Robust Pareto design of ANFIS networks for nonlinear systems with probabilistic uncertainties

In this paper, multi-objective evolutionary Pareto optimal design of Adaptive Neuro-Fuzzy Inference System (ANFIS) have been used for modeling of nonlinear systems using input-output data sets with probabilistic uncertainties. In this way, A Monte Carlo Simulation (MCS) is first performed to generate input-output data set using some probabilistic distributions. Multi-objective uniform-diversity genetic algorithms (MUGA) are then used for Pareto optimization of ANFIS networks. The important conflicting objectives of ANFIS networks that are considered in this work are, namely, the mean and variance of both Training Error (TE) and Prediction Error (PE) of such ANFIS models. It is shown that a robust ANFIS can be simply obtained using a criterion based on four values of means and variances of both TE and PE. The probabilistic evolved ANFIS model exhibits much more robustness to the uncertainties involved within the input-output data sets than that of the deterministic evolved ANFIS model. It is shown that ANFIS can be successfully applied for input-output data set with uncertainties so that a robust model can be compromisingly obtained from some non-dominated optimum ANFIS models.