Approximate Gaussian conjugacy: parametric recursive filtering under nonlinearity, multimodality, uncertainty, and constraint, and beyond

Since the landmark work of R. E. Kalman in the 1960s, considerable efforts have been devoted to time series state space models for a large variety of dynamic estimation problems. In particular, parametric filters that seek analytical estimates based on a closed-form Markov–Bayes recursion, e.g., recursion from a Gaussian or Gaussian mixture (GM) prior to a Gaussian/GM posterior (termed ‘Gaussian conjugacy’ in this paper), form the backbone for a general time series filter design. Due to challenges arising from nonlinearity, multimodality (including target maneuver), intractable uncertainties (such as unknown inputs and/or non-Gaussian noises) and constraints (including circular quantities), etc., new theories, algorithms, and technologies have been developed continuously to maintain such a conjugacy, or to approximate it as close as possible. They had contributed in large part to the prospective developments of time series parametric filters in the last six decades. In this paper, we review the state of the art in distinctive categories and highlight some insights that may otherwise be easily overlooked. In particular, specific attention is paid to nonlinear systems with an informative observation, multimodal systems including Gaussian mixture posterior and maneuvers, and intractable unknown inputs and constraints, to fill some gaps in existing reviews and surveys. In addition, we provide some new thoughts on alternatives to the first-order Markov transition model and on filter evaluation with regard to computing complexity

Robust Filtering and Smoothing with Gaussian Processes

We propose a principled algorithm for robust Bayesian filtering and smoothing in nonlinear stochastic dynamic systems when both the transition function and the measurement function are described by non-parametric Gaussian process (GP) models. GPs are gaining increasing importance in signal processing, machine learning, robotics, and control for representing unknown system functions by posterior probability distributions. This modern way of "system identification" is more robust than finding point estimates of a parametric function representation. In this article, we present a principled algorithm for robust analytic smoothing in GP dynamic systems, which are increasingly used in robotics and control. Our numerical evaluations demonstrate the robustness of the proposed approach in situations where other state-of-the-art Gaussian filters and smoothers can fail.Comment: 7 pages, 1 figure, draft version of paper accepted at IEEE Transactions on Automatic Contro

Particle filtering for EEG source localization and constrained state spaces

Particle Filters (PFs) have a unique ability to perform asymptotically optimal estimation for non-linear and non-Gaussian state-space models. However, the numerical nature of PFs cause them to have major weakness in two important areas: (1) handling constraints on the state, and (2) dealing with high-dimensional states. In the first area, handling constraints within the PF framework is crucial in dynamical systems, which are often required to satisfy constraints that arise from basic physical laws or other considerations. The current trend in constrained particle filtering is to enforce the constraints on all particles of the PF. We show that this approach leads to more stringent conditions on the posterior density that can cause incorrect state estimates. We subsequently describe a novel algorithm that restricts the mean estimate without restricting the posterior pdf, thus providing a more accurate state estimate. In the second area, we tackle the curse of dimensionality, which causes the PF to require an exponential increase in computational complexity as the dimension of the state increases. The application of interest is localization of the brain neural generators that create the Electroencephalogram (EEG) signal. Specifically, we describe a state-space model that tracks the position and moments of multiple dynamic dipoles and apply the marginalized PF, which alleviates the curse of dimensionality for tracking multiple dynamic dipoles. This modified framework allows us to consider dynamic dipoles, which were historically considered time-invariant

