Discovering Causal Structure with Reproducing-Kernel Hilbert Space $\epsilon$-Machines

Abstract

We merge computational mechanics' definition of causal states (predictively-equivalent histories) with reproducing-kernel Hilbert space (RKHS) representation inference. The result is a widely-applicable method that infers causal structure directly from observations of a system's behaviors whether they are over discrete or continuous events or time. A structural representation -- a finite- or infinite-state kernel $\epsilon$-machine -- is extracted by a reduced-dimension transform that gives an efficient representation of causal states and their topology. In this way, the system dynamics are represented by a stochastic (ordinary or partial) differential equation that acts on causal states. We introduce an algorithm to estimate the associated evolution operator. Paralleling the Fokker-Plank equation, it efficiently evolves causal-state distributions and makes predictions in the original data space via an RKHS functional mapping. We demonstrate these techniques, together with their predictive abilities, on discrete-time, discrete-value infinite Markov-order processes generated by finite-state hidden Markov models with (i) finite or (ii) uncountably-infinite causal states and (iii) continuous-time, continuous-value processes generated by thermally-driven chaotic flows. The method robustly estimates causal structure in the presence of varying external and measurement noise levels and for very high dimensional data.

23 pages, 11 figures, 64 citations; https://team.inria.fr/comcausa/continuous-causal-states/

References in corpus (5)

• Nonparametric forecasting of low-dimensional dynamical systems
• Shannon Entropy Rate of Hidden Markov Processes
• Divergent Predictive States: The Statistical Complexity Dimension of Stationary, Ergodic Hidden Markov Processes
• Spacetime Autoencoders Using Local Causal States
• Inference, Prediction, and Entropy-Rate Estimation of Continuous-time, Discrete-event Processes

Cited by in corpus (5)

• Nonequilibrium Statistical Mechanics and Optimal Prediction of Partially-Observed Complex Systems
• Memory in quantum dot blinking
• What Is a Pattern in Statistical Mechanics? Formalizing Structure and Patterns in One-Dimensional Spin Lattice Models with Computational Mechanics
• Exploring Predictive States via Cantor Embeddings and Wasserstein Distance
• Predictive complexity of quantum subsystems