Pseudo-quantile functional data clustering

This paper studies the problem of functional data clustering. Functional data have their own characteristics and contain rich information that cannot be obtained when regarding the data as multivariate data. Functional data are inherently infinite-dimensional, so classical clustering techniques for finite-dimensional data may not be suitable for functional data. There are several clustering methods for functional data based on probabilistic models or basis expansion approaches. However, most of these depend on the symmetric structure of the model or the mean response; hence, these cannot reflect characteristics of the distribution of data beyond the mean, such as behavior at the extremes. In this paper, we propose a new approach for functional data clustering based on the concept of an asymmetric norm. For this purpose, we consider pseudo-quantiles, such as M-quantiles and expectiles, and their corresponding curves that can provide rich distributional information about hidden structures in the data at various levels. Moreover, as a theoretical justification for the proposed method, a strong consistency property is investigated. Results from numerical examples, including real data analysis, demonstrate the promising empirical properties of the proposed approach.