In this book, Andrew Harvey sets out to provide a unified and comprehensive theory of structural time series models. Unlike the traditional ARIMA models, structural time series models consist explicitly of unobserved components, such as trends and seasonals, which have a direct interpretation. As a result the model selection methodology associated with structural models is much closer to econometric methodology. The link with econometrics is made even closer by the natural way in which the models can be extended to include explanatory variables and to cope with multivariate time series. From the technical point of view, state space models and the Kalman filter play a key role in the statistical treatment of structural time series models. The book includes a detailed treatment of the Kalman filter. This technique was originally developed in control engineering, but is becoming increasingly important in fields such as economics and operations research. This book is concerned primarily with modelling economic and social time series, and with addressing the special problems which the treatment of such series poses. The properties of the models and the methodological techniques used to select them are illustrated with various applications. These range from the modellling of trends and cycles in US macroeconomic time series to to an evaluation of the effects of seat belt legislation in the UK.

Forecasting, Structural Time Series Models and the Kalman Filter

On robust estimation of the location parameter

Filtering and smoothing methods are used to produce an accurate estimate of the state of a time-varying system based on multiple observational inputs (data). Interest in these methods has exploded in recent years, with numerous applications emerging in fields such as navigation, aerospace engineering, telecommunications, and medicine. This compact, informal introduction for graduate students and advanced undergraduates presents the current state-of-the-art filtering and smoothing methods in a unified Bayesian framework. Readers learn what non-linear Kalman filters and particle filters are, how they are related, and their relative advantages and disadvantages. They also discover how state-of-the-art Bayesian parameter estimation methods can be combined with state-of-the-art filtering and smoothing algorithms. The book’s practical and algorithmic approach assumes only modest mathematical prerequisites. Examples include MATLAB computations, and the numerous end-of-chapter exercises include computational assignments. MATLAB/GNU Octave source code is available for download at www.cambridge.org/sarkka, promoting hands-on work with the methods.

贝叶斯滤波与平滑 (Bayesian filtering and smoothing)

The paper proposes Metropolis adjusted Langevin and Hamiltonian Monte Carlo sampling methods defined on the Riemann manifold to resolve the shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations. The methods provide fully automated adaptation mechanisms that circumvent the costly pilot runs that are required to tune proposal densities for Metropolis-Hastings or indeed Hamiltonian Monte Carlo and Metropolis adjusted Langevin algorithms. This allows for highly efficient sampling even in very high dimensions where different scalings may be required for the transient and stationary phases of the Markov chain. The methodology proposed exploits the Riemann geometry of the parameter space of statistical models and thus automatically adapts to the local structure when simulating paths across this manifold, providing highly efficient convergence and exploration of the target density. The performance of these Riemann manifold Monte Carlo methods is rigorously assessed by performing inference on logistic regression models, log-Gaussian Cox point processes, stochastic volatility models and Bayesian estimation of dynamic systems described by non-linear differential equations. Substantial improvements in the time-normalized effective sample size are reported when compared with alternative sampling approaches. MATLAB code that is available from http://www.ucl.ac.uk/statistics/research/rmhmc allows replication of all the results reported.

Riemann manifold Langevin and Hamiltonian Monte Carlo methods

It is now over a decade since the pioneering contribution of Gordon (1993), which is commonly regarded as the first instance of modern sequential Monte Carlo (SMC) approaches. Initially focussed on applications to tracking and vision, these techniques are now very widespread and have had a significant impact in virtually all areas of signal and image processing concerned with Bayesian dynamical models. This paper is intended to serve both as an introduction to SMC algorithms for nonspecialists and as a reference to recent contributions in domains where the techniques are still under significant development, including smoothing, estimation of fixed parameters and use of SMC methods beyond the standard filtering contexts.

