Band Depth based initialization of $k$-Means for functional data clustering

TL;DR

This paper introduces FABRIk, an improved initialization method for $k$-Means clustering of functional data, utilizing band depth, B-splines, and resampling to enhance clustering accuracy especially with noisy or incomplete data.

Contribution

It extends the BRIk algorithm for longitudinal data by incorporating B-spline fitting and resampling, improving initialization effectiveness for functional data clustering.

Findings

FABRIk outperforms previous methods in clustering recovery.

The method handles noise and missing data effectively.

Simulated and real data experiments validate its superiority.

Abstract

The $k$-Means algorithm is one of the most popular choices for clustering data but is well-known to be sensitive to the initialization process. There is a substantial number of methods that aim at finding optimal initial seeds for $k$-Means, though none of them are universally valid. This paper presents an extension to longitudinal data of one of such methods, the BRIk algorithm, that relies on clustering a set of centroids derived from bootstrap replicates of the data and on the use of the versatile Modified Band Depth. In our approach we improve the BRIk method by adding a step where we fit appropriate B-splines to our observations and a resampling process that allows computational feasibility and handling issues such as noise or missing data. Our results with simulated and real data sets indicate that our $F$unctional Data $A$pproach to the BRIK method (FABRIk) is more effective than previous proposals at providing seeds to initialize $k$-Means in terms of clustering recovery.