Saikat Saha

Marginalized adaptive particle filtering for nonlinear models with unknown time-varying noise parameters

Knowledge of the noise distribution is typically crucial for the state estimation of general state-space models. However, properties of the noise process are often unknown in the majority of practical applications. The distribution of the noise may also be non-stationary or state dependent and that prevents the use of off-line tuning methods. For linear Gaussian models, Adaptive Kalman filters (AKF) estimate unknown parameters in the noise distributions jointly with the state. For nonlinear models, we provide a Bayesian solution for the estimation of the noise distributions in the exponential family, leading to a marginalized adaptive particle filter (MAPF) where the noise parameters are updated using finite dimensional sufficient statistics for each particle. The time evolution model for the noise parameters is defined implicitly as a Kullback-Leibler norm constraint on the time variability, leading to an exponential forgetting mechanism operating on the sufficient statistics. Many existing methods are based on the standard approach of augmenting the state with the unknown variables and attempting to solve the resulting filtering problem. The MAPF is significantly more computationally efficient than a comparable particle filter that runs on the full augmented state. Further, the MAPF can handle sensor and actuator offsets as unknown means in the noise distributions, avoiding the standard approach of augmenting the state with such offsets. We illustrate the MAPF on first a standard example, and then on a tire radius estimation problem on real data.

Gaussian proposal density using moment matching in SMC methods

In this article we introduce a new Gaussian proposal distribution to be used in conjunction with the sequential Monte Carlo (SMC) method for solving non-linear filtering problems. The proposal, in line with the recent trend, incorporates the current observation. The introduced proposal is characterized by the exact moments obtained from the dynamical system. This is in contrast with recent works where the moments are approximated either numerically or by linearizing the observation model. We show further that the newly introduced proposal performs better than other similar proposal functions which also incorporate both state and observations.

Particle Filtering With Dependent Noise Processes

Modeling physical systems often leads to discrete time state-space models with dependent process and measurement noises. For linear Gaussian models, the Kalman filter handles this case, as is well described in literature. However, for nonlinear or non-Gaussian models, the particle filter as described in literature provides a general solution only for the case of independent noise. Here, we present an extended theory of the particle filter for dependent noises with the following key contributions: i) The optimal proposal distribution is derived; ii) the special case of Gaussian noise in nonlinear models is treated in detail, leading to a concrete algorithm that is as easy to implement as the corresponding Kalman filter; iii) the marginalized (Rao-Blackwellized) particle filter, handling linear Gaussian substructures in the model in an efficient way, is extended to dependent noise; and, finally, iv) the parameters of a joint Gaussian distribution of the noise processes are estimated jointly with the state in a recursive way.

Marginalized particle filters for Bayesian estimation of Gaussian noise parameters

The particle filter provides a general solution to the nonlinear filtering problem with arbitrarily accuracy. However, the curse of dimensionality prevents its application in cases where the state dimensionality is high. Further, estimation of stationary parameters is a known challenge in a particle filter framework. We suggest a marginalization approach for the case of unknown noise distribution parameters that avoid both aforementioned problem. First, the standard approach of augmenting the state vector with sensor offsets and scale factors is avoided, so the state dimension is not increased. Second, the mean and covariance of both process and measurement noises are represented with parametric distributions, whose statistics are updated adaptively and analytically using the concept of conjugate prior distributions. The resulting marginalized particle filter is applied to and illustrated with a standard example from literature.

Particle filtering with dependent noise

The theory and applications of the particle filter (PF) have developed tremendously during the past two decades. However, there appear to be no version of the PF readily applicable to the case of dependent process and measurement noise. This is in contrast to the Kalman filter, where the case of correlated noise is a standard modification. Further, the fact that sampling continuous time models give dependent noise processes is an often neglected fact in literature. We derive the optimal proposal distribution in the PF for general and Gaussian noise processes, respectively. The main result is a modified prediction step. It is demonstrated that the original Bootstrap particle filter gets a particular simple and explicit form for dependent Gaussian noise. Finally, the practical importance of dependent noise is motivated in terms of sampling of continuous time models.

Estimating volatility and model parameters of stochastic volatility models with jumps using particle filter

A bearing protector employing a stator and a rotor defining a labyrinth therebetween and disposed for sealing cooperation between a housing and a rotatable shaft. The concentrically fit within one another due to relative axial movement therebetween, each is constructed as an integral continuous one-piece ring. The stator and rotor are axially secured in an assembled condition by a one-piece snap ring carried by the stator and which resiliently distorts to pass axially over the rotor and then resiliently snaps inwardly into a groove in the rotor so as to become fixed to and rotate with the rotor. This snap ring projects outwardly into a radially enlarged annular collecting chamber formed in the stator and acts, when the rotor rotates, as a slinger ring so that contaminates which enter into the labyrinth from the outside come into contact with the slinger ring and are thrown outwardly to the bottom of the collecting chamber for discharge through a drain.

Rao-Blackwellized particle filter for Markov modulated nonlinear dynamic systems

The Markov modulated (switching) state space is an important model paradigm in statistical signal processing. In this article, we specifically consider Markov modulated nonlinear state-space models and address the online Bayesian inference problem for such models. In particular, we propose a new Rao-Blackwellized particle filter for the inference task which is our main contribution here. A detailed description of the problem and an algorithm is presented.

Particle Based Smoothed Marginal MAP Estimation for General State Space Models

We consider the smoothing problem for a general state space system using sequential Monte Carlo (SMC) methods. The marginal smoother is assumed to be available in the form of weighted random particles from the SMC output. New algorithms are developed to extract the smoothed marginal maximum a posteriori (MAP) estimate of the state from the existing marginal particle smoother. Our method does not need any kernel fitting to obtain the posterior density from the particle smoother. The proposed estimator is then successfully applied to find the unknown initial state of a dynamical system and to address the issue of parameter estimation problem in state space models.

Non-parametric bayesian measurement noise density estimation in non-linear filtering

In this study, we investigate online Bayesian estimation of the measurement noise density of a given state space model using particle filters and Dirichlet process mixtures. Dirichlet processes are widely used in statistics for nonparametric density estimation. In the proposed method, the unknown noise is modeled as a Gaussian mixture with unknown number of components. The joint estimation of the state and the noise density is done via particle filters. Furthermore, the number of components and the noise statistics are allowed to vary in time. An extension of the method for the estimation of time varying noise characteristics is also introduced.

Exact Moment Matching for Efficient Importance Functions in SMC Methods

In this article we introduce a new proposal distribution to be used in conjunction with the sequential Monte Carlo (SMC) method of solving non-linear filtering problem. The proposal distribution incorporates all the information about the to be estimated current state form both the available state and observation processes. This makes it more effective than the state transition density which is more commonly used but ignores the recent observation. Because of its Gaussian nature it is also very easy to implement. We show further that the introduced proposal performs better than other similar importance functions which also incorporate both state and observations.