Quantifying Periodicity in Non-Euclidean Random Objects

TL;DR

This paper introduces a nonparametric framework for quantifying periodicity in complex non-Euclidean random objects, providing theoretical guarantees and demonstrating superior accuracy through simulations and real data applications.

Contribution

It develops a novel approach for periodicity estimation in general metric spaces, with theoretical consistency proofs and practical algorithms for diverse data types.

Findings

Accurate period estimation in non-Euclidean data demonstrated in simulations.

Theoretical guarantees for consistency and convergence of the proposed methods.

Successful application to real datasets like electricity, transportation, and water consumption.

Abstract

Time-varying non-Euclidean random objects are playing a growing role in modern data analysis, and periodicity is a fundamental characteristic of time-varying data. However, quantifying periodicity in general non-Euclidean random objects remains largely unexplored. In this work, we introduce a novel nonparametric framework for quantifying periodicity in random objects within a general metric space that lacks Euclidean structures. Our approach formulates periodicity estimation as a model selection problem and provides methodologies for period estimation, data-driven tuning parameter selection, and periodic component extraction. Our theoretical contributions include establishing the consistency of period estimation without relying on linearity properties used in the literature for Euclidean data, providing theoretical support for data-driven tuning parameter selection, and deriving uniform convergence results for periodic component estimation. Through extensive simulation studies covering three distinct types of time-varying random objects such as compositional data, networks, and functional data, we showcase the superior accuracy achieved by our approach in periodicity quantification. Finally, we apply our method to various real datasets, including U.S. electricity generation compositions, New York City transportation networks, and Germany's water consumption curves, highlighting its practical relevance in identifying and quantifying meaningful periodic patterns.