Statistical Depth Meets Machine Learning: Kernel Mean Embeddings and Depth in Functional Data Analysis

Functional data analysis (FDA) concerns the study of observations that can be represented as functions, often residing in an infinite-dimensional space. In the recent decades, FDA saw remarkable progress, with many theoretical and practical problems successfully resolved. Often, however, statistical concepts used in finite-dimensional spaces do not readily generalise to random functions. As a result, alternative definitions and desired properties have to be used [80, 27, 46, 47]. An example of a prominent tool of multivariate analysis that is difficult to generalise to functional data is statistical depth. Developed in the 1980s, statistical depth is an umbrella term for methods introducing orderings, ranks, and by extension, nonparametric statistical inference, to multivariate and more complex datasets.