Distribution and Symmetric Distribution Regression Model for Interval-Valued Variables

Abstract

Symbolic Data Analysis works with variables for which each unit or class of units takes a finite set of values/categories, an interval or a distribution (an histogram, for instance). When to each observation corresponds an empirical distribution, we have a histogram-valued variable; it reduces to the case of an interval-valued variable if each unit takes values on only one interval with probability equal to one. Distribution and Symmetric Distribution is a linear regression model proposed for histogram-valued variables that may be particularized to interval-valued variables. This model is defined for n explicative variables and is based on the distributions considered within the intervals. In this paper we study the special case where the Uniform distribution is assumed in each observed interval. As in the classical case, a goodness-of-fit measure is deduced from the model. Some illustrative examples are presented. A simulation study allows discussing interpretations of the behavior of the model for this variable type.