Robust Group Variable Selection Methods for Multiple Functional Regression Model

Abstract

With the advancements in science and ever changing technology to collect data, functional data have become common these days, especially in various fields such as neuroscience, chemometrics, e-commerce and computer science. Thus, in last two decades a vast amount of new statistical methodologies to analyze such data, so-called, functional data analysis, have been developed. Much research has been done in various areas of functional data analysis like functional linear regression, functional logistic regression, functional ANOVA, functional principal component analysis and functional outlier detection. Just as in ordinary multiple regression analysis, variable selection is an important problem in the functional regression framework. The area of functional variable selection is seldom discussed in functional data analysis. The classical existing functional variable selection methods are all based on minimizing the penalized residual sum of squares, which is non-robust in nature, in the presence of outliers. In this work, we study robust variable selection methods for functional regression model with a scalar response and functional predictors in the presence of outliers. Essentially, we consider ways that minimize the effect of outliers on the parameter estimator and selector. Since multiple parameters exist for a functional predictor group variable selection methods are used for selecting functional predictors that select grouped variables rather than individual variables. We consider the problem of selecting functional predictors using the L1 regularization in a functional linear regression model with a scalar response and functional predictors in the presence of outliers. Four estimation approaches are discussed: functional LAD- group LASSO, functional Weighted LAD- group LASSO, functional LAD- Adaptive group LASSO and functional Weighted LAD- Adaptive group LASSO. We present an extensive simulation study and a real world example to illustrate the performance of the proposed estimators.