LD filter for the state estimation of pairwise Markov models

Tsyganova J.V., Tsiganov A.V.
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The paper addresses the state estimation problem for pairwise Markov models with Gaussian noises. The class of pairwise Markov models generalizes the classical hidden Markov models. The key difference lies in the assumption that the Markov property holds not for the hidden process alone, but for the pair consisting of the state and the observation. This allows modeling more complex dependencies and, in particular, eliminates the requirement of Markovianity for the hidden process. For linear Gaussian pairwise models, Kalman filtering methods remain applicable, leading to the concept of the pairwise Kalman filter.

This work proposes a new modification of the pairwise Kalman filter based on the application of modified weighted Gram – Schmidt orthogonalization and the LD decomposition of covariance matrices. The main results are as follows: a novel LD modification of the pairwise Kalman filter (Theorem 1); a new LD-PKF algorithm for state estimation of pairwise Markov models, based on a direct procedure of modified weighted Gram–Schmidt orthogonalization and LD decomposition of covariance matrices (algorithm 2); results of comparative analysis on the numerical properties of pairwise discrete filtering algorithms.

The obtained theoretical results complement the theory of pairwise filtering in the class of linear discrete pairwise Markov models with Gaussian noises.

The developed algorithm is implemented in MATLAB. A series of numerical experiments are conducted, and the results demonstrate its effectiveness and numerical advantages over other existing modifications of the pairwise Kalman filter.

The presented results can be further used to develop new methods for parameter identification of pairwise Markov models.

Keywords: pairwise Markov model, pairwise Kalman filter, MWGS orthogonalization, LD factorization, discrete-time filtering algorithm
Citation in English: Tsyganova J.V., Tsiganov A.V. LD filter for the state estimation of pairwise Markov models // Computer Research and Modeling, 2026, vol. 18, no. 4, pp. 747-764
Citation in English: Tsyganova J.V., Tsiganov A.V. LD filter for the state estimation of pairwise Markov models // Computer Research and Modeling, 2026, vol. 18, no. 4, pp. 747-764
DOI: 10.20537/2076-7633-2026-18-4-747-764
URL: http://crm-en.ics.org.ru/journal/article/3743/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.

Copyright © 2026 Tsyganova J.V., Tsiganov A.V.

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