Shrinkage linear regression for symbolic interval-valued variables

TL;DR

This paper introduces a shrinkage-based linear regression approach for symbolic interval-valued variables, enhancing existing methods by incorporating regularization techniques and demonstrating improved performance on real datasets.

Contribution

It extends previous interval regression methods by integrating Ridge, Lasso, and Elastic Net regularization, providing a novel approach for better fitting and prediction of interval-valued data.

Findings

Improved prediction accuracy on real datasets.

Enhanced interpretability of interval regression models.

Availability of R package for implementation.

Abstract

This paper proposes a new approach to fit a linear regression for symbolic internal-valued variables, which improves both the Center Method suggested by Billard and Diday in \cite{BillardDiday2000} and the Center and Range Method suggested by Lima-Neto, E.A. and De Carvalho, F.A.T. in \cite{Lima2008, Lima2010}. Just in the Centers Method and the Center and Range Method, the new methods proposed fit the linear regression model on the midpoints and in the half of the length of the intervals as an additional variable (ranges) assumed by the predictor variables in the training data set, but to make these fitments in the regression models, the methods Ridge Regression, Lasso, and Elastic Net proposed by Tibshirani, R. Hastie, T., and Zou H in \cite{Tib1996, HastieZou2005} are used. The prediction of the lower and upper of the interval response (dependent) variable is carried out from their midpoints and ranges, which are estimated from the linear regression models with shrinkage generated in the midpoints and the ranges of the interval-valued predictors. Methods presented in this document are applied to three real data sets cardiologic interval data set, Prostate interval data set and US Murder interval data set to then compare their performance and facility of interpretation regarding the Center Method and the Center and Range Method. For this evaluation, the root-mean-squared error and the correlation coefficient are used. Besides, the reader may use all the methods presented herein and verify the results using the {\tt RSDA} package written in {\tt R} language, that can be downloaded and installed directly from {\tt CRAN} \cite{Rod2014}.