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Journal Articles Chaos: An Interdisciplinary Journal of Nonlinear Science Year : 2022

Discovering Causal Structure with Reproducing-Kernel Hilbert Space ε-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 ϵ-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–Planck 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.
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Dates and versions

hal-03029188 , version 1 (27-11-2020)
hal-03029188 , version 2 (07-02-2022)

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Nicolas Brodu, James P Crutchfield. Discovering Causal Structure with Reproducing-Kernel Hilbert Space ε-Machines. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2022, ⟨10.1063/5.0062829⟩. ⟨hal-03029188v2⟩
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