Symbolic covariance matrix for interval-valued variables and its application to principal component analysis

Abstract

In the last two decades, principal component analysis (PCA) was extended to interval-valued data; several adaptations of the classical approach are known from the literature. Our approach is based on the symbolic covariance matrix Cov for the interval-valued variables proposed by Billard (2008). Its crucial advantage, when compared to other approaches, is that it fully utilizes all the information in the data. The symbolic covariance matrix can be decomposed into a within part CovW and a between part CovB. We propose a further insight into the PCA results: the proportion of variance explained due to the within information and the proportion of variance explained due to the between information can be calculated. Additionally, we suggest PCA on CovB and CovW to be done to obtain deeper insights into the data under study. In the case study presented, the information gain when performing PCA on the intervals instead of the interval midpoints (conditionally the means) is about 45%. It turns out that, for these data, the uniformity assumption over intervals does not hold and so analysis of the data represented by histogram-valued variables is suggested.