A Bayesian network (BN) model is a multivariate statistical model that uses Directed Acyclic Graph (DAG) to represent statistical dependencies among variables and has been applied in various fields (Cooper and Herskovits, 1992, Etxeberria et al., 1997, Lam, 1998, Man Leung et al., 2002, Pearl, 1988 and Wong et al., 1999). In recent years, in DAGs, dynamic Bayesian networks (Dynamic BN:DBNs) that allow variables in the DAG of a Bayesian network to be time dependent have been proposed and applied (Fearnhead, 2006, Friedman et al., 1998, Nielsen and Nielsen, 2008, Punskaya et al., 2002, Wang et al., 2008 and Wang et al., 2011). Additionally, the application of particle filters (PFs) has been proposed for the estimation of the state from observed data in a DBN (Andrieu et al., 2004, Arulampalam et al., 2002, Doucet et al., 2001, Gustafsson et al., 2002, Tokinaga and Tan, 2010 and Wang et al., 2011). However, in the conventional method of DBNs, the allowable range of a DAG is limited to the graph shape that is known in advance, and the state probabilities belong to the pattern already known. Therefore, this conventional DBN method cannot be applied to the unknown DAG shape and state changes (Wang et al., 2011). In this paper, we deal with the estimation of state changes in system descriptions for dynamic Bayesian networks by using a genetic procedure and particle filters (PFs) (Bordley and Kadane, 1999 and Tokinaga and Ikeda, 2011).
First, we organize the relation equation between the basic DAG model and the change of the DAG shape (Fearnhead, 2006, Friedman et al., 1998, Nielsen and Nielsen, 2008, Punskaya et al., 2002, Wang et al., 2008 and Wang et al., 2011). We describe the method for estimating the DAG shape change and variable state transition by using a genetic approach (Alvarez-Diaz and Miguez, 2008, Chen and Duan, 2011, Chi and Tang, 2007, Tokinaga and Ikeda, 2011 and Wong et al., 1999). In this case, in the estimation of DAG shape change, we do not use the entire process included in the crossover procedure of Genetic Programming (GP), but use methods based on mutations, such as changing the direction of a branch of the DAG, in order to maintain consistency in the process (called Evolutionary Programming: EP in Wong et al., 1999) (Alvarez-Diaz and Miguez, 2008, Antoci et al., 2012, Chen and Duan, 2011, Chi and Tang, 2007, Tokinaga and Ikeda, 2011 and Wong et al., 1999). In addition, we introduce function fk(z) to the dynamics description, which represents the state transition, and we apply GP to estimate the description shape ( Ikeda and Tokinaga, 2007a, Ikeda and Tokinaga, 2007b, Lu et al., 2006, Lu et al., 2007 and Tokinaga and Kishikawa, 2010).
Secondly, with respect to the issue of state estimation, we focus on the change of the joint distribution of the state variables between time t + 1 and time t, and propose a state estimation method by using PFs, which are applied to state estimation in a nonlinear state equation. In the estimation of the state change by PFs, the particles to represent the DBN structure and state transition are given in multiples, so the weight of a particle representing the DAG and state transition is defined as the capability to approximate the probability distribution function obtained from a table of cases. We apply the estimation scheme of the paper to the artificially generated DBN, in which the state of the variables and the changed structure of the DAG are already known, in order to prove the applicability of the method, and discuss its applicability to real data.
In the following, we describe the problem formulation and the estimation of the DBN description change by EP・GP in Section 2. In Section 3, we explain the estimation of the state change by PFs and in Section 4, we show the application and the results. Finally, the paper concludes in Section 5.
