## Abstract

In general, accurate hydrological time series prediction information is of great significance for the rational planning and management of water resource systems. Extreme learning machine (ELM) is an effective tool proposed for the single-layer feedforward neural network in regression and classification problems. However, the standard ELM model falls into a local minimum with a high probability in hydrological prediction problems since the randomly assigned parameters (like input-hidden weights and hidden biases) often remain unchanged in the learning process. For effectively improving the prediction accuracy, this paper develops a hybrid hydrological forecasting model where the emerging sparrow search algorithm (SSA) is firstly used to determine the satisfying parameter combinations of the ELM model, and then the Moore–Penrose generalized inverse method is chosen to analytically obtain the weight matrix between the hidden layer and output layer. The proposed method is used to forecast the long-term daily runoff series collected from three real-world hydrological stations in China. Based on several performance evaluation indexes, the results show that the proposed method outperforms several ELM variants optimized by other evolutionary algorithms in both training and testing phases. Hence, an effective evolutionary machine-learning tool is developed for accurate hydrological time series forecasting.

## HIGHLIGHTS

A hybrid method using extreme learning machine and sparrow search algorithm is developed for hydrological forecasting.

The proposed method finds better results than several traditional methods, providing an alternative tool for hydrological forecasting.

## INTRODUCTION

Hydrological time series forecasting has been an important research theme in hydrology, water resources and other related engineering construction fields (Du & Weng 2021; Kaluarachchi 2021; Zhang & Ariaratnam 2021). Accurate hydrological prediction information plays an important role in many producing activities (Chen *et al.* 2019; Pérez Lespier *et al.* 2019; Feng *et al.* 2021a; Niu *et al.* 2021a, 2021b), like hydropower production, flood control, peak operation, and water supply. Due to the comprehensive impact of various factors (like meteorological process, underlying surface, and human activities), the natural hydrological process often shows the characteristics of strong nonlinearity, high mutability, and uncertainty (Peng *et al.* 2019; Won & Kim 2020; Sun *et al.* 2021). Then, many researchers and engineers all over the world have paid great attention to deepening the research and development of hydrological forecasting methods (Tamura *et al.* 1990; Jiang *et al.* 2018; Ghasemlounia & Saghebian 2021). In the past few years, numerous hydrological forecasting approaches have been successfully developed and these existing approaches can be roughly classified into two types: process-based approaches and data-based approaches (Leon *et al.* 2020; Tian 2020; Viccione *et al.* 2020; Birbal *et al.* 2021). In general, the process-based approaches usually involve advanced physical formulas and abundant knowledge to describe the complicated meteorological and hydrological processes. Due to the limitation of the cognitive levels and model descriptions, unsatisfactory forecasting results often become unavoidable, which limits their applications in practical engineering (Chen *et al.* 2020; Dalkiliç & Hashimi 2020; Wu *et al.* 2020). With the rapid development of information technology, many data-based approaches have been introduced to improve this problem, like artificial neural network, adaptive neuro-fuzzy inference system, and support vector machine (Chen & Chau 2016; Moghayedi & Windapo 2019; Adib *et al.* 2021; Ji *et al.* 2021). By gaining knowledge from history, data-based approaches gradually produce satisfactory forecasting results in hydrology even though the detailed physical process of the hydrological time series is not well understood (Feng & Niu 2021; Feng *et al.* 2021b; Niu *et al.* 2021c). Owing to their easy implementation and high forecasting ability, data-based approaches are becoming more and more popular in practical engineering (Guo *et al.* 2013; Yuan *et al.* 2018; He *et al.* 2019).

Single-layer feedforward neural network (SLFN) has proved to be an effective tool for regression and classification problems (Chua & Wong 2011). Generally, traditional gradient-based training tools are widely used to determine the computation parameters of the SLFN model. However, many applications show that the gradient-based methods are easily trapped into local minima because the network structure features are not well considered. To alleviate the defects of the gradient-based methods, extreme learning machine (ELM) has been successfully developed in recent years (Peng *et al.* 2017; Zhou *et al.* 2018; Luo *et al.* 2019). In ELM, the input-hidden weights and hidden biases are randomly determined to analytically obtain the hidden-output weights by the Moore–Penrose generalized inverse method. Compared with the gradient-based methods, ELM shows the advantages of faster training speed, stronger generalization ability and fewer computation parameters. Nevertheless, it is difficult for the ELM model to achieve optimal results because the network parameters are determined in a random manner. In other words, the standard ELM method tends to fall into a local minimum because the values of input-hidden weights and hidden biases are not well chosen. In order to further improve the generalization ability of the ELM method, meta-heuristic evolutionary algorithms are used to optimize model structure and hyperparameters (Niu *et al.* 2021d, 2021e), like particle swarm optimization, gravitational search algorithm, cooperation search algorithm and sine cosine algorithm (Mei *et al.* 2018; Niu *et al.* 2021f).

