[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85213-en":3,"doc-seo-85213-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},85213,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","TSCoNet A Two-Stage Copula CNN LSTM for Uncertainty-Aware Spatio-Temporal Forecasting","Reliable forecasting of multiple interrelated environmental variables across space and time requires accurate predictions together with trustworthy uncertainty statements. Many deep-learning models achieve good point accuracy but output little or poorly calibrated uncertainty, and forcing uncertainty via maximum likelihood can reduce accuracy under strong correlations. TSCoNet introduces a two-stage CNN–LSTM coupled with a Gaussian copula to jointly predict multivariate fields and quantify predictive uncertainty with calibrated intervals after recalibration, demonstrated on simulated and real precipitation–temperature data.","arXiv :2607 . 10410v1 [ stat .ML] 11 Jul 2026  \nTSCoNet: A Two-Stage Copula CNN–LSTM for Uncertainty-Aware Spatio-Temporal Forecasting  \nJongwook Kim 1 Jong-Min Kim2  \n1 Department of Mathematical Sciences, Ball State University, e-mail:  \n[jongwook.kim@bsu.edu](jongwook.kim@bsu.edu)  \n2 Statistics Discipline, Division of Science and Mathematics, University of  \nMinnesota-Morris, e-mail: [jongmink@morris.umn.edu](jongmink@morris.umn.edu)  \nAbstract: Reliable forecasting of several interrelated environmental variables—such as regional precipitation and temperature, or other correlated geophysical fields—across many locations calls for accurate predictions accompanied by trustworthy statements of their uncertainty.  \nModern deep-learning models forecast such variables accurately but usually report no uncertainty, and forcing them to output uncertainty through maximum likelihood tends to degrade their accuracy, especially when the variables are strongly correlated. Motivated by this tension, we develop TSCoNet, a two-stage convolutional–recurrent model coupled with a Gaussian copula that jointly forecasts multiple variables over space and time while quantifying predictive uncertainty. The method first learns accurate mean forecasts and then, holding the mean fixed, refines a shared representation to estimate the predictive variance, yielding calibrated prediction intervals after a standard recalibration, so that uncertainty is added without sacrificing point accuracy. We study the approach on simulated non-stationary spatial fields on the sphere and on a real dataset of monthly precipitation and temperature for fifty cities over 2000–2020 .  \nThe model matches the accuracy of a strong deterministic forecaster while supplying calibrated prediction intervals that the deterministic model cannot, giving a single tool that provides both accurate point forecastsand reliable uncertainty for multivariate spatio-temporal data.  \nKeywords and phrases: Spatio-temporal forecasting, Uncertainty quantification, Gaussian copula, CNN-LSTM, Intrinsic Random Functions, Multivariate time series.  \n1. Introduction  \nForecasting how interrelated environmental variables evolve across space and time—such as regional precipitation and temperature, or other geophysical fields observed across a network of locations—underlies decisions in agriculture, water and energy management, public health, and environmental risk management more broadly. Such decisions are rarely about a single variable at a single site: they involve several interrelated quantities (for instance, precipitation together with minimum and maximum temperature) at many locations and time horizons. They also depend as much on knowing how uncertain a forecast is as  \n/Uncertainty-Aware Multivariate Spatio-Temporal Forecasting 2  \non the forecast itself—an interval that is too narrow invites overconfident decisions, while one that is too wide is uninformative. Producing such forecasts is genuinely hard: the variables move together in ways that differ from place to place, their spatial dependence is non-stationary across the globe, and each carries long-term temporal structure. What practitioners need, and what remains difficult to deliver, is a forecast of all these variables jointly—over space and time—that is both accurate and accompanied by trustworthy, well-calibrated uncertainty.  \nTo address these multifaceted challenges, environmental and atmospheric modeling has traditionally relied on classical geostatistical frameworks like kriging or Gaussian Processes. While theoretically robust, these models scale poorly to high-dimensional regimes and struggle with multivariate extensions unless highly restrictive covariance structures (e.g., separability) are assumed. Modern deep learning architectures—CNN-LSTM hybrids in particular—instead excel at learning highly non-linear spatio-temporal representations, yet most operate under a deterministic framework: by indiscriminately minimizing mean sq","cbCaiilaCUKyQAAt","https://ap.wps.com/l/cbCaiilaCUKyQAAt","pdf",2249581,3,1,42,"English","en",105,"# Introduction\n## Problem motivation and challenges\n## Related modeling approaches\n## Proposed TSCoNet and training strategy","[{\"question\":\"What problem does TSCoNet address in spatio-temporal forecasting?\",\"answer\":\"It targets reliable forecasting for multiple correlated environmental variables while also producing trustworthy, well-calibrated predictive uncertainty intervals, not just point estimates.\"},{\"question\":\"Why do standard deep-learning approaches often fail at uncertainty estimation?\",\"answer\":\"Deterministic training with mean-squared-error can lead to overconfident predictions and inadequate uncertainty quantification, and forcing uncertainty through maximum likelihood may degrade accuracy when variables are strongly correlated.\"},{\"question\":\"How does TSCoNet incorporate uncertainty without sacrificing point accuracy?\",\"answer\":\"TSCoNet uses a two-stage training framework: it first learns accurate mean forecasts, then fixes the mean and refines a shared representation to estimate predictive variance via a Gaussian copula, producing calibrated prediction intervals after recalibration.\"}]",1784201782,106,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"tsconet-a-two-stage-copula-cnn-lstm-for-uncertainty-aware-spatio-temporal-forecasting","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/tsconet-a-two-stage-copula-cnn-lstm-for-uncertainty-aware-spatio-temporal-forecasting/85213/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does TSCoNet address in spatio-temporal forecasting?","Question",{"text":75,"@type":76},"It targets reliable forecasting for multiple correlated environmental variables while also producing trustworthy, well-calibrated predictive uncertainty intervals, not just point estimates.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why do standard deep-learning approaches often fail at uncertainty estimation?",{"text":80,"@type":76},"Deterministic training with mean-squared-error can lead to overconfident predictions and inadequate uncertainty quantification, and forcing uncertainty through maximum likelihood may degrade accuracy when variables are strongly correlated.",{"name":82,"@type":73,"acceptedAnswer":83},"How does TSCoNet incorporate uncertainty without sacrificing point accuracy?",{"text":84,"@type":76},"TSCoNet uses a two-stage training framework: it first learns accurate mean forecasts, then fixes the mean and refines a shared representation to estimate predictive 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