Particle Learning and Smoothing

Particle learning (PL) provides state filtering, sequential parameter learning and smoothing in a general class of state space models. Our approach extends existing particle methods by incorporating the estimation of static parameters via a fully-adapted filter that utilizes conditional sufficient statistics for parameters and/or states as particles. State smoothing in the presence of parameter uncertainty is also solved as a by-product of PL. In a number of examples, we show that PL outperforms existing particle filtering alternatives and proves to be a competitor to MCMC.Comment: Published in at http://dx.doi.org/10.1214/10-STS325 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Particle Filtering and Smoothing Using Windowed Rejection Sampling

"Particle methods" are sequential Monte Carlo algorithms, typically involving importance sampling, that are used to estimate and sample from joint and marginal densities from a collection of a, presumably increasing, number of random variables. In particular, a particle filter aims to estimate the current state $X_{n}$ of a stochastic system that is not directly observable by estimating a posterior distribution $\pi(x_{n}|y_{1},y_{2}, \ldots, y_{n})$ where the $\{Y_{n}\}$ are observations related to the $\{X_{n}\}$ through some measurement model $\pi(y_{n}|x_{n})$. A particle smoother aims to estimate a marginal distribution $\pi(x_{i}|y_{1},y_{2}, \ldots, y_{n})$ for $1 \leq i < n$. Particle methods are used extensively for hidden Markov models where $\{X_{n}\}$ is a Markov chain as well as for more general state space models. Existing particle filtering algorithms are extremely fast and easy to implement. Although they suffer from issues of degeneracy and "sample impoverishment", steps can be taken to minimize these problems and overall they are excellent tools for inference. However, if one wishes to sample from a posterior distribution of interest, a particle filter is only able to produce dependent draws. Particle smoothing algorithms are complicated and far less robust, often requiring cumbersome post-processing, "forward-backward" recursions, and multiple passes through subroutines. In this paper we introduce an alternative algorithm for both filtering and smoothing that is based on rejection sampling "in windows" . We compare both speed and accuracy of the traditional particle filter and this "windowed rejection sampler" (WRS) for several examples and show that good estimates for smoothing distributions are obtained at no extra cost

Topics in particle filtering and smoothing

Particle filtering/smoothing is a relatively new promising class of algorithms\ud to deal with the estimation problems in nonlinear and/or non-\ud Gaussian systems. Currently, this is a very active area of research and\ud there are many issues that are not either properly addressed or are still\ud open.\ud One of the key issues in particle filtering is a suitable choice of the\ud importance function. The optimal importance function which includes the\ud information from the most recent observation, is difficult to obtain in most\ud practical situations. In this thesis, we present a new Gaussian approximation\ud to this optimal importance function using the moment matching\ud method and compare it with some other recently proposed importance\ud functions.\ud In particle filtering/smoothing, the posterior is represented as a weighted\ud particle cloud. We develop a new algorithm for extracting the smoothed\ud marginal maximum a posteriori (MAP) estimate from the available particle\ud cloud of the marginal smoother, generated using either the forwardbackward\ud smoother or the two filter smoother. The smoothed marginal\ud MAP estimator is then applied to estimate the unknown initial state of a\ud dynamic system.\ud There are many approaches to deal with the unknown static system\ud parameters within particle filtering/smoothing set up. One common approach\ud is to model the parameters as a part of the state vector. This is\ud followed by adding artificial process noises to this model and then estimate\ud the parameters along with the other state variables. Although this\ud approach may work well in (certain) practical situations, the added process\ud noises may result in a unnecessary loss of accuracy of the estimated\ud parameters. Here we propose some new particle filtering/smoothing based\ud algorithms, where we avoid any effect of the artificial dynamics on the\ud estimate of the parameters

Variance estimation for Sequential Monte Carlo Algorithms: a backward sampling approach

In this paper, we consider the problem of online asymptotic variance estimation for particle filtering and smoothing. Current solutions for the particle filter rely on the particle genealogy and are either unstable or hard to tune in practice. We propose to mitigate these limitations by introducing a new estimator of the asymptotic variance based on the so called backward weights. The resulting estimator is weakly consistent and trades computational cost for more stability and reduced variance. We also propose a more computationally efficient estimator inspired by the PaRIS algorithm of Olsson & Westerborn. As an application, particle smoothing is considered and an estimator of the asymptotic variance of the Forward Filtering Backward Smoothing estimator applied to additive functionals is provided.Comment: preprin

Nonlinear state space smoothing using the conditional particle filter

To estimate the smoothing distribution in a nonlinear state space model, we apply the conditional particle filter with ancestor sampling. This gives an iterative algorithm in a Markov chain Monte Carlo fashion, with asymptotic convergence results. The computational complexity is analyzed, and our proposed algorithm is successfully applied to the challenging problem of sensor fusion between ultra-wideband and accelerometer/gyroscope measurements for indoor positioning. It appears to be a competitive alternative to existing nonlinear smoothing algorithms, in particular the forward filtering-backward simulation smoother.Comment: Accepted for the 17th IFAC Symposium on System Identification (SYSID), Beijing, China, October 201

Evaluating Structural Models for the U.S. Short Rate Using EMM and Particle Filters

We combine the efficient method of moments with appropriate algorithms from the optimal filtering literature to study a collection of models for the U.S. short rate. Our models include two continuous-time stochastic volatility models and two regime switching models, which provided the best fit in previous work that examined a large collection of models. The continuous-time stochastic volatility models fall into the class of nonlinear, non-Gaussian state space models for which we apply particle filtering and smoothing algorithms. Our results demonstrate the effectiveness of the particle filter for continuous-time processes. Our analysis also provides an alternative and complementary approach to the reprojection technique of Gallant and Tauchen (1998) for studying the dynamics of volatility.