Association measures for interval variables

TL;DR

This paper investigates how different definitions of symbolic covariance matrices in Symbolic Data Analysis may misrepresent correlations in interval data, emphasizing the importance of understanding micro-data structures for accurate analysis.

Contribution

It introduces a model linking micro-data and macro-data for interval variables, revealing limitations of existing covariance definitions and guiding better selection based on micro-data knowledge.

Findings

Existing covariance definitions can misrepresent correlations in macro-data.

Micro-data assumptions are crucial for selecting appropriate covariance measures.

There can be significant divergence between different covariance definitions in real datasets.

Abstract

Symbolic Data Analysis (SDA) is a relatively new field of statistics that extends conventional data analysis by taking into account intrinsic data variability and structure. Unlike conventional data analysis, in SDA the features characterizing the data can be multi-valued, such as intervals or histograms. SDA has been mainly approached from a sampling perspective. In this work, we propose a model that links the micro-data and macro-data of interval-valued symbolic variables, which takes a populational perspective. Using this model, we derive the micro-data assumptions underlying the various definitions of symbolic covariance matrices proposed in the literature, and show that these assumptions can be too restrictive, raising applicability concerns. We analyze the various definitions using worked examples and four datasets. Our results show that the existence/absence of correlations in the macro-data may not be correctly captured by the definitions of symbolic covariance matrices and that, in real data, there can be a strong divergence between these definitions. Thus, in order to select the most appropriate definition, one must have some knowledge about the micro-data structure.