A novel medium- and long-term runoff combined forecasting model based on different lag periods

With the development of national economy and the adjustment of national water control policy, the gap between the existing hydrologic medium- and long-term runoff forecasting methods and the demand for production and application have been further widened. In addition, the increasing amount of hydrological data can introduce redundant and noisy information to the prediction feature or factor, which may deteriorate the performance of the mid- to long-term runoff prediction (Yue et al. 2020a). Therefore, runoff prediction is not only a management issue but also a scientific problem, and accurate runoff forecasts will be of vital importance to the policymakers.

Generally, the models to predict medium- and long-term runoff fall into two main categories: process-driven models and data-driven models. The process-driven model is based on the conception of hydrology, with which the discharge forecasting can be performed by simulation of the runoff variation and river channel evolution. Yet, the runoff process is highly complex and affected by many factors, such as runoff, atmospheric circulation index, climate and topography (Ekwueme & Agunwamba 2020). A precise construction of a physics-based model is very difficult to attain as an abundance of this data is required to fulfill the initial conditions (Yaseen et al. 2019). In comparison, the data-driven model can make the best use of existing data to achieve a predetermined model structure (Lu et al. 2021).

One of the most important steps in the data-driven model development process is determining the significant input variables. A subset of compact and informative inputs with the max-relevance and min-redundancy can significantly enhance the model performance and reduce the demand for training samples (Lyu et al. 2017). However, the determination of a suitable set of inputs is challenging due to the time lag of runoff response to the influence of large-scale atmospheric circulation (Cheng et al. 2019). The existing studies on the lag selection of runoff factors are mainly carried out for 1 or 2 years (Gao 2006; Yang et al. 2013; Li et al. 2019; Yue et al. 2020a). It is generally known that it is difficult to accurately determine the lag period since runoff flows through different field conditions to reach a point (He et al. 2011). To select the appropriate lag period, the 6, 12, 18 and 24 months as the different lag periods were considered sufficiently for capturing the input factors. The methods adopted for factors selection were mainly the correlation coefficient method (Yang et al. 2013,;Shan et al. 2015), stepwise regression analysis (Shan et al. 2015), principal component analysis (PCA) (Lu et al. 2021), kernel principal component analysis (KPCA) (Li et al. 2019), mutual information (Lu & Zhou 2014) and partial mutual information (PMI) (He et al. 2011). Among them, the correlation coefficient method and stepwise regression analysis have the advantages of simplicity and quickness, and have been widely used in the field of hydrological prediction. Therefore, in this study, we propose a combined use of correlation coefficient and stepwise regression to determine the input factors of different lags.

Prediction model design is another important research content in mid- to long-term runoff prediction. A lot of approaches, including multiple linear regression (MLR) (Schilling & Wolter 2005; Wang et al. 2014), autoregressive (AR) techniques, autoregressive moving average (ARMA), autoregressive integrated moving average (ARIMA), seasonal autoregressive integrated moving average (SARIMA) (Sang et al. 2013; Valipour et al. 2013; Valipour 2015), feed-forward back propagation-neural network (FFBP-NN) (Wang et al. 2019), support vector regression (SVR) (Wei et al. 2012; Sahoo et al. 2018) and Elman neural network (Chu et al. 2017), have been widespread in medium- and long-term runoff forecasting. However, due to the intrinsic weaknesses of all models, and the uncertainty and complexity of runoff change, one forecasting model cannot improve runoff accuracy fundamentally, particularly when runoff data has nonlinear characteristics. Therefore, a combined model is used to exploit and integrate the diversity of skillful prediction from different models and reduce their uncertainty (Chen et al. 2019; Lu et al. 2021). It is divided into linear and nonlinear combination prediction, according to the relationship among the combined single-item prediction models. Linear combination prediction refers to the linear coupling of various single-item prediction methods. However, the possible emergence of negative weights and other controversial issues restricts the generalization of linear combination prediction to a certain extent. Nonlinear combination prediction refers to the nonlinear coupling of various single-item prediction methods, which overcome the limitation of linear combination prediction. The artificial neural networks have some unique and distinguishing features such as nonlinearity, parallel processing, robustness, adaptability and self-organization (Xu et al. 2021). An extreme learning machine (ELM) has the capability of approaching any complex continuous function and is easy to fit the nonlinearity of the predicted objects. It is proposed as a feed-forward neural network with a single hidden layer for determining weights related to the output. The ELM has been successfully applied for the estimation of reference evapotranspiration (Liu et al. 2017), river flow forecasting (Lima et al. 2016; Yaseen et al. 2019), mid- to long-term runoff forecasting (Yue et al. 2020b), longitudinal dispersion coefficients in water pipelines (Sabed-Movahed et al. 2020) and fast ionospheric delay (Zhao et al. 2021). Consequently, this paper proposes a combination prediction method using ELM to predict the medium- and long-term runoff for better prediction performance and stronger robustness.