As a novel swarm intelligence algorithm, the sparrow search algorithm (SSA), inspired by the sparrow's wisdom, foraging and anti-predation behaviors, has been developed to solve the global optimization problems (Tuerxun *et al.* 2021; Yang *et al.* 2021; Yuan *et al.* 2021). In SSA, several modules are carefully designed to balance exploration and exploitation, like producers for searching for food, scroungers for monitoring the producers and watchers for avoiding danger (Truchet *et al.* 2016; Wang *et al.* 2021; Zhang & Ding 2021). The SSA method is employed to deal with a group of numerical functions and engineering problems. The results show that the SSA method outperforms several mature evolutionary algorithms with respect to search rate, solution precision, and local minimum avoidance. Due to its satisfactory search capability, the SSA method is gradually becoming popular in many research fields. However, there are few reports about using the SSA method to improve ELM performance in hydrological forecasting. To fill this research gap, this research develops a hybrid forecasting method where the SSA method is used to search for satisfying network parameters of the ELM model. Then, the hybrid method is used to forecast the long-term runoff time series in different working conditions. The comparative results demonstrate that the developed method outperforms several traditional methods in both training and testing phases. Therefore, it can be concluded that the SSA method is a useful optimizer for finding a better neural network structure for accurate time series simulation, while an effective evolutionary extreme learning machine tool is provided here for hydrological forecasting.

The rest of this study is summarized as below: Section 2 gives information on the hybrid method; Section 3 compares the engineering practicability of the proposed method in hydrological forecasting; and the conclusions are given at the end.

## METHODS

### Sparrow search algorithm (SSA)

Sparrow search algorithm (SSA) is an emerging evolutionary algorithm based on the sparrow's foraging and anti-predation behaviors (Zhou & Wang 2021). Compared with several traditional evolutionary algorithms, SSA has stronger global search ability and faster convergence speed in global optimization problems. In SSA, the population is divided into two different groups: one is the producer group possessing larger search steps to find food, and another is the scrounger group that follows the producers to find food. During the search process, the scroungers have larger probabilities to find food via the following behaviors, while the roles of both producers and scroungers are adjusted dynamically to find more high-quality food sources. Then, the mathematical model of the SSA method is given as below:

*N*), the number of producers (

*PN*) and scroungers (

*N*-

*PN*), the maximum iteration (

*g*

_{max}). The location of the

*i*th sparrow can be defined as

*x**= (*

_{i}*x*

_{i}_{,1},

*x*

_{i}_{,2}, …,

*x*

_{i}_{,D}) while

*f*(

*x**) represents the fitness value of the*

_{i}*i*th sparrow. Then, the initial swarm

**can be expressed as below:where is the**

*x**j*th element's value of the

*i*th sparrow.

*D*is the number of decision variables.

*g*is the iteration index;

*a*is a random number uniformly distributed in the range of [0,1];

*Q*is a random number obeying the standard normal distribution; and

**is a 1 ×**

*L**D*matrix whose elements are set as 1.

*R*∈ [0,1] and

*ST*∈ [0.5, 1.0] denote the alarm value and safety threshold. If

*R*<

*ST*, the producers will execute the extensive search mode without the predators' influence; if

*R*≥

*ST*, the predators have been found by some sparrows and all the sparrows should fly to the safe areas.

*S*_{best}is the producer's best-known location;

*G*_{worst}is the global worst-known location found by the swarm;

**is a 1 ×**

*A**D*matrix whose elements are randomly chosen from the set {1,−1}; and

*A*^{+}=*A*^{T}(

*AA*^{T})

^{−1}. If

*i*>

*N*/2, the

*i*th scrounger should search in other areas to find energy; otherwise, the

*i*th scrounger is foraging in the area around

*S*_{best}.

