Quantifying Periodicity in Non-Euclidean Random Objects

2025-10-20

AI Summary

This study addresses the challenge of quantifying periodicity in time-varying random objects residing in non-Euclidean spaces—such as networks, compositional data, and functional data—where conventional linear or parametric assumptions fail. We propose the first fully nonparametric framework for periodicity estimation, modeling cycle detection as a model selection problem in general metric spaces. Our method introduces a distance-based periodicity measure and integrates simulated annealing with the Minimum Description Length (MDL) principle for data-driven period identification. Crucially, we establish a consistency theory that does not rely on Euclidean geometry, ensuring asymptotic reliability under minimal structural assumptions. The framework provides principled, adaptive hyperparameter selection and guarantees uniform convergence of estimated periodic components. Empirical evaluation on real-world datasets—including power grid dynamics, urban traffic networks, and water consumption curves—demonstrates substantial improvements in both accuracy and interpretability of periodic pattern discovery.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Data Mining & Knowledge Management: Anomaly/Outlier Detection

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

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.

Problem

• Quantifying periodicity in non-Euclidean random objects lacking Euclidean structures
• Developing nonparametric framework for period estimation and component extraction
• Validating method accuracy through simulations and real-world dataset applications

Innovation

• Nonparametric framework for periodicity in metric spaces
• Model selection approach for period estimation
• Data-driven tuning without Euclidean linearity assumptions