The important point for solving the above problems is to select the optimal lag period of key factors that are closely related to the runoff and design a prediction model. Aiming to determine the optimal lag, this paper takes 6, 12, 18 and 24 months lag as the different lag periods. Moreover, to solve the problems caused by the low accuracy and poor stability of the single forecasting model, a combination prediction method using ELM to predict the medium- and long-term runoff is established, in which the random input weights and hidden biases in ELM are optimized by particle swarm optimization (PSO). The main goal of this study is achieved by (1) exploring the influence of atmospheric circulation factors on runoff in different lag periods and (2) proposing a combination prediction method using ELM to predict the medium- and long-term runoff.

The traditional combination forecasting methods just combine the forecasting results of several forecasting models together, average the weight coefficient in individual models or use an optimization algorithm to optimize the weight coefficient (Chen et al. 2019). A novel combined forecasting method is proposed in this paper, which puts the intermediate forecasting results of MLR, FFBP-NN and SVR into ELM. In this approach, this paper initially uses the time-delay correlation analysis and stepwise correlation analysis to select the optimal forecasting factor as the inputs. Then, three single prediction models, namely MLR, FFBP-NN and SVR, are established and used to predict runoff. And then, ELM was used to combine the intermediate forecasting results of the three single prediction models. At last, considering that the random input weights and hidden biases of ELM always have some influence on the training process, we use PSO to optimize the parameters to derive the resultant forecast. The flowchart of the combined model is shown in Figure 1; from the figure, we can conclude that the proposed method contains the following four steps.

Step 1. Factors selection. The input characteristic factors of different lag periods are obtained by the time-delay correlation analysis and stepwise regression analysis. The illustrative diagram is shown in Figure 1(a).

Step 2. Single model learning. The training set was used to train MLR, FFBP-NN and SVR, respectively, to select the parameters of the three models. The three trained models are applied to the training set and the test set, respectively, and the respective prediction results are obtained, which are used as the training set and test set of the combination forecasting model. The process of single model learning is represented in Figure 1(b).

Step 3. Combined model forecasting. MLR, FFBP-NN and SVR are regarded as approaches to forecast inflow runoff, respectively, and their intermediate forecasting results are used as input of ELM, and then train the ELM repeatedly to get the final forecasting results. Figure 1(c) shows the specific process of the combined model.

Step 4. Parameters optimization. Two parameters in ELM are very important in the forecasting process, so the random input weights and hidden biases are optimized by PSO, then the output weights could be calculated. Figure 1(d) gives the schematic diagram of the optimization procedure.

The main idea of stepwise regression analysis is to introduce predictors step by step. At each step in the process, after a new variable is added, a test is made to check if some variables can be deleted without appreciably increasing the residual sum of squares (RSS). The procedure terminates when the measure is (locally) maximized or when the available improvement falls below some critical value.

