Unsupervised Learning via Mixtures of Skewed Distributions with Hypercube Contours

Mixture models whose components have skewed hypercube contours are developed via a generalization of the multivariate shifted asymmetric Laplace density. Specifically, we develop mixtures of multiple scaled shifted asymmetric Laplace distributions. The component densities have two unique features: they include a multivariate weight function, and the marginal distributions are also asymmetric Laplace. We use these mixtures of multiple scaled shifted asymmetric Laplace distributions for clustering applications, but they could equally well be used in the supervised or semi-supervised paradigms. The expectation-maximization algorithm is used for parameter estimation and the Bayesian information criterion is used for model selection. Simulated and real data sets are used to illustrate the approach and, in some cases, to visualize the skewed hypercube structure of the components

Factor Analysis of Data Matrices: New Theoretical and Computational Aspects With Applications

The classical fitting problem in exploratory factor analysis (EFA) is to find estimates for the factor loadings matrix and the matrix of unique factor variances which give the best fit to the sample covariance or correlation matrix with respect to some goodness-of-fit criterion. Predicted factor scores can be obtained as a function of these estimates and the data. In this thesis, the EFA model is considered as a specific data matrix decomposition with fixed unknown matrix parameters. Fitting the EFA model directly to the data yields simultaneous solutions for both loadings and factor scores. Several new algorithms are introduced for the least squares and weighted least squares estimation of all EFA model unknowns. The numerical procedures are based on the singular value decomposition, facilitate the estimation of both common and unique factor scores, and work equally well when the number of variables exceeds the number of available observations. Like EFA, noisy independent component analysis (ICA) is a technique for reduction of the data dimensionality in which the interrelationships among the observed variables are explained in terms of a much smaller number of latent factors. The key difference between EFA and noisy ICA is that in the latter model the common factors are assumed to be both independent and non-normal. In contrast to EFA, there is no rotational indeterminacy in noisy ICA. In this thesis, noisy ICA is viewed as a method of factor rotation in EFA. Starting from an initial EFA solution, an orthogonal rotation matrix is sought that minimizes the dependence between the common factors. The idea of rotating the scores towards independence is also employed in three-mode factor analysis to analyze data sets having a three-way structure. The new theoretical and computational aspects contained in this thesis are illustrated by means of several examples with real and artificial data

K-Tensors: Clustering Positive Semi-Definite Matrices

This paper introduces a novel self-consistency clustering algorithm ($K$-Tensors) designed for {partitioning a distribution of} positive-semidefinite matrices based on their eigenstructures. As positive semi-definite matrices can be represented as ellipsoids in $\mathbb R^p$, $p \ge 2$, it is critical to maintain their structural information to perform effective clustering. However, traditional clustering algorithms {applied to matrices} often {involve vectorization of} the matrices, resulting in a loss of essential structural information. To address this issue, we propose a distance metric {for clustering} that is specifically based on the structural information of positive semi-definite matrices. This distance metric enables the clustering algorithm to consider the differences between positive semi-definite matrices and their projections onto {a} common space spanned by \thadJulyTen{orthonormal vectors defined from a set of} positive semi-definite matrices. This innovative approach to clustering positive semi-definite matrices has broad applications in several domains including financial and biomedical research, such as analyzing functional connectivity data. By maintaining the structural information of positive semi-definite matrices, our proposed algorithm promises to cluster the positive semi-definite matrices in a more meaningful way, thereby facilitating deeper insights into the underlying data in various applications