Statistics for differential topological properties between datasets with an application to reservoir computers

Area of Science:

• Dynamical Systems Theory
• Topology
• Data Analysis

Background:

• Analyzing multidimensional time series from coupled systems is challenging due to data complexity.
• Simple graphical methods are insufficient for identifying fundamental relationships between subsystems.
• Pre-existing relationships like continuity and smoothness are critical for the success of further data analysis.

Purpose of the Study:

• To develop novel statistical tests for fundamental relationships in multidimensional time series data.
• To provide data-driven methods for assessing continuity, differentiability, and embeddings between system components.
• To enhance the analysis of dynamical systems, particularly reservoir computing models.

Main Methods:

• Developed data-driven statistics based on topological concepts.
• Applied concepts including continuity, differentiability, point set distance comparisons, diffeomorphisms, and embeddings.
• Created statistical tests for fundamental relationships in multidimensional time series.

Main Results:

• Introduced new statistical tests to rigorously assess fundamental relationships in complex time series.
• Demonstrated the utility of these tests in identifying critical data properties like continuity and embeddings.
• Showcased the application in analyzing the dynamics of reservoir computing systems.

Conclusions:

• The developed topological statistics are essential for pre-processing and understanding complex dynamical systems.
• These methods ensure the validity and meaningfulness of subsequent data fitting and analysis.
• The approach offers significant advancements for the analysis of coupled systems and reservoir computing.