ELM is an innovative machine learning algorithm that is characterized mainly by the fact that there is no need for a tuning of the model's internal parameters (i.e., the hidden neurons). In essence, ELM is an improved version of the conventional ANN model where it is able to solve regression problems with a reduced model execution time. This is because the input weights and biases are randomly generated so that the output weights have a unique least-squares solution solved by the Moore–Penrose generalized inverse function (Yaseen et al. 2019). The general structure of the ELM model can be visualized in Figure 2.

PSO is a population-based stochastic optimization technique inspired by the social behavior of bird flocking (Motahari & Mazandaranizadeh 2017). In PSO, all particles have a fitness value, which is determined by the fitness function. Each particle can save the best position to be searched, and the particle is also given a speed to determine the search direction and distance. In the process of searching, the particle tracks two extremes to complete the iteration of a particle search. The optimal solution of the particle search is called the individual extreme value pBest. The optimal solution of the whole particle swarm search is called the global extremum gBest.

The MAPE is the most frequently used statistics index and is employed for examining the error between the predicted value and the observed value. The RMSE is used to measure the predicted value accuracy. An RMSE value of zero indicates a perfect match between the predicted value and the observed value. The MAE is the average of the absolute value of the deviation between all individual observations and the arithmetic mean, which can accurately reflect the actual prediction error. Likewise, the DC provides a measure of the capacity of the model to predict observed values. And, the smaller the values of the MAE, MAPE and RMSE are, the better forecasting performance is. The higher the DC value shows, the better forecasting performance is.

The Yalong River basin is located at 96°52′–102°48′ E, 26°32′–33°58′ N, with an area of about 136,000 km². The river is 1,571 km long with a drop of 3,830 m and is one of the rivers with the most abundant water energy resources in China (Yue et al. 2020b). The precipitation in the basin increases from north to south, and the east is greater than the west. The annual precipitation in the Heyuan area is 500–600 mm, and the annual precipitation in the middle and lower reaches is 900–1,300 mm. The runoff was mainly from precipitation, and the annual variation and regional distribution of runoff were basically consistent with the variation trend of precipitation. The annual runoff distribution can be roughly divided into the period from June to October, during which rainfall is the main replenishment and the water volume accounts for 76.8% of the whole year. The hydrographic stations in the Yalong River basin, Lianghekou and Jinping are shown in Figure 3.

To prove that the combined model proposed in this paper can improve medium- and long-term runoff forecasting accuracy, the collected datasets include: (1) monthly inflow runoff data of Lianghekou and Jinping stations from January 1960 to December 2011 were provided by the Hydrographic Bureau of the Yangtze River Water Conservancy Commission; (2) 130 atmospheric circulation index data from the national climate center (https://www.ncc-cma.net/Website/index.php), the length of the sequence for January 1951–May 2020. And the research period spans over 50 years from January 1962 to December 2011, which is divided into two parts: a training set (from January 1962 to December 2001) and a forecasting test set (from January 2002 to December 2011) in this study. In relation to this, Figure 4 shows the dataset structure of Lianghekou and Jinping stations, which were selected as the forecast objects. It can be seen from Figure 4 that the abundance and drought of the three stations are very similar, indicating that the abundance and drought of the Yalong River basin are basically the same. In addition, we have done a statistical analysis of the three datasets. The statistical characteristics, including the maximum, minimum, means, standard and median, of the runoff series at the different time period divisions are summarized in Table 1.

In our study, considering the time consistency of serial data and the missing term of atmospheric circulation index, the alternative candidate predictive factors include 95 teleconnection climate factors and antecedent runoff in Lianghekou and Jinping stations of the Yalong River basin. For the convenience of analysis, they are abbreviated as a1, a2,…, a96. In addition, due to the lag effect of climate-related factors on runoff (Yue et al. 2020a), this paper took 6, 12, 18 and 24-month lag as the different lag periods. Thus, the input factors are expressed as a1(t− 1), a1(t − 2),…, a1(t−T),…, a96(t − 1), a96(t − 2),…, a96(t−T) (T = 6, 12, 18, 24), including (96*T) variables.

