Selecting embedding delays: An overview of embedding techniques and a new method using persistent homology

Area of Science:

• Dynamical Systems and Time Series Analysis
• Computational Topology
• Machine Learning for Scientific Data

Background:

• Delay embedding is fundamental for reconstructing dynamical systems from time series data.
• Parameter selection, particularly embedding lag, significantly influences analysis outcomes.
• Existing methods for non-uniform delay embedding lack robust dynamical interpretability.

Purpose of the Study:

• To provide a foundational overview of delay embedding theory.
• To critically evaluate current methods for selecting embedding lags.
• To introduce and validate a novel method for embedding lag selection.

Main Methods:

• Review of uniform and non-uniform delay embedding lag selection techniques.
• Development of Significant Times on Persistent Strands (SToPS) using persistent homology.
• Construction of a characteristic time spectrum for quantifying lag significance.
• Testing SToPS on periodic, chaotic, and fast-slow time series data.

Main Results:

• SToPS demonstrates comparable performance to existing automated non-uniform embedding methods.
• Embeddings generated using SToPS improve n-step prediction accuracy for fast-slow time series.
• The method integrates dynamical and topological insights for lag selection.

Conclusions:

• The SToPS method offers a more dynamically interpretable approach to embedding lag selection.
• SToPS provides a robust alternative for time series analysis and prediction, especially for complex systems.
• This work advances the practical application of topological data analysis in time series forecasting.