Subpattern matching persistent homology for coupling complexity analysis of multivariate time series

Area of Science:

• Complex Systems
• Dynamical Systems Theory
• Computational Topology

Background:

• Multivariate time series analysis often requires methods to quantify the complex interactions between components.
• Existing methods, such as ordinal patterns, provide a basis for analyzing time series complexity but may lack flexibility.
• Persistent homology offers a powerful framework for analyzing the topological features of data.

Purpose of the Study:

• To develop a new persistent homology approach for quantifying the coupling complexity of multivariate time series.
• To generalize existing ordinal pattern-based methods by allowing customizable pattern selection.
• To investigate the behavior of this complexity measure in the context of random Boolean networks.

Main Methods:

• Construction of filtered simplicial complexes using user-defined patterns that capture relationships within multivariate time series.
• Application of persistent homology to analyze the topological structure of these complexes.
• Utilizing binary patterns for binary multivariate time series generated by random Boolean networks.

Main Results:

• The proposed persistent homology approach subsumes existing ordinal pattern methods.
• Total persistence of the filtered simplicial complexes is proposed as a measure of coupling complexity.
• The average coupling complexity measure peaks near the criticality of dynamical stability in random Boolean networks.

Conclusions:

• The developed persistent homology method provides a flexible and powerful tool for studying coupling complexity in multivariate time series.
• The findings suggest a link between coupling complexity and critical dynamics, with potential implications for understanding complex systems.
• The method's ability to capture relations among time series components offers new avenues for time series analysis.