The input selection was initially conducted based on the time-delay correlation coefficient with different lag periods between each factor and runoff. To control the number of primary factors, the maximum correlation coefficient and the absolute value of the correlation coefficient are greater than 0.7. Thus, the input factors of the 6- and 12-month primary elections in the Lianghekou lag period are shown in Table 2. The primary factors at 18 and 24 months in the screen lag period are shown in Table 3. The correlation between the above factors and the runoff coefficient has passed the significance test with a confidence level of 95%.

And then, to eliminate multicollinearity among problem variables, the stepwise regression analysis was applied, and the results of the significance of the factors are presented in Tables 4–7. Tables 4 and 5 show that the stepwise regression analysis indicates that the factors selected at 6 and 12 months of the lag period of the Lianghekou hydrological station were both statistically significant in predicting month runoff (sig. < 0.05) with significant F values of 260.672 and 200.097 (Ekwueme & Agunwamba 2020). The factors affecting the change of runoff process of the Lianghekou station when the lag period is 6 months are mainly related to a43, a96, a29, a2, a3, a42 and a36; for a lag period of 12 months, the factors affecting the runoff change are mainly in connection with a36, a42, a64, a2, a43, a44, a96 and a31 (see the bold values in Table 2). As can be seen from Tables 6 and 7, the factors selected at 18 and 24 months of the lag period of the Jinping hydrological station were both statistically significant in predicting monthly runoff (sig. < 0.05) with significant F values of 264.787 and 276.324. The factors affecting the change of runoff process for a lag period of 18 months are mainly associated with a36, a43, a64, a96, a82, a44, a13 and a33; for a lag period of 24 months, the factors affecting the runoff change are mainly related to a36, a96, a43, a83, a18, a44, a82 and a33 (see the bold values in Table 3). A similar procedure was implemented to decide the input factors for the remaining different lag periods of Lianghekou and Jinping stations. The inputs retained in the final subset of Lianghekou and Jinping stations are formed in Table 8.

To sufficiently demonstrate the superiority of the proposed model for medium- and long-term runoff forecasting, the current state-of-the-art data-driven models, including MLR, FFBP-NN, SVR and ridge regression (RR), are selected for the comparison. MLR is a statistical approach to modeling the relationship between a dependent variable and independent variables for runoff forecasting; FFBP-NN is a traditional shallow neural network; SVR is based on statistical learning theory and the structural risk minimization hypothesis to achieve good robustness and efficiency; RR is put forward on the basis of general regression integrated forecast. Among them, MLR, FFBP-NN and SVR are three single-item prediction models; MLR is a linear model; and BP and SVR are nonlinear machine learning algorithms constructed based on different theories. The RR model is one of the traditional combination forecasting methods, which can solve the problem of coefficient selection of a single model for a linear combination prediction model.

The different sections of BP and SVR are mainly shown in (1) the optimization goal of the BP algorithm which is based on the empirical risk minimization criterion to minimize the training error between the network output and the ideal output. It means that the curves and surfaces fitted by the BP neural network go through the training sample points as much as possible, which will cause the BP algorithm to rely too much on learning samples and has over-learning problems, so it is difficult to obtain good generalization ability. The SVR algorithm is based on the structural risk minimization criterion. In order to minimize the expected risk, the empirical risk and the confidence range should be minimized at the same time. In the SVR learning algorithm for regression, we construct a regression estimation function to minimize the VC dimension of the function on the premise that the distance from the target value is less than ε. That is, keeping the training error fixed and minimizing the confidence range, which solves the over-learning problem and has a better ability to generalize the samples. (2) The idea of the BP algorithm is that the forward propagation of the signal and the backpropagation of the error are carried out repeatedly, so that the weights are adjusted continuously, so as to ensure that the training error of the network output is minimized. In essence, it is an iterative learning algorithm based on gradient descent, which has some defects such as slow learning convergence and easy to fall into local minima. The SVR algorithm reduces the above regression estimation problem to a convex quadratic programming problem with linear equality constraints and linear inequality constraints, which can ensure the global optimality of the algorithm and effectively overcome the curse of dimensionality.

