This dissertation deals primarily with the moments and distribution H̅ = - Σ n_i/N logn_i/N. Some techniques of obtaining multivariate moments, in particular multinomial moments, are given. I The approach used In obtaining the moments of H̅ was through the probability generating function of the multinomial distribution. A series of rather simple mathematical operations will produce the E(H̅) as an Integral and Var(H̅) as a double Integral. These Integrale are evaluated exactly thus giving the exact mean and variance of H̅. The mean and variance Is also given In series form. The series for the mean of H̅ appears to be divergent. Several charts are given which Indicate the percent error incurred when the series are used. The combinatorial approach was used In finding the asymptotic distribution of H̅. The IBM 1130 and the IBM 360 model 65 were used to do this work. The results is that H̅ is asymptotically normal In the general case and H̅ is asymptotically chi-square In the equiprobable case. Tables are given for the mean and variance of H̅ in the general case and In the equiprobable case. Two methods are given for finding multivariate moments. The Q-Product Method due to Shenton, Bowman, and Reinfelds [36th Session of the International Statistical Institute,, 1967] and the Small Sample Method. There is every indication that these methods can be completely automated. A table of the first fourteen binomial moments is given and a table through order six of the multinomial moments is given. / Ph. D.

This thesis discusses the development and performance of a novel evolutionary algorithm, the Targeted Estimation of Distribution Algorithm (TEDA). TEDA takes the concept of targeting, an idea that has previously been shown to be effective as part of a Genetic Algorithm (GA) called Fitness Directed Crossover (FDC), and introduces it into a novel hybrid algorithm that transitions from a GA to an Estimation of Distribution Algorithm (EDA). Targeting is a process for solving optimisation problems where there is a concept of control points, genes that can be said to be active, and where the total number of control points found within a solution is as important as where they are located. When generating a new solution an algorithm that uses targeting must first of all choose the number of control points to set in the new solution before choosing which to set. The hybrid approach is designed to take advantage of the ability of EDAs to exploit patterns within the population to effectively locate the global optimum while avoiding the tendency of EDAs to prematurely converge. This is achieved by initially using a GA to effectively explore the search space before transitioning into an EDA as the population converges on the region of the global optimum. As targeting places an extra restriction on the solutions produced by specifying their size, combining it with the hybrid approach allows TEDA to produce solutions that are of an optimal size and of a higher quality than would be found using a GA alone without risking a loss of diversity. TEDA is tested on three different problem domains. These are optimal control of cancer chemotherapy, network routing and Feature Subset Selection (FSS). Of these problems, TEDA showed consistent advantage over standard EAs in the routing problem and demonstrated that it is able to find good solutions faster than untargeted EAs and non evolutionary approaches at the FSS problem. It did not demonstrate any advantage over other approaches when applied to chemotherapy. The FSS domain demonstrated that in large and noisy problems TEDA’s targeting derived ability to reduce the size of the search space significantly increased the speed with which good solutions could be found. The routing domain demonstrated that, where the ideal number of control points is deceptive, both targeting and the exploitative capabilities of an EDA are needed, making TEDA a more effective approach than both untargeted approaches and FDC. Additionally, in none of the problems was TEDA seen to perform significantly worse than any alternative approaches.

