Algorithmic Differentiation of the MWGS-Based Arrays for Computing the Information Matrix Sensitivity Equations within the Problem of Parameter Identification

• Published in: Mathematics (Basel) Vol. 10; no. 1; p. 126
• Main Authors: Tsyganov, Andrey, Tsyganova, Julia
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.01.2022
• Subjects: algorithmic differentiation; Algorithms; array algorithms; Arrays; Computation; Decomposition; Difference equations; Differentiation; Discrete time systems; information filter; information matrix; Kalman filters; Mathematics; Matrices (mathematics); MWGS-orthgonalization; Noise; Parameter identification; Parameter sensitivity; Robustness (mathematics); Roundoff error; Sensitivity; sensitivity equation; State space models; Stochastic systems; Transformations (mathematics)
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math10010126

The paper considers the problem of algorithmic differentiation of information matrix difference equations for calculating the information matrix derivatives in the information Kalman filter. The equations are presented in the form of a matrix MWGS (modified weighted Gram–Schmidt) transformation. The solution is based on the usage of special methods for the algorithmic differentiation of matrix MWGS transformation of two types: forward (MWGS-LD) and backward (MWGS-UD). The main result of the work is a new MWGS-based array algorithm for computing the information matrix sensitivity equations. The algorithm is robust to machine round-off errors due to the application of the MWGS orthogonalization procedure at each step. The obtained results are applied to solve the problem of parameter identification for state-space models of discrete-time linear stochastic systems. Numerical experiments confirm the efficiency of the proposed solution.