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Data assimilation is the process of estimating the state of dynamic systems (linear or nonlinear, Gaussian or non-Gaussian) as accurately as possible from noisy observational data. Although the Three Dimensional Variational (3D-VAR) methods, Four Dimensional Variational (4D-VAR) methods and Ensemble Kalman filter (EnKF) methods are widely used and effective for linear and Gaussian dynamics, new methods of data assimilation are required for the general situation, that is, nonlinear non-Gaussian dynamics. General Bayesian recursive estimation theory is reviewed in this thesis. The Bayesian estimation approach provides a rather general and powerful framework for handling nonlinear, non-Gaussian, as well as linear, Gaussian estimation problems. Despite a general solution to the nonlinear estimation problem, there is no closed-form solution in the general case. Therefore, approximate techniques have to be employed. In this thesis, the sequential Monte Carlo (SMC) methods, commonly referred to as the particle filter, is presented to tackle non-linear, non-Gaussian estimation problems. In this thesis, we use the SMC methods only for the nonlinear state estimation problem, however, it can also be used for the nonlinear parameter estimation problem. In order to demonstrate the new methods in the general nonlinear non-Gaussian case, we compare Sequential Monte Carlo (SMC) methods with the Ensemble Kalman Filter (EnKF) by performing data assimilation in nonlinear and non-Gaussian dynamic systems. The models used in this study are referred to as state-space models. The Lorenz 1963 and 1966 models serve as test beds for examining the properties of these assimilation methods when used in highly nonlinear dynamics. The application of Sequential Monte Carlo methods to different fixed parameters in dynamic models is considered. Four different scenarios in the Lorenz 1063 [sic] model and three different scenarios in the Lorenz 1996 model are designed in this study for both the SMC methods and EnKF method with different filter size from 50 to 1000. The comparison results show that the SMC methods have theoretical advantages and also work well with highly nonlinear Lorenz models for state estimation in practice. Although Ensemble Kalman Filter (EnKF) computes only the mean and the variance of the state, which is based on linear state-space models with Gaussian noise, the SMC methods do not outperform EnKF in practical applications as we expected in theoretical insights. The main drawback of Sequential Monte Carlo (SMC) methods is that it requires much computational power, which is the obstacle to extend SMC methods to high dimensional atmospheric and oceanic models. We try to interpret the reason why the SMC data assimilation result is similar to the EnKF data assimilation result in these experiments and discuss the potential future application for high dimensional realistic atmospheric and oceanic models in this thesis.
In these notes, we introduce particle filtering as a recursive importance sampling method that approximates the minimum-mean-square-error (MMSE) estimate of a sequence of hidden state vectors in scenarios where the joint probability distribution of the states and the observations is non-Gaussian and, therefore, closed-form analytical expressions for the MMSE estimate are generally unavailable. We begin the notes with a review of Bayesian approaches to static (i.e., time-invariant) parameter estimation. In the sequel, we describe the solution to the problem of sequential state estimation in linear, Gaussian dynamic models, which corresponds to the well-known Kalman (or Kalman-Bucy) filter. Finally, we move to the general nonlinear, non-Gaussian stochastic filtering problem and present particle filtering as a sequential Monte Carlo approach to solve that problem in a statistically optimal way. We review several techniques to improve the performance of particle filters, including importance function optimization, particle resampling, Markov Chain Monte Carlo move steps, auxiliary particle filtering, and regularized particle filtering. We also discuss Rao-Blackwellized particle filtering as a technique that is particularly well-suited for many relevant applications such as fault detection and inertial navigation. Finally, we conclude the notes with a discussion on the emerging topic of distributed particle filtering using multiple processors located at remote nodes in a sensor network. Throughout the notes, we often assume a more general framework than in most introductory textbooks by allowing either the observation model or the hidden state dynamic model to include unknown parameters. In a fully Bayesian fashion, we treat those unknown parameters also as random variables. Using suitable dynamic conjugate priors, that approach can be applied then to perform joint state and parameter estimation. Table of Contents: Introduction / Bayesian Estimation of Static Vectors / The Stochastic Filtering Problem / Sequential Monte Carlo Methods / Sampling/Importance Resampling (SIR) Filter / Importance Function Selection / Markov Chain Monte Carlo Move Step / Rao-Blackwellized Particle Filters / Auxiliary Particle Filter / Regularized Particle Filters / Cooperative Filtering with Multiple Observers / Application Examples / Summary
In the study of data assimilation, people focus on estimating state variables and parameters of dynamical models, and make predictions forward in time, using given observations. It is a method that has been applied to many different fields, such as numerical weather prediction and neurobiology. To make successful estimations and predictions using data assimilation methods, there are a few difficulties that are often encountered. First is the quantity and quality of the data. In some of the typical problems in data assimilation, the number of observations are usually a few order of magnitude smaller than the number of total variables. Considering this and the fact that almost all the data gathered are noisy, how to estimate the observed and unobserved state variables and make good predictions using the noisy and incomplete data is one of the key challenge in data assimilation. Another issue arises from the dynamical model. Most of the interesting models are non-linear, and usually chaotic, which means that a small error in the estimation will grow exponentially over time. This property of the chaotic system addresses the necessity of accurate estimations of variables. In this thesis, I will start with an overview of data assimilation, by formulating the problem that data assimilation tries to solve, and introducing several widely used methods. Then I will explain the Precision Annealing Monte Carlo method that has been developed in the group, as well as its variation using Hamiltonian Monte Carlo. Finally I will demonstrate a few example problems that can be solved using data assimilation methods, varying from a simple but instructional 20-dimension Lorenz 96 model, to a complicated ocean model named Regional Ocean Modeling System.