State Estimation Based on Nested Particle Filters

In reality many processes are nonlinear and in order to have a knowledge about the true process conditions, it is important to make decisions based on the state of the system. Process measurements such as pressure, temperature, and pH, are available at time instances and this information is necessary in order to obtain the state of the system. Filtering is a state estimation technique by which the estimate is obtained at a time instant, given the process measurements at their respective time instances. Several lters have been developed so far for the estimation of the states of the system. Kalman lters are the optimal lter algorithms used for linear state and measurement models. Approximations are made to this algorithm in order to apply to non-linear systems. Particle lter (PF) is one such approximation made to the Kalman ltering technique. It involves drawing a set of samples or particles from the state of the system. It works on the principle of importance sampling, where, the samples are derived from a probability density function which is similar to the state model. The particles are resampled according to their weights in order to determine the estimate. Taking into account the diculties in particle ltering technique, a nested particles lter (NPF) was developed. NPF works in such a way that there is a set of particles under each sample of the original particle lter, and from these nest of samples the transition prior is updated using an extended Kalman particle lter (EKPF). The idea of nested particle lter was developed from the unscented particles lter (UPF), which uses the concept of local linearization to develop the importance density. Better importance densities are formulated in this case through which better posteriori are obtained. It is important to note that the update of the NPF can be done with any suboptimal nonlinear lter available. This thesis work is based on developing the NPF with a direct sampling particle lter (DSPF) update. Some modications are made to the unscented particl

Bayesian inference for dynamic pose estimation using directional statistics

The dynamic pose of an object, where the object can represent a spacecraft, aircraft, or mobile robot, among other possibilities, is defined to be the position, velocity, attitude, and angular velocity of the object. A new method to perform dynamic pose estimation is developed that leverages directional statistics and operates under the Bayesian estimation framework, as opposed to the minimum mean square error (MMSE) framework that conventional methods employ. No small attitude uncertainty assumption is necessary using this method, and, therefore, a more accurate estimate of the state can be obtained when the attitude uncertainty is large. Two new state densities, termed the Gauss-Bingham and Bingham-Gauss mixture (BGM) densities, are developed that probabilistically represent a state vector comprised of an attitude quaternion and other Euclidean states on their natural manifold, the unit hypercylinder. When the Euclidean states consist of position, velocity, and angular velocity, the state vector represents the dynamic pose. An uncertainty propagation scheme is developed for a Gauss-Bingham-distributed state vector, and two demonstrations of this uncertainty propagation scheme are presented that show its applicability to quantify the uncertainty in dynamic pose, especially when the attitude uncertainty becomes large. The BGM filter is developed, which is an approximate Bayesian filter in which the true temporal and measurement evolution of the BGM density, as quantified by the Chapman-Kolmogorov equation and Bayes\u27 rule, are approximated by a BGM density. The parameters of the approximating BGM density are found via integral approximation on a component-wise basis, which is shown to be the Kullback-Leibler divergence optimal parameters of each component. The BGM filter is then applied to three simulations in order to compare its performance to a multiplicative Kalman filter and demonstrate its efficacy in estimating dynamic pose. The BGM filter is shown to be more statistically consistent than the multiplicative Kalman filter when the attitude uncertainty is large --Abstract, page iii

Distributed Particle Filtering over Sensor Networks for Autonomous Navigation of UAVs

State estimation and control over sensor networks is a problem met in several applications such as surveillance and condition monitoring of large-scale systems, multi-robot systems and cooperating UAVs. In sensor networks the simplest kind of architecture is centralized. Distributed sensors send measurement data to a central processing unit which provides th

Moving Horizon Estimation with Dynamic Programming

Moving Horizon Estimation(MHE) is a optimization based strategy to state estimation. It involves computation of arrival cost, a penalty term, based on the MHE cost function. Minimization of this arrival cost is done through various methods. All these methods use nonlinear programming optimization technique which gives the estimate. The main idea of MHE revolves around minimizing the estimation cost function. The cost function is dependent on prediction error computation from data and arrival cost summarization. The major issue that hampers the MHE is choosing the arrival cost for ensuring stability of the overall estimation and computational time. In order to attain this stability, this thesis incorporates dynamic programming algorithm to estimate MHE cost function. Dynamic programming is an algorithm for solving complex problems. The MHE cost function algorithm has been modied based on dynamic programming algorithm in order to ensure stability of the overall estimation. In order to apply this algorithm, a specic non-linear lter, particle lter is used for the initialization of MHE. The reason of using particle lter for initialization of MHE is due to fact that dynamic programming algorithm works on principle of samples and particle lter provides the samples. A comparison of mean squared error(MSE) using the nonlinear programming optimization and dynamic programming optimization is veried for the proposed theory of using dynamic programming algorithm in estimation of cost functio