stage Estimation Method --- p.18 / Chapter Chapter 4 --- Computation of the Estimates --- p.23 / Chapter §4.1 --- Maximum Likelihood Estimates in Stage I --- p.23 / Chapter §4.2 --- Generalized Least Squares Estimates in Stage II --- p.27 / Chapter §4.3 --- Approximation for the weight matrix W --- p.28 / Chapter Chapter 5 --- Some Illustrative Examples --- p.31 / Chapter §5.1 --- Single Population --- p.31 / Chapter §5.2 --- Multisample --- p.37 / Chapter PART II ： --- ANALYSIS OF CONTINUOUS AND POLYTOMOUS VARIABLES --- p.42 / Chapter Chapter 6 --- Introduction --- p.42 / Chapter Chapter 7 --- Several Populations Structural Equation Models with Continuous and Polytomous Variables --- p.44 / Chapter §7.1 --- The Model --- p.44 / Chapter §7.2 --- Analysis of the Model --- p.45 / Chapter Chapter 8 --- Analysis of Structural Equation Models of Polytomous and Continuous Variables with Incomplete Data by Multisample Technique --- p.54 / Chapter §8.1 --- Motivation --- p.54 / Chapter §8.2 --- The Model --- p.55 / Chapter §8.3 --- The Method --- p.56 / Chapter Chapter 9 --- Computation of the Estimates --- p.60 / Chapter §9.1 --- Optimization Procedure --- p.60 / Chapter §9.2 --- Derivatives --- p.61 / Chapter Chapter 10 --- Some Illustrative Examples --- p.65 / Chapter §10.1 --- Multisample Example --- p.65 / Chapter §10.2 --- Incomplete Data Example --- p.67 / Chapter §10.3 --- The LISREL Program --- p.69 / Chapter Chapter 11 --- Conclusion --- p.71 / Tables --- p.73 / Appendix --- p.85 / References --- p.89

by Terry Shing-fong Lew. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 40-42). / Chapter Chapter 1 --- Introduction --- p.Page1 / Chapter Chapter 2 --- Measures of Association for Polytomous Variables --- p.Page5 / Chapter §2.1 --- "Notations," --- p.5 / Chapter §2.2 --- "Pearson Product-moment Correlation Coefficient," --- p.6 / Chapter §2.3 --- "Spearman Rank Correlation Coefficient," --- p.7 / Chapter §2.4 --- "Kendall's Tau-b," --- p.9 / Chapter §2.5 --- "Polychoric Correlation Coefficient," --- p.9 / Chapter Chapter 3 --- Monte Carlo Study of Measures of Association for Polytomous Variables with Multivariate Normal Distribution --- p.Page 13 / Chapter §3.1 --- "Design," --- p.13 / Chapter §3.2 --- "Results and Findings," --- p.18 / Chapter §3.3 --- "Discussion," --- p.23 / Chapter §3.4 --- "Implications," --- p.26 / Chapter Chapter 4 --- Monte Carlo Studies for Polytomous Variables with Non-normal Distribution --- p.Page 27 / Chapter §4.1 --- "Elliptica1-t Distribution," --- p.27 / Chapter §4.2 --- "Design," --- p.28 / Chapter §4.3 --- "Results and Findings," --- p.30 / Chapter §4.4 --- "Discussion," --- p.33 / Chapter §4.5 --- "Implications," --- p.34 / Chapter Chapter 5 --- Conclusion --- p.Page36 / References --- p.Page40 / Figures --- p.Page43 / Tables --- p.Page51

by King-hong Leung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 110-111). / Chapter Chapter 1 --- Introduction --- p.1 / Chapter Chapter 2 --- Prediction Problem of Factor Scores --- p.5 / Chapter 2.1 --- The Basic Model --- p.5 / Chapter 2.2 --- Regression Formula in Predicting Factor Scores --- p.7 / Chapter 2.3 --- The Model with Polytomous Variables --- p.9 / Chapter Chapter 3 --- Prediction Methods of Factor Scores --- p.11 / Chapter 3.1 --- Model with Continuous and Polytomous Variables --- p.11 / Chapter 3.2 --- Model with Polytomous Variables --- p.16 / Chapter Chapter 4 --- Monte-Carlo Study --- p.20 / Chapter 4.1 --- Model with Continuous and Polytomous Variables --- p.20 / Chapter 4.1.1 --- Design of the Monte-Carlo Study --- p.20 / Chapter 4.1.2 --- Results of the Monte-Carlo Study --- p.24 / Chapter 4.2 --- Model with Polytomous Variables --- p.30 / Chapter 4.2.1 --- Design of the Monte-Carlo Study --- p.30 / Chapter 4.2.2 --- Results of the Monte-Carlo Study --- p.33 / Chapter Chapter 5 --- Summary and Conclusion --- p.38 / Tables --- p.41 / Figures --- p.56 / References --- p.110

