Increasingly, for many application areas, it is becoming important to include elements of nonlinearity and non-Gaussianity in order to model accurately the underlying dynamics of a physical system. Moreover, it is typically crucial to process data on-line as it arrives, both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics. In this paper, we review both optimal and suboptimal Bayesian algorithms for nonlinear/non-Gaussian tracking problems, with a focus on particle filters. Particle filters are sequential Monte Carlo methods based on point mass (or "particle") representations of probability densities, which can be applied to any state-space model and which generalize the traditional Kalman filtering methods. Several variants of the particle filter such as SIR, ASIR, and RPF are introduced within a generic framework of the sequential importance sampling (SIS) algorithm. These are discussed and compared with the standard EKF through an illustrative example.

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A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking

The extended Kalman filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, more than 35 years of experience in the estimation community has shown that is difficult to implement, difficult to tune, and only reliable for systems that are almost linear on the time scale of the updates. Many of these difficulties arise from its use of linearization. To overcome this limitation, the unscented transformation (UT) was developed as a method to propagate mean and covariance information through nonlinear transformations. It is more accurate, easier to implement, and uses the same order of calculations as linearization. This paper reviews the motivation, development, use, and implications of the UT.

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Unscented filtering and nonlinear estimation

In this paper, we present a new nonlinear filter for high-dimensional state estimation, which we have named the cubature Kalman filter (CKF) The heart of the CKF is a spherical-radial cubature rule, which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear Bayesian filter Specifically, we derive a third-degree spherical-radial cubature rule that provides a set of cubature points scaling linearly with the state-vector dimension The CKF may therefore provide a systematic solution for high-dimensional nonlinear filtering problems The paper also includes the derivation of a square-root version of the CKF for improved numerical stability The CKF is tested experimentally in two nonlinear state estimation problems In the first problem, the proposed cubature rule is used to compute the second-order statistics of a nonlinearly transformed Gaussian random variable The second problem addresses the use of the CKF for tracking a maneuvering aircraft The results of both experiments demonstrate the improved performance of the CKF over conventional nonlinear filters

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Cubature Kalman Filters

From the Publisher:
Kalman filtering is a well-established topic in the field of control and signal processing and represents by far the most refined method for the design of neural networks. This book takes a nontraditional nonlinear approach and reflects the fact that most practical applications are nonlinear. The book deals with important applications in such fields as control, financial forecasting, and idle speed control.

Kalman Filtering and Neural Networks

The development of the Internet in recent years has made it possible and useful to access many different information systems anywhere in the world to obtain information. While there is much research on the integration of heterogeneous information systems, most commercial systems stop short of the actual integration of available data. Data fusion is the process of fusing multiple records representing the same real-world object into a single, consistent, and clean representation.This article places data fusion into the greater context of data integration, precisely defines the goals of data fusion, namely, complete, concise, and consistent data, and highlights the challenges of data fusion, namely, uncertain and conflicting data values. We give an overview and classification of different ways of fusing data and present several techniques based on standard and advanced operators of the relational algebra and SQL. Finally, the article features a comprehensive survey of data integration systems from academia and industry, showing if and how data fusion is performed in each.

Data fusion

This paper points out the flaws in using the extended Kalman filter (EKE) and introduces an improvement, the unscented Kalman filter (UKF), proposed by Julier and Uhlman (1997). A central and vital operation performed in the Kalman filter is the propagation of a Gaussian random variable (GRV) through the system dynamics. In the EKF the state distribution is approximated by a GRV, which is then propagated analytically through the first-order linearization of the nonlinear system. This can introduce large errors in the true posterior mean and covariance of the transformed GRV, which may lead to sub-optimal performance and sometimes divergence of the filter. The UKF addresses this problem by using a deterministic sampling approach. The state distribution is again approximated by a GRV, but is now represented using a minimal set of carefully chosen sample points. These sample points completely capture the true mean and covariance of the GRV, and when propagated through the true nonlinear system, captures the posterior mean and covariance accurately to the 3rd order (Taylor series expansion) for any nonlinearity. The EKF in contrast, only achieves first-order accuracy. Remarkably, the computational complexity of the UKF is the same order as that of the EKF. Julier and Uhlman demonstrated the substantial performance gains of the UKF in the context of state-estimation for nonlinear control. Machine learning problems were not considered. We extend the use of the UKF to a broader class of nonlinear estimation problems, including nonlinear system identification, training of neural networks, and dual estimation problems. In this paper, the algorithms are further developed and illustrated with a number of additional examples.