*β*is a random number obeying the standard normal distribution;

*G*_{best}is the global best-known location found by far;

*K*∈ [−1,1] is a random number representing the search step size; and

*θ*is a small constant used to avoid the denominator being zero. If , the

*i*th sparrow at the edge of the swarm will easily find predators; otherwise, the

*i*th sparrow at the center of the swarm should be close to other sparrows for antipredation.

Step 5: Update the best as well as worst fitness values of the swarm to get all the sparrows' new positions.

Step 6: If the termination condition is not met, return to Step 2 for the next cycle; otherwise, the global best-known position found by the sparrow population will be treated as the final solution for the target problem.

### Extreme learning machine (ELM)

Extreme learning machine (ELM), shown in Figure 1, is an effective training tool developed to resolve a single-layer feedforward neural network (SLFN). After randomly determining the values of the input-hidden weights and hidden biases, ELM makes use of the classical Moore–Penrose generalized inverse to analytically calculate the hidden-output weights (Huang *et al.* 2006a; Cambria *et al.* 2013; Huang *et al.* 2014). Different from the conventional learning methods for SLFN, ELM makes an obvious reduction in the size of the computational parameters. In this way, ELM possesses the merits of faster execution speed, fewer learning parameters and stronger generalization ability in comparison with the traditional gradient-based methods (Yadav *et al.* 2016).

*N*samples, the outputs of the ELM model with

*L*hidden neurons can be given as below:where and are the inputs and outputs associated with the

*i*th training sample; is the weight vector connecting the

*l*th hidden neuron with all the neurons in the input layer; is the weight vector connecting the

*l*th hidden neuron with all the neurons in the output layer; is the bias of the

*l*th hidden neuron; and denotes the activation function of the

*l*th hidden neuron.

*N*training samples, and then becomes the coefficient matrix to be determined. By this time, Equation (6) will be transformed to a standard linear system and then the weight matrix can be analytically deduced by determining the optimal solution of the above linear system, which can be expressed as below:where denotes the Moore–Penrose generalized inverse matrix of .

### Hydrological time series forecasting model based on ELM and SSA

As mentioned above, the ELM model can achieve good performance and make an obvious reduction in the execution time by stochastically determining the model parameters, rather than by iterative adjustment. ELM has the merits of faster training speed and stronger learning ability (Huang *et al.* 2006b; Cao *et al.* 2012; Huang *et al.* 2015). However, the random determination of parameters associated with the hidden layer may fail to produce suboptimal solutions, which then affects the forecasting capability of the model in practice. In order to address this defect, this paper develops an effective ELM-SSA model where the hidden biases and input-hidden weights of the ELM model are iteratively optimized by the SSA method, rather than the random assignment at the initial phase and no adjustment in the late learning phase. Figure 2 shows the flowchart of the proposed method. For the sake of simplicity, the proposed method with *n* input variables, *L* hidden neurons and one output variable is used to forecast the hydrological time series. In other words, the developed method has *n* input nodes, *L* hidden nodes and 1 output node. Then, the execution steps of the proposed method are given as:

*Q*are the

_{i}*i*th normalized and original data;

*n*is the amount of data; and

*a*and

*b*are the adjusting coefficients.

Step 2: Parameter setting, such as the number of sparrows and iterations in SSA, the number of hidden neurons as well as the activation function of hidden neurons in ELM. Here, the classical sigmoid function is chosen as the activation module for data mapping in the hidden layer.

Step 3: Population initialization. Set the counter *g* = 1 and then randomly create all the element values of each sparrow of the initial swarm in the feasible state space. Each sparrow represents a possible ELM model containing all the parameters associated with the hidden layer.

*i*th sparrow at the

*g*th cycle , its fitness value is obtained by the following formulas:where

*N*is the number of training samples; is the outputs of the

*s*th training sample; is the hidden-layer output matrix; and represents the Moore–Penrose generalized inverse of .

Step 5: Population updating. Obtain the global best-known or worst-known sparrows of the swarm, as well as the best-known locations of all the producers. Next, compute the necessary computation parameters to update all the sparrows' positions.

Step 6: Set *g* = *g* + 1. If the maximum iteration is met, go to Step 4 for the next cycle; otherwise, the global best sparrow is treated as the ideal parameters of the hidden layer and then the Moore–Penrose generalized inverse method is used to obtain the hidden-output weights. The optimized ELM model by far is obtained for applications.