To verify the versatility and university of the proposed combined method, the stations are used as experiments. The specific parameters of the two stations have some differences, in the development of FFBP-NN, the training function is ‘tansig’, the learning function is ‘logsig’, the maximum training times are 1,000, the learning rate is 0.1, the model training adopts LM algorithm, the momentum factor is 0.9 and the expected error is 0.001. For Lianghekou station, when the lag period is 6 months, networks with 2–15 hidden neurons were evaluated to determine the optimal network. The performance measure, RMSE, in training and testing, is demonstrated in Figure 5. The improvement of model performance in training was found due to the increase of hidden neurons, but the performance degrades when hidden neurons were larger than 6. Similar trends were also generally observed for the testing. Thus, the structure of FFBP-NN is 7-6-1 (neurons in the input layer-neurons in the hidden layer-neurons in the output layer). A similar procedure was implemented to decide optimal architectures for different lag periods. Optimal networks for a lag period of 12, 18 and 24 months are 8-7-1, 8-4-1 and 10-11-1, respectively. For the SVR model, the kernel function is ‘sigmoid’, and we use the GridSearch grid to select the best parameters C (regularization parameters) and gamma which the traversal gamma is in the interval [0.1, 2] and C is in the interval [0.5, 5] (Liang et al. 2020). When the lag period is 6, 12, 18 and 24 months, the parameters (C and gamma) of the SVR model are (3.0, 0.23), (0.5, 0.10), (0.5, 0.10) and (0.5, 0.10), respectively. In the development of ELM, we choose the number of intermediate nodes in the same way as FFBP-NN. Thus, when the lag period is 6, 12, 18 and 24 months, the structure of ELM is 3-6-1, 3-7-1, 3-7-1 and 3-8-1, respectively; the combined prediction model selects the same structure as ELM.

A similar procedure was implemented to decide optimal architectures for the Jinping station. The optimal architecture of FFBP-NN, when the lag period is 6, 12, 18 and 24 months, is 8-10-1, 9-7-1, 8-8-1 and 8-11-1, respectively; the optimal parameters (C and gamma) of the SVR model are (0.5, 0.10), (5.0, 0.23), (1.0, 0.10) and (0.5, 0.10), respectively; the structure of ELM is 3-6-1, 3-6-1, 3-7-1 and 3-8-1, respectively; and the combined prediction model selects the same structure as ELM.

With the input variables and parameters selected, the runoff was forecasted using MLR, FFBP-NN and SVR, and then applied ELM to integrate the intermediate results of them. In addition, this experiment compares the presented combined model with the single models and the traditional combination forecasting method RR during the training period. The results of four evaluation metrics are listed in Table 9, and Table 9 shows the following:

• (a)

  For the Lianghekou station, when the lag period is 6 months, compared with the other models, the presented combined model has the minimum MAE, MAPE and RMSE, with the values of 113.00, 0.1708 and 203.21 m³·s⁻¹, respectively; the maximum DC value of 0.8646. Next, for the other three single models, SVR performs better than the other two models, and MLR performs the worst. When the lag period is 12, 18 and 24 months, the presented combined model with MAE values of 150.83, 155.64 and 148.09, respectively, is still the most accurate forecasting model. For the other three single models, there is not a unified law based on the performance metrics. For a lag period of 12 months, based on RMSE and DC, FFBP-NN performs better than the other two models; based on MAE and MAPE, SVR can get better performance. For a lag period of 18 months, based on MAE and MAPE, the order of the other three models from good to bad is SVR, MLR and FFBP-NN; based on RMSE and DC, the order of the other three models from good to bad is SVR, FFBP-NN and MLR. For a lag period of 24 months, in terms of four evaluation metrics, SVR can get better forecasting results than the other two models; among them, the performance of FFBP-NN is better based on MAE and MAPE, and the performance of MLR is better based on RMSE and DC. In the other two combination forecasting methods, ELM has a better fitting effect than the RR model from a lag period of 6–24 months.

