Constrained variable clustering and the best basis problem in functional data analysis

Functional data analysis involves data described by regular functions rather than by a finite number of real valued variables. While some robust data analysis methods can be applied directly to the very high dimensional vectors obtained from a fine grid sampling of functional data, all methods benefit from a prior simplification of the functions that reduces the redundancy induced by the regularity. In this paper we propose to use a clustering approach that targets variables rather than individual to design a piecewise constant representation of a set of functions. The contiguity constraint induced by the functional nature of the variables allows a polynomial complexity algorithm to give the optimal solution.

💡 Research Summary

The paper addresses the pervasive problem in functional data analysis (FDA) of handling extremely high‑dimensional vectors that arise when continuous functions are sampled on a fine grid. While generic high‑dimensional techniques can be applied, they often ignore the intrinsic redundancy caused by the smoothness of the underlying functions. The authors propose a novel variable‑centric clustering approach that groups adjacent sampling points (variables) into contiguous blocks and represents each block by a simple statistic, typically the block mean. This yields a piecewise‑constant approximation of every function in the data set.

A key innovation is the explicit contiguity constraint: clusters must consist of consecutive indices because the variables correspond to ordered points on the function’s domain. Imposing this constraint transforms the otherwise NP‑hard variable‑clustering problem into a tractable dynamic‑programming (DP) formulation. The authors define a cost for any interval as the sum of squared deviations of the original values from the interval mean. The DP recurrence C(j,k)=min_{i<j}