Wong Chun Kit. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves [132-143]). / Abstracts in English and Chinese. / List of Tables --- p.i / List of Figures --- p.iv / List of Symbols --- p.v / Chapter CHAPTER ONE: --- INTRODUCTION --- p.1 / Chapter 1.1 --- Objectives and Significance of the Study --- p.4 / Chapter 1.2 --- Physical Setting of Hong Kong --- p.5 / Chapter 1.3 --- Climate of Hong Kong --- p.9 / Chapter 1.4 --- Data Acquisition --- p.11 / Chapter 1.4.1 --- Raingauges in Hong Kong --- p.11 / Chapter 1.4.2. --- Database for the Spatial Variation Analyses --- p.14 / Chapter 1.4.2.1. --- Data Selection for the Analyses for Factors Affecting Rainfall ´ؤ Elevation and Aspect --- p.15 / Chapter 1.4.2.2. --- Data Selection for the Classification of Stations and Inter-station Correlation Analysis --- p.17 / Chapter 1.4.3 --- Database for the Temporal Variation Analyses --- p.20 / Chapter CHAPTER TWO : --- LITERATURE REVIEW --- p.22 / Chapter 2.1 --- Spatial Variation of Rainfall --- p.22 / Chapter 2.2 --- Detection of Temporal Changes in Rainfall --- p.28 / Chapter 2.3 --- Urban Influence on Rainfall --- p.29 / Chapter 2.4 --- Studies in Hong Kong --- p.33 / Chapter CHAPTER THREE : --- METHODOLOGY --- p.33 / Chapter 3.1 --- Preliminary Processing of the Data --- p.38 / Chapter 3.2 --- Data Analysis --- p.40 / Chapter 3.2.1 --- General Pattern of Rainfall Distribution --- p.40 / Chapter 3.2.2 --- Data Analyses of Spatial Variation --- p.41 / Chapter 3.2.2.1 --- Correlation between Rainfall and Elevation --- p.41 / Chapter 3.2.2.2 --- Correlation between Rainfall and Aspect --- p.42 / Chapter 3.2.2.3 --- Classification of Stations --- p.43 / Chapter 3.2.2.4 --- Inter-Station Correlation Analysis --- p.46 / Chapter 3.2.3 --- Data Analysis of Temporal Variation --- p.46 / Chapter 3.2.3.1 --- The Running Mean Method --- p.47 / Chapter 3.2.3.2 --- The 'Standard Error of the Difference' Test --- p.49 / Chapter CHAPTER FOUR: --- RESULTS AND DISCUSSION --- p.50 / Chapter 4.1 --- Graphical Representation of Spatial Rainfall Pattern --- p.50 / Chapter 4.1.1 --- Annual Rainfall Pattern --- p.50 / Chapter 4.1.2 --- Monthly Rainfall Pattern --- p.56 / Chapter 4.1.3 --- Frequency Distribution of Raindays --- p.59 / Chapter 4.1.4 --- Pentade Rainfall Pattern --- p.64 / Chapter 4.1.5 --- Diurnal Rainfall Pattern --- p.67 / Chapter 4.1.6 --- Implications of the Spatial Rainfall Pattern --- p.70 / Chapter 4.2 --- Analyses of Spatial Variation in Rainfall --- p.78 / Chapter 4.2.1 --- Relationship between Rainfall and Elevation --- p.78 / Chapter 4.2.2 --- Relationship between Rainfall and Aspect --- p.82 / Chapter 4.2.3 --- Classification of Stations --- p.85 / Chapter 4.2.3.1 --- Principal Components Interpretation --- p.87 / Chapter 4.2.3.2 --- Result of Classification --- p.90 / Chapter 4.2.4 --- Inter-Station Correlation Analysis --- p.98 / Chapter 4.2.5 --- Discussion of the Rainfall Spatial Variation --- p.103 / Chapter 4.3 --- Analyses of Temporal Variation in Rainfall --- p.107 / Chapter 4.3.1 --- Annual Rainfall --- p.107 / Chapter 4.3.2 --- Monthly Rainfall --- p.110 / Chapter 4.3.3 --- Pentade Rainfall --- p.112 / Chapter 4.3.4 --- Diurnal Rainfall --- p.117 / Chapter 4.3.5 --- Discussion of the Rainfall Temporal Variation --- p.118 / Chapter CHAPTER FIVE: --- CONCLUSIONS AND RECOMMENDATIONS --- p.126 / Chapter 5.1 --- Summary of Findings --- p.126 / Chapter 5.2 --- Limitation of this Research --- p.129 / Chapter 5.3 --- Prospects of this Research --- p.130 / Bibliography

by Foon-Yip Ng. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 111-115). / Abstract also in Chinese. / Chapter Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Main Problems --- p.1 / Chapter 1.2 --- The Basic Concept of Decision Theory --- p.4 / Chapter 1.3 --- The Class of Orthogonally Invariant Estimators --- p.6 / Chapter 1.4 --- Related Works --- p.8 / Chapter 1.5 --- Summary --- p.10 / Chapter Chapter 2 --- Estimation of the Scale Matrix in a Wishart Distribution --- p.12 / Chapter 2.1 --- Review of the Previous Works --- p.13 / Chapter 2.2 --- Some Useful Statistical and Mathematical Results --- p.15 / Chapter 2.3 --- Improved Estimation of Σ under the Loss L1 --- p.18 / Chapter 2.4 --- Simulation Study for Wishart Distribution under the Loss L1 --- p.22 / Chapter 2.5 --- Improved Estimation of Σ under the Loss L2 --- p.25 / Chapter 2.6 --- Simulation Study for Wishart Distribution under the Loss L2 --- p.28 / Chapter Chapter 3 --- Estimation of the Scale Matrix in a Multivariate F Distribution --- p.31 / Chapter 3.1 --- Review of the Previous Works --- p.32 / Chapter 3.2 --- Some Useful Statistical and Mathematical Results --- p.35 / Chapter 3.3 --- Improved Estimation of Δ under the Loss L1____ --- p.38 / Chapter 3.4 --- Simulation Study for Multivariate F Distribution under the Loss L1 --- p.42 / Chapter 3.5 --- Improved Estimation of Δ under the Loss L2 ________ --- p.46 / Chapter 3.6 --- Relationship between Wishart Distribution and Multivariate F Distribution --- p.51 / Chapter 3.7 --- Simulation Study for Multivariate F Distribution under the Loss L2 --- p.52 / Chapter Chapter 4 --- Estimation of the Scale Matrix in an Elliptically Contoured Matrix Distribution --- p.57 / Chapter 4.1 --- Some Properties of Elliptically Contoured Matrix Distributions --- p.58 / Chapter 4.2 --- Review of the Previous Works --- p.60 / Chapter 4.3 --- Some Useful Statistical and Mathematical Results --- p.62 / Chapter 4.4 --- Improved Estimation of Σ under the Loss L3 --- p.63 / Chapter 4.5 --- Simulation Study for Multivariate-Elliptical t Distributions under the Loss L3 --- p.67 / Chapter 4.5.1 --- Properties of Multivariate-Elliptical t Distribution --- p.67 / Chapter 4.5.2 --- Simulation Study for Multivariate- Elliptical t Distributions --- p.70 / Chapter 4.6 --- Simulation Study for ε-Contaminated Normal Distributions under the Loss L3 --- p.74 / Chapter 4.6.1 --- Properties of ε-Contaminated Normal Distributions --- p.74 / Chapter 4.6.2 --- Simulation Study for 2-Contaminated Normal Distributions --- p.76 / Chapter 4.7 --- Discussions --- p.79 / APPENDIX --- p.81 / BIBLIOGRAPHY --- p.111