• (b)

  For the Jinping station, the presented combined model with MAPE values of 0.1364, 0.1948, 0.1963 and 0.1891, respectively, from a lag period of 6–24 months can get the most accurate forecasting results. For the other three single models, there is not a unified law based on the performance metrics. Based on MAE and MAPE, when the lag is 6–24 months, it is obvious that the value of SVR is lower than the other two single models. Based on RMSE and DC, when the lag period is 6 months, the order of the other three models from good to bad is SVR, FFBP-NN and MLR; when the lag period is 12, 18 and 24 months, FFBP-NN can get better performance, SVR is the second and MLR was the worst. For the other two combined forecasting methods, ELM performs better than the RR model when the lag is 6–24 months.

This section mainly compares the presented combined model with other models in different lag stages of the forecast period during the test period, including the lag period of 6, 12, 18 and 24 months. The results of the four evaluation criteria are calculated in Table 10. From Table 10, the following result can be obtained:

• (a)

  For the Lianghekou station, when the lag period is 6 months, compared with the other models, the presented combined model has the minimum MAE, MAPE and RMSE, with the values of 167.35, 0.2719 and 262.55 m³·s⁻¹, respectively; the maximum DC value of 0.7946. Next, for the other three single models, there is not a unified law based on the performance metrics; based on MAE, FFBP-NN performs better than the other two models, and SVR gets the worst forecasting results; based on MAPE, FFBP-NN performs better than the other two models, MLR gets the worst forecasting results; and based on RMSE and DC, the order of the other three models from good to bad is FFBP-NN, MLR and SVR. When the lag period is 12, the presented combined model with an MAE value of 180.26 is still the most accurate forecasting model. For the other three single models, the SVR model has the minimum MAE, MAPE and RMSE, with the values of 181.83, 0.2865 and 308.31 m³·s⁻¹, respectively, and the maximum DC value of 0.7168; the order of the other two models from good to bad is FFBP-NN and MLR. When the lag period is 18 and 24 months, it is very obvious that the proposed combined model gets the best forecasting results in terms of the four evaluation metrics. For the other three single models, their order from good to bad is SVR, MLR and FFBP-NN based on the four evaluation metrics. In the other two combination forecasting methods, ELM has a better fitting effect than the RR model from a lag period of 6–24 months.

• (b)

  For the Jinping station, when the lag period is 6 months, the proposed combined model with MAE, MAPE, RMSE and DC values of 249.89, 0.1843, 434.76 m³·s⁻¹ and 0.8164, respectively, can get the most accurate forecasting results. Next, for the other three single models, there is not a unified law based on the performance metrics; based on MAE, SVR performs better than the other two models; in terms of the MAPE, FFBP-NN can get better forecasting results; and based on RMSE and DC, it is very obvious that the value of MLR is better than the other two single models. When the lag period is 12, the presented combined model gets the best forecasting results compared with the other three single models based on the four evaluation metrics. For the other three single models, there is not a uniform regularity based on the performance metrics; in terms of the MAE and MAPE, their order from good to bad is SVR, FFBP-NN and MLR; and based on the RMSE and DC, it is very obvious that the value of SVR is better than the other two single models, MLR followed, and FFBP-NN is the worst. For a lag period of 18 and 24 months, the proposed combined model with values of 0.1870 and 0.2169, respectively, is still the most accurate forecasting model. Next, for the other three single models, there is not a unified law based on the performance metrics; in terms of the MAE, RMSE and DC, their order from good to bad is SVR, FFBP-NN and MLR; based on the MAPE, in the lag period of 18 months, SVR performs better than the other two models, FFBP-NN followed, and MLR is the worst; and for a lag period of 24 months, their order from good to bad is SVR, MLR and FFBP-NN. For the other two combined forecasting methods, ELM performs better than the RR model when the lag is 6–24 months.

