Generative Modeling of Approximately Periodic Time Series by a Posterior-Weighted Gaussian Process

2026-05-13

AI Summary

This work addresses the challenge of modeling approximately periodic time series in industrial and cyber-physical systems, which exhibit strong structural regularity alongside inter-cycle variability that neither purely periodic nor aperiodic models can adequately capture. The authors propose a two-stage stochastic generative model based on Gaussian processes, introducing a novel posterior-weighted periodic kernel that explicitly decouples shared structural patterns from individual variations under a common mean function. This approach is the first within the Gaussian process framework to effectively separate the common mode of approximately periodic signals from their smooth deviations, enabling high-quality trajectory generation. Experiments on synthetic data demonstrate that the model generates realistic and structurally coherent approximately periodic sequences, confirming its efficacy in jointly modeling both structural consistency and cycle-to-cycle variability.

Abstract

Discrete automated processes in industrial and cyber-physical systems often exhibit a repetitive structure in which successive repetitions follow a common trajectory while differing in duration, amplitude, and fine-scale dynamics. Such \emph{approximately periodic} behavior poses a challenge for Gaussian Processes (GP) modeling: strictly periodic models suppress inter-repetition variability, while non-periodic models fail to capture the strong structural regularities required for generation. In this work, we propose a stochastic generative model for approximately periodic time series. The model is based on a GP whose posterior is modulated by a novel kernel. Our approach decouples intra-repetition structure from inter-repetition variability through a two-stage construction which yields a generative distribution with a identical mean function across repetitions, while allowing smooth variation between repetitions. The modeling choices are supported by an implementation in which realistic synthetic trajectories are generated from toy datasets.

Problem

• approximately periodic time series
• Gaussian Processes
• generative modeling
• inter-repetition variability
• temporal structure

Innovation

• approximately periodic time series
• generative modeling
• Gaussian Process
• posterior-weighted kernel
• inter-repetition variability