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The result is a widely applicable method that infers causal structure directly from very different kinds of observations and systems. Here, we expand this method to explicitly introduce the causal diffusion components it produces. These encode the kernel causal state estimates as a set of coordinates in a reduced dimension space. We show how each component extracts predictive features from data and demonstrate their application on four examples: first, a simple pendulum\u2014an exactly solvable system; second, a molecular-dynamic trajectory of n-butane\u2014a high-dimensional system with a well-studied energy landscape; third, the monthly sunspot sequence\u2014the longest-running available time series of direct observations; and fourth, multi-year observations of an active crop field\u2014a set of heterogeneous observations of the same ecosystem taken for over a decade. In this way, we demonstrate that the empirical kernel causal state algorithm robustly discovers predictive structures for systems with widely varying dimensionality and stochasticity.<\/jats:p>","DOI":"10.1063\/5.0242981","type":"journal-article","created":{"date-parts":[[2025,3,31]],"date-time":"2025-03-31T16:37:45Z","timestamp":1743439065000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":["Inferring kernel \u03f5-machines: Discovering structure in complex systems"],"prefix":"10.1063","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5920-6674","authenticated-orcid":false,"given":"Alexandra M.","family":"Jurgens","sequence":"first","affiliation":[{"name":"INRIA Bordeaux Sud Ouest , 33405 Talence Cedex,","place":["France"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7364-8850","authenticated-orcid":false,"given":"Nicolas","family":"Brodu","sequence":"additional","affiliation":[{"name":"INRIA Bordeaux Sud Ouest , 33405 Talence Cedex,","place":["France"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2025,3,31]]},"reference":[{"key":"2025033112093708100_c1","doi-asserted-by":"publisher","first-page":"712","DOI":"10.1103\/PhysRevLett.45.712","article-title":"Geometry from a time series","volume":"45","year":"1980","journal-title":"Phys. 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