No abstract

This paper introduces symbolic data analysis, explaining how it extends the classical data models to take into account more complete and complex information. Several examples motivate the approach, before the modeling of variables assuming new types of realizations are formally presented. Some methods for the (multivariate) analysis of symbolic data are presented and discussed. This is however far from being exhaustive, given the present dynamic development of this new field of research. 

Clustering, Symbolic Data Analysis, Interval Data, Standardization, Cluster Interpretation,

Histogram-valued variables are a particular kind of variables studied in Symbolic Data Analysis where to each entity under analysis corresponds a distribution that may be represented by a histogram or by a quantile function. Linear regression models for this type of data are necessarily more complex than a simple generalization of the classical model: the parameters cannot be negative; still the linear relation between the variables must be allowed to be either direct or inverse. In this work, we propose a new linear regression model for histogram-valued variables that solves this problem, named Distribution and Symmetric Distribution Regression Model. To determine the parameters of this model, it is necessary to solve a quadratic optimization problem, subject to non-negativity constraints on the unknowns; the error measure between the predicted and observed distributions uses the Mallows distance. As in classical analysis, the model is associated with a goodness-of-fit measure whose values range between 0 and 1. Using the proposed model, applications with real and simulated data are presented.