Doctor of Philosophy / Department of Statistics / Haiyan Wang / This dissertation considers different methods to improve the performance of the Prediction Analysis of Microarrays (PAM). PAM is a popular algorithm for high-dimensional classification. However, it has a drawback of retaining too many features even after multiple runs of the algorithm to perform further feature selection. The average number of selected features is 2611 from the application of PAM to 10 multi-class microarray human cancer datasets. Such a large number of features make it difficult to perform follow up study. This drawback is the result of the soft thresholding method used in the PAM algorithm and the thresholding parameter estimate of PAM. In this dissertation, we extend the PAM algorithm with two other thresholding methods (hard and order thresholding) and a deep search algorithm to achieve better thresholding parameter estimate. In addition to the new proposed algorithms, we derived an approximation for the probability of misclassification for the hard thresholded algorithm under the binary case. Beyond the aforementioned work, this dissertation considers the heteroscedastic case in which the variances for each feature are different for different classes. In the PAM algorithm the variance of the values for each predictor was assumed to be constant across different classes. We found that this homogeneity assumption is invalid for many features in most data sets, which motivates us to develop the new heteroscedastic version algorithms. The different thresholding methods were considered in these algorithms. All new algorithms proposed in this dissertation are extensively tested and compared based on real data or Monte Carlo simulation studies. The new proposed algorithms, in general, not only achieved better cancer status prediction accuracy, but also resulted in more parsimonious models with significantly smaller number of genes.

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A measure of the quality of Voronoi tessellations resulting from various mesh generators founded on feature-driven models is introduced in this work. A planar tessellation covers an image with polygons of various shapes and sizes. Tessellations have potential utility due to their geometry and the opportunity to derive useful information from them for object recognition, image processing and classification. Problem domains including images are generally feature-endowed, non-random domains. Generators modeled otherwise may easily guarantee quality of meshes but certainly bear no reference to features of the meshed problem domain. They are therefore unsuitable in point pattern identification, characterization and subsequently the study of meshed regions. We therefore found generators on features of the problem domain. This provides a basis for element quality studies and improvement based on quality criteria. The resulting polygonal meshes tessellating an n-dimensional digital image into convex regions are of varying element qualities. Given several types of mesh generating sets, a measure of overall solution quality is introduced to determine their effectiveness. Given a tessellation of general and mixed shapes, this presents a challenge in quality improvement. The Centroidal Voronoi Tessellation (CVT) technique is developed for quality improvement and guarantees of mixed, general-shaped elements and to preserve the validity of the tessellations. Mesh quality indicators and entropies introduced are useful for pattern studies, analysis, recognition and assessing information. Computed features of tessellated spaces are explored for image information content assessment and cell processing to expose detail using information theoretic methods. Tessellated spaces also furnish information on pattern structure and organization through their quality distributions. Mathematical and theoretical results obtained from these spaces help in understanding Voronoi diagrams as well as for their successful applications. Voronoi diagrams expose neighbourhood relations between pattern units. Given this realization, the foundation of near sets is developed for further applications. / February 2017