State estimation of stochastic temporal Boolean control networks

In this paper, the state estimation problem for stochastic temporal Boolean control networks (STBCNs) is investigated. The STBCNs evolve according to a temporal Boolean model affected by process noise, while the measurements are corrupted by observation noise. Based on the available input and output sequences, optimal state and state sequence estimation methods that minimize the mean-square error are developed. First, leveraging the semi-tensor product (STP) of matrices, the state-space representation of STBCNs is formulated. A Boolean Bayesian filtering method is then proposed, and two recursive matrix-based procedures are designed to compute the conditional probability distributions of the system states and state sequences in vector form, respectively. Furthermore, by employing the STP framework, these probability distributions are transformed into Boolean-valued expectations for each node. In addition, for a fixed observation window, a forward–backward estimation technique is introduced to obtain the state probability distribution vector at each time instant, which is also transformed into Boolean-valued expectations for each node. Based on these expectations, the optimal state and state sequence estimates that minimize the mean-square error are derived. Finally, the effectiveness of the proposed approach is demonstrated using the Escherichia coli Boolean model.