{"id":210,"date":"2019-08-10T13:18:25","date_gmt":"2019-08-10T17:18:25","guid":{"rendered":"https:\/\/blogs.harvard.edu\/siams\/?p=210"},"modified":"2019-08-10T13:18:25","modified_gmt":"2019-08-10T17:18:25","slug":"notes-on-particle-filters-for-high-dimensional-geoscience-applications-a-review-van-leeuwen-et-al-2019","status":"publish","type":"post","link":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/08\/10\/notes-on-particle-filters-for-high-dimensional-geoscience-applications-a-review-van-leeuwen-et-al-2019\/","title":{"rendered":"Notes on “Particle filters for high dimensional geoscience applications: a review”. van Leeuwen et al 2019"},"content":{"rendered":"
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Notes on<\/p>\n

Peter Jan van Leeuwen,\u00a0Hans R. K\u00fcnsch,\u00a0<\/span>Lars Nerger,\u00a0<\/span>Roland Potthast,\u00a0<\/span>Sebastian Reich. \u00a0<\/span>Q J R <\/span>Meteorol Soc. 2019;1\u201331. \u00a0<\/span>Particle filters for high-dimensional geoscience applications: A review<\/span><\/p>\n

This paper is focusing on the problem of “weight degeneracy” for the weighting of particles in a particle filter.<\/p>\n

Introduction:<\/p>\n

“the linear data assimilation problem” is hard in a high dimensions. \u00a0Numerical weather prediction has 10^9 state variables and 10^7 observations every 6-12 hours. \u00a0Two currently methods are\u00a04DVar, Ensemble Kalman Filter (EnKF). \u00a0Hybrids of these evidently do okay but need “ad hoc fixes like localization and inflation”. \u00a0These methods are harder to use when there’s underlying advective flow, as well. \u00a0The linear problem is evidently hard, but actual problems are nonlinear, and these methods don’t work well.<\/p>\n

“Variational methods can easily fail when the cost function is multimodal, and are hampered by the assumption that the prior probability density function (pdf) of the state is assumed to be Gaussian.” \u00a0EnKFs also have a Gaussian prior. \u00a0<\/span><\/p>\n

Evidently particle filters don’t have assumptions on the prior or the likelihood. \u00a0(“Particle filters hold the promise of fully nonlinear data assimilation without any assumption on prior or likelihood, and recent textbooks like Reich and Cotter (2015), Nakamura and Potthast (2015), and van Leeuwen et al. (2015) provide useful introductions to data assimilation in general, and particle filters in particular.”)<\/span><\/p>\n

There are also MCMC methods that are fully nonlinear.<\/p>\n

Description of a particle filter (“standard or bootstrap”):<\/p>\n

    \n
  • choose N model states (“particles”, x_n-1). \u00a0These are sampled from the prior pdf.<\/li>\n
  • propagate the particles forward in time to the next observation time using the (nonlinear) model (x_n). \u00a0Include a random forcing with the propagation (there’s an assumption that physics is missing from the model and the random forcing compensates for that).<\/li>\n
  • there’s an observation (y_n) with random measurement errors at this time (with known characteristics)<\/li>\n
  • “assimilate” the observations. \u00a0Multiply the prior pdf by the likelihood (how likely the current observation is given a current model state, p(y_n | x_n): this hinges on our measurement error. \u00a0Is this measurement of y_n possible given the actual state was x_n? ). The product is proportional to p(x_n | y_n), the posterior pdf (the probability of each x_n given the observation that we have). \u00a0This step is using Bayes’ theorem.<\/li>\n
  • create a weight (related to the probability) for each particle proportional to p(y_n | x_n_i)<\/li>\n
  • resampling is common because weight might concentrate in just a few particles. \u00a0“This duplicates high weight particles and abandons low weight particles”. \u00a0“for particle tilts to work, we need to ensure that their weights remain similar”.<\/li>\n<\/ul>\n

    The review deals with “weight degeneracy”.<\/p>\n

      \n
    1. “proposal-density freedom” and “equal-weight particle filters”<\/li>\n
    2. one-step transformations from particles in the prior to particles in the posterior<\/li>\n
    3. use localization<\/li>\n
    4. combine particle filters with EnKFs<\/li>\n<\/ol>\n