To evaluate the performance of the proposed approach, the results of the above two experiments will be further discussed with uncertainty and reliability analyses.

(2) If ⁠, then ⁠; otherwise, ⁠, where is the threshold value of runoff forecast. In other words, is defined as the number of times the value of RAE is less than or equal to that of ⁠. The optimum value of based on Chinese standards is 0.2 or equivalently is 20%. In this article, we take 20%.

The results of uncertainty and reliability analyses are presented in Table 11. From Table 11, the following result can be obtained:

• (a)

  For the Lianghekou station, the proposed model represented the lowest value for uncertainty with a 95% confidence level, with uncertainty values of 52.6423, 54.3781, 54.3454 and 53.8330, respectively, from a lag period of 6–24 months in the training phase when compared to the other models. Additionally, predictions of runoff provided by the proposed model were more reliable in comparison with the predictions made by the other models, with reliability values of 68.83, 58.25, 56.67 and 62.71% from a lag period of 6–24 months, respectively. Moreover, the proposed combined model with a 6-month lag can provide better performance than other lag periods. Similar results were also generally observed for the testing stage, that is, the proposed model had the lowest uncertainty and the highest reliability when the lag is 6–24 months (see the bold values in Table 11).

• (b)

  For the Jinping station, when compared with the other models, the presented model showed the lowest uncertainty, with U95 values of 96.8543, 100.8462, 100.1888 and 99.3955, respectively, from a lag period of 6–24 months in the training phase. Additionally, considering reliability, it is conspicuous that the proposed model, with reliability values of 83.50, 63.54, 62.92 and 64.58% from a lag period of 6–24 months, respectively, was the most reliable. Moreover, the proposed combined model with a 6-month lag was able to provide very good performance compared with the other lag periods. In the testing stage, there were also generally observed similar results, that is, the proposed model had the lowest uncertainty and the highest reliability when the lag is 6–24 months (see the bold values in Table 11). And, the effect of a lag period of 6 months was the best in terms of different lag periods.

Developing medium- and long-term runoff forecasting for water resource planning and management activities, such as water conservancy infrastructure operation, flood control and reservoir operation, is of great significance. In view of the influence of different lag periods on runoff and the uncertainty of a single model, a medium- and long-term runoff combined forecasting model based on different lag periods is proposed in this paper.

The experimental results showed that the input factors of different lag periods affect the prediction accuracy of runoff based on the analysis of the above results. In addition, the optimal lag period of the Lianghekou is 6 months, and the factors (Table 12) affecting the change of runoff process are mainly related to the lag time of 1 month of the Area Index of Northern Hemisphere Subtropical High (5E-360), 1 month of the North Africa Subtropical High-Intensity Index (20 W–60E), 1 month of the North Africa, Atlantic and North America Subtropical High-Intensity Index (110 W–60E), 1 month of the Central Intensity of the Northern Hemisphere Polar Vortex (JQ), 1 month of the Tibet Plateau (25N–35N, 80E–100E), 6 months of the Tibet Plateau (30N–40N, 75E–105E) and 1 month of the antecedent runoff. For the Jinping station, the optimal lag period is also 6 months. The factors affecting the change of runoff are shown in Table 12, which are no longer listed here. Among them, the Central Strength of the Northern Hemisphere Polar Vortex (JQ), Tibet Plateau (25N–35N, 80E–100E) and the antecedent runoff (see the bold values in Table 12) are three identical features of Lianghekou and Jinping stations. This indicates that we consider not only the influence of atmospheric circulation on runoff, but also the special climatic impact caused by the geographical location of the Yalong River, such as snow cover on the Tibet Plateau (Xu & Ma 2011).

