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

We present a stable control strategy for groups of vehicles to move and reconfigure cooperatively in response to a sensed, distributed environment. Each vehicle in the group serves as a mobile sensor and the vehicle network as a mobile and reconfigurable sensor array. Our control strategy decouples, in part, the cooperative management of the network formation from the network maneuvers. The underlying coordination framework uses virtual bodies and artificial potentials. We focus on gradient climbing missions in which the mobile sensor network seeks out local maxima or minima in the environmental field. The network can adapt its configuration in response to the sensed environment in order to optimize its gradient climb.

Cooperative control of mobile sensor networks:Adaptive gradient climbing in a distributed environment

One of the major obstacles to using Bayesian methods for pattern recognition has been its computational expense. This thesis presents an approximation technique that can perform Bayesian inference faster and more accurately than previously possible. This method, “Expectation Propagation,” unifies and generalizes two previous techniques: assumed-density filtering, an extension of the Kalman filter, and loopy belief propagation, an extension of belief propagation in Bayesian networks. The unification shows how both of these algorithms can be viewed as approximating the true posterior distribution with simpler distribution, which is close in the sense of KL-divergence. Expectation Propagation exploits the best of both algorithms: the generality of assumed-density filtering and the accuracy of loopy belief propagation. 
Loopy belief propagation, because it propagates exact belief states, is useful for limited types of belief networks, such as purely discrete networks. Expectation Propagation approximates the belief states with expectations, such as means and variances, giving it much wider scope. Expectation Propagation also extends belief propagation in the opposite direction—propagating richer belief states which incorporate correlations between variables. 
This framework is demonstrated in a variety of statistical models using synthetic and real-world data. On Gaussian mixture problems, Expectation Propagation is found, for the same amount of computation, to be convincingly better than rival approximation techniques: Monte Carlo, Laplace's method, and variational Bayes. For pattern recognition, Expectation Propagation provides an algorithm for training Bayes Point Machine classifiers that is faster and more accurate than any previously known. The resulting classifiers outperform Support Vector Machines on several standard datasets, in addition to having a comparable training time. Expectation Propagation can also be used to choose an appropriate feature set for classification, via Bayesian model selection. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)

A family of algorithms for approximate bayesian inference

Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Lie-group structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Lie-group structure, highlighting theory, algorithmic issues and a number of applications.

Lie-group methods

An error bound for multidimensional quadrature is derived that includes the Koksma-Hlawka inequality as a special case. This error bound takes the form of a product of two terms. One term, which depends only on the integrand, is defined as a generalized variation. The other term, which depends only on the quadrature rule, is defined as a generalized discrepancy. The generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the L p -star discrepancy and P α that arises in the study of lattice rules.

https://www.ams.org/mcom/1998-67-221/S0025-5718-98-00894-1/S0025-5718-98-00894-1.pdf

A generalized discrepancy and quadrature error bound

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Methods of Numerical Integration

In this paper we examine the problem of testing for hetero geneity and heterosckedasticity in a Bayesian framework. We first show that a model with random coefficients is identical to a model with heteroskedastic residuals. We then consider two approaches for testing. The first one is concer ned with the point of view of misspecification. The original model is homoskedastic. One is willing to detect any departure from homoskedasticity. We propose diagnostics based on the examination of the Bayesian residuals, after stressing the differences between classical and Bayesian residuals. In the second approach, the starting point is a precise form of the alternative hypothesis, and the model for inference is heteroskedastic. A test for homoskedasticity is then a test for a parameter restriction. This can be done by looking at highest posterior probability regions or by the use of the posterior odds ratio. As a joint product, we develop the posterior analysis of a heteroskedastic regression model for several classes of prior distributions.

Bayesian diagnostics for heterogeneity

Dynamic Regression Models

When a linear econometric model is expressed in logarithms, the presence of identities entails a strong non-linearity for the simulation stage of the complete model. This aspect can be damageable for the simulation algorithms which are implemented on microcomputers. A traditional solution consists in linearizing the identity by a Taylor expansion around the mean of the variables. In this paper we propose an approximation based on an error correction mecanism which takes into account the statistical properties of the series. The quality of the approximation is shown on a three equation econometric model estimated on french quartely data covering the period 1963-1985.

Michel Lubrano

Papers

Methods of Numerical Integration

Bayesian diagnostics for heterogeneity

Dynamic Regression Models

Approximation des identites comptables dans les mo e es specifies en Iogarithme

Unit Root Inference