Bayesian Inference of Stochastic Dynamical Models PDF Download

Author: Peter Guang Yi Lu
Publisher:
ISBN:
Category :
Languages : en
Pages : 175

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Book Description
A new methodology for Bayesian inference of stochastic dynamical models is developed. The methodology leverages the dynamically orthogonal (DO) evolution equations for reduced-dimension uncertainty evolution and the Gaussian mixture model DO filtering algorithm for nonlinear reduced-dimension state variable inference to perform parallelized computation of marginal likelihoods for multiple candidate models, enabling efficient Bayesian update of model distributions. The methodology also employs reduced-dimension state augmentation to accommodate models featuring uncertain parameters. The methodology is applied successfully to two high-dimensional, nonlinear simulated fluid and ocean systems. Successful joint inference of an uncertain spatial geometry, one uncertain model parameter, and [Omicron](105) uncertain state variables is achieved for the first. Successful joint inference of an uncertain stochastic dynamical equation and [Omicron](105) uncertain state variables is achieved for the second. Extensions to adaptive modeling and adaptive sampling are discussed.

Author: David Insua
Publisher: John Wiley & Sons
ISBN: 1118304039
Category : Mathematics
Languages : en
Pages : 315

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Book Description
Bayesian analysis of complex models based on stochastic processes has in recent years become a growing area. This book provides a unified treatment of Bayesian analysis of models based on stochastic processes, covering the main classes of stochastic processing including modeling, computational, inference, forecasting, decision making and important applied models. Key features: Explores Bayesian analysis of models based on stochastic processes, providing a unified treatment. Provides a thorough introduction for research students. Computational tools to deal with complex problems are illustrated along with real life case studies Looks at inference, prediction and decision making. Researchers, graduate and advanced undergraduate students interested in stochastic processes in fields such as statistics, operations research (OR), engineering, finance, economics, computer science and Bayesian analysis will benefit from reading this book. With numerous applications included, practitioners of OR, stochastic modelling and applied statistics will also find this book useful.

Author: Mike West
Publisher: Springer Science & Business Media
ISBN: 1475793650
Category : Mathematics
Languages : en
Pages : 720

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Book Description
In this book we are concerned with Bayesian learning and forecast ing in dynamic environments. We describe the structure and theory of classes of dynamic models, and their uses in Bayesian forecasting. The principles, models and methods of Bayesian forecasting have been developed extensively during the last twenty years. This devel opment has involved thorough investigation of mathematical and sta tistical aspects of forecasting models and related techniques. With this has come experience with application in a variety of areas in commercial and industrial, scientific and socio-economic fields. In deed much of the technical development has been driven by the needs of forecasting practitioners. As a result, there now exists a relatively complete statistical and mathematical framework, although much of this is either not properly documented or not easily accessible. Our primary goals in writing this book have been to present our view of this approach to modelling and forecasting, and to provide a rea sonably complete text for advanced university students and research workers. The text is primarily intended for advanced undergraduate and postgraduate students in statistics and mathematics. In line with this objective we present thorough discussion of mathematical and statistical features of Bayesian analyses of dynamic models, with illustrations, examples and exercises in each Chapter.

Author: Luc Bauwens
Publisher: Oxford University Press
ISBN: 0198773137
Category : Business & Economics
Languages : en
Pages : 370

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Book Description
This book contains an up-to-date coverage of the last twenty years advances in Bayesian inference in econometrics, with an emphasis on dynamic models. It shows how to treat Bayesian inference in non linear models, by integrating the useful developments of numerical integration techniques basedon simulations (such as Markov Chain Monte Carlo methods), and the long available analytical results of Bayesian inference for linear regression models. It thus covers a broad range of rather recent models for economic time series, such as non linear models, autoregressive conditionalheteroskedastic regressions, and cointegrated vector autoregressive models. It contains also an extensive chapter on unit root inference from the Bayesian viewpoint. Several examples illustrate the methods.

