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 -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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February 2022
Research Article|
February 01 2022
Discovering causal structure with reproducing-kernel Hilbert space ε-machines
Nicolas Brodu
;
Nicolas Brodu
a)
1
Geostat Team—Geometry and Statistics in Acquisition Data, INRIA Bordeaux Sud Ouest
, 200 rue de la Vieille Tour, 33405 Talence Cedex, France
a)Author to whom correspondence should be addressed: [email protected]
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James P. Crutchfield
James P. Crutchfield
b)
2
Complexity Sciences Center and Department of Physics and Astronomy, University of California at Davis
, One Shields Avenue, Davis, California 95616, USA
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a)Author to whom correspondence should be addressed: [email protected]
b)
Electronic mail: [email protected]
Citation
Nicolas Brodu, James P. Crutchfield; Discovering causal structure with reproducing-kernel Hilbert space ε-machines. Chaos 1 February 2022; 32 (2): 023103. https://doi.org/10.1063/5.0062829
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