Structural classification based correlation and its application to principal component analysis for high-dimension low-sample size data

Abstract

This paper proposes a structural classification based correlation and application to principal component analysis (PCA) for high-dimension low-sample size (HDLSS) data. The structural classification based correlation consists of two kinds of correlations; correlation of objects over variables and correlation of classification structures of objects over clusters. Therefore, this correlation can measure not only the similarity of objects but also the similarity of classification structures. We exploit this correlation to PCA whose target data is HDLSS data in which the number of variables is much larger than the number of objects. Since it is known that we cannot obtain correct solutions as the eigen-values of the covariance matrix of variables for HDLSS data and the result of ordinary PCA is based on eigen-values of the covariance matrix of variables, if we apply the ordinary PCA for HDLSS data, we cannot obtain the correct result. In order to solve this problem, we exploit the proposed structural classification based correlation with respect to variables. Since this correlation includes the correlation of classification structures, we can solve this problem and obtain a similarity relationship of objects in a lower dimensional space spanned by the obtained principal components. From several numerical examples, we show the effectiveness of our proposed principal component analysis using the structural classification based correlation for the HDLSS data.

Cluster‐scaled principal component analysis

“…The cluster-scaled PCA utilizes this advantage of fuzzy clustering. A numerical example shows a better performance for the cluster-scaled PCA, and other examples are shown in many pieces of literature (Sato-Ilic, 2010, 2011a, 2011b, 2012a, 2012b.…”

“…In particular, for the high‐dimension and low‐sample size data, conventional PCA theoretically cannot obtain correct solutions (Ahn et al, 2007; Baik et al, 2005; Hall et al, 2005; Tong et al, 2014; Welsh et al, 2001). In order to solve this problem, we have proposed several cluster‐scaled PCA for high‐dimension and low‐sample size data (Sato‐Ilic, 2011a, 2011b, 2012a, 2012b) and obtained a better performance using an idea of symbolic data (Billard & Diday, 2007; Bock & Diday, 2000; Diday, 2016).…”

“…In order to solve these problems, in this article, we introduce the simple and reduced complexity of driving the solutions (this means that we do not need a large number of parameters) by using an idea of a cluster-scaled metric. This metric is essentially utilizing the obtained cluster after the clustering of data as a metric to use another method, which is the PCA in this article, and obtain a more accurate result or result which we cannot obtain by using an ordinary method (Sato-Ilic, 2010, 2011a, 2011b, 2012a, 2012b. This article's structure is as follows: in Section 2, we describe fuzzy clustering, which we will use in this article as a clustering method.…”

Unsupervised Adaptation for High-Dimensional with Limited-Sample Data Classification Using Variational Autoencoder

“…Many researchers considered PCA in the classification and clustering of biological data in the context of HDLSS, among them are [26,27,28,29]. In fact, PCA reduces the dimensionality of the data linearly, and it may not extract some nonlinear relationships of the data.…”