The Shuffled Complex Evolution-University of Arizona (SCE-UA) algorithm is a powerful optimization method commonly used in hydrological and environmental modeling. It's based on the idea of simulating the complex behavior of nature’s evolutionary process.

By imitating the natural evolution, SCE-UA algorithm efficiently explores a wide range of solution space to find the optimal solution for complex problems such as parameter calibration, uncertainty analysis, and sensitivity analysis. It has been applied in various fields like water resources management, climate modeling, and ecosystem analysis.

And the algorithm proposed by Muttil and Jayawardena (2008), called SCE-MJ model calibrating algorithm, aims to enhance robustness and efficiency in hydrological processes. It utilizes a shuffled complex approach, which involves multiple complexes, each with a set of parameters.

The SCE-MJ algorithm dynamically shuffles the parameter sets between complexes to explore the search space more effectively. This helps in finding the optimal parameter values for calibrating hydrological models. By introducing a crossover process and parameter adaptation, the algorithm improves the efficiency of the optimization process.

The FASCE process is a refined advancement over its predecessor, the SCE-UA method. Detailed comprehension of the SCE-UA model can be acquired through references such as Duan et al., 1992, Duan and Gupta, 1993; Li et al., 2013.

A strategic method offered by Muttil and Jayawardena in 2008 is adopted for determining the initial population in our research. This is geared towards ensuring the SCE-UA algorithm does not become trapped in the local optimality within feasible spatial dimensions. The said strategy is imperative in preserving diversity within the population group. Interested parties can gain more knowledge on this strategy by referring to Muttil and Jayawardena 2008.

Moreover, to elevate the potency of the algorithm’s search capabilities, an innovative, adaptive simplex search mechanism is introduced within this study. This operator is utilized during the reflection or contraction phases to instigate the inception of a new point and shift adaptively towards the optimal point amidst the simplex.

This adaptive procedure is a significant enhancement against the standard SCE-UA method. Benefits include the ability to employ both the optimal point’s data and the information derived during the process of generation. The advent of novel points, courtesy of this adaptive simplex search operator, is dictated by Eqs 1–4. X n e w = θ X b + 1 − θ X r e f (1) X n e w = θ X b + 1 − θ X c o n (2) S = 1 y ¯ 1 n ∑ i = 1 n y i − y ¯ 2 (3) θ = 1 2 π e − S 2 2 , S > 0.05 S , S ≤ 0.05 (4)

Where X _new denotes the new point, X _ref denotes the reflected point, X _con denotes the contracted point, X _b is the best point of the simplex, θ is the adaptive shift coefficient, n is the number of current population excluding the worst point, y _i is the objective function value of the ith point, y ¯ is the mean objective function value of the current population excluding the worst point, S denotes the fluctuation degree of the current population excluding the worst point.

Besides, the FASCE also uses two stopping criteria: Parameters distribution and Maximum number of function evaluation, which are the same with SCE-UA. This means that the algorithm can early stop when the parameters distribution is too narrow to search.

In the SCE-MJ algorithm, the shift coefficient must be preset in advance. And this coefficient is always determined by artificial experience. Generally, for some relatively simple test functions (such as Goldstein-Price function, Six-hump function, Griewank function, and so on), the number of local optima is small, and the shift coefficient is set bigger, the shuffle operator can make the generated new point move fast to the global optimum, and the convergence speed is enhanced significantly; and for some complicated test functions (such as Restrigin function, Rastrigin (10D) function, Rosenbrock (10D) function, and so on), the number of local optima is relatively large, and the shift coefficient is set smaller. Thus, to avoid missing the global optimum, the shuffle operator can make the generated new point move slowly to the global optimum, and then the convergence speed improvement becomes small relatively.

However, the complexity of practical problems in engineering usually is not known. Therefore, it is very difficult to set reasonable shift coefficient. Besides, it is not reasonable to set a constant shift coefficient during the whole evolutionary computation process of the algorithm.

To overcome these shortcomings, we propose a novel algorithm FASCE with an adaptive shift operator. And this adaptive shift operator has no parameter needed predefined beforehand and can set adaptive shift coefficient according to the evolutionary computation process. There are three stage in the algorithm evolution as follows:

(1) At the initial stage of evolution, the fluctuation degree of the population is usually big. Then, the adaptive shift operator adopts small shift coefficient to avoid the algorithm missing global optimum.

The differences between these three algorithms are shown in Figure 1.

In this paper, we conducted a study on the Changjiang (Yangtze) River basin. To calibrate and validate our model, we used the observed streamflow data from Yichang Station between 1 January 2005, and 31 December 2007. The data is divided into two sets: the first 730 streamflow data points were used for model calibration, and the remaining data points were used for model validation.

