A runoff probability density prediction method based on B-spline quantile regression and kernel density estimation

•Probability density prediction can measure uncertainties of runoff.•B-spline quantile regression is proposed to predict the probability density of runoff.•Different evaluation criteria of probabilistic predictions is adopted.•The new method is used in Shigu station in Jinsha River of China. Exact and dependable runoff forecasting plays a vital role in water resources management and utilization. This paper proposes a B-spline quantile regression probability density prediction method to predict future runoff and quantify the uncertainty of prediction. The method includes three steps. First, the B-spline function is used to perform spline processing on the runoff data. Secondly, the spline-processed training data is input into the quantile regression model to calculate the parameters of the B-spline quantile regression model, and the successfully constructed B-spline quantile regression model is combined with kernel density estimation to construct a probability density prediction method. Finally, the constructed B-spline quantile regression probability density prediction method is used to forecast future runoff flow, and quantitatively analyze the relevant prediction uncertainty. Six evaluation indicators are constructed, among which the root mean square error, the deterministic coefficient, the pass rate are the evaluation metrics of point predictions based on probability mean, median and mode; the prediction interval coverage probability, prediction interval normalized average width and continuous ranked probability score are the evaluation criteria of interval predictions and probabilistic forecasting. The presented method is able to depict the probability density curve of future runoff flow, and obtain more comprehensive information than point forecasting and interval predictions. As a case study, this method is applicable to the Shigu station of the Jinsha River in China. The results show that the results of the proposed method are better than that of existing some state-of-the-art methods. From the perspective of application, the pass rate and the deterministic coefficient of the method have reached the grade A of accuracy. Therefore, the B-spline quantile regression model provides an alternative scheme to runoff prediction.