Discussion contribution "Functional models for time-varying random objects'' by Dubey and M\"uller (to appear in JRSS-B)

TL;DR

This paper discusses extending PCA to metric-valued functions, proposing a kernel PCA approach that relates closely to Dubey and Müller's method, offering a potentially simpler alternative.

Contribution

It introduces a kernel PCA method for time-varying random objects, providing a simpler framework compared to Dubey and Müller's complex object FPCs and Fréchet scores.

Findings

Kernel PCA closely relates to DM's approach.

Kernel PCA offers a simpler implementation.

The method effectively captures time-varying object variability.

Abstract

In an inspiring paper Dubey and M\"uller (DM) extend PCA to the case that observations are metric-valued functions. As an alternative, we develop a kernel PCA approach, which we show is closely related to the DM approach. While kernel principal components (kPCs) are simply defined, DM require added complexity in the form of "object FPCs'' and "Fr\'echet scores".