Integrated forecasting method of medium-and long-term runoff by ridge regression based on optimal sub-model selection

With the development of the 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 (Ai et al. 2022). Therefore, as the most important task in hydrology forecasting, accurate and reliable medium- and long-term runoff forecasting is essential to improve response efficiency of flood disaster preparedness and disaster resistance (Lv et al. 2020).

At present, many forecasting models have been proposed and widely applied in medium-and long-term runoff, which include multiple linear regression (MLR) (Maniquiz et al. 2010; Ahani et al. 2018), autoregressive moving average (ARMA) (Valipour et al. 2013), back propagation neural network (BPNN) (Chang & Li 2017), wavelet neural network (WNN) (Shoaib et al. 2018), support vector regression (SVR) (Kisi & Parmar 2016; Adnan et al. 2022), extreme learning machine (ELM) (Yaseen et al. 2019), deep belief networks (DBN) (Yue et al. 2023), long–short term memory network (LSTM) (Gao et al. 2020; Xiang et al. 2020), etc. Although these models have merits on their own, due to the intrinsic weaknesses of all models, and the uncertainty and complexity of runoff change, one forecasting model cannot improve runoff accuracy fundamentally.

However, to combine information and disperse errors from different models, combination forecasting model was proposed, and many studies have proposed that it can improve the forecast performance by a combination of multiple models. For instance, Xu et al. (2013) applied the adaptive federated filtering algorithm to combine the forecasting models, and demonstrated that multi-model information fusion can enhance the stability and accuracy of prediction. Chu et al. (2017) developed a Bayesian model averaging (BMA)-based multi-model, and performed better than those of the other models. Ai et al. (2022) proposed a combination prediction method using ELM to predict the medium- and long-term runoff for better prediction performance and stronger robustness. While they proved that forecast combination improves accuracy, there is limited research on the sub-models selection in the medium-and long-term runoff combination forecasting. Due to the introduction of inappropriate sub-models in the combined model, the prediction accuracy will be reduced. Thus, how to select the optimal sub-models from the available individual models became important in combination forecast.

Recently, the methods adopted for sub-models selection were mainly mutual information (MI) (Cang & Yu 2014), max-linear-relevance and min-linear-redundancy (Che 2015), the nearness degree (Su et al. 2019) and neighborhood MI with a maximum relevance and minimum redundancy algorithm (Xiao et al. 2019). Among them, the nearness degree has the advantages of simplicity and quickness, and has been widely used in the selection of individual and combination approaches. Therefore, in this study, we propose the nearness degree to select the optimal sub-models for the medium- and long-term runoff forecasting. In addition, the ridge regression (RR) is used as the combination method to combine optimal sub-models, because it has been verified to be feasible in terms of weather (Feng & Wu 1985), ozone concentration (Ji & Cheng 2018), infrared spectrum (Ding 2019), etc.

Aiming to solve the above problems, this paper proposes a integrated forecasting method of medium- and long-term runoff by RR based on optimal sub-models selection. For this reason, the nearness degree firstly is used to select the optimal sub-models. Second, based on the optimal sub-models, a combination prediction method using RR to predict the medium-and long-term runoff is established. Third, the method is applied to the Guandi hydropower station.

In this section, the methods and data are described in detail, including the data source, the data normalization, the nearness degree of optimal sub-models selection, the RR integrated forecasting, and the framework of the proposed model.

The Yalong River basin is located at 96°52′–102°480 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. 2020). The Guandi hydropower station, located in the lower reaches of the Yalong River, is a large hydropower hub dominated by power generation, with a total storage capacity of 760 million m³ and a normal water level of 1,330.00 m.

Sub-models mean that the number of single models participating in the forecast is at least 2, that is, ⁠. If an inappropriate single forecast model is introduced when combining them, the forecast accuracy may be reduced (Ling & Zhang 2019). Therefore, in order to improve the prediction accuracy, it is necessary to select the optimal sub-models before combining them. In this paper, the optimal prediction accuracy vector will be constructed by the relative error, and the sub-models will be selected by the close degree between the vectors. The best prediction accuracy vector is defined in the following (Su et al. 2019).