Author: Dani Gamerman
Publisher: CRC Press
ISBN: 9781584885870
Category : Mathematics
Languages : en
Pages : 352

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Book Description
While there have been few theoretical contributions on the Markov Chain Monte Carlo (MCMC) methods in the past decade, current understanding and application of MCMC to the solution of inference problems has increased by leaps and bounds. Incorporating changes in theory and highlighting new applications, Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition presents a concise, accessible, and comprehensive introduction to the methods of this valuable simulation technique. The second edition includes access to an internet site that provides the code, written in R and WinBUGS, used in many of the previously existing and new examples and exercises. More importantly, the self-explanatory nature of the codes will enable modification of the inputs to the codes and variation on many directions will be available for further exploration. Major changes from the previous edition: · More examples with discussion of computational details in chapters on Gibbs sampling and Metropolis-Hastings algorithms · Recent developments in MCMC, including reversible jump, slice sampling, bridge sampling, path sampling, multiple-try, and delayed rejection · Discussion of computation using both R and WinBUGS · Additional exercises and selected solutions within the text, with all data sets and software available for download from the Web · Sections on spatial models and model adequacy The self-contained text units make MCMC accessible to scientists in other disciplines as well as statisticians. The book will appeal to everyone working with MCMC techniques, especially research and graduate statisticians and biostatisticians, and scientists handling data and formulating models. The book has been substantially reinforced as a first reading of material on MCMC and, consequently, as a textbook for modern Bayesian computation and Bayesian inference courses.

Author: Kostas Triantafyllopoulos
Publisher: Springer Nature
ISBN: 303076124X
Category : Mathematics
Languages : en
Pages : 503

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Book Description
Bayesian Inference of State Space Models: Kalman Filtering and Beyond offers a comprehensive introduction to Bayesian estimation and forecasting for state space models. The celebrated Kalman filter, with its numerous extensions, takes centre stage in the book. Univariate and multivariate models, linear Gaussian, non-linear and non-Gaussian models are discussed with applications to signal processing, environmetrics, economics and systems engineering. Over the past years there has been a growing literature on Bayesian inference of state space models, focusing on multivariate models as well as on non-linear and non-Gaussian models. The availability of time series data in many fields of science and industry on the one hand, and the development of low-cost computational capabilities on the other, have resulted in a wealth of statistical methods aimed at parameter estimation and forecasting. This book brings together many of these methods, presenting an accessible and comprehensive introduction to state space models. A number of data sets from different disciplines are used to illustrate the methods and show how they are applied in practice. The R package BTSA, created for the book, includes many of the algorithms and examples presented. The book is essentially self-contained and includes a chapter summarising the prerequisites in undergraduate linear algebra, probability and statistics. An up-to-date and complete account of state space methods, illustrated by real-life data sets and R code, this textbook will appeal to a wide range of students and scientists, notably in the disciplines of statistics, systems engineering, signal processing, data science, finance and econometrics. With numerous exercises in each chapter, and prerequisite knowledge conveniently recalled, it is suitable for upper undergraduate and graduate courses.

Author: Jing Lin (Ph. D.)
Publisher:
ISBN:
Category :
Languages : en
Pages : 567

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Book Description
The rapidly-growing computational power and the increasing capability of uncertainty quantification, statistical inference, and machine learning have opened up new opportunities for utilizing data to assist, identify and refine physical models. In this thesis, we focus on Bayesian learning for a particular class of models: high-dimensional nonlinear dynamical systems, which have been commonly used to predict a wide range of transient phenomena including fluid flows, heat transfer, biogeochemical dynamics, and other advection-diffusion-reaction-based transport processes. Even though such models often express the differential form of fundamental laws, they commonly contain uncertainty in their initial and boundary values, parameters, forcing and even formulation. Learning such components from sparse observation data by principled Bayesian inference is very challenging due to the systems’ high-dimensionality and nonlinearity. We systematically study the theoretical and algorithmic properties of a Bayesian learning methodology built upon previous efforts in our group to address this challenge. Our systematic study breaks down into the three hierarchical components of the Bayesian learning and we develop new numerical schemes for each. The first component is on uncertainty quantification for stochastic dynamical systems and fluid flows. We study dynamic low-rank approximations using the dynamically orthogonal (DO) equations including accuracy and computational costs, and develop new numerical schemes for re-orthonormalization, adaptive subspace augmentation, residual-driven closure, and stochastic Navier-Stokes integration. The second part is on Bayesian data assimilation, where we study the properties of and connections among the different families of nonlinear and non-Gaussian filters. We derive an ensemble square-root filter based on minimal-correction second-moment matching that works especially well under the adversity of small ensemble size, sparse observations and chaotic dynamics. We also obtain a localization technique for filtering with high-dimensional systems that can be applied to nonlinear non-Gaussian inference with both brute force Monte Carlo (MC) and reduced subspace modeling in a unified way. Furthermore, we develop a mutual-information-based adaptive sampling strategy for filtering to identify the most informative observations with respect to the state variables and/or parameters, utilizing the sub-modularity of mutual information due to the conditional independence of observation noise. The third part is on active Bayesian model learning, where we have a discrete set of candidate dynamical models and we infer the model formulation that best explains the data using principled Bayesian learning. To predict the observations that are most useful to learn the model formulation, we further extend the above adaptive sampling strategy to identify the data that are expected to be most informative with respect to both state variables and the uncertain model identity. To investigate and showcase the effectiveness and efficiency of our theoretical and numerical advances for uncertainty quantification, Bayesian data assimilation, and active Bayesian learning with stochastic nonlinear high-dimensional dynamical systems, we apply our dynamic data-driven reduced subspace approach to several dynamical systems and compare our results against those of brute force MC and other existing methods. Specifically, we analyze our advances using several drastically different dynamical regimes modeled by the nonlinear Lorenz-96 ordinary differential equations as well as turbulent bottom gravity current dynamics modeled by the 2-D unsteady incompressible Reynolds-averaged Navier-Stokes (RANS) partial differential equations. We compare the accuracy, efficiency, and robustness of different methodologies and algorithms. With the Lorenz- 96 system, we show how the performance differs under periodic, weakly chaotic, and very chaotic dynamics and under different observation layouts. With the bottom gravity current dynamics, we show how model parameters, domain geometries, initial fields, and boundary forcing formulations can be identified and how the Bayesian methodology performs when the candidate model space does not contain the true model. The results indicate that our active Bayesian learning framework can better infer the state variables and dynamical model identity with fewer observations than many alternative approaches in the literature.

