State Estimation for Stochastic Time-Varying Boolean Networks

In this article, a general theoretical framework is developed for the state estimation problem of stochastic time-varying Boolean networks (STVBNs). The STVBN consists of a system model describing the evolution of the Boolean states and a model relating the noisy measurements to the Boolean states. Both the process noise and the measurement noise are characterized by sequences of mutually independent Bernoulli distributed stochastic variables taking values of 1 or 0, which imply that the state/measurement variables may be flipped with certain probabilities. First, an algebraic representation of the STVBNs is derived based on the semitensor product. Then, based on Bayes’ theorem, a recursive matrix-based algorithm is obtained to calculate the one-step prediction and estimation of the forward–backward state probability distribution vectors. Owing to the Boolean nature of the state variables, the Boolean Bayesian filter is designed that can be utilized to provide the minimum MSE state estimate for the STVBNs. The fixed-interval smoothing filter is also obtained by resorting to the forward–backward technique. Finally, a simulation experiment is carried out for the context estimation problem of the <inline-formula><tex-math notation="LaTeX">$p53$</tex-math></inline-formula>-<inline-formula><tex-math notation="LaTeX">$MDM2$</tex-math></inline-formula> negative-feedback gene regulatory network.