       <\/p>\n<\/div>\n

      Terms to learn about:<\/p>\n

      4DVar, ensemble Kalman filter, cost function, likelihood, localization<\/p>\n

       <\/p>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

      Notes on Peter Jan van Leeuwen,\u00a0Hans R. K\u00fcnsch,\u00a0Lars Nerger,\u00a0Roland Potthast,\u00a0Sebastian Reich. \u00a0Q J R Meteorol Soc. 2019;1\u201331. \u00a0Particle filters for high-dimensional geoscience applications: A review This paper is focusing on the problem of “weight degeneracy” for the weighting of particles in a particle filter. Introduction: “the linear data assimilation problem” is hard in a high […]<\/p>\n","protected":false},"author":8032,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"jetpack_post_was_ever_published":false},"categories":[157891],"tags":[],"class_list":["post-210","post","type-post","status-publish","format-standard","hentry","category-research-paper"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7E5LF-3o","jetpack-related-posts":[{"id":217,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/09\/17\/notes-on-maybeck-stochastic-models-estimation-and-control\/","url_meta":{"origin":210,"position":0},"title":"Notes on Maybeck: Stochastic Models, Estimation, and Control","author":"siams","date":"17 September 2019","format":false,"excerpt":"Notes on Chapter 1 of Maybeck 1979,\u00a0Stochastic Models, Estimation, and Control. 1.1: why stochastic models, estimation, and control? A math model isn't perfect, and parameters are not known absolutely. \u00a0Sensors don't provide perfect data either. \u00a0Given uncertainties, you still want to be able to estimate quantities of interest and control\u2026","rel":"","context":"Similar post","block_context":{"text":"Similar post","link":""},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":193,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/07\/22\/notes-on-calculus-blue-volume-1-chapter-3-and-4\/","url_meta":{"origin":210,"position":1},"title":"Notes on Calculus Blue Volume 1, Chapters 3, 4, 5, 6","author":"siams","date":"22 July 2019","format":false,"excerpt":"More of Calculus Blue by Prof Ghrist Math. Chapter 3 is on coordinates and distance (see section 12.1 of the 6th edition of Hughes-Hallett). 01.03 (0:36) \"Coordinates: intro\". \u00a0Review coordinates and see it in data. 01.03.01 (2:16) \"coordinates & many dimensions\". \u00a0from curves and surfaces we'll head on. \u00a0plane, then\u2026","rel":"","context":"In "Math"","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":170,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/07\/10\/meiss-differential-dynamical-systems-chaos\/","url_meta":{"origin":210,"position":2},"title":"Meiss: Differential Dynamical Systems (chaos)","author":"siams","date":"10 July 2019","format":false,"excerpt":"I am reading James Meiss' text Differential Dynamical Systems (SIAM). \u00a0I am specifically interested in how he tells the story of chaos. In the Preface, he mentions the following: That\u00a0\u00a0Chapter 5 focuses on invariant manifolds: stable and unstable sets heteroclinic orbits stable manifolds local stable manifold theorem global stable manifolds\u2026","rel":"","context":"In "Dynamical Systems"","block_context":{"text":"Dynamical Systems","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/dynamical-systems\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":146,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/25\/courant-and-john-1965-differential-equations-chapter-9\/","url_meta":{"origin":210,"position":3},"title":"Courant (and John) 1965, Differential Equations: Chapter 9.","author":"siams","date":"25 June 2019","format":false,"excerpt":"In the intro to Chapter 9 they note that we've already seen differential equations in Chapter 3, p. 223, and on p.312, and in Chapter 4 (see p 405).\u00a0 So I'll start there. Section 3.4: First encounter: in \"Some Applications of the Exponential Function\", y' = ay is introduced.\u00a0 \"Since\u2026","rel":"","context":"In "Differential equations"","block_context":{"text":"Differential equations","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/differential-equations-math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":150,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/25\/varburg-and-purcell-7th-edition-differential-equations-mainly-chapter-18\/","url_meta":{"origin":210,"position":4},"title":"Varburg and Purcell 7th edition. Differential equations (mainly chapter 18)","author":"siams","date":"25 June 2019","format":false,"excerpt":"Section 5.2: What is a diff eq?\u00a0 Provides an example and two solution methods before defining diff eq (and doesn't define a solution...).\u00a0 Then presents separation of variables via an example.\u00a0 Then a falling body example and an escape velocity example. Section 7.5: exponential growth and decay.\u00a0 They motivate y'\u2026","rel":"","context":"In "Differential equations"","block_context":{"text":"Differential equations","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/differential-equations\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":220,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/09\/18\/notes-on-ljung-system-identification\/","url_meta":{"origin":210,"position":5},"title":"Notes on Ljung: System Identification","author":"siams","date":"18 September 2019","format":false,"excerpt":"Reading Ljung. \u00a0System Identification: theory for the user. 1: Introduction. Goal: infer a model from observations. \u00a0\"Model\" refers to the set of relationships between variables in the system. \u00a0System identification involves analyzing input and output signals from the system. Example: assume a linear difference equation relates inputs to outputs. \u00a0Use\u2026","rel":"","context":"Similar post","block_context":{"text":"Similar post","link":""},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]}],"_links":{"self":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/210","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/users\/8032"}],"replies":[{"embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/comments?post=210"}],"version-history":[{"count":3,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/210\/revisions"}],"predecessor-version":[{"id":213,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/210\/revisions\/213"}],"wp:attachment":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/media?parent=210"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/categories?post=210"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/tags?post=210"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}