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Population Symbolic Covariance Matrices for Interval Data

Symbolic Data Analysis (SDA) is a relatively new field of statistics that extends classical data analysis by taking into account intrinsic data variability and structure. As SDA has been mainly approached from a sampling perspective, we introduce population formulations of the symbolic mean, variance, covariance, correlation, covariance matrix and correlation matrix for interval-valued symbolic variables, providing a theoretical framework that gives support to interval-valued SDA. Moreover, we provide an interpretation of the various definitions of covariance and correlation matrices according to the structure of micro-data, which allows selecting the model that best suits specific datasets. Our results are illustrated using two datasets. Specifically, we select the most appropriate model for each dataset using goodness-of-fit tests and quantile-quantile plots, and provide an explanation of the micro-data based on the covariance matrix.