Category Archives: Bayesian Filtering

Kalman Filter as the extreme case of finite impulse response filters as the horizon increases in length

Shunyi Zhao, Biao Huang, Yuriy S. Shmaliy, Bayesian state estimation on finite horizons: The case of linear state–space model,Automatica, Volume 85, 2017, Pages 91-99, DOI: 10.1016/j.automatica.2017.07.043.

The finite impulse response (FIR) filter and infinite impulse response filter including the Kalman filter (KF) are generally considered as two different types of state estimation methods. In this paper, the sequential Bayesian philosophy is extended to a filter design using a fixed amount of most recent measurements, with the aim of exploiting the FIR structure and unifying some basic FIR filters with the KF. Specifically, the conditional mean and covariance of the posterior probability density functions are first derived to show the FIR counterpart of the KF. To remove the dependence on initial states, the corresponding likelihood is further maximized and realized iteratively. It shows that the maximum likelihood modification unifies the existing unbiased FIR filters by tuning a weighting matrix. Moreover, it converges to the Kalman estimate with the increase of horizon length, and can thus be considered as a link between the FIR filtering and the KF. Several important properties including stability and robustness against errors of noise statistics are illustrated. Finally, a moving target tracking example and an experiment with a three degrees-of-freedom helicopter system are introduced to demonstrate effectiveness.

Dealing with nonlinearities in Kalman filters through Monte Carlo modelling for minimizing divergence

S. Gultekin and J. Paisley, Nonlinear Kalman Filtering With Divergence Minimization, IEEE Transactions on Signal Processing, vol. 65, no. 23, pp. 6319-6331, DOI: 10.1109/TSP.2017.2752729.

We consider the nonlinear Kalman filtering problem using Kullback-Leibler (KL) and α-divergence measures as optimization criteria. Unlike linear Kalman filters, nonlinear Kalman filters do not have closed form Gaussian posteriors because of a lack of conjugacy due to the nonlinearity in the likelihood. In this paper, we propose novel algorithms to approximate this posterior by optimizing the forward and reverse forms of the KL divergence, as well as the α-divergence that contains these two as limiting cases. Unlike previous approaches, our algorithms do not make approximations to the divergences being optimized, but use Monte Carlo techniques to derive unbiased algorithms for direct optimization. We assess performance on radar and sensor tracking, and options pricing, showing general improvement over the extended, unscented, and ensemble Kalman filters, as well as competitive performance with particle filtering.

The problem of the interdependence among particles in PF after the resampling step, and an approach to solve it

R. Lamberti, Y. Petetin, F. Desbouvries and F. Septier, Independent Resampling Sequential Monte Carlo Algorithms, IEEE Transactions on Signal Processing, vol. 65, no. 20, pp. 5318-5333, DOI: 10.1109/TSP.2017.2726971.

Sequential Monte Carlo algorithms, or particle filters, are Bayesian filtering algorithms, which propagate in time a discrete and random approximation of the a posteriori distribution of interest. Such algorithms are based on importance sampling with a bootstrap resampling step, which aims at struggling against weight degeneracy. However, in some situations (informative measurements, high-dimensional model), the resampling step can prove inefficient. In this paper, we revisit the fundamental resampling mechanism, which leads us back to Rubin’s static resampling mechanism. We propose an alternative rejuvenation scheme in which the resampled particles share the same marginal distribution as in the classical setup, but are now independent. This set of independent particles provides a new alternative to compute a moment of the target distribution and the resulting estimate is analyzed through a CLT. We next adapt our results to the dynamic case and propose a particle filtering algorithm based on independent resampling. This algorithm can be seen as a particular auxiliary particle filter algorithm with a relevant choice of the first-stage weights and instrumental distributions. Finally, we validate our results via simulations, which carefully take into account the computational budget.

Varying the number of particles in a PF in order to improve the speed of convergence, with a short related work about adapting the number of particles for other goals

V. Elvira, J. Míguez and P. M. Djurić, “Adapting the Number of Particles in Sequential Monte Carlo Methods Through an Online Scheme for Convergence Assessment,” in IEEE Transactions on Signal Processing, vol. 65, no. 7, pp. 1781-1794, April1, 1 2017. DOI: 10.1109/TSP.2016.2637324.

