Jiří Ajgl

Differential entropy estimation by particles

Abstract The paper deals with the estimation of the differential entropy of the probability density function of a stochastic system. A nonparametric entropy estimator based on particles with weights is proposed. The estimator provides asymptotically unbiased estimates. For small number of particles, the bias originates from the difference between the state density and the sampling density. A simulation study is provided. The approach to the differential entropy estimation by particles is a pillar of the fusion problem.

Conservativeness of estimates given by probability density functions: Formulation and aspects

Abstract In state estimation, the output of a filter can consist of a vector estimate with an associated quality matrix or it can be given by a probability density function. Although the first option prevails in tracking, in many problems, it is necessary to cope with multiple hypotheses, i.e. with multiple vector–matrix pairs. The pairs are called conservative, if the quality matrices do not undervalue a measure of uncertainty of the vector estimates. However, no conservativeness definition for multiple pairs or for densities has been coined yet. The paper proposes such a definition for densities, provides a sufficient condition, explores some aspects and gives several special cases and numerical examples.

On Linear Estimation Fusion under Unknown Correlations of Estimator Errors

Abstract The linear fusion of estimators is widely used in decentralised state estimation. Because the maintaining of estimation error cross-correlations between local estimators is not affordable in large-scale problems, approaches dealing with unknown correlations were developed. The Covariance Intersection fusion is a linear fusion of estimators and it provides a fused estimator quality matrix that does not undervalue the mean square error matrix. This paper derives the matrix of the fused estimator quality for arbitrary weights of the linear fusion rule that considers the unknown correlations. It also shows that there can exist better matrices of the fused estimator quality than the ones proposed by the Covariance Intersection fusion rule.

On Nonlinearity Measuring Aspects of Stochastic Integration Filter

The paper deals with Bayesian state estimation of nonlinear stochastic dynamic systems. The focus is aimed at the stochastic integration filter, which is based on a stochastic integration rule. It is shown that the covariance matrix of the integration error calculated as a byproduct of the rule can be used as a measure of nonlinearity. The measure informs the user about validity of the assumptions of Gaussianity, which is adopted by the stochastic integration filter. It is also demonstrated how to use this information for a prediction of the number of remaining iterations of the rule. The paper also focuses on utilization of the integration error covariance matrix for improving estimates of the mean square error of the estimates, which is produced by the filter.

Stochastic Integration Filter with Improved State Estimate Mean-Square Error Computation

The paper deals with the Bayesian state estimation of nonlinear stochastic dynamic systems. The focus is aimed at the stochastic integration filter, which represents the Gaussian filters with the state and measurement prediction moments calculated by the stochastic integration rule. Besides the value of the integral, the rule also provides the covariance matrix of the integral value error. In the paper an improved mean-square error of the state estimate is proposed based on utilization of the integral error covariance matrix. The improved calculation is illustrated using two numerical examples for the stochastic integration filter of the third and fifth degrees.

Fusion of Multiple Estimates by Covariance Intersection: Why and Howit Is Suboptimal

Abstract The fusion under unknown correlations tunes a combination of local estimates in such a way that upper bounds of the admissible mean square error matrices are optimised. Based on the recently discovered relation between the admissible matrices and Minkowski sums of ellipsoids, the optimality of existing algorithms is analysed. Simple examples are used to indicate the reasons for the suboptimality of the covariance intersection fusion of multiple estimates. Further, an extension of the existing family of upper bounds is proposed, which makes it possible to get closer to the optimum, and a general case is discussed. All results are obtained analytically and illustrated graphically.

Design of a robust fusion of probability densities

The paper deals with the fusion of probability densities. A selection of the weights of the weighted geometric mean of densities is justified, as well as the selection of the geometric mean itself, from a more general perspective. It is shown that the Chernoff fusion provides the density that minimises the greatest Kullback-Leibler divergence to the densities that are being fused. The interpretation of the densities is discussed and finally, illustrative examples are provided.

Marginal marginalised particle filter

This paper deals with filters that combine the analytical Kalman filtering and the Monte Carlo simulation based particle filtering. Since the particles are related to the state trajectories from the initial time up to the current time rather than to the state at the last time only, these filters cannot be directly used in fusion of probability densities of the last state. Therefore, marginalisation of the outdated parts of the state trajectories is proposed in the paper. In order to obtain a reproducible probability density, at least theoretically, the Gaussian sum description of random variables is also newly considered in the problem formulation.