Inferring kernel ϵ-machines: Discovering structure in complex systems.

Area of Science:

• Complex Systems Science
• Dynamical Systems Theory
• Machine Learning and Data Analysis

Background:

• Computational mechanics defines causal states as predictively equivalent trajectory classes for stochastic dynamical systems.
• Previous work successfully mapped these causal states into a reproducing kernel Hilbert space, enabling broad applicability.
• Existing methods infer causal structure from diverse observations and systems but lack explicit dimensionality reduction.

Purpose of the Study:

• To expand the kernel causal state method by introducing explicit causal diffusion components.
• To encode kernel causal state estimates as coordinates in a reduced-dimension space.
• To demonstrate the extraction of predictive features and the algorithm's robustness across varied systems.

Main Methods:

• Development and application of causal diffusion components within the kernel causal state framework.
• Encoding causal state estimates into a reduced-dimensional coordinate system.
• Empirical validation using diverse examples: pendulum, n-butane molecular dynamics, sunspot time series, and crop field observations.

Main Results:

• Causal diffusion components effectively encode kernel causal state estimates in a lower-dimensional space.
• Each component demonstrably extracts predictive features from observational data.
• The empirical kernel causal state algorithm shows robust discovery of predictive structures in systems of varying dimensionality and stochasticity.

Conclusions:

• The enhanced kernel causal state algorithm with causal diffusion components offers a powerful tool for uncovering system dynamics.
• This method provides a robust framework for inferring causal structure from heterogeneous and high-dimensional data.
• The approach is broadly applicable to scientific domains dealing with complex stochastic dynamical systems.