It is assumed that is the minimum relative error of p prediction models at time t, and is called the best prediction accuracy vector. The smaller the value is, the higher the prediction accuracy of the model is. If and e are very close, it indicates that the prediction accuracy of the forecast model is high. Therefore, the nearness between and e can be used to determine the prediction accuracy of the forecast model. The concept of proximity is as follows:

Based on the optimal sub-models selection, RR is adopted for integrated forecasting. The principle is as follows (Feng & Wu 1985):

In this paper, before using RR to establish integrated forecasting, sub-models are sorted by using proximity degree. Then, according to the priority ranking, a single model participates in the RR integrated forecast, and the constructed RR integrated forecast model is optimized, in order to determine the optimal RR integrated forecast model and sub-models. The detailed steps are as follows:

• S1. The relative error vector is obtained by calculating the of sub-models F;

• S2. Determine the best prediction accuracy vector ⁠;

• S3. Calculate the approximation degree ⁠;

• S4. The is sorted from large to small, and if ⁠, the corresponding model is sorted as According to the principle that the forecasting effect of the forecasting model ahead of the priority is better than that of randomly selecting the same number of forecasting models (Ling & Zhang 2019), RR is used to do the integrated forecast according to the ⁠., which is recorded as ⁠;

• S5. By using the method of subsample estimation, the estimated value of RR is obtained according to the Formulas (4)–(6), and the RR integrated forecast equation is obtained by bringing it in (3);

• S6. On the basis of obtaining the integrated prediction results, the nearness degree is calculated according to step S1–S3, and the model is optimized according to its value. Where is the best prediction accuracy vector of RR model, and is the relative error vector of the j th RR model;

• S7. If ⁠, RR-j is the optimal integrated forecast model, and is the optimal single forecast model participating in the integration.

The RMSE is utilized to assess the accuracy of the simulated values, with a closer value to zero indicating a better match between the simulated and observed values. The closer the DC value is to 1, the more consistent the proposed forecast model is with the actual situation. The higher the QR value, the higher the accuracy of the proposed forecast model. The MAE represents the average absolute deviation between individual observations and the arithmetic mean, providing an accurate reflection of the actual prediction error. The MAPE, being the most frequently used statistical index, is employed to examine the error between the predicted and observed values (Ai et al. 2022). Among them, smaller values of RMSE, MAE, and MAPE indicate better forecasting performance, whereas higher values of DC and QR reflect improved overall model performance.

Due to the lag effect of climate-related factors on runoff (Cheng et al. 2019), this paper takes 2a lag as the term to apply 2,304 (96*24) alternative input variables to 96 climatic factors of the Guandi hydropower station and normalizes the data according to Equation (1). Based on the method of combining correlation coefficient and stepwise regression (Ai et al. 2022), reasonable input factors are selected, and the results are shown in Table 2.

Based on the literature review, an individual prediction model cannot perform optimally in any environment for precise and stable mid- to long-term runoff prediction. Therefore, we consider as many sub-models as possible for the adaptive sub-model selection to ensure the prediction accuracy of the integrated forecasting method. In this study, four individual forecast models are used, including MLR, BP, WNN, and SVR; in addition, there are three combination methods, simple average (SA), GRI and RR, are selected for the comparison. To determine the most acceptable model, it is necessary to find the suitable parameters for each of the above algorithms, as follows.

For the combination forecasting models, the SA combination method can be expressed as where ⁠, is the forecast value (output) from the th single forecasting model and is the combined forecast model at time t, M is the total number of individual forecasting models. Thus, the weight W of SA in this paper is 1/3, 1/3, and 1/3, respectively. According to the Equation (3), the parameters of GRI is ⁠. And, the ridge estimate of RR is 0.8790, 0.0384, and 0.1007, respectively.

Based on the steps S1–S3 and formula (3), the nearness of sub-models (i = 1,2,3,4) and e in the verification period are 0.5463, 0.6121, 0.5761, and 0.7002, respectively. In terms of the nearness value, the SVR exhibits the highest prediction accuracy, followed by BPNN and WNN, while MLR shows the worst performance. This outcome can be attributed to the numerous factors influencing medium-and long-term hydrological processes, most of which are expressed through complex nonlinear relationships. The SVR model is well-suited for addressing nonlinear problems by providing an effective mapping between input and output data in a higher-dimensional feature space, thereby enhancing forecasting accuracy. Moreover, the SVR algorithm operates based on the structural risk minimization criterion. To minimize the expected risk, it is crucial to simultaneously minimize the empirical risk and the confidence range. This approach involves maintaining a fixed training error while minimizing the confidence range, effectively addressing over-learning issues and enhancing the model's ability to generalize across samples. Given the linear nature of MLR, it is considered less suitable for capturing these complexities. Therefore, BPNN, WNN, and SVR are considered as the individually optimal nonlinear models.

