Efficient Spectral Differentiation in Grid-Based Continuous State Estimation

This paper deals with the state estimation of stochastic models with continuous dynamics. The aim is to incorporate spectral differentiation methods into the solution to the Fokker-Planck equation in grid-based state estimation routine, while taking into account the specifics of the field, such as probability density function (PDF) features, moving grid, zero boundary conditions, etc. The spectral methods, in general, achieve very fast convergence rate of O(c^N )(O < c < 1) for analytical functions such as the probability density function, where N is the number of grid points. This is significantly better than the standard finite difference method (or midpoint rule used in discrete estimation) typically used in grid-based filter design with convergence rate O( 1 / N^2 ). As consequence, the proposed spectral method based filter provides better state estimation accuracy with lower number of grid points, and thus, with lower computational complexity.

Stochastic Integration Based Estimator: Robust Design and Stone Soup Implementation

This paper deals with state estimation of nonlinear stochastic dynamic models. In particular, the stochastic integration rule, which provides asymptotically unbiased estimates of the moments of nonlinearly transformed Gaussian random variables, is reviewed together with the recently introduced stochastic integration filter (SIF). Using SIF, the respective multi-step prediction and smoothing algorithms are developed in full and efficient square-root form. The stochastic-integration-rule-based algorithms are implemented in Python (within the Stone Soup framework) and in MATLAB and are numerically evaluated and compared with the well-known unscented and extended Kalman filters using the Stone Soup defined tracking scenario.

Data-Augmented Numerical Integration in State Prediction: Rule Selection

This paper deals with the state prediction of nonlinear stochastic dynamic systems. The emphasis is laid on a solution to the integral Chapman-Kolmogorov equation by a deterministic-integration-rule-based point-mass method. A novel concept of reliable data-augmented, i.e., mathematics- and data-informed, integration rule is developed to enhance the point-mass state predictor, where the trained neural network (representing data contribution) is used for the selection of the best integration rule from a set of available rules (representing mathematics contribution). The proposed approach combining the best properties of the standard mathematics-informed and novel data-informed rules is thoroughly discussed.

Efficient Point Mass Predictor for Continuous and Discrete Models with Linear Dynamics

This paper deals with state estimation of stochastic models with linear state dynamics, continuous or discrete in time. The emphasis is laid on a numerical solution to the state prediction by the time-update step of the grid-point-based point-mass filter (PMF), which is the most computationally demanding part of the PMF algorithm. A novel way of manipulating the grid, leading to the time-update in form of a convolution, is proposed. This reduces the PMF time complexity from quadratic to log-linear with respect to the number of grid points. Furthermore, the number of unique transition probability values is greatly reduced causing a significant reduction of the data storage needed. The proposed PMF prediction step is verified in a numerical study.