Selecting embedding delays: An overview of embedding techniques and a new method using persistent homology
Abstract
Delay embedding methods are a staple tool in the field of time series analysis and prediction. However, the selection of embedding parameters can have a big impact on the resulting analysis. This has led to the creation of a large number of methods to optimise the selection of parameters such as embedding lag. This paper aims to provide a comprehensive overview of the fundamentals of embedding theory for readers who are new to the subject. We outline a collection of existing methods for selecting embedding lag in both uniform and non-uniform delay embedding cases. Highlighting the poor dynamical explainability of existing methods of selecting non-uniform lags, we provide an alternative method of selecting embedding lags that includes a mixture of both dynamical and topological arguments. The proposed method, {\em Significant Times on Persistent Strands} (SToPS), uses persistent homology to construct a characteristic time spectrum that quantifies the relative dynamical significance of each time lag. We test our method on periodic, chaotic and fast-slow time series and find that our method performs similar to existing automated non-uniform embedding methods. Additionally, -step predictors trained on embeddings constructed with SToPS was found to outperform other embedding methods when predicting fast-slow time series.
Cite
@article{arxiv.2302.03447,
title = {Selecting embedding delays: An overview of embedding techniques and a new method using persistent homology},
author = {Eugene Tan and Shannon Algar and Débora Corrêa and Michael Small and Thomas Stemler and David Walker},
journal= {arXiv preprint arXiv:2302.03447},
year = {2025}
}