Time-penalized trees: consistency results and simulations

This paper extends the study of Time-penalized Trees (TpT), a novel tree-based algorithm designed for longitudinal data with time-varying covariates and responses. TpT introduces an exponential time-penalty into the splitting criterion, enabling joint partitioning of covariate space and time. We provide a theoretical analysis showing that, under mild assumptions on subject durations and the distribution of splitting gains, TpTs can achieve consistency. Specifically, we prove that (i) as the global sample size tends to infinity, the number of observations in each timebased subtree also tends to infinity, and (ii) these subtrees can grow unboundedly in depth, leveraging classical decision trees consistency results. We complement these findings with a simulation study illustrating how, for finite-support or subexponential duration distributions, the number of subjects reaching any given time node increases without bound, while suitably chosen time subtrees continue to split indefinitely if impurity gains follow a distribution with a nontrivial supremum. Although a thorough experimental comparison with existing longitudinal models, as well as a more flexible penalty framework remain avenues for future work, our results highlight the potential of TpT as an interpretable and computationally efficient method for dynamic decisionmaking in data-intensive longitudinal settings.