Discovering interpretable structure in longitudinal predictors via coefficient trees

We consider the regression setting in which the response variable is not longitudinal (i.e., it is observed once for each case), but it is assumed to depend functionally on a set of predictors that are observed longitudinally, which is a specific form of functional predictors. In this situation, we often expect that the same predictor observed at nearby time points are more likely to be associated with the response in the same way. In such situations, we can exploit those aspects and discover groups of predictors that share the same (or similar) coefficient according to their temporal proximity. We propose a new algorithm called coefficient tree regression for data in which the non-longitudinal response depends on longitudinal predictors to efficiently discover the underlying temporal characteristics of the data. The approach results in a simple and highly interpretable tree structure from which the hierarchical relationships between groups of predictors that affect the response in a similar manner based on their temporal proximity can be observed, and we demonstrate with a real example that it can provide a clear and concise interpretation of the data. In numerical comparisons over a variety of examples, we show that our approach achieves substantially better predictive accuracy than existing competitors, most likely due to its inherent form of dimensionality reduction that is automatically discovered when fitting the model, in addition to having interpretability advantages and lower computational expense.