Measure inducing classification and regression trees for functional data

We propose a tree-based algorithm (μCART) for classification and regression problems in the context of functional data analysis, which allows to leverage measure learning and multiple splitting rules at the node level, with the objective of reducing error while retaining the interpretability of a tree. For each internal node, our main contribution is the idea of learning a weighted functional (Formula presented.) space by means of constrained convex optimization, which is then used to extract multiple weighted integral features from the functional predictors, in order to determine the binary split. The approach is designed to manage multiple functional predictors and/or responses, by defining suitable splitting rules and loss functions that can depend on the specific problem and can also be combined with additional scalar and categorical predictors, as the tree is grown with the original greedy CART algorithm. We focus on the case of scalar-valued functional predictors defined on unidimensional domains and illustrate the effectiveness of our method in both classification and regression tasks, through a simulation study and four real-world applications.