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

TL;DR

This paper introduces a novel Gaussian Process-based generative model for approximately periodic time series, effectively capturing both repetitive structure and variability in industrial and cyber-physical systems.

Contribution

It proposes a two-stage GP model with a new kernel that separates intra-repetition structure from inter-repetition variability, enabling realistic synthetic data generation.

Findings

Successfully generates realistic synthetic trajectories from toy datasets.

Decouples intra-repetition structure from inter-repetition variability.

Supports modeling of approximately periodic behaviors in complex systems.

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.