A Novel Hybrid Dimension Reduction Technique for Undersized High Dimensional Gene Expression Data Sets Using Information Complexity Criterion for Cancer Classification

Gene expression data typically are large, complex, and highly noisy. Their dimension is high with several thousand genes (i.e., features) but with only a limited number of observations (i.e., samples). Although the classical principal component analysis (PCA) method is widely used as a first standard step in dimension reduction and in supervised and unsupervised classification, it suffers from several shortcomings in the case of data sets involving undersized samples, since the sample covariance matrix degenerates and becomes singular. In this paper we address these limitations within the context of probabilistic PCA (PPCA) by introducing and developing a new and novel approach using maximum entropy covariance matrix and its hybridized smoothed covariance estimators. To reduce the dimensionality of the data and to choose the number of probabilistic PCs (PPCs) to be retained, we further introduce and develop celebrated Akaike's information criterion (AIC), consistent Akaike's information criterion (CAIC), and the information theoretic measure of complexity (ICOMP) criterion of Bozdogan. Six publicly available undersized benchmark data sets were analyzed to show the utility, flexibility, and versatility of our approach with hybridized smoothed covariance matrix estimators, which do not degenerate to perform the PPCA to reduce the dimension and to carry out supervised classification of cancer groups in high dimensions.