Scalable Nonlinear Cox Modeling via Random Fourier Features with Analytic Uncertainty

Scalable Nonlinear Cox Modeling via Random Fourier Features with Analytic Uncertainty | 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 Scalable Nonlinear Cox Modeling via Random Fourier Features with Analytic Uncertainty Fahrettin KAYA This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9108239/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 14 You are reading this latest preprint version Abstract Background: The Cox proportional hazards model often fails to capture complex biomedical risk structures, such as U-shaped biomarker associations, due to its assumption of linearity between the log-hazard and covariates. While existing kernel-based generalizations offer the necessary flexibility, their 0 ( n 3 ) computational complexity limits applicability in large-scale cohort studies. Furthermore, most non-linear machine learning methods lack closed-form analytical measures of uncertainty for individual predictions. Methods: We developed a novel Random Fourier Features-based Cox regression approach (RFF-Cox) to model non-linear risk relationships within a scalable framework. By mapping stationary kernels into a finite-dimensional explicit feature space, the method reduces computational complexity to 0(nm 2 ) . Model parameters are estimated via the Newton–Raphson algorithm on a ridge-regularized partial likelihood, while the bandwidth parameter (σ) is automatically optimized using a marginal likelihood criterion based on the Laplace approximation. A distinguishing feature of our approach is the stabilization of the Fisher information matrix via eigen-decomposition, enabling the generation of analytical 95% confidence intervals for individual survival estimates through the delta method and log–log transformation. Performance was evaluated using controlled simulations and six real-world datasets with sample sizes ranging from 432 to 9,105. Results: In simulation scenarios, the RFF-Cox model demonstrated a marked accuracy advantage over the classical Cox model in capturing U-shaped risk functions (C-index: 0.84 vs. 0.68). In real-world applications, the model exhibited discriminatory power competitive with Random Survival Forests and Gradient Boosting methods while showing superior computational efficiency; for instance, training time on the SUPPORT2 dataset was reduced from 126 seconds to 1.4 seconds. IPCW-weighted calibration analyses yielded low Integrated Calibration Error (ICI < 0.05) across all time horizons, confirming the reliability of probability estimates. Moreover, uncertainty in individual predictions, quantified via analytical confidence intervals, varied significantly across risk groups. Conclusions: RFF-Cox provides a practical survival analysis framework that combines automatic hyperparameter selection, computational efficiency, and transparent reporting of statistical uncertainty. The method overcomes the limitations of classical linear models while offering the speed and interpretability required to serve as a viable alternative to machine learning algorithms in large-scale data settings. Survival Analysis Cox Model Random Fourier Features Kernel Methods Scalability Uncertainty Quantification Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Revision Version 1 posted Editorial decision: Revision requested 29 Apr, 2026 Reviews received at journal 25 Apr, 2026 Reviews received at journal 20 Apr, 2026 Reviews received at journal 17 Apr, 2026 Reviews received at journal 17 Apr, 2026 Reviewers agreed at journal 15 Apr, 2026 Reviewers agreed at journal 15 Apr, 2026 Reviewers agreed at journal 08 Apr, 2026 Reviewers agreed at journal 08 Apr, 2026 Reviewers invited by journal 08 Apr, 2026 Editor invited by journal 18 Mar, 2026 Editor assigned by journal 16 Mar, 2026 Submission checks completed at journal 16 Mar, 2026 First submitted to journal 12 Mar, 2026 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-9108239","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":608918434,"identity":"65c196af-d3fc-4a80-9714-7fd9507b06fe","order_by":0,"name":"Fahrettin KAYA","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAwElEQVRIiWNgGAWjYBAC9h4QWZEA5kgQpYXnDIg8A9diQKQWxjaStPAcfibxc16avcEB5oO3eRj+5BPWwttmJtm7LSdxwwG2ZGseBgPLBkJa7PkZzCR4t1UkGBzgMZMGaiHsMh5+9m+Sf+dUAB3G/41ILbw9ZtK8DTmMGw7wsBGphedMsbXMsbTEmYfZjC3nGBgToyV94803Ncn2fMebH954UyFHRCgzMLBAIp0ZRBClAaj2A3HqRsEoGAWjYMQCAAt1Mi8QuET/AAAAAElFTkSuQmCC","orcid":"","institution":"Kahramanmaraş Sütçü İmam University","correspondingAuthor":true,"prefix":"","firstName":"Fahrettin","middleName":"","lastName":"KAYA","suffix":""}],"badges":[],"createdAt":"2026-03-12 20:53:35","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9108239/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9108239/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":105035921,"identity":"398ca769-a34b-4f53-be41-3cb8f0617ad1","added_by":"auto","created_at":"2026-03-20 07:26:57","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":748685,"visible":true,"origin":"","legend":"","description":"","filename":"ManuscriptRev1.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9108239/v1_covered_372b0e21-ec35-423f-8d7c-6a8d3c9dac12.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Scalable Nonlinear Cox Modeling via Random Fourier Features with Analytic Uncertainty","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"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":"bmc-medical-research-methodology","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"bmrm","sideBox":"Learn more about [BMC Medical Research Methodology](http://bmcmedresmethodol.biomedcentral.com/)","snPcode":"","submissionUrl":"https://www.editorialmanager.com/bmrm/default.aspx","title":"BMC Medical Research Methodology","twitterHandle":"BMC_series","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"em","reportingPortfolio":"BMC Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Survival Analysis, Cox Model, Random Fourier Features, Kernel Methods, Scalability, Uncertainty Quantification","lastPublishedDoi":"10.21203/rs.3.rs-9108239/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9108239/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003e\u003cstrong\u003eBackground:\u003c/strong\u003e The Cox proportional hazards model often fails to capture complex biomedical risk structures, such as U-shaped biomarker associations, due to its assumption of linearity between the log-hazard and covariates. While existing kernel-based generalizations offer the necessary flexibility, their \u003cem\u003e0\u003c/em\u003e(\u003cem\u003en\u003c/em\u003e\u003csup\u003e\u003cem\u003e3\u003c/em\u003e\u003c/sup\u003e) computational complexity limits applicability in large-scale cohort studies. Furthermore, most non-linear machine learning methods lack closed-form analytical measures of uncertainty for individual predictions.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMethods:\u003c/strong\u003e We developed a novel Random Fourier Features-based Cox regression approach (RFF-Cox) to model non-linear risk relationships within a scalable framework. By mapping stationary kernels into a finite-dimensional explicit feature space, the method reduces computational complexity to \u003cem\u003e0(nm\u003c/em\u003e\u003csup\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sup\u003e\u003cem\u003e)\u003c/em\u003e. Model parameters are estimated via the Newton–Raphson algorithm on a ridge-regularized partial likelihood, while the bandwidth parameter (σ) is automatically optimized using a marginal likelihood criterion based on the Laplace approximation. A distinguishing feature of our approach is the stabilization of the Fisher information matrix via eigen-decomposition, enabling the generation of analytical 95% confidence intervals for individual survival estimates through the delta method and log–log transformation. Performance was evaluated using controlled simulations and six real-world datasets with sample sizes ranging from 432 to 9,105.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eResults:\u003c/strong\u003e In simulation scenarios, the RFF-Cox model demonstrated a marked accuracy advantage over the classical Cox model in capturing U-shaped risk functions (C-index: 0.84 vs. 0.68). 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