Increasingly, for many application areas, it is becoming important to include elements of nonlinearity and non-Gaussianity in order to model accurately the underlying dynamics of a physical system. Moreover, it is typically crucial to process data on-line as it arrives, both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics. In this paper, we review both optimal and suboptimal Bayesian algorithms for nonlinear/non-Gaussian tracking problems, with a focus on particle filters. Particle filters are sequential Monte Carlo methods based on point mass (or "particle") representations of probability densities, which can be applied to any state-space model and which generalize the traditional Kalman filtering methods. Several variants of the particle filter such as SIR, ASIR, and RPF are introduced within a generic framework of the sequential importance sampling (SIS) algorithm. These are discussed and compared with the standard EKF through an illustrative example.

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A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking

Any series of observations ordered along a single dimension, such as time, may be thought of as a time series. The emphasis in time series analysis is on studying the dependence among observations at different points in time. What distinguishes time series analysis from general multivariate analysis is precisely the temporal order imposed on the observations. Many economic variables, such as GNP and its components, price indices, sales, and stock returns are observed over time. In addition to being interested in the contemporaneous relationships among such variables, we are often concerned with relationships between their current and past values, that is, relationships over time.

Time Series Analysis

计量经济分析 = Econometric analysis

Monte Carlo methods are revolutionizing the on-line analysis of data in fields as diverse as financial modeling, target tracking and computer vision. These methods, appearing under the names of bootstrap filters, condensation, optimal Monte Carlo filters, particle filters and survival of the fittest, 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, including convergence results and applications to tracking, guidance, automated target recognition, aircraft navigation, robot navigation, econometrics, financial modeling, neural networks, optimal control, optimal filtering, communications, reinforcement learning, signal enhancement, model averaging and selection, computer vision, semiconductor design, population biology, dynamic Bayesian networks, and time series analysis. This will be of great value to students, researchers and practitioners, who have some basic knowledge of probability. Arnaud Doucet received the Ph. D. degree from the University of Paris-XI Orsay in 1997. From 1998 to 2000, he conducted research at the Signal Processing Group of Cambridge University, UK. He is currently an assistant professor at the Department of Electrical Engineering of Melbourne University, Australia. His research interests include Bayesian statistics, dynamic models and Monte Carlo methods. Nando de Freitas obtained a Ph.D. degree in information engineering from Cambridge University in 1999. He is presently a research associate with the artificial intelligence group of the University of California at Berkeley. His main research interests are in Bayesian statistics and the application of on-line and batch Monte Carlo methods to machine learning. Neil Gordon obtained a Ph.D. in Statistics from Imperial College, University of London in 1993. He is with the Pattern and Information Processing group at the Defence Evaluation and Research Agency in the United Kingdom. His research interests are in time series, statistical data analysis, and pattern recognition with a particular emphasis on target tracking and missile guidance.

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Sequential Monte Carlo methods in practice

In this article, we present an overview of methods for sequential simulation from posterior distributions. These methods are of particular interest in Bayesian filtering for discrete time dynamic models that are typically nonlinear and non-Gaussian. A general importance sampling framework is developed that unifies many of the methods which have been proposed over the last few decades in several different scientific disciplines. Novel extensions to the existing methods are also proposed. We show in particular how to incorporate local linearisation methods similar to those which have previously been employed in the deterministic filtering literatures these lead to very effective importance distributions. Furthermore we describe a method which uses Rao-Blackwellisation in order to take advantage of the analytic structure present in some important classes of state-space models. In a final section we develop algorithms for prediction, smoothing and evaluation of the likelihood in dynamic models.

/pdf/on-sequential-monte-carlo-sampling-methods-for-bayesian-3bit26usft.pdf

On sequential Monte Carlo sampling methods for Bayesian filtering

 The system allows access by UK based students to full-time undergraduate education in the UK irrespective of ability to pay, subject to getting a place at a university. This should be kept.  The economic basis of the current system: o Education creates positive externalities so education should be supported by the state; o Education creates private benefit so graduates should contribute towards the cost of their tuition; o Returns to education are highly uncertain so someone (e.g. the state) should provide insurance (income contingency) in case the graduate’s earnings turn out to be modest; makes sense. This should be kept.

Submission to the review on “Higher Education Funding and Student Finance”

In this paper we obtain a closed form expression for the convergence rate of the Gibbs sampler applied to an AR(1) plus noise model in terms of the parameters of the model. We also provide evidence that a ``centered'' parameterisation of a state space model is preferable for the performance of the Gibbs sampler. These two results provide guidance when the Gaussianity or linearity of the state space form is lost. We illustrate this by examining the performance of a Markov Chain Monte Carlo sampler for the Stochastic Volatility model.

Analytic convergence rates and parameterisation issues for the Gibbs sampler applied to state space models

Here we develop a method for performing nonparametric Bayesian inference on quantiles Relying on geometric measure theory and employing a Hausdorff base measure, we are able to specify meaningful priors for the quantile while treating the distribution of the data otherwise nonparametrically We further extend the method to a hierarchical model for quantiles of subpopulations, linking subgroups together solely through their quantiles Our approach is computationally straightforward, allowing for censored and noisy data We demonstrate the proposed methodology on simulated data and an applied problem from sports statistics, where it is observed to stabilize and improve inference and prediction

Nonparametric hierarchical Bayesian quantiles

This paper introduces a new class of multivariate volatility models that utilizes high-frequency data. We discuss the models' dynamics and highlight their differences from multivariate GARCH models. We also discuss their covariance targeting specification and provide closed-form formulas for multi-step forecasts. Estimation and inference strategies are outlined. Empirical results suggest that the HEAVY model outperforms the multivariate GARCH model out-of-sample, with the gains being particularly significant at short forecast horizons. Forecast gains are obtained for both forecast variances and correlations.

Neil Shephard

Papers

Submission to the review on “Higher Education Funding and Student Finance”

Analytic convergence rates and parameterisation issues for the Gibbs sampler applied to state space models

Nonparametric hierarchical Bayesian quantiles

Multivariate High-Frequency-Based Volatility (HEAVY) Models

Time Series Experiments and Causal Estimands