{"id":1919,"date":"2013-01-05T11:02:50","date_gmt":"2013-01-05T16:02:50","guid":{"rendered":"http:\/\/mseas.mit.edu\/?p=1919"},"modified":"2021-07-06T13:19:53","modified_gmt":"2021-07-06T17:19:53","slug":"data-assimilation-with-gaussian-mixture-models-using-the-dynamically-orthogonal-field-equations-part-ii-applications","status":"publish","type":"post","link":"https:\/\/mseas.mit.edu\/?p=1919","title":{"rendered":"Data Assimilation with Gaussian Mixture Models using the Dynamically Orthogonal Field Equations. Part II: Applications"},"content":{"rendered":"The properties and capabilities of the GMM-DO filter are assessed and exemplified by applications\r\nto two dynamical systems: (1) the Double Well Diffusion and (2) Sudden Expansion flows; both\r\nof which admit far-from-Gaussian statistics. The former test case, or twin experiment, validates\r\nthe use of the EM algorithm and Bayesian Information Criterion with Gaussian Mixture Models\r\nin a filtering context; the latter further exemplifies its ability to efficiently handle state vectors of\r\nnon-trivial dimensionality and dynamics with jets and eddies. For each test case, qualitative and\r\nquantitative comparisons are made with contemporary filters. The sensitivity to input parameters\r\nis illustrated and discussed. Properties of the filter are examined and its estimates are described,\r\nincluding: the equation-based and adaptive prediction of the probability densities; the evolution\r\nof the mean field, stochastic subspace modes and stochastic coefficients; the fitting of Gaussian\r\nMixture Models; and, the efficient and analytical Bayesian updates at assimilation times and the\r\ncorresponding data impacts. The advantages of respecting nonlinear dynamics and preserving\r\nnon-Gaussian statistics are brought to light. For realistic test cases admitting complex distributions\r\nand with sparse or noisy measurements, the GMM-DO filter is shown to fundamentally improve the\r\nfiltering skill, outperforming simpler schemes invoking the Gaussian parametric distribution.","protected":false},"excerpt":{"rendered":"<p>The properties and capabilities of the GMM-DO filter are assessed and exemplified by applications to two dynamical systems: (1) the Double Well Diffusion and (2) Sudden Expansion flows; both of which admit far-from-Gaussian statistics. The former test case, or twin experiment, validates the use of the EM algorithm and Bayesian Information Criterion with Gaussian Mixture [&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-1919","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\/1919","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=1919"}],"version-history":[{"count":13,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/1919\/revisions"}],"predecessor-version":[{"id":2673,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=\/wp\/v2\/posts\/1919\/revisions\/2673"}],"wp:attachment":[{"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1919"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1919"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mseas.mit.edu\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1919"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}