Fitting Trees to Functional Data, with an Application to Time-of-Day Patterns

Decision trees often give simple descriptions of complex, nonlinear relationships between several predictors and a univariate or multivariate response. But if the response is a high-dimensional vector that can be thought of as a discretized function, then fitting a multivariate regression tree may be unsuccessful. This article explores two ways to fit trees to functional data. Both first reduce the dimensionality of the data and then fit a standard multivariate tree to the reduced response. In the first approach, each individual's response curve is represented as a linear combination of spline basis functions, penalizing for roughness, and then a multivariate regression tree is fit to the coefficients of the basis functions. In the second, a multivariate regression tree is fit to the first several principal component scores for the responses. The two methods are illustrated with time-of-day patterns for customers who place international calls.