Shape-Preserving Minimum Trace (SP-MinT): A Regularized Forecast Reconciliation Method for Hierarchical Time Series

Shape-Preserving Minimum Trace (SP-MinT): A Regularized Forecast Reconciliation Method for Hierarchical Time Series | 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 Article Shape-Preserving Minimum Trace (SP-MinT): A Regularized Forecast Reconciliation Method for Hierarchical Time Series Mauro Gonzalez-Sierra, Jorge I. Velez, Adriana Arango-Manrique This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9161917/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 7 You are reading this latest preprint version Abstract Forecast reconciliation has become the standard for ensuring coherence in hierarchical time series. However, state-of-the-art methods like Minimum Trace (MinT) prioritize the minimization of error variance, often at the expense of distorting the temporal morphology of the forecast. This paper reframes forecast reconciliation as a multi-objective problem, showing that variance-optimal coherence is insufficient for operational decision-making, and proposing a shape-aware reconciler that explicitly encodes temporal structure. We introduce Shape-Preserving Minimum Trace (SP-MinT), a novel framework that regularizes the optimization process with domain-informed priors constructed from historical day-of-week profiles. We validate the method using a rigorous rolling cross-validation on real-world electricity demand data from Victoria, Australia. The results demonstrate that SP-MinT outperforms the standard MinT-WLS benchmark by reducing the Root Mean Squared Error (RMSE) by 30.5% and the Shape Error (Dynamic Time Warping) by 74%. By bridging the gap between statistical optimality and morphological fidelity, SP-MinT offers grid operators hierarchically coherent forecasts that respect physical ramping constraints. Physical sciences/Energy science and technology Physical sciences/Engineering Physical sciences/Mathematics and computing Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviews received at journal 29 Mar, 2026 Reviewers agreed at journal 26 Mar, 2026 Reviewers invited by journal 26 Mar, 2026 Editor assigned by journal 26 Mar, 2026 Editor invited by journal 24 Mar, 2026 Submission checks completed at journal 21 Mar, 2026 First submitted to journal 21 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. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-9161917","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":613792482,"identity":"145d24f7-6e07-48e5-b4c5-927706ded96f","order_by":0,"name":"Mauro Gonzalez-Sierra","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+ElEQVRIiWNgGAWjYHAD5gYGhgoGBgMgU4JILYxALWcYJEjUwthGhBbd9jNmHz78ssmTbz/Y+PDnvMN15gzMB2/zMNjJ4dJidibHeObMvrRixp7EZgPJbYclLBvYkq15GJKNcWo5kGPMzNtzOLGZIbFNwhCoxeAAj5k0D8OBxAZcWs6/gWhp43/Y/iNxDkgL/zf8Wm4AbeH5cTixRyKxjeFgA9gWNgJanhUzzmxIS5wh8bBZsuFYuuTOZjZjyzkGePxyPnkzw4c/Nonz+5MPfvxRY81vzt788MabCtwhBgbA6EACzCDCAK8GIPhDSMEoGAWjYBSMaAAAqqNWIGD0WW8AAAAASUVORK5CYII=","orcid":"","institution":"Universidad Tecnológica de Bolívar","correspondingAuthor":true,"prefix":"","firstName":"Mauro","middleName":"","lastName":"Gonzalez-Sierra","suffix":""},{"id":613792483,"identity":"e9be8dd0-3625-46eb-9a25-4fabcccf4e02","order_by":1,"name":"Jorge I. 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