Novel Rapid Approach for Adaptive Gaussian Kernel Density Estimation: Gridpoint-wise Propagation of Anisotropic Diffusion Equation

Novel Rapid Approach for Adaptive Gaussian Kernel Density Estimation: Gridpoint-wise Propagation of Anisotropic Diffusion Equation | 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 Novel Rapid Approach for Adaptive Gaussian Kernel Density Estimation: Gridpoint-wise Propagation of Anisotropic Diffusion Equation Christian Sustay Martinez, Patrick K. Quoika, Martin Zacharias This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8038701/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 9 You are reading this latest preprint version Abstract Anyone who works with data is regularly faced with the issue of how to represent the distribution of the collected data. Nowadays histogramming is still the standard method of choice. Yet, more sophisticated methods exist, such as Kernel Density Estimation (KDE). In comparison, KDE enables smoother and more accurate representation of the data distribution. However, some challenges remain, most of which center around the optimal choice of kernel bandwidth, and the adaptiveness of this bandwidth. Here, we propose a novel method for Gaussian KDE(GKDE) that improves upon classical approaches in terms of accuracy and computational efficiency. Our approach is parallelizable and therefore fast. This parallelizability enables implementation on modern computational hardware, such as GPU, which is a great advantage incomparison to other methods, especially if large amounts of data need to be processed. Furthermore, it automatically chooses the kernel bandwidth based on the collected data and is bydesign adaptive, i.e., the bandwidth varies across the data range. This allows smooth representation of broad features, without oversmoothing sharp features in the distribution. Moreover, ourmethod is applicable in an arbitrary number of dimensions. The approach is—similar to other novel methods—based on the propagation of the heat equation. We propose a novel measure for the detection of the optimal bandwidth, which is the variance to mean ratio of the density estimate on the grid points, for different bootstrapped samples of the original data. Apart from the introduction of the method itself, we show some illustrating examples on the performance of the approach. To this end, we evaluate various different data distributions in one and two dimensions, observing good accuracy across the board. We call this promising new approach Gridpoint-wise Adaptive Density Propagation KDE (GradePro). Full Text Additional Declarations No competing interests reported. Supplementary Files supplementaryinfo.pdf Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 12 Mar, 2026 Reviews received at journal 12 Mar, 2026 Reviewers agreed at journal 12 Mar, 2026 Reviews received at journal 13 Jan, 2026 Reviewers agreed at journal 28 Dec, 2025 Reviewers invited by journal 24 Nov, 2025 Editor assigned by journal 23 Nov, 2025 Submission checks completed at journal 06 Nov, 2025 First submitted to journal 05 Nov, 2025 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. 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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-8038701","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":541075793,"identity":"fab7816a-0522-4ec7-8e8c-45fcb2628616","order_by":0,"name":"Christian Sustay Martinez","email":"","orcid":"","institution":"Technical University of Munich","correspondingAuthor":false,"prefix":"","firstName":"Christian","middleName":"Sustay","lastName":"Martinez","suffix":""},{"id":541075796,"identity":"70f24b7a-7e9b-40fe-a58a-75332d3e88a6","order_by":1,"name":"Patrick K. 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