Interpretable Solar-Cycle Forecasting via Causal TrendPriors and Residual Attention

Interpretable Solar-Cycle Forecasting via Causal TrendPriors and Residual Attention | 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 Interpretable Solar-Cycle Forecasting via Causal TrendPriors and Residual Attention Varun Soman, Amaranathan Sabarinath This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7832781/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 Forecasting the solar cycle remains challenging because the monthly Sunspot Number (SSN) exhibits irregular amplitudes, variable timings, and quasi-periodic, non-stationary behavior. We present an interpretable hybrid frame-work that decomposes the signal into a smooth causal trend and a short-term residual component. The trend is estimated with a one-sided (causal), robust locally weighted smoothing (LOWESS) filter so that only past data influence each estimate, preventing information leakage. Residual variations are modeled by a lightweight attention network adapted for scientific time-series forecasting. The residual model uses physically and temporally meaningful inputs— solar-cycle phase, odd–even parity, the 10.7 cm solar radio flux (F10.7), the planetary geomagnetic index (Ap), and sunspot-group area. Training with a quantile loss produces predictive quantiles, which are refined by a phase-aware conformal calibration procedure to obtain reliable prediction intervals (PIs). On a strictly chronological hindcast of Solar Cycle 24, the model achieves a median mean absolute error (MAE) of about 14 and a root-mean-square error (RMSE) of about 19, outperforming a persistence-with-drift baseline (MAE ≈65) and a gradient-boosting regressor. Calibrated 80 % PIs attain empirical coverage close to the nominal level. Feature-removal tests show that explicit cycle phase is the main driver of forecast skill. Once the causal trend and phase are included, standard physical covariates add little marginal benefit; removing them leaves accuracy essentially unchanged and can even improve it slightly, yielding a more parsimonious model. Forecasts initialized at the official December 2019 minimum provide plausible trajectories for Solar Cycles 25 and 26 with calibrated uncertainty bands. This causal-trend plus attention-residual framework yields consistent, interpretable, and well-calibrated forecasts, complementing review studies of solar-cycle variability (e.g. Hathaway 2015) and employing established tools for smoothing (Cleveland 1979), attention (Vaswani et al. 2017), and interval calibration (Romano, Patterson, and Candes 2019). Sunspot Number (SSN) Solar cycle Forecasting LOWESS Attention Quantile regression Conformal prediction Uncertainty F10.7 Geomagnetic index Ap 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-7832781","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":533559043,"identity":"4846e9ad-dbbd-41a6-afcf-f4cb114e20c7","order_by":0,"name":"Varun Soman","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/0lEQVRIiWNgGAWjYDADfgYGNhDNDOVLENYi2UCyFoMDEC1EqDx+9pl0wZ9tecbXDj978HOPHTt//wHGDz8YLPJwajmTbiY9s+12sdntNHPDnmfJzBI3EpglexgkinF7IY1NmrfhduK22wlmEjwHmJkZbjAwSAP9ktiAS0v/MzZpnj+3EzfPTv8m+edAPbP8+QPMv/Fp4ZcA2sLDdjtxg3SOmTTPgcPMBgcS2PDawi/xjNmat+124ozbOWXSMgeOMxveSGyz7DHArYWNP43xNshh/bPTt0m+OVCdLHf+8OEbPyrqcGrBAMkMDIxAxQbEqgcCOxLUjoJRMApGwQgBAHGLUgl8lKlOAAAAAElFTkSuQmCC","orcid":"","institution":"Maharaja's College Ernakulam","correspondingAuthor":true,"prefix":"","firstName":"Varun","middleName":"","lastName":"Soman","suffix":""},{"id":533559045,"identity":"5b1357db-ac06-41ed-910c-ffc19630428a","order_by":1,"name":"Amaranathan Sabarinath","email":"","orcid":"","institution":"Vikram Sarabhai Space Centre","correspondingAuthor":false,"prefix":"","firstName":"Amaranathan","middleName":"","lastName":"Sabarinath","suffix":""}],"badges":[],"createdAt":"2025-10-11 07:23:16","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7832781/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7832781/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":94820779,"identity":"31a54d3e-c8c5-41af-aa75-9105f6639ce0","added_by":"auto","created_at":"2025-10-31 06:02:54","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":5405,"visible":true,"origin":"","legend":"","description":"","filename":"683fe32bf02c4457bdc4c3b05533bc57.json","url":"https://assets-eu.researchsquare.com/files/rs-7832781/v1/a3d02e147449dcdbf4f5e7f0.json"},{"id":95657045,"identity":"704cab81-da7c-4b6b-a3e7-7ec2e71b8693","added_by":"auto","created_at":"2025-11-11 16:20:02","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":636808,"visible":true,"origin":"","legend":"","description":"","filename":"SpringerIMLRevisedV2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7832781/v1_covered_c6462b12-d2de-40ea-be56-904ac67c532a.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Interpretable Solar-Cycle Forecasting via Causal TrendPriors and Residual Attention","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"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":"Sunspot Number (SSN), Solar cycle, Forecasting, LOWESS, Attention, Quantile regression, Conformal prediction, Uncertainty, F10.7, Geomagnetic index Ap","lastPublishedDoi":"10.21203/rs.3.rs-7832781/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7832781/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Forecasting the solar cycle remains challenging because the monthly Sunspot Number (SSN) exhibits irregular amplitudes, variable timings, and quasi-periodic, non-stationary behavior. We present an interpretable hybrid frame-work that decomposes the signal into a smooth causal trend and a short-term residual component. The trend is estimated with a one-sided (causal), robust locally weighted smoothing (LOWESS) filter so that only past data influence each estimate, preventing information leakage. Residual variations are modeled by a lightweight attention network adapted for scientific time-series forecasting. The residual model uses physically and temporally meaningful inputs— solar-cycle phase, odd–even parity, the 10.7 cm solar radio flux (F10.7), the planetary geomagnetic index (Ap), and sunspot-group area. Training with a quantile loss produces predictive quantiles, which are refined by a phase-aware conformal calibration procedure to obtain reliable prediction intervals (PIs). 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