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These are common fundamental questions that should be answered before diving into function fitting and other data handling schemes since lacking continuity and, possibly, smoothness in data relations guarantees that other more specific data fitting attempts will fail or be meaningless. We develop useful testing statistics by creating data-driven versions of basic topological concepts including continuity, differentiability, point set distance comparisons, diffeomorphisms, and embeddings. As an application we show how such statistics can aid in the analysis of the dynamics of reservoir computer systems, many of which rely on there being specific relations (like embeddings) between the drive and the reservoir computer.<\/jats:p>","DOI":"10.1063\/5.0269914","type":"journal-article","created":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T20:29:48Z","timestamp":1753907388000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":5,"title":["Statistics for differential topological properties between datasets with an application to reservoir computers"],"prefix":"10.1063","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3075-0913","authenticated-orcid":false,"given":"Louis","family":"Pecora","sequence":"first","affiliation":[{"name":"Institute for Research in Electronics and Applied Physics, University of Maryland 1 , College Park, MD, 20742,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2371-2049","authenticated-orcid":false,"given":"Thomas","family":"Carroll","sequence":"additional","affiliation":[{"name":"US Naval Research Laboratory 2 , Washington, DC 20375,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2025,7,30]]},"reference":[{"key":"2025073016294410800_c1","volume-title":"Differential Topology with a View to Applications","year":"1978"},{"key":"2025073016294410800_c2","volume-title":"Detecting Strange Attractors in Turbulence","year":"1981"},{"key":"2025073016294410800_c3","doi-asserted-by":"publisher","first-page":"579","DOI":"10.1007\/BF01053745","article-title":"Embedology","volume":"65","year":"1991","journal-title":"J. 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