Generating probabilistic predictions using mean-variance estimation and echo state network

Shapley value-driven superior subset selection algorithm for carbon price interval forecast combination

Interval prediction requires not only accuracy but also the consideration of interval width and coverage, making model selection complex. However, research rarely addresses this challenge in interval combination forecasting. To address this issue, this study introduces a model selection for interval forecast combination based on the Shapley value (MSIFC-SV). This algorithm calculates Shapley values to measure each model's marginal contribution and establishes a redundancy criterion on the basis of changes in interval scores. If the removal of a model does not decrease the interval score, it is considered redundant and excluded. The selection process starts with all the models and ranks them by their Shapley values. Models are then assessed for retention or removal according to the redundancy criterion, which continues until all redundant models are excluded. The remaining subset is used to generate interval forecast combinations through interval Bayesian weighting. Empirical analysis of carbon price shows that MSIFC-SV outperforms individual models and derived subsets across metrics such as prediction interval coverage probability (PICP), mean prediction interval width (MPIW), coverage width criterion (CWC), and interval score (IS). Comparisons with benchmark methods further demonstrate the superiority of MSIFC-SV. Furthermore, MSIFC-SV is also successfully extended to the public dataset-housing price dataset, this indicates its universality. In summary, MSIFC-SV provides reliable model selection and delivers high-quality interval forecasts.

Water Quality Indicator Interval Prediction in Wastewater Treatment Process Based on the Improved BES-LSSVM Algorithm

This paper proposes a novel interval prediction method for effluent water quality indicators (including biochemical oxygen demand (BOD) and ammonia nitrogen (NH3-N)), which are key performance indices in the water quality monitoring and control of a wastewater treatment plant. Firstly, the effluent data regarding BOD/NH3-N and their necessary auxiliary variables are collected. After some basic data pre-processing techniques, the key indicators with high correlation degrees of BOD and NH3-N are analyzed and selected based on a gray correlation analysis algorithm. Next, an improved IBES-LSSVM algorithm is designed to predict the BOD/NH3-N effluent data of a wastewater treatment plant. This algorithm relies on an improved bald eagle search (IBES) optimization algorithm that is used to find the optimal parameters of least squares support vector machine (LSSVM). Then, an interval estimation method is used to analyze the uncertainty of the optimized LSSVM model. Finally, the experimental results demonstrate that the proposed approach can obtain high prediction accuracy, with reduced computational time and an easy calculation process, in predicting effluent water quality parameters compared with other existing algorithms.

Landslide displacement interval prediction using lower upper bound estimation method with pre-trained random vector functional link network initialization

Interval prediction is an efficient approach to quantifying the uncertainties associated with landslide evolution. In this paper, a novel method, termed lower upper bound estimation (LUBE), of constructing prediction intervals (PIs) based on neural networks (NNs) is applied and extended to landslide displacement prediction. A random vector functional link network (RVFLN) is adopted as the NN used in the improved LUBE. A hybrid evolutionary algorithm, termed PSOGSA, that combines particle swarm optimization (PSO) and gravitational search algorithm (GSA) is utilized to train LUBE. The loss function of LUBE is redesigned by considering the quality of PI centre, which allows for a more comprehensive evaluation of PIs. The population initialization in the training process of LUBE is implemented by transferring the weights of a series of pre-trained RVFLNs. The performance of the improved LUBE method is validated by considering a comprehensive set of cases using seven benchmark datasets. In addition, a hybrid method that integrates ensemble empirical mode decomposition (EEMD) with the improved LUBE is proposed for the special case of landslide displacement prediction. Six real-world reservoir-induced landslides are considered to validate the capability and merit of the proposed hybrid method.

Forecasting of Landslide Displacement Using a Probability-Scheme Combination Ensemble Prediction Technique

Data-driven models have been extensively employed in landslide displacement prediction. However, predictive uncertainty, which consists of input uncertainty, parameter uncertainty, and model uncertainty, is usually disregarded in deterministic data-driven modeling, and point estimates are separately presented. In this study, a probability-scheme combination ensemble prediction that employs quantile regression neural networks and kernel density estimation (QRNNs-KDE) is proposed for robust and accurate prediction and uncertainty quantification of landslide displacement. In the ensemble model, QRNNs serve as base learning algorithms to generate multiple base learners. Final ensemble prediction is obtained by integration of all base learners through a probability combination scheme based on KDE. The Fanjiaping landslide in the Three Gorges Reservoir area (TGRA) was selected as a case study to explore the performance of the ensemble prediction. Based on long-term (2006–2018) and near real-time monitoring data, a comprehensive analysis of the deformation characteristics was conducted for fully understanding the triggering factors. The experimental results indicate that the QRNNs-KDE approach can perform predictions with perfect performance and outperform the traditional backpropagation (BP), radial basis function (RBF), extreme learning machine (ELM), support vector machine (SVM) methods, bootstrap-extreme learning machine-artificial neural network (bootstrap-ELM-ANN), and Copula-kernel-based support vector machine quantile regression (Copula-KSVMQR). The proposed QRNNs-KDE approach has significant potential in medium-term to long-term horizon forecasting and quantification of uncertainty.