Factoring Dynamic Bayesian Networks using Possible Conflicts Carlos J. Alonso-Gonzalez 1 , Noemi Moya 1 , and Gautam Biswas 2 1 Department of ComputerScience University of Valladolid, Valladolid, 47011, Spain calonso@infor.uva.es noemi@infor.uva.es 2 Institute for Software Integrated Systems , Nashville, TN 37235, USA gautam.biswas@vanderbilt.edu ABSTRACT Dynamic Bayesian Networks (DBNs) are tem- poral probabilistic graphical models that repre- sent in a very compact way dynamic systems. They have been used for model based diagno- sis of complex systems because they naturally cope with uncertainties in the diagnosis process, particularly sensor uncertainty in noisy environ- ments. A caveat of DBN is the complexity of the inference procedure which is usually performed with Particle Filtering algorithms. Recently, fac- toring has been proposed to decompose a DBN into subsystems, distributing the diagnosis pro- cess and reducing the computational burden. This paper proposes decomposing a system with Possible Conflicts (PCs) and, afterwards, build- ing a DBN factor from each resultant PC. The method can be systematically applied to a state space representation of a dynamic system to ob- tain minimal observable subsystems with analyti- cal redundancy. Assuming single fault hypothesis and known fault modes, the method allows per- forming consistency based fault detection, isola- tion and identification with the unifying formal- ism of DBN. The three tank system benchmark has been used to illustrate the approach. Two fault scenarios are discussed and a comparison of the behaviors of a DBN of the complete system with the DBN factors is also included. 1 INTRODUCTION The increasing complexity of current engineering sys- tems, together with the increasing demand on their safe and reliable operation even in the presence of system faults, makes fault diagnosis an essential tool. Faults must be detected, and if possible isolated and identi- fied, close to their onset (Narasimhan, 2007) so that quick action can be taken to minimize the effects of the fault and thus prevent damage. Due to the com- plexity of these systems, formal methods are required for systematic design, analysis, and implementation of system diagnosers. Model-based diagnosis provides a formal framework to achieve these objectives. Main approaches to model based diagnosis of con- tinuous systems are consistency based, control the- ory based, and stochastic based (Narasimhan, 2007). Stochastic approaches have promoted the use of prob- abilistic methods for fault diagnosis. This is motivated by the uncertainty in the diagnosis process. The main sources of uncertainty are the models and the sensors, particularly in noisy environments. Among stochas- tic approaches, Dynamic Bayesian Networks (DBNs) (Murphy, 2002) play an important role. DBNs have been applied (Dearden and Clancy, 2001; Koller and Lerner, 2001) to fault diagnosis be- cause they allow estimating state variables of a dy- namic system without the usual Gaussian assumption for noise and modeling errors, which no longer ap- ply when faults occur (Arulampalam et al., 2002). Its major drawbacks are computational complexity of learning and inference procedures. In model based diagnosis, network structure and coefficients may be obtained from models, particularly from Temporal Causal Graphs (TCGs) (Lerner et al., 2000; Roy- choudhury et al., 2008). Real time inference has been tackled with Particle Filtering (Arulampalam et al., 2002). A problem with Particle Filtering is ’sample im- poverishment’: less weighted samples tend to disap- pear. Importance sampling may reduce this effect that is especially harmful for diagnosis: faulty states have small probabilities.(Roychoudhury et al., 2008) pro- poses solving this problem using multiple DBNs: a nominal DBN to track the system in normal oper- ation and, under single fault hypothesis, a DBN to model each fault. When a fault is detected, the TRAN- SCEND method (Mosterman and Biswas, 1999) is used to generate fault hypotheses. The fault hypothe- ses are tracked in parallel by their associated faulty DBN. Eventually, the DBN which best fits observa- tions provides fault isolation and fault identification. The major drawback of this proposal is the compu- tational complexity of hypotheses tracking, because each DBN models the whole systems plus the hypoth- esized fault. To reduce computational complexity of inference, factoring DBNs has been recently proposed. In 1