arXiv subjects

Simo Särkkä

Publications and source records attributed to Simo Särkkä.

117 records · Page 7Linked to original sources

arXiv2015

Combining Particle MCMC with Rao-Blackwellized Monte Carlo Data Association for Parameter Estimation in Multiple Target Tracking

We consider state and parameter estimation in multiple target tracking problems with data association uncertainties and unknown number of targets. We show how the problem can be recast into a conditionally linear Gaussian state-space model with unknown parameters and present an algorithm for computationally efficient inference on the resulting model. The proposed algorithm is based on combining the Rao-Blackwellized Monte Carlo data association algorithm with particle Markov chain Monte Carlo algorithms to jointly estimate both parameters and data associations. Both particle marginal Metropolis-Hastings and particle Gibbs variants of particle MCMC are considered. We demonstrate the performance of the method both using simulated data and in a real-data case study of using multiple target tracking to estimate the brown bear population in Finland.

stat.ME↗

arXiv2014

Batch Nonlinear Continuous-Time Trajectory Estimation as Exactly Sparse Gaussian Process Regression

In this paper, we revisit batch state estimation through the lens of Gaussian process (GP) regression. We consider continuous-discrete estimation problems wherein a trajectory is viewed as a one-dimensional GP, with time as the independent variable. Our continuous-time prior can be defined by any nonlinear, time-varying stochastic differential equation driven by white noise; this allows the possibility of smoothing our trajectory estimates using a variety of vehicle dynamics models (e.g., `constant-velocity'). We show that this class of prior results in an inverse kernel matrix (i.e., covariance matrix between all pairs of measurement times) that is exactly sparse (block-tridiagonal) and that this can be exploited to carry out GP regression (and interpolation) very efficiently. When the prior is based on a linear, time-varying stochastic differential equation and the measurement model is also linear, this GP approach is equivalent to classical, discrete-time smoothing (at the measurement times); when a nonlinearity is present, we iterate over the whole trajectory to maximize accuracy. We test the approach experimentally on a simultaneous trajectory estimation and mapping problem using a mobile robot dataset.

cs.RO↗

arXiv2014

Sparse approximations of fractional Matérn fields

We consider a fast approximation method for a solution of a certain stochastic non-local pseudodifferential equation. This equation defines a Matérn class random field. The approximation method is based on the spectral compactness of the solution. We approximate the pseudodifferential operator with a Taylor expansion. By truncating the expansion, we can construct an approximation with Gaussian Markov random fields. We show that the solution of the truncated version can be constructed with an over-determined system of stochastic matrix equations with sparse matrices. We solve the system of equations with a sparse Cholesky decomposition. We consider the convergence of the discrete approximation of the solution to the continuous one. Finally numerical examples are given.

math.ST↗

arXiv2014

Adaptive Metropolis Algorithm Using Variational Bayesian Adaptive Kalman Filter

Markov chain Monte Carlo (MCMC) methods are powerful computational tools for analysis of complex statistical problems. However, their computational efficiency is highly dependent on the chosen proposal distribution, which is generally difficult to find. One way to solve this problem is to use adaptive MCMC algorithms which automatically tune the statistics of a proposal distribution during the MCMC run. A new adaptive MCMC algorithm, called the variational Bayesian adaptive Metropolis (VBAM) algorithm, is developed. The VBAM algorithm updates the proposal covariance matrix using the variational Bayesian adaptive Kalman filter (VB-AKF). A strong law of large numbers for the VBAM algorithm is proven. The empirical convergence results for three simulated examples and for two real data examples are also provided.

math.ST↗

arXiv2014

Moment Conditions for Convergence of Particle Filters with Unbounded Importance Weights