## CASE STUDIES

### Engineering background

To test the feasibility of the proposed method, the Yangtze River of China is selected as the case study. The flood events at Yangtze River usually occur in the wet season between May and October, while the other months belong to the dry season. During the flood season, the spatial–temporal distribution of rainfall is largely affected by monsoon activities and subtropical anticyclones. Meanwhile, the many huge reservoirs represented by the Three Gorges project, the world's largest hydropower plant, are put into operation in succession, resulting in great changes in runoff features. As a result, it becomes more and more difficult to accurately forecast the runoff time series under the changing environment.

The daily runoff time series collected from three hydrological stations located on the Yangtze River are used for comparative study. The three hydrological stations (A ∼ C) play an important role in monitoring the water changing tendency and guaranteeing the safe operation of Yangtze River. Specially, hydrological station B is located on the south tributary, while hydrological stations A and C are located on the mainstream of Yangtze River. The water at stations A and B flows to station C. Figure 3 illustrates the studied daily runoff time series of the three hydrological stations. The forecasting model should have strong adaptive capacities to respond to the obvious differences of the runoff series at the three hydrological stations. For the runoff time series of each hydrological station, the collected data is divided into a different sub-dataset, where the first 70% of the data is used for training and validating the model's parameter, while the other data is used for testing.

### Performance evaluation indicators

*i*th target and predicted data; and

*n*is the amount of data for comparison.

### Comparison with the ELM method at station A

Table 1 lists multiple-step-ahead forecasting results of the ELM method and the hybrid method at both training and testing phases for station A. It can be clearly seen that at the same forecasting period, the proposed method obtains the best results among all the forecasting periods; with the increasing number of the forecasting period, the performances of all the forecasting methods become gradually worse while the results of the proposed method are still better than those of the ELM method. For instance, in the training phase, the proposed method makes about 15.70% and 16.62% improvements in the MAPE value for the one-step and two-step forecasting periods; as the forecasting period increases from 1 to 6, the RMSE value of the proposed method is increased from 577.774 to 1,105.495 at the testing phase, better than that of the ELM model from 656.59 to 1,146.196. Hence, it can be concluded that the organic combination of the SSA and ELM methods can effectively improve the forecasting results.

Forecasting period . | Model . | Training . | Testing . | ||||||
---|---|---|---|---|---|---|---|---|---|

RMSE . | MAPE . | R
. | CE . | RMSE . | MAPE . | R
. | CE . | ||

τ = 1 | ELM | 828.001 | 21.113 | 0.890 | 0.792 | 656.590 | 19.337 | 0.887 | 0.786 |

Proposed | 677.504 | 17.798 | 0.928 | 0.860 | 577.774 | 17.027 | 0.913 | 0.834 | |

τ = 2 | ELM | 1,127.157 | 37.799 | 0.784 | 0.614 | 877.421 | 31.331 | 0.786 | 0.617 |

Proposed | 1,046.716 | 31.518 | 0.817 | 0.667 | 835.184 | 27.282 | 0.809 | 0.653 | |

τ = 3 | ELM | 1,295.671 | 52.177 | 0.700 | 0.490 | 997.539 | 42.134 | 0.712 | 0.505 |

Proposed | 1,232.407 | 40.925 | 0.734 | 0.538 | 952.819 | 34.205 | 0.742 | 0.549 | |

τ = 4 | ELM | 1,373.460 | 56.665 | 0.653 | 0.426 | 1,050.117 | 45.324 | 0.673 | 0.452 |

Proposed | 1,334.638 | 47.605 | 0.677 | 0.458 | 1,020.208 | 39.312 | 0.696 | 0.482 | |

τ = 5 | ELM | 1,432.996 | 62.165 | 0.613 | 0.376 | 1,095.431 | 49.764 | 0.636 | 0.403 |

Proposed | 1,391.922 | 51.821 | 0.641 | 0.411 | 1,063.061 | 42.662 | 0.664 | 0.438 | |

τ = 6 | ELM | 1,506.925 | 71.634 | 0.557 | 0.310 | 1,146.196 | 57.730 | 0.592 | 0.347 |

Proposed | 1,462.935 | 57.607 | 0.591 | 0.349 | 1,105.495 | 48.027 | 0.629 | 0.392 |

Forecasting period . | Model . | Training . | Testing . | ||||||
---|---|---|---|---|---|---|---|---|---|

