Factor Analysis of Interval Data

TL;DR

This paper introduces a factor analysis model for interval-valued symbolic data, estimating underlying unobservable factors using principal component or principal axis methods, with applications to climate and automotive data.

Contribution

It develops a novel factor analysis approach for interval data, including explicit formulas and methods for estimating factor scores based on the Mallows distance.

Findings

Effective extraction of underlying factors demonstrated on climate and car data.

Method accurately captures correlation structures in synthetic and real interval data.

Provides explicit formulas for quantile functions and distances under Triangular distributions.

Abstract

This paper presents a factor analysis model for symbolic data, focusing on the particular case of interval-valued variables. The proposed method describes the correlation structure among the measured interval-valued variables in terms of a few underlying, but unobservable, uncorrelated interval-valued variables, called \textit{common factors}. Uniform and Triangular distributions are considered within each observed interval. We obtain the corresponding sample mean, variance and covariance assuming a general Triangular distribution. In our proposal, factors are extracted either by Principal Component or by Principal Axis Factoring, performed on the interval-valued variables correlation matrix. To estimate the values of the common factors, usually called \textit{factor scores}, two approaches are considered, which are inspired in methods for real-valued data: the Bartlett and the Anderson-Rubin methods. In both cases, the estimated values are obtained solving an optimization problem that minimizes a function of the weighted squared Mallows distance between quantile functions. Explicit expressions for the quantile function and the squared Mallows distance are derived assuming a general Triangular distribution. The applicability of the method is illustrated using two sets of data: temperature and precipitation in cities of the United States of America between the years 1971 and 2000 and measures of car characteristics of different makes and models. Moreover, the method is evaluated on synthetic data with predefined correlation structures.