Quantifying Periodicity in Non-Euclidean Object-Valued Time Series

Abstract

Non-Euclidean object-valued time series are playing a growing role in modern data analysis, and periodicity is a fundamental characteristic of many time series. 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 structure. 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 deriving uniform convergence rates for general Fr\'echet regression under temporal dependence, 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 excellent accuracy achieved by our approach in periodicity quantification. Finally, we apply our method to various real datasets, including compositional data arising in U.S. electricity generation, New York City transportation networks, and Germany's water consumption curves, highlighting its practical relevance in identifying and quantifying meaningful periodic patterns.