Theorems for Inferring Canonical Environmental Signature from Causal Response

Theorems for Inferring Canonical Environmental Signature from Causal Response | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Theorems for Inferring Canonical Environmental Signature from Causal Response Ahmad Muhammad, Salim Jibrin Danbatta, Muhammad Abubakar Isah, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8916580/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract We present a model-free framework for inferring environmental persistence signatures from a single causal response signal R(t). The approach is based on seven estimators of persistence timescales centroid, step-response, autocorrelation, envelope, energy, and spectral-pole timescales whose agreements and disagreements encode diagnostic information about underlying system dynamics. Four theorems establish necessary and sufficient conditions for canonical regimes: (1) τ (s) = τ (2) for step-type responses with finite lag area, (2) τ (s) = τ pole for minimum-phase LTI systems, (3) The diagnostic parameter ρ 13 = τ (s) /τ (3) = 1 if and only if memory is exponential, and (4) The diagnostic parameter ∆ su → −1 and ∆ env 23 → 0.5 if and only if the system is a linear underdamped oscillator. We validate the framework on canonical systems including RC circuits, critically damped and underdamped mechanical systems, and pharmacokinetics. The capacity of these signatures under realistic measurement noise was analyzed. The framework enables passive, excitation-agnostic classification of dynamical regimes monotonic vs. oscillatory, Markovian vs. non-Markovian, linear vs. nonlinear damping from R(t) alone, without parametric assumptions or excitation knowledge. We made available all implementation codes (open-source) in the name QSignature library on Github. causal inference environmental signature persistence timescales system identification passive sensing response analysis Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8916580","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":597101346,"identity":"367b84c4-5134-425a-9ddd-39f61aac94bb","order_by":0,"name":"Ahmad Muhammad","email":"","orcid":"","institution":"Qatar University","correspondingAuthor":false,"prefix":"","firstName":"Ahmad","middleName":"","lastName":"Muhammad","suffix":""},{"id":597101347,"identity":"d1bd8655-15ed-49ef-ba03-dcc6f4864c41","order_by":1,"name":"Salim Jibrin Danbatta","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+klEQVRIiWNgGAWjYFAC5oYDDGwMDAZgTgVIgLmBgBZGmBZmxgaGMyAtjIS1MMC1MLbBRPAA+faDjYdulNlFm7OfP/7g57zaaP52oJYfFdtwajE4k9hwOOdccu7OnmTGxt5tx3NnHAba1nPmNm4tDEAtuW3MuRsOJDM28G47ltsA1MLM2IZbi3z/Q5CW+twN5x8zNv6dcyx3PiEtDDfAthzO3XAjmbGZt6EmdwMhLQY3HoL8cjx354zHhrNljh3I3QjUchCfX+T7kw9/zimrzt3On/jg45uautx55w8ffPCjAo/D0MBhMHmAaPVAUEeK4lEwCkbBKBghAAA62WgNwXh9FwAAAABJRU5ErkJggg==","orcid":"","institution":"Üsküdar University","correspondingAuthor":true,"prefix":"","firstName":"Salim","middleName":"Jibrin","lastName":"Danbatta","suffix":""},{"id":597101348,"identity":"246b173e-4d2e-407c-b7f8-76287f5b0d41","order_by":2,"name":"Muhammad Abubakar Isah","email":"","orcid":"","institution":"Istanbul Commerce University","correspondingAuthor":false,"prefix":"","firstName":"Muhammad","middleName":"Abubakar","lastName":"Isah","suffix":""},{"id":597101349,"identity":"8305bf01-a10e-4766-b859-a520f4d36848","order_by":3,"name":"Ibrahim Yahaya Muhammad","email":"","orcid":"","institution":"King Mongkut's University of Technology Thonburi","correspondingAuthor":false,"prefix":"","firstName":"Ibrahim","middleName":"Yahaya","lastName":"Muhammad","suffix":""}],"badges":[],"createdAt":"2026-02-19 10:38:57","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8916580/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8916580/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104400261,"identity":"4fb23730-7d5b-45df-91b0-ec6077a5d0e8","added_by":"auto","created_at":"2026-03-11 12:09:23","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2737150,"visible":true,"origin":"","legend":"","description":"","filename":"ANPManuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8916580/v1_covered_eef9399d-90b6-4967-9aac-e7162751214c.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Theorems for Inferring Canonical Environmental Signature from Causal Response","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"causal inference, environmental signature, persistence timescales, system identification, passive sensing, response analysis","lastPublishedDoi":"10.21203/rs.3.rs-8916580/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8916580/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We present a model-free framework for inferring environmental persistence signatures from a single causal response signal R(t). 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