Change Detection: A functional analysis perspective

TL;DR

This paper introduces a novel change detection method for stochastic processes using tensor product representations and eigenspaces, capable of identifying local and global behavior changes across various topologies.

Contribution

It presents a new functional analysis-based approach for change detection in stochastic signals, leveraging eigenspaces from covariance operators for flexible, multilevel analysis.

Findings

Effective in detecting local and global changes

Applicable to various topologies including spherical domains

Demonstrated on real and simulated data

Abstract

We develop a new approach for detecting changes in the behavior of stochastic processes and random fields based on tensor product representations such as the Karhunen-Lo\`{e}ve expansion. From the associated eigenspaces of the covariance operator a series of nested function spaces are constructed, allowing detection of signals lying in orthogonal subspaces. In particular this can succeed even if the stochastic behavior of the signal changes either in a global or local sense. A mathematical approach is developed to locate and measure sizes of extraneous components based on construction of multilevel nested subspaces. We show examples in $\mathbb{R}$ and on a spherical domain $\mathbb{S}^{2}$. However, the method is flexible, allowing the detection of orthogonal signals on general topologies, including spatio-temporal domains.