According to the step S4 and sub-models sequencing, in this paper, RR is used for integrated forecasting according to the methods of SVR-BPNN, SVR-BPNN-WNN, and SVR-BPNN-WNN-MLR, denoted as RR-j (j= 1, 2, 3, 4). On the basis of the integrated prediction results, according to steps S1–S3 and formula (3), the nearness degrees of and of RR-2, RR-3 and RR-4 integrated prediction models are 0.9223, 0.9299 and 0.9077, respectively. It can be seen that RR-3 model has the highest prediction accuracy, followed by RR-2 and RR-4 models. The main reason should be that the prediction accuracy of MLR in the single model involved in integration is the worst, which leads to the unsatisfactory prediction results of the integrated forecast, indicating that it is necessary to select the single model involved in integration when constructing the integrated forecast model (Ling & Zhang 2019).

It is known that SVR, BPNN and WNN are the optimal sub-models, and RR-3 is the optimal integrated forecast model. The specific integration process of the RR-3 model is as follows. Based on obtaining SVR(f₁), BPNN(f₂), WNN(f₃) prediction results, the ridge estimate of the RR-3 integrated prediction model is obtained according to step S5. It can be concluded that SVR model has the best prediction effect, followed by WNN and BPNN. Then, by substituting into Equation (4), the equation of integrated prediction RR-3 is obtained as ⁠.

Based on the comprehensive comparison and analysis of the forecasting effects of BPNN, WNN, SVR, SA, GRI, and RR-3 models during the training period and verification period, it is found that the RR-3 can get best forecasting performance. This is mainly because the BPNN model has a strong nonlinear mapping ability and can reflect the nonlinear characteristics of runoff. WNN model combines the advantages of wavelet analysis and BPNN model, and has high prediction accuracy, but its qualified rate decreases obviously in the verification period, indicating that WNN model has the phenomenon of over-fitting and the reliability of the model is reduced. After training, BPNN and WNN models establish a network model based on empirical risk minimization, which has some shortcomings such as local minimum and instability. For example, during the model verification period, the evaluation indexes RMSE and DC of BPNN model perform relatively well, but its qualified rate QR is relatively poor. The SVR model utilizes kernel functions and employs structural risk minimization as the guiding principle, ultimately yielding a unique solution that can address some of the shortcomings of the aforementioned models (Liang et al. 2020), resulting in a better simulation prediction effect compared to BPNN and WNN.

On the other hand, in the SA combination model, each individual predictive model contributes equally (with the same weight) to the combined value, but is less reliable. The GRI combination model is a kind of unequal weight method which finds the weight by linear fitting. However, if the prediction results of single models are highly correlated, the mean square error of weights will be very large and unstable. For example, during the testing period, the prediction effect of GRI combination model is worse than that of SA. Conversely, the RR-3 combination model avoids the above problems by adding ridge parameters, obtaining appropriate weights, and demonstrating better prediction performance. To sum up, the RR-3 model integrates the advantages of each single model and improves the accuracy of medium-and long-term runoff forecasting.

To consider the influence of sub-models on combination models, a medium- and long-term runoff integrated forecasting method based on optimal sub-models selection was proposed in this paper. And, it is applied to the medium- and long-term runoff forecast of the Guandi hydropower station in Yalong River Basin. The main findings can be briefly concluded as follows. First, SVR, BPNN and WNN are identified as the optimal sub-models, with RR-3 being the optimal integrated forecasting model composed of the aforementioned three single forecasting models. Second, through the comparative analysis of the runoff forecasting results, it is evident that the RR-3 integrated forecasting model demonstrates a good forecasting effect, which proves that the sub-models can affect the accuracy of runoff combination forecasting.

Based on the accomplished results, there are still some areas that need to be improved. In the future, a nonlinear combination method will be considered to improve the accuracy of medium- and long-term runoff forecasting. Besides, for the state-of-the-art single model in the combination model, we will adopt deep learning methods, such as LSTM and Convolutional Neural Network (CNN). Moreover, the latest data would be updated and supplemented, including human activity, evaporation, temperature, and so on.

This work was supported by ‘Accurate Extraction Research and Product Realization of Water Information About Impervious Surface from High-resolution Images’(Grant No. NJPI-RC-2023-06) and ‘Jiangsu Province Vocational Education Teaching Reform Key Project Funding’ (Grant No. ZZZ18).

C.B. wrote the original draft, methodology, formal analysis. C.Z. was involved in conceptualization, writing-reviewing, and funding acquisition. S.C. performed methodology and data curation. S.Y. collected resources, and was involved in data curation, writing – review and editing.

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

The authors declare there is no conflict.