Two Efficient Twin ELM Methods With Prediction Interval

In the operational optimization and scheduling problems of actual industrial processes, such as iron and steel, and microelectronics, the operational indices and process parameters usually need to be predicted. However, for some input and output variables of these prediction models, there may exist a lot of uncertainties coming from themselves, the measurement error, the rough representation, and so on. In such cases, constructing a prediction interval (PI) for the output of the corresponding prediction model is very necessary. In this paper, two twin extreme learning machine (TELM) models for constructing PIs are proposed. First, we propose a regularized asymmetric least squares extreme learning machine (RALS-ELM) method, in which different weights of its squared error loss function are set according to whether the error of the model output is positive or negative in order that the above error can be differentiated in the parameter learning process, and Tikhonov regularization is introduced to reduce overfitting. Then, we propose an asymmetric Bayesian extreme learning machine (AB-ELM) method based on the Bayesian framework with the asymmetric Gaussian distribution (AB-ELM), in which the weights of its likelihood function are determined as the same method in RALS-ELM, and the type II maximum likelihood algorithm is derived to learn the parameters of AB-ELM. Based on RALS-ELM and AB-ELM, we use a pair of weights following the reciprocal relationship to obtain two nonparallel regressors, including a lower-bound regressor and an upper-bound regressor, respectively, which can be used for calculating the PIs. Finally, some discussions are given, about how to adjust the weights adaptively to meet the desired PI, how to use the proposed TELMs for nonlinear quantile regression, and so on. Results of numerical comparison on data from one synthetic regression problem, three University of California Irvine benchmark regression problems, and two actual industrial regression problems show the effectiveness of the proposed models.