Basic statistics for distributional symbolic variables: a new metric-based approach

In data mining it is usual to describe a group of measurements using summary statistics or through empirical distribution functions. Symbolic data analysis (SDA) aims at the treatment of such kinds of data, allowing the description and the analysis of conceptual data or of macrodata summarizing classical data. In the conceptual framework of SDA, the paper aims at presenting new basic statistics for distribution-valued variables, i.e., variables whose realizations are distributions. The proposed measures extend some classical univariate (mean, variance, standard deviation) and bivariate (covariance and correlation) basic statistics to distribution-valued variables, taking into account the nature and the variability of such data. The novel statistics are based on a distance between distributions: the \(\ell _2\) Wasserstein distance. A comparison with other univariate and bivariate statistics presented in the literature points out some relevant properties of the proposed ones. An application on a clinic dataset shows the main differences in terms of interpretation of results.