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A central problem in numerous applications is estimating the unknown parameters of a system of ordinary differential equations (ODEs) from noisy measurements of a function of some of the states at discrete times. Formulating this dynamic inverse problem in a Bayesian statistical framework, state and parameter estimation can be performed using sequential Monte Carlo (SMC) methods, such as particle filters (PFs) and ensemble Kalman filters (EnKFs).Addressing the issue of particle retention in PF-SMC, we propose to solve ODE systems within a PF framework with higher order numerical integrators which can handle stiffness and to base the choice of the innovation variance on estimates of discretization errors. Using linear multistep method (LMM) numerical solvers in this context gives a handle on the stability and accuracy of propagation, and provides a natural and systematic way to rigorously estimate the innovation variance via well-known local error estimates.We explore computationally efficient implementations of LMM PF-SMC by considering parallelized and vectorized formulations. While PF algorithms are known to be amenable to parallelization due to the independent propagation of each particle, by formulating the problem in a vectorized fashion, it is possible to arrive at an implementation of the method which takes full advantage of multiple processors.We employ a variation of LMM PF-SMC in estimating unknown parameters of a tracer kinetics model from sequences of real positron emission tomography scan data. A combination of optimization and statistical inference is utilized: nonlinear least squares finds optimal starting values, which then act as hyperparameters in the Bayesian framework. The LMM PF-SMC algorithm is modified to allow variable time steps to accommodate the increase in time interval length between data measurements from beginning to end of the procedure, keeping the time step the same for each particle.We also apply the idea of linking innovation variance with numerical integration error estimates to EnKFs by employing a stochastic interpretation of the discretization error in numerical integrators, extending the technique to deterministic, large-scale nonlinear evolution models. The resulting algorithm, which introduces LMM time integrators into the EnKF framework, proves especially effective in predicting unmeasured system components.
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.
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.
A unified Bayesian treatment of the state-of-the-art filtering, smoothing, and parameter estimation algorithms for non-linear state space models.
In various scientific and industrial fields, stochastic simulations are taking on a new importance. This is due to the increasing power of computers and practitioners’ aim to simulate more and more complex systems, and thus use random parameters as well as random noises to model the parametric uncertainties and the lack of knowledge on the physics of these systems. The error analysis of these computations is a highly complex mathematical undertaking. Approaching these issues, the authors present stochastic numerical methods and prove accurate convergence rate estimates in terms of their numerical parameters (number of simulations, time discretization steps). As a result, the book is a self-contained and rigorous study of the numerical methods within a theoretical framework. After briefly reviewing the basics, the authors first introduce fundamental notions in stochastic calculus and continuous-time martingale theory, then develop the analysis of pure-jump Markov processes, Poisson processes, and stochastic differential equations. In particular, they review the essential properties of Itô integrals and prove fundamental results on the probabilistic analysis of parabolic partial differential equations. These results in turn provide the basis for developing stochastic numerical methods, both from an algorithmic and theoretical point of view. The book combines advanced mathematical tools, theoretical analysis of stochastic numerical methods, and practical issues at a high level, so as to provide optimal results on the accuracy of Monte Carlo simulations of stochastic processes. It is intended for master and Ph.D. students in the field of stochastic processes and their numerical applications, as well as for physicists, biologists, economists and other professionals working with stochastic simulations, who will benefit from the ability to reliably estimate and control the accuracy of their simulations.