A joint optimization method on parameter and structure for belief-rule-based systems

The belief-rule-based system (BRBS) is one of the most visible and fastest growing branches of decision support systems. As the knowledge base in the BRBS, the belief-rule-base (BRB) is required to be equipped with the optimal parameters and structure, which means the optimal value and number of parameters, respectively. Several optimization methods were therefore proposed in the past decade. However, these methods presented different limitations, such as the use of the incomplete parameter optimization model, lack of structure optimization, and so on. Moreover, it is impracticable to determine the optimal parameters and structure of a BRB using the training error because of over-fitting. The present work is focused on the joint optimization on parameter and structure for the BRB. Firstly, a simple example is utilized to illustrate and analyze the generalization capability of the BRBS under different numbers of rules, which unveils the underlying information that the BRBS with a small training error may not have superior approximation performances. Furthermore, by using the Hoeffding inequality theorem in probability theory, it is a constructive proof that the generalization error could be a better choice of criterion and measurement to determine the optimal parameters and structure of a BRB. Based on the above results, a heuristic strategy to optimize the structure of the BRB is proposed, which is followed by a parameter optimization method using the differential evolution (DE) algorithm. Finally, a joint optimization method is introduced to optimize the parameters and structure of the BRB simultaneously. In order to verify the generality and effectiveness of the proposed method, two practical case studies, namely oil pipeline leak detection and bridge risk assessment, are examined to demonstrate how the proposed method can be implemented in the BRB under disjunctive and conjunctive assumptions along with their performance comparative analysis.