Author: Dani Gamerman
Publisher: CRC Press
ISBN: 9780412818202
Category : Mathematics
Languages : en
Pages : 264

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Book Description
Bridging the gap between research and application, Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference provides a concise, and integrated account of Markov chain Monte Carlo (MCMC) for performing Bayesian inference. This volume, which was developed from a short course taught by the author at a meeting of Brazilian statisticians and probabilists, retains the didactic character of the original course text. The self-contained text units make MCMC accessible to scientists in other disciplines as well as statisticians. It describes each component of the theory in detail and outlines related software, which is of particular benefit to applied scientists.

Author: Michail D. Vrettas
Publisher:
ISBN:
Category :
Languages : en
Pages :

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Book Description
This thesis is concerned with approximate inference in dynamical systems, from a variational Bayesian perspective. When modelling real world dynamical systems, stochastic differential equations appear as a natural choice, mainly because of their ability to model the noise of the system by adding a variant of some stochastic process to the deterministic dynamics. Hence, inference in such processes has drawn much attention. Here two new extended frameworks are derived and presented that are based on basis function expansions and local polynomial approximations of a recently proposed variational Bayesian algorithm. It is shown that the new extensions converge to the original variational algorithm and can be used for state estimation (smoothing). However, the main focus is on estimating the (hyper- ) parameters of these systems (i.e. drift parameters and diffusion coefficients). The new methods are numerically validated on a range of different systems which vary in dimensionality and non-linearity. These are the Ornstein-Uhlenbeck process, for which the exact likelihood can be computed analytically, the univariate and highly non-linear, stochastic double well and the multivariate chaotic stochastic Lorenz '63 (3-dimensional model). The algorithms are also applied to the 40 dimensional stochastic Lorenz '96 system. In this investigation these new approaches are compared with a variety of other well known methods such as the ensemble Kalman filter / smoother, a hybrid Monte Carlo sampler, the dual unscented Kalman filter (for jointly estimating the systems states and model parameters) and full weak-constraint 4D-Var. Empirical analysis of their asymptotic behaviour as a function of observation density or length of time window increases is provided.

Author: Christos H Skiadas
Publisher: World Scientific
ISBN: 9814474479
Category : Mathematics
Languages : en
Pages : 669

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Book Description
This volume presents the most recent applied and methodological issues in stochastic modeling and data analysis. The contributions cover various fields such as stochastic processes and applications, data analysis methods and techniques, Bayesian methods, biostatistics, econometrics, sampling, linear and nonlinear models, networks and queues, survival analysis, and time series. The volume presents new results with potential for solving real-life problems and provides novel methods for solving these problems by analyzing the relevant data. The use of recent advances in different fields is emphasized, especially new optimization and statistical methods, data warehouse, data mining and knowledge systems, neural computing, and bioinformatics.