Identifying influential observations in concurrent functional regression with weighted bootstrap

Abstract

Metrics such as DFBETAS, DFFITS, and Cook’s Distance are used in ordinary linear regression to assess the influence each individual observation has on the fitted model. Here we quantify the influence of functional observations in the concurrent functional linear model where both predictor and response are functional objects, presenting four measures to identify which functional observations are the most influential on the results. We introduce a novel weighted bootstrapping with perturbations method to identify when an observation is significantly influential on the fitted functional regression model. We show this method’s validity using a simulation study and two real data examples. In the first example, our new measures are deployed on paired river heights during ten past flood events. Our method identifies which of the ten past flood observations has the most impact (and whether it is significant) on the functional linear model. These results also show which flood event most alters the relationship between the two river heights. The second example involves concurrently observed air and water temperature functions at 35 locations across the United States coastline. Our method identifies the location(s) with significant influence on the functional linear model.