Principal component analysis for probabilistic symbolic data: a more generic and accurate algorithm

In the symbolic data framework, probabilistic symbolic data are considered as those whose components are random variables with general probability distributions. Intervals (or uniform distributions), histograms (or empirical distributions), Gaussian distribution and Chi-squared distribution are all the special cases of them. The existing approaches devoted to the subject have a common shortcoming since they can not obtain the distributions of linear combinations (i.e., principal components) of random variables especially for not identically distributed ones. This paper will overcome the shortcoming by providing an exact probability density function for each principal component by using the inversion theorem. Further, the paper defines a covariance matrix for probabilistic symbolic data and presents a new principal component analysis based on this variance–covariance structure. The effectiveness of the proposed method is illustrated by a simulated numerical experiment, and two real-life cases including clustering of oils and fats data, and evaluation of indexed journals of Science Citation Index.