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An Overview of Existing Methods and Recent Advances in Sequential Monte Carlo

Recent developments have demonstrated that particle filtering is an emerging and powerful methodology, using Monte Carlo methods, for sequential signal processing with a wide range of applications in science and engineering. It has captured the attention of many researchers in various communities, including those of signal processing, statistics and econometrics. Based on the concept of sequential importance sampling and the use of Bayesian theory, particle filtering is particularly useful in dealing with difficult nonlinear and non-Gaussian problems. The underlying principle of the methodology is the approximation of relevant distributions with random measures composed of particles (samples from the space of the unknowns) and their associated weights. First, we present a brief review of particle filtering theory; and then we show how it can be used for resolving many problems in wireless communications. We demonstrate its application to blind equalization, blind detection over flat fading channels, multiuser detection, and estimation and detection of space-time codes in fading channels.

Particle filtering

We present particle filtering algorithms for tracking a single target using data from binary sensors. The sensors transmit signals that identify them to a central unit if the target is in their neighborhood; otherwise they do not transmit anything. The central unit uses a model for the target movement in the sensor field and estimates the target's trajectory, velocity, and power using the received data. We propose and implement the tracking by employing auxiliary particle filtering and cost-reference particle filtering. Unlike auxiliary particle filtering, cost-reference particle filtering does not rely on any probabilistic assumptions about the dynamic system. In the paper, we also extend the method to include estimation of constant parameters, and we derive the posterior Cramer-Rao bounds (PCRBs) for the states. We show the performances of the proposed methods by extensive computer simulations and compare them to the derived bounds.

http://www.ece.sunysb.edu/~monica/Publications_files/journal08b.pdf

Target Tracking by Particle Filtering in Binary Sensor Networks

A fundamental problem in signal processing is the estimation of unknown parameters or functions from noisy observations. Important examples include localization of objects in wireless sensor networks [1] and the Internet of Things [2]; multiple source reconstruction from electroencephalograms [3]; estimation of power spectral density for speech enhancement [4]; or inference in genomic signal processing [5]. Within the Bayesian signal processing framework, these problems are addressed by constructing posterior probability distributions of the unknowns. The posteriors combine optimally all of the information about the unknowns in the observations with the information that is present in their prior probability distributions. Given the posterior, one often wants to make inference about the unknowns, e.g., if we are estimating parameters, finding the values that maximize their posterior or the values that minimize some cost function given the uncertainty of the parameters. Unfortunately, obtaining closed-form solutions to these types of problems is infeasible in most practical applications, and therefore, developing approximate inference techniques is of utmost interest.

Adaptive Importance Sampling: The past, the present, and the future

Particle filtering is a sequential signal processing methodology that uses discrete random measures composed of particles and weights to approximate probability distributions of interest. The quality of approximation depends on many factors including the number of particles used for filtering and the way new particles are generated by the filter. The problem of good approximation becomes increasingly challenging as the dimension of the state space increases. In this paper, we address a possible solution for improved particle filtering in high dimensional cases by using a set of particle filters operating on partitioned subspaces of the complete state space. We provide simulation results that show the feasibility of the proposed approach.

Multiple Particle Filtering

In recent years, particle filtering has become a powerful tool for tracking signals and time-varying parameters of random dynamic systems. These methods require a mathematical representation of the dynamics of the system evolution, together with assumptions of probabilistic models. In this paper, we present a new class of particle filtering methods that do not assume explicit mathematical forms of the probability distributions of the noise in the system. As a consequence, the proposed techniques are simpler, more robust, and more flexible than standard particle filters. Apart from the theoretical development of specific methods in the new class, we provide computer simulation results that demonstrate the performance of the algorithms in the problem of autonomous positioning of a vehicle in a 2-dimensional space.

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Monica F. Bugallo

Papers

Particle filtering

Target Tracking by Particle Filtering in Binary Sensor Networks

Adaptive Importance Sampling: The past, the present, and the future

Multiple Particle Filtering

A new class of particle filters for random dynamic systems with unknown statistics