Maximum-Likelihood State Estimators in Probabilistic Boolean Control Networks.

Mitsuru Toyoda,Yuhu Wu

doi:10.1109/tcyb.2021.3127880

Abstract

This study addresses state estimation problems for probabilistic Boolean control networks (PBCNs). Compared with deterministic Boolean networks, PBCNs have the stochastic switching in logical update functions in the state equation. Consequently, statistical analysis is required to estimate unavailable states, which induces an optimization problem called maximum-likelihood estimation. This article mainly focuses on two scenarios: 1) state estimation from partially measured state and 2) state estimation from output data, meaning observer design. The resulting optimization problems are solved using efficient algorithms based on dynamic programming. Concurrently, Dijkstra-type algorithms, which solve equivalent shortest path problems, are also proposed using best-first search. Furthermore, both the proposed algorithms derive novel observer design methods for PBCNs. The proposed algorithms are evaluated with practical estimation problems aiming to the sensor reduction and applied to gene regulatory networks of apoptosis and Lac operon.

Highlights

B OOLEAN control network (BCN) [1] is a simple and tractable model to represent logical systems comprising Boolean control and state variables
This study addresses the gene regulatory networks of the apoptosis and the lac operon
3) By expanding the state estimation problem summarized above, further results for the observer design just using the output of probabilistic BCN (PBCN) are obtained

Summary

INTRODUCTION

B OOLEAN control network (BCN) [1] is a simple and tractable model to represent logical systems comprising Boolean control and state variables. Existing reports expanding concepts of BCNs to PBCNs have required complicated and detailed discussions [18], [22]–[24] Similar to these existing reports on PBCNs, deterministic calculations as summarized in the previous section are not applicable because of the stochasticity in PBCNs, and the adaptation of the deterministic techniques with BCNs approximation cannot work, as discussed in the motivating examples of Section III. In Examples 4 and 5 of Section VI, the estimation problem using outputs of a system can address a situation with partially measured states, as illustrated in application to two well-known gene regulatory networks, an apoptosis network and Lac operon. 3) By expanding the state estimation problem summarized above, further results for the observer design just using the output of PBCNs are obtained. The notation “ ” in an expression A B is omitted and written as AB

PROBLEM FORMULATION

OPTIMIZATION ALGORITHMS FOR MAXIMUM-LIKELIHOOD ESTIMATION

SHORTEST PATH-BASED ESTIMATION ALGORITHM

FURTHER RESULTS FOR STATE OBSERVER DESIGN

CONCLUSION

VIII. FUTURE RECOMMENDATIONS