Prediction Interval Construction and Optimization for Adaptive Neurofuzzy Inference Systems

The performance of an adaptive neurofuzzy inference system (ANFIS) significantly drops when uncertainty exists in the data or system operation. Prediction intervals (PIs) can quantify the uncertainty associated with ANFIS point predictions. This paper first presents a methodology to adapt the delta technique for the construction of PIs for outcomes of the ANFIS models. As the ANFIS models are linear in their consequent part, the ANFIS-based PIs are computationally less expensive than neural network (NN)-based PIs. Second, this paper proposes a method to optimize ANFIS-based PIs. A new PI-based cost function is developed for the training of the ANFIS models. A simulated annealing-based algorithm is applied to minimize the new nonlinear cost function and adjust the premise and consequent parameters of the ANFIS model. Using three real-world case studies, it is shown that ANFIS-based PIs are computationally less expensive than NN-based PIs. The application of the proposed optimization algorithm leads to better quality PIs than optimized NN-based PIs.