Monte Carlo methods are revolutionizing the on-line analysis of data in many fileds. They have made it possible to solve numerically many complex, non-standard problems that were previously intractable. This book presents the first comprehensive treatment of these techniques.
This book contains two review articles on nonlinear data assimilation that deal with closely related topics but were written and can be read independently. Both contributions focus on so-called particle filters. The first contribution by Jan van Leeuwen focuses on the potential of proposal densities. It discusses the issues with present-day particle filters and explorers new ideas for proposal densities to solve them, converging to particle filters that work well in systems of any dimension, closing the contribution with a high-dimensional example. The second contribution by Cheng and Reich discusses a unified framework for ensemble-transform particle filters. This allows one to bridge successful ensemble Kalman filters with fully nonlinear particle filters, and allows a proper introduction of localization in particle filters, which has been lacking up to now.
Data assimilation (DA) has been recognized as one of the core techniques for modern forecasting in various earth science disciplines including meteorology, oceanography, and hydrology. Since early 1990s DA has been an important s- sion topic in many academic meetings organized by leading societies such as the American Meteorological Society, American Geophysical Union, European G- physical Union, World Meteorological Organization, etc. nd Recently, the 2 Annual Meeting of the Asia Oceania Geosciences Society (AOGS), held in Singapore in June 2005, conducted a session on DA under the - tle of “Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications.” nd This rst DA session in the 2 AOGS was a great success with more than 30 papers presented and many great ideas exchanged among scientists from the three different disciplines. The scientists who participated in the meeting suggested making the DA session a biennial event. th Two years later, at the 4 AOGS Annual Meeting, Bangkok, Thailand, the DA session was of cially named “Sasaki Symposium on Data Assimilation for At- spheric, Oceanic and Hydrologic Applications,” to honor Prof. Yoshi K. Sasaki of the University of Oklahoma for his life-long contributions to DA in geosciences.
This book reviews popular data-assimilation methods, such as weak and strong constraint variational methods, ensemble filters and smoothers. The author shows how different methods can be derived from a common theoretical basis, as well as how they differ or are related to each other, and which properties characterize them, using several examples. Readers will appreciate the included introductory material and detailed derivations in the text, and a supplemental web site.
Data Assimilation (DA) is a method through which information is extracted from measured quantities and with the help of a mathematical model is transferred through a probability distribution to unknown or unmeasured states and parameters characterizing the system of study. With an estimate of the model paramters, quantitative predictions may be made and compared to subsequent data. Many recent DA efforts rely on an probability distribution optimization that locates the most probable state and parameter values given a set of data. The procedure developed and demonstrated here extends the optimization by appending a biased random walk around the states and parameters of high probability to generate an estimate of the structure in state space of the probability density function (PDF). The estimate of the structure of the PDF will facilitate more accurate estimates of expectation values of means, standard deviations and higher moments of states and parameters that characterize the behavior of the system of study. The ability to calculate these expectation values will allow for an error bar or tolerance interval to be attached to each estimated state or parameter, in turn giving significance to any results generated. The estimation method's merits will be demonstrated on a simulated well known chaotic system, the Lorenz 96 system, and on a toy model of a neuron. In both situations the model system provides unique challenges for estimation: In chaotic systems any small error in estimation generates extremely large prediction errors while in neurons only one of the (at minimum) four dynamical variables can be measured leading to a small amount of data with which to work. This thesis will conclude with an exploration of the equivalence of machine learning and the formulation of statistical DA. The application of previous DA methods are demonstrated on the classic machine learning problem: the characterization of handwritten images from the MNIST data set. The results of this work are used to validate common assumptions in machine learning work such as the dependence of the quality of results on the amount of data presented and the size of the network used. Finally DA is proposed as a method through which to discern an 'ideal' network size for a set of given data which optimizes predictive capabilities while minimizing computational costs.
This book provides a general introduction to Sequential Monte Carlo (SMC) methods, also known as particle filters. These methods have become a staple for the sequential analysis of data in such diverse fields as signal processing, epidemiology, machine learning, population ecology, quantitative finance, and robotics. The coverage is comprehensive, ranging from the underlying theory to computational implementation, methodology, and diverse applications in various areas of science. This is achieved by describing SMC algorithms as particular cases of a general framework, which involves concepts such as Feynman-Kac distributions, and tools such as importance sampling and resampling. This general framework is used consistently throughout the book. Extensive coverage is provided on sequential learning (filtering, smoothing) of state-space (hidden Markov) models, as this remains an important application of SMC methods. More recent applications, such as parameter estimation of these models (through e.g. particle Markov chain Monte Carlo techniques) and the simulation of challenging probability distributions (in e.g. Bayesian inference or rare-event problems), are also discussed. The book may be used either as a graduate text on Sequential Monte Carlo methods and state-space modeling, or as a general reference work on the area. Each chapter includes a set of exercises for self-study, a comprehensive bibliography, and a “Python corner,” which discusses the practical implementation of the methods covered. In addition, the book comes with an open source Python library, which implements all the algorithms described in the book, and contains all the programs that were used to perform the numerical experiments.
This book reviews popular data-assimilation methods, such as weak and strong constraint variational methods, ensemble filters and smoothers. The author shows how different methods can be derived from a common theoretical basis, as well as how they differ or are related to each other, and which properties characterize them, using several examples. Readers will appreciate the included introductory material and detailed derivations in the text, and a supplemental web site.