In this paper, we derive moment conditions for particle filter importance weights, which ensure that the particle filter estimates of the expectations of bounded Borel functions converge in mean square and $L^4$ sense, and that the empirical measure of the particle filter converges weakly to the true filtering measure. The result extends the previously derived conditions by not requiring the boundedness of the importance weights, but only boundedness of second or fourth order moments. We show that the boundedness of the second order moments of the weights implies the convergence of the estimates bounded functions in the mean square sense, and the $L^4$ convergence as well as the empirical measure convergence are assured by the boundedness of the fourth order moments of the weights. We also present an example class of models and importance distributions where the moment conditions hold, but the boundedness does not. The unboundedness in these models is caused by point-singularities in the weights which still leave the weight moments bounded. We show by using simulated data that the particle filter for this kind of model also performs well in practice.

math.DS↗

arXiv2014

Gaussian filtering and variational approximations for Bayesian smoothing in continuous-discrete stochastic dynamic systems

The Bayesian smoothing equations are generally intractable for systems described by nonlinear stochastic differential equations and discrete-time measurements. Gaussian approximations are a computationally efficient way to approximate the true smoothing distribution. In this work, we present a comparison between two Gaussian approximation methods. The Gaussian filtering based Gaussian smoother uses a Gaussian approximation for the filtering distribution to form an approximation for the smoothing distribution. The variational Gaussian smoother is based on minimizing the Kullback-Leibler divergence of the approximate smoothing distribution with respect to the true distribution. The results suggest that for highly nonlinear systems, the variational Gaussian smoother can be used to iteratively improve the Gaussian filtering based smoothing solution. We also present linearization and sigma-point methods to approximate the intractable Gaussian expectations in the Variational Gaussian smoothing equations. In addition, we extend the variational Gaussian smoother for certain class of systems with singular diffusion matrix.

math.DS↗

arXiv2014

Series Expansion Approximations of Brownian Motion for Non-Linear Kalman Filtering of Diffusion Processes

In this paper, we describe a novel application of sigma-point methods to continuous-discrete filtering. In principle, the nonlinear continuous- discrete filtering problem can be solved exactly. In practice, the solution contains terms that are computationally intractible. Assumed density filtering methods attempt to match statistics of the filtering distribution to some set of more tractible probability distributions. We describe a novel method that decomposes the Brownian motion driving the signal in a generalised Fourier series, which is truncated after a number of terms. This approximation to Brownian can be described using a relatively small number of Fourier coefficients, and allows us to compute statistics of the filtering distribution with a single application of a sigma-point method. Assumed density filters that exist in the literature usually rely on discretisation of the signal dynamics followed by iterated application of a sigma point transform (or a limiting case thereof). Iterating the transform in this manner can lead to loss of information about the filtering distri- bution in highly nonlinear settings. We demonstrate that our method is better equipped to cope with such problems.

stat.CO↗

arXiv2013

Infinite-dimensional Bayesian filtering for detection of quasi-periodic phenomena in spatio-temporal data

This paper introduces a spatio-temporal resonator model and an inference method for detection and estimation of nearly periodic temporal phenomena in spatio-temporal data. The model is derived as a spatial extension of a stochastic harmonic resonator model, which can be formulated in terms of a stochastic differential equation (SDE). The spatial structure is included by introducing linear operators, which affect both the oscillations and damping, and by choosing the appropriate spatial covariance structure of the driving time-white noise process. With the choice of the linear operators as partial differential operators, the resonator model becomes a stochastic partial differential equation (SPDE), which is compatible with infinite-dimensional Kalman filtering. The resulting infinite-dimensional Kalman filtering problem allows for a computationally efficient solution as the computational cost scales linearly with measurements in the temporal dimension. This framework is applied to weather prediction and to physiological noise elimination in fMRI brain data.

stat.CO↗

arXiv2008

Application of Girsanov Theorem to Particle Filtering of Discretely Observed Continuous-Time Non-Linear Systems

This article considers the application of particle filtering to continuous-discrete optimal filtering problems, where the system model is a stochastic differential equation, and noisy measurements of the system are obtained at discrete instances of time. It is shown how the Girsanov theorem can be used for evaluating the likelihood ratios needed in importance sampling. It is also shown how the methodology can be applied to a class of models, where the driving noise process is lower in the dimensionality than the state and thus the laws of state and noise are not absolutely continuous. Rao-Blackwellization of conditionally Gaussian models and unknown static parameter models is also considered.

stat.ME↗