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.

Noise Covariances Identification by MDM: Weighting, Recursion, and Implementation

The problem of noise covariance matrix identification of stochastic linear time-varying state-space models is addressed. The measurement difference method (MDM) is generalized to time-varying dimensions of the measurement and control. Three MDM identification techniques that differ in weighting used in the underlying least squares method are proposed. The techniques differ in estimate quality and computational complexity. In addition, recursive forms are designed for two techniques. The performance of the proposed techniques is analyzed using two numerical examples. The implementation of techniques is enclosed with the paper.

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.

State Noise Density Identification of LTV System by Kernel Deconvolution

This paper focuses on identification of the state noise density of a linear time-varying system described by the state-space model with the known measurement noise density. For this purpose, a novel method extending the capabilities of the measurement difference method (MDM) is proposed. The proposed method is based on the enhanced MDM residue calculation being a sum of the state and measurement noise, and on the construction of the residue sample kernel density. The state noise density is then estimated by the density deconvolution algorithm utilising the Fourier transform. The developed method is supplemented with automatic selection of the deconvolution user-defined parameters based on the proposed method of the noise moment equality. The state noise density estimation performance is evaluated in numerical examples and supplemented with the MALAB example implementation.

AI-Aided Kalman Filters

The Kalman filter (KF) and its variants are among the most celebrated algorithms in signal processing. These methods are used for state estimation of dynamic systems by relying on mathematical representations in the form of simple state-space (SS) models, which may be crude and inaccurate descriptions of the underlying dynamics. Emerging data-centric artificial intelligence (AI) techniques tackle these tasks using deep neural networks (DNNs), which are model-agnostic. Recent developments illustrate the possibility of fusing DNNs with classic Kalman-type filtering, obtaining systems that learn to track in partially known dynamics. This article provides a tutorial-style overview of design approaches for incorporating AI in aiding KF-type algorithms. We review both generic and dedicated DNN architectures suitable for state estimation, and provide a systematic presentation of techniques for fusing AI tools with KFs and for leveraging partial SS modeling and data, categorizing design approaches into task-oriented and SS model-oriented. The usefulness of each approach in preserving the individual strengths of model-based KFs and data-driven DNNs is investigated in a qualitative and quantitative study, whose code is publicly available, illustrating the gains of hybrid model-based/data-driven designs. We also discuss existing challenges and future research directions that arise from fusing AI and Kalman-type algorithms.