Ensemble Stochastic Configuration Networks for Estimating Prediction Intervals: A Simultaneous Robust Training Algorithm and Its Application.

Obtaining accurate point prediction of industrial processes' key variables is challenging due to the outliers and noise that are common in industrial data. Hence the prediction intervals (PIs) have been widely adopted to quantify the uncertainty related to the point prediction. In order to improve the prediction accuracy and quantify the level of uncertainty associated with the point prediction, this article estimates the PIs by using ensemble stochastic configuration networks (SCNs) and bootstrap method. The estimated PIs can guarantee both the modeling stability and computational efficiency. To encourage the cooperation among the base SCNs and improve the robustness of the ensemble SCNs when the training data are contaminated with noise and outliers, a simultaneous robust training method of the ensemble SCNs is developed based on the Bayesian ridge regression and M-estimate. Moreover, the hyperparameters of the assumed distributions over noise and output weights of the ensemble SCNs are estimated by the expectation-maximization (EM) algorithm, which can result in the optimal PIs and better prediction accuracy. Finally, the performance of the proposed approach is evaluated on three benchmark data sets and a real-world data set collected from a refinery. The experimental results demonstrate that the proposed approach exhibits better performance in terms of the quality of PIs, prediction accuracy, and robustness.