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The unscented Kalman filter for nonlinear estimation

Over the last 20-30 years, the extended Kalman filter (EKF) has become the algorithm of choice in numerous nonlinear estimation and machine learning applications. These include estimating the state of a nonlinear dynamic system as well estimating parameters for nonlinear system identification (eg, learning the weights of a neural network). The EKF applies the standard linear Kalman filter methodology to a linearization of the true nonlinear system. This approach is sub-optimal, and can easily lead to divergence. Julier et al. (1997), proposed the unscented Kalman filter (UKF) as a derivative-free alternative to the extended Kalman filter in the framework of state estimation. This was extended to parameter estimation by Wan and Van der Merwe et al., (2000). The UKF consistently outperforms the EKF in terms of prediction and estimation error, at an equal computational complexity of (OL/sup 3/)/sup l/ for general state-space problems. When the EKF is applied to parameter estimation, the special form of the state-space equations allows for an O(L/sup 2/) implementation. This paper introduces the square-root unscented Kalman filter (SR-UKF) which is also O(L/sup 3/) for general state estimation and O(L/sup 2/) for parameter estimation (note the original formulation of the UKF for parameter-estimation was O(L/sup 3/)). In addition, the square-root forms have the added benefit of numerical stability and guaranteed positive semi-definiteness of the state covariances.

The square-root unscented Kalman filter for state and parameter-estimation

A probabilistic framework, called Sigma-point Kalman Filters (SPKF) was applied to the problem domain addressed by the extended Kalman Filter (EKF). SPKF methods are superior to the standard EKF based estimation approaches, as an SPKF achieves second-order or higher accuracy. The SPKF has also been applied to the integrated navigation problem as it relates to unmanned aerial vehicle (UAV) autonomy. The SPKF-based sensor latency compensation technique is used to demonstrate the lagged GPS measurements.

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Sigma-Point Kalman Filters for Nonlinear Estimation and Sensor Fusion: Applications to Integrated Navigation

For sequential probabilistic inference in nonlinear non-Gaussian systems, approximate solutions must be used. We present a novel recursive Bayesian estimation algorithm that combines an importance sampling based measurement update step with a bank of sigma-point Kalman filters for the time-update and proposal distribution generation. The posterior state density is represented by a Gaussian mixture model that is recovered from the weighted particle set of the measurement update step by means of a weighted EM algorithm. This step replaces the resampling stage needed by most particle filters and mitigates the "sample depletion" problem. We show that this new approach has an improved estimation performance and reduced computational complexity compared to other related algorithms.

Gaussian mixture sigma-point particle filters for sequential probabilistic inference in dynamic state-space models

Behavioral flexibility is key to survival in a dynamic environment. While flexible, goal-directed behaviors are initially dependent on dorsomedial striatum, they become dependent on lateral striatum as behaviors become inflexible. Similarly, lesions of dopamine terminals in lateral striatum disrupt the development of inflexible habits. This work suggests that dopamine release in lateral striatum may drive inflexible behaviors, though few studies have investigated a causative role of subpopulations of striatal dopamine terminals in reversal learning, a measure of flexibility. Here, we performed two optogenetic experiments to activate dopamine terminals in dorsal medial (DMS), dorsal lateral (DLS), or ventral (NAc) striatum in DAT-Cre mice that expressed channelrhodopsin-2 via viral injection (Experiment I) or through transgenic breeding with an Ai32 reporter line (Experiment II) to determine how specific dopamine subpopulations impact reversal learning. Mice performed a reversal task in which they self-stimulated DMS, DLS, or NAc dopamine terminals by pressing one of two levers before action-outcome lever contingencies were reversed. Largely consistent with presumed ventromedial/lateral striatal function, we found that mice self-stimulating medial dopamine terminals reversed lever preference following contingency reversal, while mice self-stimulating NAc showed parial flexibility, and DLS self-stimulation resulted in impaired reversal. Impairments in DLS mice were characterized by more regressive errors and reliance on lose-stay strategies following reversal, as well as reduced within-session learning, suggesting reward insensitivity and overreliance on previously learned actions. This study supports a model of striatal function in which dorsomedial and ventral dopamine facilitate goal-directed responding, and dorsolateral dopamine supports more inflexible responding.

https://onlinelibrary.wiley.com/doi/pdfdirect/10.1111/ejn.15910

R. van der Merwe

Papers

The unscented Kalman filter for nonlinear estimation

The square-root unscented Kalman filter for state and parameter-estimation

Sigma-Point Kalman Filters for Nonlinear Estimation and Sensor Fusion: Applications to Integrated Navigation

Gaussian mixture sigma-point particle filters for sequential probabilistic inference in dynamic state-space models

Characterization of striatal dopamine projections across striatal subregions in reversal learning.