Typescript (photocopy). / Digitized by Kansas Correctional Industries

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND AERO. / Includes bibliographical references. / by Shinsuke Endoh. / M.S.

by Wai-Yin Chan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 42-45). / Chapter Chapter 1 --- Introduction / Chapter 1.1 --- Main Problems --- p.1 / Chapter 1.2 --- Class of Regularized Estimator --- p.4 / Chapter 1.3 --- Preliminaries --- p.6 / Chapter 1.4 --- Related Works --- p.9 / Chapter 1.5 --- Brief Summary --- p.10 / Chapter Chapter 2 --- Estimation of the Covariance Matrix and its Eigenvalues in a Wishart Distribution / Chapter 2.1 --- Significance of The Problem --- p.12 / Chapter 2.2 --- Review of the Previous Work --- p.13 / Chapter 2.3 --- Properties of the Wishart Distribution --- p.18 / Chapter 2.4 --- Main Results --- p.20 / Chapter 2.5 --- Simulation Study --- p.23 / Chapter Chapter 3 --- Estimation of the Scale Matrix and its Eigenvalues in a Multivariate F Distribution / Chapter 3.1 --- Formulation and significance of the Problem --- p.26 / Chapter 3.2 --- Review of the Previous Works --- p.28 / Chapter 3.3 --- Properties of Multivariate F Distribution --- p.30 / Chapter 3.4 --- Main Results --- p.33 / Chapter 3.5 --- Simulation Study --- p.38 / Chapter Chapter 4 --- Further research --- p.40 / Reference --- p.42 / Appendix --- p.46

Recently the independent multinomial selections model (IMS) with the multinomial logit link has been suggested as an analysis tool for radio-telemetry habitat selection data. This model assumes independence between animals, independence between sightings within an animal, and identical multinomial habitat selection probabilities for all animals. We propose two generalizations to the IMS model. The first generalization is to allow a Markov chain dependence between consecutive sightings of the same animal. This generalization allows for both positive correlation (individuals persisting in the same habitat class in which they were previously sighted) and negative correlation (individual vacating the habitat class in which they were previously sighted). The second generalization is to allow for heterogeneity. Here, a hierarchical Dirichlet-multinomial distribution is used to allow for variability in selection probabilities between animals. This generalization accounts for over-dispersion of selection probabilities and allows for inference to the population of animals, assuming that the animals studied constitute a random sample from that population.. Both generalizations are one parameter extensions to the multinomial logit model and allow for testing the assumptions of identical multinomial selection probabilities and independence. These tests are performed using the score, Wald, and asymptotic likelihood ratio statistics. Estimates of model parameters are obtained using maximum likelihood techniques, and habitat characteristics are tested using drop-in-deviance statistics. Using example data, we show that persistence and heterogeneity exist in habitat selection data and illustrate the difference in analysis results between the IMS model and the persistence and heterogeneity models. Through simulation, we show that analyzing persistence data assuming independence between sightings within an animal gives liberal tests of significance for habitat characteristics when the data are generated with positive correlation and conservative tests of significance when the data are generated with negative correlation. Similarly, we show that analyzing heterogeneous data, assuming identical multinomial selection probabilities, gives liberal tests of significance for habitat characteristics. / Graduation date: 2001

Reprinted from IEEE transactions on information theory, vol. IT-16, no.2, March 1970. / Bibliography: p. 184.

by Philip M. Morse.