A Kernel Two-Sample Test for Functional Data
This provides a method for statisticians and data analysts working with functional data, but it is incremental as it extends existing MMD-based tests to functional spaces.
The authors tackled the problem of testing whether two samples of functions come from the same distribution by proposing a nonparametric two-sample test based on Maximum Mean Discrepancy (MMD) using kernels on function spaces, with theoretical properties established and demonstrated on synthetic and real-world datasets.
We propose a nonparametric two-sample test procedure based on Maximum Mean Discrepancy (MMD) for testing the hypothesis that two samples of functions have the same underlying distribution, using kernels defined on function spaces. This construction is motivated by a scaling analysis of the efficiency of MMD-based tests for datasets of increasing dimension. Theoretical properties of kernels on function spaces and their associated MMD are established and employed to ascertain the efficacy of the newly proposed test, as well as to assess the effects of using functional reconstructions based on discretised function samples. The theoretical results are demonstrated over a range of synthetic and real world datasets.