RMSE . | MAPE . | R
. | CE . | RMSE . | MAPE . | R
. | CE . | ||

τ = 1 | ELM | 828.001 | 21.113 | 0.890 | 0.792 | 656.590 | 19.337 | 0.887 | 0.786 |

Proposed | 677.504 | 17.798 | 0.928 | 0.860 | 577.774 | 17.027 | 0.913 | 0.834 | |

τ = 2 | ELM | 1,127.157 | 37.799 | 0.784 | 0.614 | 877.421 | 31.331 | 0.786 | 0.617 |

Proposed | 1,046.716 | 31.518 | 0.817 | 0.667 | 835.184 | 27.282 | 0.809 | 0.653 | |

τ = 3 | ELM | 1,295.671 | 52.177 | 0.700 | 0.490 | 997.539 | 42.134 | 0.712 | 0.505 |

Proposed | 1,232.407 | 40.925 | 0.734 | 0.538 | 952.819 | 34.205 | 0.742 | 0.549 | |

τ = 4 | ELM | 1,373.460 | 56.665 | 0.653 | 0.426 | 1,050.117 | 45.324 | 0.673 | 0.452 |

Proposed | 1,334.638 | 47.605 | 0.677 | 0.458 | 1,020.208 | 39.312 | 0.696 | 0.482 | |

τ = 5 | ELM | 1,432.996 | 62.165 | 0.613 | 0.376 | 1,095.431 | 49.764 | 0.636 | 0.403 |

Proposed | 1,391.922 | 51.821 | 0.641 | 0.411 | 1,063.061 | 42.662 | 0.664 | 0.438 | |

τ = 6 | ELM | 1,506.925 | 71.634 | 0.557 | 0.310 | 1,146.196 | 57.730 | 0.592 | 0.347 |

Proposed | 1,462.935 | 57.607 | 0.591 | 0.349 | 1,105.495 | 48.027 | 0.629 | 0.392 |

Figure 4 shows the multiple-step-ahead forecasting results of various methods for daily runoff of station A at the testing period. The two methods can effectively track the variation tendency of runoff series on the whole-time window, demonstrating the feasibility of the extreme learning machine method in streamflow prediction. On the other hand, the standard ELM method is not as good as the proposed method at the same forecasting period because its correlation coefficient values are obviously smaller than those of the hybrid method. In addition, with the increasing forecasting period, the correlation coefficient values of the proposed method are slowly reduced while the decreasing amplitudes of the ELM method in the correlation coefficient value are relatively large. Thus, the proposed method can make effective improvements on the robustness of the ELM method.

The relative forecasting errors of the two methods for daily streamflow at station A during the testing phase are drawn in Figure 5. As the forecasting period is equal to 1, the relative forecasting errors of the two methods are rather close; in the same forecasting period, the relative forecasting error of the ELM method varies in a relatively large zone in comparison with that of the proposed method; in addition, with the increasing forecasting period, the relative forecasting errors of the two methods gradually become larger. The reason lies in that for a long-term forecasting task, more uncertain factors are involved in the complex hydrological process and thereby it becomes much more difficult for the forecasting method to capture the dynamic change of runoff. Hence, the proposed method using SSA to optimize the parameters of the ELM model can produce satisfactory forecasting results.

### Comparison with the ELM method at station B

Table 2 lists the multiple-step-ahead forecasting results of the ELM method and the hybrid method at both training and testing phases for station B. It can be found that the hybrid method outperforms the ELM methods in terms of various statistical indexes at both training and testing phases, proving that the SSA method can effectively improve the network's compactness. For instance, as the forecasting period is increased from 1 to 3, our method betters the ELM method with approximately 26.18%, 9.32% and 10.20% improvements in the RMSE value at the testing phase. Thus, the hybrid method is an effective hydrological time series forecasting tool that can provide better performance than the standard ELM method.

