Point-mass filters solve Bayesian recursive relations by approximating probability density functions of a system state over grids of discrete points. The approach suffers from the curse of dimensionality. The exponential increase of the number of the grid points can be mitigated by application of low-rank approximations of multidimensional arrays. Tensor train decompositions represent individual values by the product of matrices. This paper focuses on selected issues that are substantial in state estimation. Namely, the contamination of the density approximations by negative values is discussed first. Functional decompositions of quadratic functions are compared with decompositions of discretised Gaussian densities next. In particular, the connection of correlation with tensor train ranks is explored. Last, the consequences of interpolating the density values from one grid to a new grid are analysed.

Non-linear filters consider probability density functions in various non-parametric representations. They often suffer from the curse of dimensionality. Computation of weights over a grid of points becomes infeasible even for low dimensions. Filters processing data produced in different sensor nodes provide their own probability densities. Combination of such densities is desired. A favourite paradigm is to construct a fused density as a weighted arithmetic or geometric mean of the individual densities. This paper prospects the fusion for tensor train representation of densities produced by point-mass filters. In this representation, the weights are neither evaluated for a whole grid nor fully stored in the memory of the filters. Aspects of tensor-train-based fusion are discussed, such as computation of auxiliary characteristics and experience with numerical examples.

Subjective Bayesian estimation perceives probability density functions as expert opinions. Among various rules for combining the opinions, the product and the weighted geometric mean of densities are prominent. Nevertheless, closed-form representations are scarce and non-parametric approaches often suffer from the curse of dimensionality. This paper prospects the fusion of densities represented by non-parametric marginal densities and a parametric Gaussian copula. The explicit reconstruction of the joint densities followed by an optimisation step is avoided. A cheap approximate combination is proposed instead. The combination of marginal densities is tuned by a Gaussian term, while the proposed copula parameter uses moments of the marginal densities. The presented examples illustrate the approximative nature of the approach for non-Gaussian densities and highlight some numerical issues.

Active fault diagnosis (AFD) of stochastic large-scale systems in multiple model framework involves two stages: offline and online. In the offline stage, an excitation input generator is designed based on a Bellman function. In the online stage, the generator is utilized together with an estimator of the model indices. A similar estimator is used in the offline stage for the Bellman function calculation using the value iteration technique. However, due to the high dimensions of information states of the associated perfect state information problem, the estimator in the offline stage must involve approximations. The paper provides the relations for the estimate calculation using the Bayesian recursive relations, proposes four algorithms, and studies effects of such approximations on the AFD decisions. In particular, the quality of the model index estimates is analyzed using a power network model.

Linear fusion of estimates is a basic tool for combining probabilistic data. If the correlation of estimation errors is unknown, the fusion performance is evaluated with respect to the worst case. Inverse Covariance Intersection fusion is a rule for combining two estimates with partially known crosscorrelation matrix. This paper generalises the rule to fusing multiple estimates. First, the generalised assumption and the essential theory are presented. A suboptimal solution with a simple parametrisation is derived next and it is shown to be better than the solution for unknown correlation. Finally, a recursive fusion of multiple estimates is designed.

A recently proposed algorithm of fusion under partially known correlations of estimation errors has been proved to outperform the classic Covariance Intersection algorithm, which was proposed for the case of no knowledge of correlation. This paper shows that the assumptions of the recently proposed algorithm are rather strict with respect to the classic one. Namely, the mean square error (MSE) matrices of the two state estimates cannot be upper-bounded arbitrarily. A relaxation of the assumption that the matrices have to be known exactly is discussed, as well as an iterative fusion of multiple estimates, and several examples dealing with dependent errors are provided.

In decentralised estimation, locally measured data are processed locally and the local filters are unaware of the other ones. Due to the lack of the global knowledge, the fusion of the local estimates cannot utilise the correlations of the local estimate errors in the computation of the fused mean square error matrix. For this reason, algorithms of fusion under unknown correlations have been designed to provide upper bounds of these matrices. This paper reveals a fundamental relation between the upper bounds and Minkowski sums of ellipsoids. The obtained insight improves the comprehension of the fusion algorithms. Application of the ellipsoidal calculus is illustrated using elemental examples.

The paper introduces a novel approach to an estimator design, the cooperative filter design, for state estimation of nonlinear systems. The approach is based on the idea of combining estimates of several different approximate (and thus sub-optimal) nonlinear filters, which are configured to perform the same task. Within the concept, two strategies are proposed, namely the cooperative estimation and cooperative monitoring. The strategies have the potential of an improvement of the estimation performance in terms of accuracy and consistency, which was confirmed by a numerical illustration.