Akaike Information Criterion-based conjunctive belief rule base learning for complex system modeling

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Nonlinear complex system modeling has become the basis of many theoretical and practical problems, which requires balancing the correlations between the modeling accuracy and the modeling complexity. However, the two objectives may not be consistent with each other under many practical conditions, especially for complex systems with multiple influential factors. The belief rule base (BRB) has shown advantages in nonlinear complex system modeling under uncertainty. However, most of current works on BRB has focused only on the modeling accuracy. As such, an Akaike Information Criterion (AIC)-based objective, AICBRB, is deduced to represent both the modeling accuracy (denoted by the Mean Square Error (MSE)) and the modeling complexity (denoted by the number of the parameters). Based on the proposed AICBRB, a bi-level optimization model and a corresponding bi-level optimization algorithm are developed. Moreover, an empirical optimization path search strategy is proposed for upper-level optimization. The optimization path is comprised of multiple solutions with optimal performance. After the BRB learning process, both the structure and the parameters of BRB are optimized, which identifies the best decision structure of BRB. A numerical multi-extreme function case and a practical pipeline leak detection case are studied. The results show that an optimization path could be identified with a series of optimal solutions. With AICBRB as the objective, the jointly optimized BRB in the best decision structure can be obtained with an improved modeling accuracy as well as reduced modeling complexity.