{"id":1916,"date":"2013-01-05T10:59:59","date_gmt":"2013-01-05T15:59:59","guid":{"rendered":"http:\/\/mseas.mit.edu\/?p=1916"},"modified":"2021-07-06T13:20:14","modified_gmt":"2021-07-06T17:20:14","slug":"data-assimilation-with-gaussian-mixture-models-using-the-dynamically-orthogonal-field-equations-part-i-theory-and-scheme","status":"publish","type":"post","link":"https:\/\/mseas.mit.edu\/?p=1916","title":{"rendered":"Data Assimilation with Gaussian Mixture Models using the Dynamically Orthogonal Field Equations. Part I: Theory and Scheme"},"content":{"rendered":"This work introduces and derives an efficient, data-driven assimilation scheme, focused on a\r\ntime-dependent stochastic subspace, that respects nonlinear dynamics and captures non-Gaussian\r\nstatistics as it occurs. The motivation is to obtain a filter that is applicable to realistic geophysical\r\napplications but that also rigorously utilizes the governing dynamical equations with information\r\ntheory and learning theory for efficient Bayesian data assimilation. Building on the foundations of\r\nclassical filters, the underlying theory and algorithmic implementation of the new filter are developed\r\nand derived. The stochastic Dynamically Orthogonal (DO) field equations and their adaptive\r\nstochastic subspace are employed to predict prior probabilities for the full dynamical state, effectively\r\napproximating the Fokker-Planck equation. At assimilation times, the DO realizations are fit to\r\nsemiparametric Gaussian mixture models (GMMs) using the Expectation-Maximization algorithm\r\nand the Bayesian Information Criterion. Bayes\u2019 Law is then efficiently carried out analytically within\r\nthe evolving stochastic subspace. The resulting GMM-DO filter is illustrated in a very simple example.\r\nVariations of the GMM-DO filter are also provided along with comparisons with related schemes.","protected":false},"excerpt":{"rendered":"<p>This work introduces and derives an efficient, data-driven assimilation scheme, focused on a time-dependent stochastic subspace, that respects nonlinear dynamics and captures non-Gaussian statistics as it occurs. The motivation is to obtain a filter that is applicable to realistic geophysical applications but that also rigorously utilizes the governing dynamical equations with information theory and learning [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[31,182,33,34,5,62,64],"tags":[],"class_list":["post-1916","post","type-post","status-publish","format-standard","hentry","category-uncertainty-quantification-and-reduced-order-modeling","category-learning-and-data-assimilation","category-uncertainty-quantification-and-predictions","category-data-assimilation","category-publications","category-papers-in-refereed-journals-data-assimilation","category-papers-in-refereed-journals-uncertainty-quantification-and-predictions"],"_links":{"self":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/1916","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1916"}],"version-history":[{"count":11,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/1916\/revisions"}],"predecessor-version":[{"id":2672,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/1916\/revisions\/2672"}],"wp:attachment":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1916"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1916"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1916"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}