An alternative objective function for fitting regression trees to functional response variables

Abstract

Analyses of systems that can be represented by functional responses are becoming common in many scientific disciplines. Functional regression trees (FRT) provide a methodology for modelling such systems. Recent work has focused on fitting models where the response variable is a probability density function, using a splitting criterion that is based on the sum of dissimilarities between the densities. We suggest a different criterion based on deviations of the densities from their mean. We provide motivation and justification for this criterion, and demonstrate its superior performance using an extensive simulation exercise. We discuss the computational aspects of the FRT procedure and show that substantial speed gains can be made through use of a dissimilarity matrix. Our results show that the proposed splitting criterion outperforms both the original and a splitting criterion based on Euclidean distance. Pointwise standard error curves for a predicted functional response can be generated through the fitting procedure, which we demonstrate in a case study with a forestry data set. Supplementary materials are available.