Quantile-Based Hydrological Modelling

Predictive uncertainty in hydrological modelling is quantified by using post-processing or Bayesian-based methods. The former methods are not straightforward and the latter ones are not distribution-free (i.e., assumptions on the probability distribution of the hydrological model’s output are necessary). To alleviate possible limitations related to these specific attributes, in this work we propose the calibration of the hydrological model by using the quantile loss function. By following this methodological approach, one can directly simulate pre-specified quantiles of the predictive distribution of streamflow. As a proof of concept, we apply our method in the frameworks of three hydrological models to 511 river basins in the contiguous US. We illustrate the predictive quantiles and show how an honest assessment of the predictive performance of the hydrological models can be made by using proper scoring rules. We believe that our method can help towards advancing the field of hydrological uncertainty.

Additive Ensemble Neural Network with Constrained Weighted Quantile Loss for Probabilistic Electric-Load Forecasting

This work proposes a quantile regression neural network based on a novel constrained weighted quantile loss (CWQLoss) and its application to probabilistic short and medium-term electric-load forecasting of special interest for smart grids operations. The method allows any point forecast neural network based on a multivariate multi-output regression model to be expanded to become a quantile regression model. CWQLoss extends the pinball loss to more than one quantile by creating a weighted average for all predictions in the forecast window and across all quantiles. The pinball loss for each quantile is evaluated separately. The proposed method imposes additional constraints on the quantile values and their associated weights. It is shown that these restrictions are important to have a stable and efficient model. Quantile weights are learned end-to-end by gradient descent along with the network weights. The proposed model achieves two objectives: (a) produce probabilistic (quantile and interval) forecasts with an associated probability for the predicted target values. (b) generate point forecasts by adopting the forecast for the median (0.5 quantiles). We provide specific metrics for point and probabilistic forecasts to evaluate the results considering both objectives. A comprehensive comparison is performed between a selection of classic and advanced forecasting models with the proposed quantile forecasting model. We consider different scenarios for the duration of the forecast window (1 h, 1-day, 1-week, and 1-month), with the proposed model achieving the best results in almost all scenarios. Additionally, we show that the proposed method obtains the best results when an additive ensemble neural network is used as the base model. The experimental results are drawn from real loads of a medium-sized city in Spain.

Unified Quantile Regression Deep Neural Network with Time-Cognition for Probabilistic Residential Load Forecasting

Residential load forecasting is important for many entities in the electricity market, but the load profile of single residence shows more volatilities and uncertainties. Due to the difficulty in producing reliable point forecasts, probabilistic load forecasting becomes more popular as a result of catching the volatility and uncertainty by intervals, density, or quantiles. In this paper, we propose a unified quantile regression deep neural network with time-cognition for tackling this challenging issue. At first, a convolutional neural network with multiscale convolution is devised for extracting more behavioral features from the historical load sequence. In addition, a novel periodical coding method marks the model to enhance its ability of capturing regular load pattern. Then, features generated from both subnetworks are fused and fed into the forecasting model with an end-to-end manner. Besides, a globally differentiable quantile loss function constrains the whole network for training. At last, forecasts of multiple quantiles are directly generated in one shot. With ablation experiments, the proposed model achieved the best results in the AQS, AACE, and inversion error, and especially the average of the AACE is grown by 34.71%, 75.22%, and 32.44% compared with QGBRT, QCNN, and QLSTM, respectively, indicating that our method has excellent reliability and robustness rather than the state-of-the-art models obviously. Meanwhile, great performances of efficient time response demonstrate that our proposed work has promising prospects in practical applications.

VARIABLE SELECTION FOR PARTIALLY LINEAR VARYING COEFFICIENT QUANTILE REGRESSION MODEL

In this paper, we propose a variable selection procedure for partially linear varying coefficient model under quantile loss function with adaptive Lasso penalty. The functional coefficients are estimated by B-spline approximations. The proposed procedure simultaneously selects significant variables and estimates unknown parameters. The major advantage of the proposed procedures over the existing ones is easy to implement using existing software, and it requires no specification of the error distributions. Under the regularity conditions, we show that the proposed procedure can be as efficient as the Oracle estimator, and derive the optimal convergence rate of the functional coefficients. A simulation study and a real data application are undertaken to assess the finite sample performance of the proposed variable selection procedure.