Robust Function-on-Function Regression

TL;DR

This paper introduces a robust functional linear regression model that effectively handles outliers in functional data, improves model selection, and enhances outlier detection, demonstrated through simulations and jet engine sensor data analysis.

Contribution

It proposes a Fisher-consistent robust regression model with a new model selection procedure and outlier detection, addressing limitations of classical methods.

Findings

Effectively captures regression behavior with outliers

Accurately identifies outliers in functional data

Demonstrates improved performance on real sensor data

Abstract

Functional linear regression is a widely used approach to model functional responses with respect to functional inputs. However, classical functional linear regression models can be severely affected by outliers. We therefore introduce a Fisher-consistent robust functional linear regression model that is able to effectively fit data in the presence of outliers. The model is built using robust functional principal component and least squares regression estimators. The performance of the functional linear regression model depends on the number of principal components used. We therefore introduce a consistent robust model selection procedure to choose the number of principal components. Our robust functional linear regression model can be used alongside an outlier detection procedure to effectively identify abnormal functional responses. A simulation study shows our method is able to effectively capture the regression behaviour in the presence of outliers, and is able to find the outliers with high accuracy. We demonstrate the usefulness of our method on jet engine sensor data. We identify outliers that would not be found if the functional responses were modelled independently of the functional input, or using non-robust methods.