Particle filters are broadly used to approximate posterior distributions of hidden states in state-space models by means of sets of weighted particles. While the convergence of the filter is guaranteed when the number of particles tends to infinity, the quality of the approximation is usually unknown but strongly dependent on the number of particles. In this paper, we propose a novel method for assessing the convergence of particle filters in an online manner, as well as a simple scheme for the online adaptation of the number of particles based on the convergence assessment. The method is based on a sequential comparison between the actual observations and their predictive probability distributions approximated by the filter. We provide a rigorous theoretical analysis of the proposed methodology and, as an example of its practical use, we present simulations of a simple algorithm for the dynamic and online adaptation of the number of particles during the operation of a particle filter on a stochastic version of the Lorenz 63 system.

A new Kalman Filter that is more robust under certain deviations of the gaussian hypothesis

Badong Chen, Xi Liu, Haiquan Zhao, Jose C. Principe, Maximum correntropy Kalman filter, Automatica, Volume 76, February 2017, Pages 70-77, ISSN 0005-1098, DOI: 10.1016/j.automatica.2016.10.004.

Traditional Kalman filter (KF) is derived under the well-known minimum mean square error (MMSE) criterion, which is optimal under Gaussian assumption. However, when the signals are non-Gaussian, especially when the system is disturbed by some heavy-tailed impulsive noises, the performance of KF will deteriorate seriously. To improve the robustness of KF against impulsive noises, we propose in this work a new Kalman filter, called the maximum correntropy Kalman filter (MCKF), which adopts the robust maximum correntropy criterion (MCC) as the optimality criterion, instead of using the MMSE. Similar to the traditional KF, the state mean vector and covariance matrix propagation equations are used to give prior estimations of the state and covariance matrix in MCKF. A novel fixed-point algorithm is then used to update the posterior estimations. A sufficient condition that guarantees the convergence of the fixed-point algorithm is also given. Illustration examples are presented to demonstrate the effectiveness and robustness of the new algorithm.

Bayesian estimation when computing the likelihood is hard

Kirthevasan Kandasamy, Jeff Schneider, Barnabás Póczos, Query efficient posterior estimation in scientific experiments via Bayesian active learning, Artificial Intelligence, Volume 243, February 2017, Pages 45-56, ISSN 0004-3702, DOI: 10.1016/j.artint.2016.11.002.

A common problem in disciplines of applied Statistics research such as Astrostatistics is of estimating the posterior distribution of relevant parameters. Typically, the likelihoods for such models are computed via expensive experiments such as cosmological simulations of the universe. An urgent challenge in these research domains is to develop methods that can estimate the posterior with few likelihood evaluations.In this paper, we study active posterior estimation in a Bayesian setting when the likelihood is expensive to evaluate. Existing techniques for posterior estimation are based on generating samples representative of the posterior. Such methods do not consider efficiency in terms of likelihood evaluations. In order to be query efficient we treat posterior estimation in an active regression framework. We propose two myopic query strategies to choose where to evaluate the likelihood and implement them using Gaussian processes. Via experiments on a series of synthetic and real examples we demonstrate that our approach is significantly more query efficient than existing techniques and other heuristics for posterior estimation.

Partially observed boolean dynamic systems

M. Imani and U. M. Braga-Neto, “Maximum-Likelihood Adaptive Filter for Partially Observed Boolean Dynamical Systems,” in IEEE Transactions on Signal Processing, vol. 65, no. 2, pp. 359-371, Jan.15, 15 2017.DOI: 10.1109/TSP.2016.2614798.

We present a framework for the simultaneous estimation of state and parameters of partially observed Boolean dynamical systems (POBDS). Simultaneous state and parameter estimation is achieved through the combined use of the Boolean Kalman filter and Boolean Kalman smoother, which provide the minimum mean-square error state estimators for the POBDS model, and maximum-likelihood (ML) parameter estimation; in the presence of continuous parameters, ML estimation is performed using the expectation-maximization algorithm. The performance of the proposed ML adaptive filter is demonstrated by numerical experiments with a POBDS model of gene regulatory networks observed through noisy next-generation sequencing (RNA-seq) time series data using the well-known p53-MDM2 negative-feedback loop gene regulatory model.