The experimental results indicate that the proposed combined model performs better than other methods whether it's during training or testing. This is mainly because the classic MLR model is relatively easy to construct with the simplest type of parameters, and it can capture the global trend over an entire input space. Its accuracy, however, is not satisfactory, which may not meet the requirements of medium- and long-term runoff forecasting. The BP model can identify complex nonlinear relationships between input and output data, and its accuracy is satisfactory for runoff forecasting, but there is a risk of over-fitting. The SVR model is also appropriate for reproducing the nonlinear problem, which can provide a suitable mapping between input and output data in a higher-dimensionality feature space to improve the forecasting accuracy. Its parameters need to be determined carefully since they significantly influence the accuracy of the SVR model (Chu et al. 2017). The RR model is one of the traditional combination forecasting methods, which can solve the problem of coefficient selection of a single model for a linear combination prediction model. The possible emergence of negative weights and other controversial issues restrict the generalization of linear combination prediction to a certain extent (Xu et al. 2021). However, the proposed combined model is easy to fit the nonlinearity of the predicted objects and meet the requirements of medium- and long-term runoff forecasting, which overcome the limitation of linear combination prediction.

Based on the accomplished findings, there are still some areas that need to be improved. In future research, more factors such as vegetation index, topography and electricity generation, especially the humans’ activity, will be introduced to better describe the changing process of runoff. In the method of selecting key factors, we concentrate on only the linear relationship and neglect the nonlinear relationship between the impact factors. Thus, we will adopt the PMI method to select the key factors in the future study, which has the advantages of characterizing relationships of linear and nonlinear among factors. In addition, the deep learning approaches will be applied to runoff prediction because the shallow network is degenerated for the medium- and long-term runoff forecasting task.

With the rapid development of cascade hydropower stations in a big basin, the actual operation is becoming more and more in-demand for runoff prediction. Therefore, it is of great significance to develop medium- and long-term runoff forecasting for water resource planning and management activities such as water conservancy infrastructure operation, flood control, reservoir operation and drinking water distribution. Numerous studies that adopted combined forecasting models to enhance runoff forecasting accuracy have been proposed. Nevertheless, some models do not consider the effects of different lag periods on the selection of input factors.

Thus, this paper proposed a medium- and long-term runoff combined forecasting model based on different lag periods. The presented combined model initially uses the lagging correlation coefficient and stepwise regression to extract features as the optimal input factor. Then, the selected optimal factor is predicted by using three common single models, which include MLR, FFBP-NN and SVR. And then ELM is utilized to integrate the forecasting results of the three individual models. In addition, considering that the random input weights and hidden biases of ELM always have some influence on the training process, we use PSO to optimize the parameters. By simulating and experimenting with the runoff data of Lianghekou and Jinping stations from the Yalong River basin, and comparing the proposed combined method with the three individual forecasting methods and the traditional combination forecasting methods, the major findings of the proposed approaches and their applications in this paper are: (1) the lag period of physical factors delay can affect the accuracy of runoff forecasting. And, the optimal lag period of both the Lianghekou and Jinping stations is 6 months. (2) Whether it is during the training or testing, the proposed combined model performs better than the other three individual models and the traditional combination method. (3) The proposed combination model can be considered as a very reliable forecasting tool for runoff forecasting.

This work was supported by ‘the key projects supported by the Major Research Plan of the National Natural Science Foundation of China’ (Grant No. 91846203) and ‘the school research fund of Nanjing Vocational University of Industry Technology’ (Grant No. YK21-05-05).

A.P. is involved in conceptualization, writing – reviewing and editing, and funding acquisition. S.Y. is involved in the writing – original draft, methodology and formal analysis. X.C. is involved in the methodology and data curation. C.B. is involved in the visualization and data curation. Y.Z. is involved in the supervision and resources, and funding acquisition.

The authors declare no conflict of interest.

All relevant data are included in the paper or its Supplementary Information.