Testing Homogeneity: The Trouble with Sparse Functional Data

TL;DR

This paper introduces a new permutation-based test for homogeneity in functional data, effective for both densely and sparsely measured samples, with proven consistency and demonstrated success on synthetic and real data.

Contribution

It proposes the first guaranteed consistent test for homogeneity in sparse functional data using an Energy distance-based statistic.

Findings

Test performs well on synthetic data

Effective on real-world datasets

Proven convergence and consistency

Abstract

Testing the homogeneity between two samples of functional data is an important task. While this is feasible for intensely measured functional data, we explain why it is challenging for sparsely measured functional data and show what can be done for such data. In particular, we show that testing the marginal homogeneity based on point-wise distributions is feasible under some constraints and propose a new two sample statistic that works well with both intensively and sparsely measured functional data. The proposed test statistic is formulated upon Energy distance, and the critical value is obtained via the permutation test. The convergence rate of the test statistic to its population version is derived along with the consistency of the associated permutation test. To the best of our knowledge, this is the first paper that provides guaranteed consistency for testing the homogeneity for sparse functional data. The aptness of our method is demonstrated on both synthetic and real data sets.