A Kernel Two-Sample Test for Functional Data

Nonparametric two-sample tests for equality of distributions are widely studied in statistics, driven by applications in goodness-of-fit tests, anomaly and change-point detection and clustering. Classical examples of such tests include the Kolmogorov-Smirnov test [41, 69, 62] and Wald-Wolfowitz runs test [84] with subsequent multivariate extensions [25]. Due to advances in the ability to collect large amounts of real time or spatially distributed data there is a need to develop statistical methods appropriate for functional data, where each data sample is a discretised function. Such data has been studied for decades in the Functional Data Analysis (FDA) literature [32, 35] particularly in the context of analysing populations of time series, or in statistical shape analysis [45]. More recently, due to this modern abundance of functional data, increased study has been made in the machine learning literature for algorithms suited to such data [7, 15, 37, 12, 88].