Forecasting period . | Model . | Training . | Testing . | ||||||
---|---|---|---|---|---|---|---|---|---|

RMSE . | MAPE . | R
. | CE . | RMSE . | MAPE . | R
. | CE . | ||

τ = 1 | ELM | 2,524.510 | 9.206 | 0.964 | 0.929 | 2,156.296 | 8.104 | 0.964 | 0.929 |

Proposed | 1,848.003 | 6.521 | 0.981 | 0.962 | 1,591.868 | 6.081 | 0.981 | 0.962 | |

τ = 2 | ELM | 3,733.924 | 14.214 | 0.919 | 0.844 | 3,204.370 | 12.996 | 0.919 | 0.844 |

Proposed | 3,391.754 | 12.544 | 0.934 | 0.872 | 2,905.577 | 11.244 | 0.934 | 0.872 | |

τ = 3 | ELM | 4,475.360 | 24.004 | 0.881 | 0.777 | 3,931.205 | 19.754 | 0.875 | 0.765 |

Proposed | 4,078.582 | 15.834 | 0.903 | 0.814 | 3,530.244 | 14.615 | 0.901 | 0.811 | |

τ = 4 | ELM | 4,750.504 | 24.434 | 0.865 | 0.748 | 4,188.029 | 20.857 | 0.857 | 0.734 |

Proposed | 4,584.702 | 20.083 | 0.875 | 0.766 | 4,032.888 | 18.434 | 0.868 | 0.753 | |

τ = 5 | ELM | 4,868.630 | 24.747 | 0.858 | 0.736 | 4,308.272 | 21.549 | 0.848 | 0.718 |

Proposed | 4,722.937 | 21.117 | 0.867 | 0.751 | 4,154.468 | 19.970 | 0.860 | 0.738 | |

τ = 6 | ELM | 5,025.318 | 26.161 | 0.848 | 0.718 | 4,447.283 | 22.855 | 0.837 | 0.700 |

Proposed | 4,858.154 | 22.350 | 0.858 | 0.737 | 4,284.134 | 21.268 | 0.850 | 0.721 |

Forecasting period . | Model . | Training . | Testing . | ||||||
---|---|---|---|---|---|---|---|---|---|

RMSE . | MAPE . | R
. | CE . | RMSE . | MAPE . | R
. | CE . | ||

τ = 1 | ELM | 2,524.510 | 9.206 | 0.964 | 0.929 | 2,156.296 | 8.104 | 0.964 | 0.929 |

Proposed | 1,848.003 | 6.521 | 0.981 | 0.962 | 1,591.868 | 6.081 | 0.981 | 0.962 | |

τ = 2 | ELM | 3,733.924 | 14.214 | 0.919 | 0.844 | 3,204.370 | 12.996 | 0.919 | 0.844 |

Proposed | 3,391.754 | 12.544 | 0.934 | 0.872 | 2,905.577 | 11.244 | 0.934 | 0.872 | |

τ = 3 | ELM | 4,475.360 | 24.004 | 0.881 | 0.777 | 3,931.205 | 19.754 | 0.875 | 0.765 |

Proposed | 4,078.582 | 15.834 | 0.903 | 0.814 | 3,530.244 | 14.615 | 0.901 | 0.811 | |

τ = 4 | ELM | 4,750.504 | 24.434 | 0.865 | 0.748 | 4,188.029 | 20.857 | 0.857 | 0.734 |

Proposed | 4,584.702 | 20.083 | 0.875 | 0.766 | 4,032.888 | 18.434 | 0.868 | 0.753 | |

τ = 5 | ELM | 4,868.630 | 24.747 | 0.858 | 0.736 | 4,308.272 | 21.549 | 0.848 | 0.718 |

Proposed | 4,722.937 | 21.117 | 0.867 | 0.751 | 4,154.468 | 19.970 | 0.860 | 0.738 | |

τ = 6 | ELM | 5,025.318 | 26.161 | 0.848 | 0.718 | 4,447.283 | 22.855 | 0.837 | 0.700 |

Proposed | 4,858.154 | 22.350 | 0.858 | 0.737 | 4,284.134 | 21.268 | 0.850 | 0.721 |

Figure 6 shows the multiple-step-ahead forecasting results of various methods for daily runoff of station B in the testing period. For the two methods, the variation tendency of runoff time series is well simulated in the testing phase, while the standard ELM method is inferior to the proposed method due to its smaller correlation coefficient value. Therefore, it can be concluded that the generalization ability of the standard ELM method can be sharply improved by the SSA method.

Figure 7 shows the peak flows of the various methods for station B at the testing phase. It can be clearly seen that the proposed method can obtain better forecasting results than the control methods. For the first forecasting period, the hybrid method makes about 14.95% underestimation in the forecasting error of the peak flow, smaller than the 26.27% of ELM. Hence, our method can effectively yield reliable hydrological forecasting information to provide strong technical support for the decision-making process of the water resources system.

