Simo Särkkä

On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems

This paper considers the application of the unscented Kalman filter (UKF) to continuous-time filtering problems, where both the state and measurement processes are modeled as stochastic differential equations. The mean and covariance differential equations which result in the continuous-time limit of the UKF are derived. The continuous-discrete UKF is derived as a special case of the continuous-time filter, when the continuous-time prediction equations are combined with the update step of the discrete-time UKF. The filter equations are also transformed into sigma-point differential equations, which can be interpreted as matrix square root versions of the filter equations.

Rao-Blackwellized particle filter for multiple target tracking

In this article we propose a new Rao-Blackwellized particle filtering based algorithm for tracking an unknown number of targets. The algorithm is based on formulating probabilistic stochastic process models for target states, data associations, and birth and death processes. The tracking of these stochastic processes is implemented using sequential Monte Carlo sampling or particle filtering, and the efficiency of the Monte Carlo sampling is improved by using Rao-Blackwellization.

Unscented Rauch--Tung--Striebel Smoother

This note considers the application of the unscented transform to optimal smoothing of nonlinear state-space models. In this note, a new Rauch-Tung-Striebel type form of the fixed-interval unscented Kalman smoother is derived. The new smoother differs from the previously proposed two-filter-formulation-based unscented Kalman smoother in the sense that it is not based on running two independent filters forward and backward in time. Instead, a separate backward smoothing pass is used, which recursively computes corrections to the forward filtering result. The smoother equations are derived as approximations to the formal Bayesian optimal smoothing equations. The performance of the new smoother is demonstrated with a simulation.

Recursive Noise Adaptive Kalman Filtering by Variational Bayesian Approximations

This article considers the application of variational Bayesian methods to joint recursive estimation of the dynamic state and the time-varying measurement noise parameters in linear state space models. The proposed adaptive Kalman filtering method is based on forming a separable variational approximation to the joint posterior distribution of states and noise parameters on each time step separately. The result is a recursive algorithm, where on each step the state is estimated with Kalman filter and the sufficient statistics of the noise variances are estimated with a fixed-point iteration. The performance of the algorithm is demonstrated with simulated data.

Phase-Based UHF RFID Tracking With Nonlinear Kalman Filtering and Smoothing

In this paper, we present an UHF RFID location tracking system, which is based on measuring the phases of back scattered signals from RFID tag using multiple spatially distributed antennas at a single carrier frequency. The wavelength ambiguity of the phase measurements is resolved by using the extended Kalman filter (EKF) and the Rauch-Tung-Striebel (RTS) smoother, where the state includes the position, velocity and the phase offsets of antennas. The performance of the method is experimentally verified at 890 MHz using a commercially available RFID reader.

Kalman filtering and smoothing solutions to temporal Gaussian process regression models

In this paper, we show how temporal (i.e., time-series) Gaussian process regression models in machine learning can be reformulated as linear-Gaussian state space models, which can be solved exactly with classical Kalman filtering theory. The result is an efficient non-parametric learning algorithm, whose computational complexity grows linearly with respect to number of observations. We show how the reformulation can be done for Matérn family of covariance functions analytically and for squared exponential covariance function by applying spectral Taylor series approximation. Advantages of the proposed approach are illustrated with two numerical experiments.

On Gaussian Optimal Smoothing of Non-Linear State Space Models

In this note we shall present a new Gaussian approximation based framework for approximate optimal smoothing of non-linear stochastic state space models. The approximation framework can be used for efficiently solving non-linear fixed-interval, fixed-point and fixed-lag optimal smoothing problems. We shall also numerically compare accuracies of approximations, which are based on Taylor series expansion, unscented transformation, central differences and Gauss-Hermite quadrature.

Spatiotemporal Learning via Infinite-Dimensional Bayesian Filtering and Smoothing: A Look at Gaussian Process Regression Through Kalman Filtering

Gaussian process-based machine learning is a powerful Bayesian paradigm for nonparametric nonlinear regression and classification. In this article, we discuss connections of Gaussian process regression with Kalman filtering and present methods for converting spatiotemporal Gaussian process regression problems into infinite-dimensional state-space models. This formulation allows for use of computationally efficient infinite-dimensional Kalman filtering and smoothing methods, or more general Bayesian filtering and smoothing methods, which reduces the problematic cubic complexity of Gaussian process regression in the number of time steps into linear time complexity. The implication of this is that the use of machine-learning models in signal processing becomes computationally feasible, and it opens the possibility to combine machine-learning techniques with signal processing methods.

Dynamic retrospective filtering of physiological noise in BOLD fMRI: DRIFTER

In this article we introduce the DRIFTER algorithm, which is a new model based Bayesian method for retrospective elimination of physiological noise from functional magnetic resonance imaging (fMRI) data. In the method, we first estimate the frequency trajectories of the physiological signals with the interacting multiple models (IMM) filter algorithm. The frequency trajectories can be estimated from external reference signals, or if the temporal resolution is high enough, from the fMRI data. The estimated frequency trajectories are then used in a state space model in combination of a Kalman filter (KF) and Rauch-Tung-Striebel (RTS) smoother, which separates the signal into an activation related cleaned signal, physiological noise, and white measurement noise components. Using experimental data, we show that the method outperforms the RETROICOR algorithm if the shape and amplitude of the physiological signals change over time.

Gaussian filtering and smoothing for continuous-discrete dynamic systems

This paper is concerned with Bayesian optimal filtering and smoothing of non-linear continuous-discrete state space models, where the state dynamics are modeled with non-linear Ito-type stochastic differential equations, and measurements are obtained at discrete time instants from a non-linear measurement model with Gaussian noise. We first show how the recently developed sigma-point approximations as well as the multi-dimensional Gauss-Hermite quadrature and cubature approximations can be applied to classical continuous-discrete Gaussian filtering. We then derive two types of new Gaussian approximation based smoothers for continuous-discrete models and apply the numerical methods to the smoothers. We also show how the latter smoother can be efficiently implemented by including one additional cross-covariance differential equation to the filter prediction step. The performance of the methods is tested in a simulated application.