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Shrinkage linear regression for symbolic interval-valued variables
In MODULAD 2018, vol. Modulad 45, pp.19-38
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 [2] and
the Center and Range Method suggested by Lima-Neto, E.A. and De Carvalho, F.A.T. in
[9, 10]. 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 [12, 8] 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 RSDA package written in R language, that can be downloaded and installed
directly from CRAN [14],