A variant of particle filters that uses feedback to model how particles move towards the real posterior

T. Yang, P.~G. Mehta, S.~P. Meyn, Feedback particle filter, IEEE Transactions on Automatic Control, 58 (10) (2013), pp. 2465â–2480, DOI: 10.1109/TAC.2013.2258825.

The feedback particle filter introduced in this paper is a new approach to approximate nonlinear filtering, motivated by techniques from mean-field game theory. The filter is defined by an ensemble of controlled stochastic systems (the particles). Each particle evolves under feedback control based on its own state, and features of the empirical distribution of the ensemble. The feedback control law is obtained as the solution to an optimal control problem, in which the optimization criterion is the Kullback-Leibler divergence between the actual posterior, and the common posterior of any particle. The following conclusions are obtained for diffusions with continuous observations: 1) The optimal control solution is exact: The two posteriors match exactly, provided they are initialized with identical priors. 2) The optimal filter admits an innovation error-based gain feedback structure. 3) The optimal feedback gain is obtained via a solution of an Euler-Lagrange boundary value problem; the feedback gain equals the Kalman gain in the linear Gaussian case. Numerical algorithms are introduced and implemented in two general examples, and a neuroscience application involving coupled oscillators. In some cases it is found that the filter exhibits significantly lower variance when compared to the bootstrap particle filter.

A gentle introduction to Box-Particle Filters

A. Gning, B. Ristic, L. Mihaylova and F. Abdallah, An Introduction to Box Particle Filtering [Lecture Notes], in IEEE Signal Processing Magazine, vol. 30, no. 4, pp. 166-171, July 2013. DOI: 10.1109/MSP.2013.225460.

Resulting from the synergy between the sequential Monte Carlo (SMC) method [1] and interval analysis [2], box particle filtering is an approach that has recently emerged [3] and is aimed at solving a general class of nonlinear filtering problems. This approach is particularly appealing in practical situations involving imprecise stochastic measurements that result in very broad posterior densities. It relies on the concept of a box particle that occupies a small and controllable rectangular region having a nonzero volume in the state space. Key advantages of the box particle filter (box-PF) against the standard particle filter (PF) are its reduced computational complexity and its suitability for distributed filtering. Indeed, in some applications where the sampling importance resampling (SIR) PF may require thousands of particles to achieve accurate and reliable performance, the box-PF can reach the same level of accuracy with just a few dozen box particles. Recent developments [4] also show that a box-PF can be interpreted as a Bayes? filter approximation allowing the application of box-PF to challenging target tracking problems [5].

Using the Bingham distribution of probability, which is defined on a d-dimensional sphere to be antipodally symmetric, to address the problem of angle periodicity in [0,2pi] when estimating orientation in a recursive filter

Gilitschenski, I.; Kurz, G.; Julier, S.J.; Hanebeck, U.D., Unscented Orientation Estimation Based on the Bingham Distribution, in Automatic Control, IEEE Transactions on , vol.61, no.1, pp.172-177, Jan. 2016, DOI: 10.1109/TAC.2015.2423831.

In this work, we develop a recursive filter to estimate orientation in 3D, represented by quaternions, using directional distributions. Many closed-form orientation estimation algorithms are based on traditional nonlinear filtering techniques, such as the extended Kalman filter (EKF) or the unscented Kalman filter (UKF). These approaches assume the uncertainties in the system state and measurements to be Gaussian-distributed. However, Gaussians cannot account for the periodic nature of the manifold of orientations and thus small angular errors have to be assumed and ad hoc fixes must be used. In this work, we develop computationally efficient recursive estimators that use the Bingham distribution. This distribution is defined on the hypersphere and is inherently more suitable for periodic problems. As a result, these algorithms are able to consistently estimate orientation even in the presence of large angular errors. Furthermore, handling of nontrivial system functions is performed using an entirely deterministic method which avoids any random sampling. A scheme reminiscent of the UKF is proposed for the nonlinear manifold of orientations. It is the first deterministic sampling scheme that truly reflects the nonlinear manifold of orientations.