Diagnostics for functional regression via residual processes

Methods of regression diagnostics for functional regression models are developed which relate a functional response to predictor variables that can be multivariate vectors or random functions. For this purpose, a residual process is defined by subtracting the predicted from the observed response functions. This residual process is expanded into functional principal components (FPC), and the corresponding FPC scores are used as natural proxies for the residuals in functional regression models. For the case of a univariate covariate, a randomization test is proposed based on these scores to examine if the residual process depends on the covariate. If this is the case, it indicates lack of fit of the model. Graphical methods based on the FPC scores of observed and fitted functions can be used to complement more formal tests. The methods are illustrated with data from a recent study of Drosophila fruit flies regarding life-cycle gene expression trajectories as well as functional data from a dose-response experiment for Mediterranean fruit flies (Ceratitis capitata).