A Partition Dirichlet Process Model for Functional Data Analysis

Recently, extensions of the Dirichlet process to the functional domain have been presented in the literature. These processes can be classified based on the type of labeling they induce across different arguments in the domain, i.e., marginal or joint labeling. We show that marginal labeling processes have undesirable properties if functional observations are assumed to be almost surely continuous. Joint labeling processes avoid this undesirable behavior. They are specified through finite dimensional distributions for locations across the entire domain. They range from a common label for all arguments (relatively easy to fit) to joint local label selection at every argument (computationally very demanding). Here, we offer a middle ground - a joint labeling process which partitions the domain of the stochastic process and assign labels to individual partition elements. We call the proposed model a partition functional Dirichlet process and show that it can outperform both of the foregoing extremes of joint labeling. Given data that is a sample from each function in a collection of independent functions over the given domain, we employ this process as a prior for the true functions. We show results from simulation studies as well as a dataset arising from reflectance curves to demonstrate the performance of this process and make comparison with the two extreme labeling models.