Localization processes for functional data analysis

TL;DR

This paper introduces a novel local approximation method for functional data analysis that improves estimation, classification, and outlier detection by using piecewise neighboring curves and a stabilization-based distance.

Contribution

It develops a new local approximation technique for functional data, providing asymptotic theory and demonstrating superior performance over existing benchmarks.

Findings

Method is competitive with existing approaches.

Provides asymptotic results for large data samples.

Improves estimation of unobserved segments and classification accuracy.

Abstract

We propose an alternative to $k$-nearest neighbors for functional data whereby the approximating neighboring curves are piecewise functions built from a functional sample. Using a locally defined distance function that satisfies stabilization criteria, we establish pointwise and global approximation results in function spaces when the number of data curves is large enough. We exploit this feature to develop the asymptotic theory when a finite number of curves is observed at time-points given by an i.i.d. sample whose cardinality increases up to infinity. We use these results to investigate the problem of estimating unobserved segments of a partially observed functional data sample as well as to study the problem of functional classification and outlier detection. For such problems, our methods are competitive with and sometimes superior to benchmark predictions in the field.