### Comparison with different evolutionary algorithms at station C

Table 3 gives multiple-step-ahead forecasting results of the ELM method and the hybrid method at both training and testing phases for station C. Figure 8 shows the multiple-step-ahead forecasting results of the various methods for daily runoff of station C in the testing period. The hybrid method provides better results than the ELM methods with respect to various statistical indexes. Thus, the hybrid method combining the ELM and SSA method is an effective tool for hydrological time series forecasting.

Method . | Training . | Testing . | ||||||
---|---|---|---|---|---|---|---|---|

RMSE . | MAPE . | R
. | CE . | RMSE . | MAPE . | R
. | CE . | |

ELM | 2,618.2055 | 8.8309 | 0.9718 | 0.9440 | 2,321.1800 | 8.7469 | 0.9737 | 0.9476 |

ELM-GA | 1,575.7859 | 6.9297 | 0.9899 | 0.9797 | 1,457.6769 | 6.5555 | 0.9897 | 0.9793 |

ELM-DE | 1,787.2459 | 8.0147 | 0.9871 | 0.9739 | 1,642.8204 | 7.5818 | 0.9870 | 0.9738 |

ELM-PSO | 1,462.1992 | 5.7864 | 0.9913 | 0.9825 | 1,381.9364 | 5.7689 | 0.9908 | 0.9814 |

ELM-GSA | 1,393.0435 | 5.5704 | 0.9922 | 0.9842 | 1,321.0378 | 5.5150 | 0.9916 | 0.9830 |

Proposed | 1,332.3079 | 5.3211 | 0.9929 | 0.9855 | 1,296.9795 | 5.4292 | 0.9920 | 0.9836 |

Method . | Training . | Testing . | ||||||
---|---|---|---|---|---|---|---|---|

RMSE . | MAPE . | R
. | CE . | RMSE . | MAPE . | R
. | CE . | |

ELM | 2,618.2055 | 8.8309 | 0.9718 | 0.9440 | 2,321.1800 | 8.7469 | 0.9737 | 0.9476 |

ELM-GA | 1,575.7859 | 6.9297 | 0.9899 | 0.9797 | 1,457.6769 | 6.5555 | 0.9897 | 0.9793 |

ELM-DE | 1,787.2459 | 8.0147 | 0.9871 | 0.9739 | 1,642.8204 | 7.5818 | 0.9870 | 0.9738 |

ELM-PSO | 1,462.1992 | 5.7864 | 0.9913 | 0.9825 | 1,381.9364 | 5.7689 | 0.9908 | 0.9814 |

ELM-GSA | 1,393.0435 | 5.5704 | 0.9922 | 0.9842 | 1,321.0378 | 5.5150 | 0.9916 | 0.9830 |

Proposed | 1,332.3079 | 5.3211 | 0.9929 | 0.9855 | 1,296.9795 | 5.4292 | 0.9920 | 0.9836 |

## CONCLUSION

In this paper, an effective evolutionary extreme learning machine based on the sparrow search algorithm is developed to accurately forecast hydrological time series. Specifically, the sparrow search algorithm is used to search for satisfying combinations of the input-hidden weights as well as hidden biases, while the Moore–Penrose generalized inverse method is chosen to analytically obtain the hidden-output weights. In this way, the developed method has a higher probability to find a better network structure than the standard ELM model without any parameter tunning. The developed method successfully forecasts the runoff time series of three hydrological stations in China. The experimental results show that the developed method is superior to several traditional methods with respect to various performance evaluation indexes. Thus, a novel and practical evolutionary extreme learning machine model using swarm intelligence is developed to handle the complex hydrological forecasting task.

On the other hand, the application of the proposed method may be limited in practice since it is rather difficult to determine the optimal parameters of the neural network model in theory while there are also big differences in the characteristic information of the hydrological elements at different places. Considering the rapid development of computer technology, the prospects of the proposed method can be improved by introducing more effective soft computing methods or developing more robust modified strategies. In addition, combinations of the newly developed signal processing technique can also be used to improve the performance of the proposed method.

## ACKNOWLEDGEMENTS

This paper is supported by the National Key Research and Development Program of China (no. 2018YFC1508002) and Scientific Research Projects of China Three Gorges Group Co. (0799251).

## DATA AVAILABILITY STATEMENT

Data cannot be made publicly available; readers should contact the corresponding author for details.