by Siu Chi Sang Obadiah. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 66-68). / Abstracts in English and Chinese. / Chapter 1 --- Pseudorandom Number Generator --- p.5 / Chapter 1.1 --- Introduction --- p.5 / Chapter 1.2 --- Statistical Indistingushible and Entropy --- p.7 / Chapter 1.3 --- Example of PNG --- p.9 / Chapter 2 --- Basic Knowledge of Cellular Automata --- p.12 / Chapter 2.1 --- Introduction --- p.12 / Chapter 2.2 --- Elementary and Totalistic Cellular Automata --- p.14 / Chapter 2.3 --- Four classes of Cellular Automata --- p.17 / Chapter 2.4 --- Entropy --- p.20 / Chapter 3 --- Theoretical analysis of the CA PNG --- p.26 / Chapter 3.1 --- The Generator --- p.26 / Chapter 3.2 --- Global Properties --- p.27 / Chapter 3.3 --- Stability Properties --- p.31 / Chapter 3.4 --- Particular Initial States --- p.33 / Chapter 3.5 --- Functional Properties --- p.38 / Chapter 3.6 --- Computational Theoretical Properties --- p.42 / Chapter 3.7 --- Finite Size Behaviour --- p.44 / Chapter 3.8 --- Statistical Properties --- p.51 / Chapter 3.8.1 --- statistical test used --- p.54 / Chapter 4 --- Practical Implementation of the CA PNG --- p.56 / Chapter 4.1 --- The implementation of the CA PNG --- p.56 / Chapter 4.2 --- Applied to the set of integers --- p.58 / Chapter 5 --- Application to Cryptography --- p.61 / Chapter 5.1 --- Stream Cipher --- p.61 / Chapter 5.2 --- One Time Pad --- p.62 / Chapter 5.3 --- Probabilistic Encryption --- p.63 / Chapter 5.4 --- Probabilistic Encryption with RSA --- p.64 / Chapter 5.5 --- Prove yourself --- p.65 / Bibliography

Tang Fung Chu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 79-81). / Abstracts in English and Chinese. / Chapter Chapter 1. --- Introduction --- p.1 / Chapter Chapter 2. --- The Multivariate Model and Bayesian Analysis of Stochastic Prior Information --- p.4 / Chapter 2.1 --- The Model --- p.4 / Chapter 2.2 --- Identification of the Model --- p.5 / Chapter 2.3 --- Bayesian Analysis of Stochastic Prior Information --- p.8 / Chapter 2.4 --- Computational Procedure --- p.10 / Chapter 2.4.1 --- Optimization Procedures --- p.11 / Chapter 2.4.2 --- Analytical Expressions --- p.12 / Chapter Chapter 3. --- Example and Simulation Study --- p.18 / Chapter 3.1 --- Example --- p.18 / Chapter 3.2 --- Simulation Study --- p.19 / Chapter 3.2.1 --- Designs --- p.20 / Chapter 3.2.2 --- Results --- p.23 / Chapter Chapter 4. --- Conclusion --- p.26 / Tables --- p.29 / References --- p.79

Chung Chi Keung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 80-84). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Partition Maximum Likelihood Estimation of the General Model --- p.5 / Chapter 2.1 --- Introduction --- p.5 / Chapter 2.2 --- Model --- p.5 / Chapter 2.3 --- The Partition Maximum Likelihood Procedure --- p.8 / Chapter 2.3.1 --- PML estimation of pa --- p.9 / Chapter 2.3.2 --- PML estimation of pab --- p.13 / Chapter 2.3.3 --- Asymptotic properties of the first-stage PML estimates --- p.15 / Chapter 3 --- Bayesian Analysis of Stochastic Prior Information --- p.19 / Chapter 3.1 --- Introduction --- p.19 / Chapter 3.2 --- Bayesian analysis of the Model --- p.20 / Chapter 3.2.1 --- "Case 1, Γ = σ2I" --- p.21 / Chapter 3.2.2 --- Case 2，Г as diagonal matrix with different diagonal el- ements --- p.24 / Chapter 3.2.3 --- "Case 3, Г as a general positive definite matrix" --- p.26 / Chapter 4 --- Simulation Design and Numerical Example --- p.29 / Chapter 4.1 --- Simulation Design --- p.29 / Chapter 4.1.1 --- Model --- p.29 / Chapter 4.1.2 --- Methods of evaluation --- p.32 / Chapter 4.1.3 --- Data analysis --- p.33 / Chapter 4.2 --- Numerical Example --- p.34 / Chapter 4.2.1 --- Model --- p.35 / Chapter 5 --- Conclusion and Discussion --- p.42 / APPENDIX I to V --- p.44-50 / TABLES 1 to 10 --- p.51-77 / FIGURES 1 to 3 --- p.78-79 / REFERENCE --- p.80-84

by Lai-seung Chan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 58-59). / Abstract also in Chinese. / Chapter Chapter 1. --- Introduction --- p.1 / Chapter Chapter 2. --- The Bivariate Model and Maximum Likelihood Estimation --- p.5 / Chapter 2.1 --- The Model --- p.5 / Chapter 2.2 --- Likelihood function of the model --- p.7 / Chapter 2.3 --- Derivatives of likelihood equations --- p.8 / Chapter 2.4 --- Asymptotic properties --- p.11 / Chapter 2.5 --- Optimization procedures --- p.12 / Chapter Chapter 3. --- Generalization to Multivariate Model --- p.14 / Chapter 3.1 --- The Model --- p.14 / Chapter 3.2 --- The Partition Maximum Likelihood (PML) Estimation --- p.15 / Chapter 3.3 --- Asymptotic properties of the PML estimates --- p.19 / Chapter 3.4 --- Optimization procedures --- p.21 / Chapter Chapter 4. --- Simulation Study --- p.22 / Chapter 4.1 --- Designs --- p.22 / Chapter 4.2 --- Results --- p.26 / Chapter Chapter 5. --- Conclusion --- p.30 / Tables --- p.32 / References --- p.58