  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.
