Doctor of Philosophy / Department of Statistics / Kun Chen and Weixin Yao / This proposal contains two projects that are related to robust mixture models. In the robust project, we propose a new robust mixture of regression models (Bai et al., 2012). The existing methods for tting mixture regression models assume a normal distribution for error and then estimate the regression param- eters by the maximum likelihood estimate (MLE). In this project, we demonstrate that the MLE, like the least squares estimate, is sensitive to outliers and heavy-tailed error distributions. We propose a robust estimation procedure and an EM-type algorithm to estimate the mixture regression models. Using a Monte Carlo simulation study, we demonstrate that the proposed new estimation method is robust and works much better than the MLE when there are outliers or the error distribution has heavy tails. In addition, the proposed robust method works comparably to the MLE when there are no outliers and the error is normal. In the second project, we propose a new robust mixture of linear mixed-effects models. The traditional mixture model with multiple linear mixed effects, assuming Gaussian distribution for random and error parts, is sensitive to outliers. We will propose a mixture of multiple linear mixed t-distributions to robustify the estimation procedure. An EM algorithm is provided to and the MLE under the assumption of t- distributions for error terms and random mixed effects. Furthermore, we propose to adaptively choose the degrees of freedom for the t-distribution using profile likelihood. In the simulation study, we demonstrate that our proposed model works comparably to the traditional estimation method when there are no outliers and the errors and random mixed effects are normally distributed, but works much better if there are outliers or the distributions of the errors and random mixed effects have heavy tails.

With the advent of high-throughput genomic and proteomic technologies, in conjunction with the difficulty in obtaining even moderately sized samples, small-sample classifier design has become a major issue in the biological and medical communities. Training-data error estimation becomes mandatory, yet none of the popular error estimation techniques have been rigorously designed via statistical inference or optimization. In this investigation, we place classifier error estimation in a framework of minimum mean-square error (MMSE) signal estimation in the presence of uncertainty, where uncertainty is relative to a prior over a family of distributions. This results in a Bayesian approach to error estimation that is optimal and unbiased relative to the model. The prior addresses a trade-off between estimator robustness (modeling assumptions) and accuracy. Closed-form representations for Bayesian error estimators are provided for two important models: discrete classification with Dirichlet priors (the discrete model) and linear classification of Gaussian distributions with fixed, scaled identity or arbitrary covariances and conjugate priors (the Gaussian model). We examine robustness to false modeling assumptions and demonstrate that Bayesian error estimators perform especially well for moderate true errors. The Bayesian modeling framework facilitates both optimization and analysis. It naturally gives rise to a practical expected measure of performance for arbitrary error estimators: the sample-conditioned mean-square error (MSE). Closed-form expressions are provided for both Bayesian models. We examine the consistency of Bayesian error estimation and illustrate a salient application in censored sampling, where sample points are collected one at a time until the conditional MSE reaches a stopping criterion. We address practical considerations for gene-expression microarray data, including the suitability of the Gaussian model, a methodology for calibrating normal-inverse-Wishart priors from unused data, and an approximation method for non-linear classification. We observe superior performance on synthetic high-dimensional data and real data, especially for moderate to high expected true errors and small feature sizes. Finally, arbitrary error estimators may be optimally calibrated assuming a fixed Bayesian model, sample size, classification rule, and error estimation rule. Using a calibration function mapping error estimates to their optimally calibrated values off-line, error estimates may be calibrated on the fly whenever the assumptions apply.

In this thesis a machine vision based indoor navigation system is presented. This is achieved by using rotationally independent optimized color reference labels and a geometrical camera calibration model which determines a set of camera parameters. All reference labels carry one byte of information (0 to 255), which can be designed for different values. An algorithm in Matlab has been developed so that a machine vision system for N number of symbols can recognize the symbols at different orientations. A camera calibration model describes the mapping between the 3-D world coordinates and the 2-D image coordinates. The reconstruction system uses the direct linear transform (DLT) method with a set of control reference labels in relation to the camera calibration. The least-squares adjustment method has been developed to calculate the parameters of the machine vision system. In these experiments it has been demonstrated that the pose of the camera can be calculated, with a relatively high precision, by using the least-squares estimation.

Several studies have examined the effect of remittances on economic growth,poverty, education, and governance, among other factors, in developing countrieswith inconclusive results. Using annual panel data of 99 developing countries invarious empirical models, this study aim to answer the question how remittances affect a broader aspect of development using the Human Development Index asdependent variable. The findings indicate that there is a positive relationship between remittances and the level of human development in developing countries.

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The aim of this work is to introduce a new type of fuzzy system AnYa. This system, unlike the classical fuzzy systems Takagi-Sugeno and Mamdani, uses a type of antecendent based on real data distribution. As part of the work there will be mentioned system programmed and its functionality will be verified on testing data.

The aim of this work is to introduce a new type of fuzzy system AnYa. This system, unlike the classical fuzzy systems Takagi-Sugeno and Mamdani, uses a type of antecendent based on real data distribution. As part of the work there will be mentioned system programmed and its functionality will be verified on testing data.