Before we could begin calibrating our model, it was crucial to select an appropriate model structure. In this study, the SVM is chosen for one-day-ahead streamflow forecasting. This means that our model focuses on predicting future streamflow. We used the streamflow data from the past few days as inputs for the model, which is a common approach in similar studies (Aqil et al., 2007; Kisi, 2009; Wang et al., 2009). To determine which previous flow values to include as inputs, we utilized autocorrelation function (ACF) and partial autocorrelation function (PACF) analyses. You can find the ACF and PACF plots of the streamflow data in Figure 3.

Examining the results shown in Figure 3, it can be noted that:

1) The ACF exhibited a correlation coefficient higher than 0.8 at lag 5.

Based on these findings, we concluded that the 5 antecedent flow values serve as suitable inputs for our daily streamflow forecasting model.

Here, the SVM algorithm is employed for streamflow forecasting. The primary objective (as Eq. 13) of using SVM is to discover a function that can accurately predict streamflow. SVM works by identifying a mathematical function that maps the input variables to the output variable, in this case, the streamflow.

By incorporating SVM into our approach, we can better forecast future streamflow levels, providing valuable insights for water resource management and planning. max ∑ i = 1 l α i + − α i − y i − ε ∑ i = 1 l α i + + α i − − 1 2 ∑ i , j α i + − α i − α j + − α j − K X i , X j (13)

Subject to: 0 ≤ α i + ≤ C , 0 ≤ α i − ≤ C ∑ i = 1 l α i + − α i − = 0 i = 1,2 , … , l Where the K X i , X j is the Kernel Trick, e.g., if the kernel function is Radial Basis Kernel, then K X i , X j = ⁡ exp − X i − X j 2 2 σ 2 . C is the penalty factor. ε is the tolerance coefficient. σ is the kernel parameter. α ⁺ and α ⁻ are Lagrange coefficients.

The utilized equations clearly demonstrate that within the SVM-based model for daily streamflow prediction, there exist three specific parameters: the penalty factor C, tolerance level ε, and the kernel parameter σ. It's crucial that these parameters are accurately calibrated. Illustrating the varying ranges of parameters, Table 2 reflects the same parameter values as stipulated in the study by Gill et al., in 2006.

Furthermore, the model’s training objective is primarily based on the Root Mean Square Error (RMSE), a statistically significant metric devised to quantify prediction error. The mathematical expression defining RMSE is demonstrated in the equation presented as Eq. 14. RMSE = 1 n ∑ i = 1 n Q f o r e i − Q m e a s i 2 (14)

Where n denotes the length of streamflow; Q f o r e i denotes the ith forecasting streamflow value; Q m e a s i denotes the ith measured streamflow value.

A series of test trials are executed prior to initiating the actual training phase, with the primary goal of identifying the optimal success threshold. The model has undergone several iterations of preliminary testing, leading to the observation that the optimal RMSE value achieved via the FASCE algorithm is approximately 3,270 m³/s. Hence, this specific value is assigned as the success threshold determinant. In essence, a trial is considered successful if the best function value yielded by the algorithm is less than 3,270 m³/s. On the contrary, a trial is deemed a failure if it hits the maximum limit of function evaluations. To moderate the potential impact of any inherent random element within the previously discussed three algorithms, each one is operated independently 20 times. The experimental outcomes are consolidated and presented in Tables 3, 4.

Within the context of this paper, two key metrics have been selected as the estimate indices for forecasting results: the Root Mean Square Error (RMSE) and Mean Error (ME). ME can be calculated by Eq. 15. The mathematical interpretation of ME is explained in the subsequent equation. Lower RMSE and ME values are indicative of superior performance. ME = 1 n ∑ i = 1 n Q f o r e i − Q m e a s i (15) Where n denotes the length of streamflow; Q f o r e i denotes the ith forecasting streamflow value; Q m e a s i denotes the ith measured streamflow value.

Based on the analysis shown in Table 3, it is evident that three algorithms exhibit nearly identical RMSE values, thereby indicating the suitability of the threshold value of 3,270 m³/s. To further evaluate the efficiency of these algorithms, Table 4 provides a summary of their performance. Clearly, the FASCE algorithm outperforms the other two algorithms in terms of achieving the same level of performance. Not only does FASCE demonstrate significantly faster speed compared to SCE-UA and SCE-MJ, but it also exhibits similar search accuracy when compared to SCE-MJ. One possible reason for this could be that the FASCE algorithm primarily focuses on enhancing the search speed of SCE-MJ, without modifying its local searching operator.

Figure 4 illustrates the results obtained by the FASCE algorithm during the calibration and validation period. In this figure, dots represent the observed streamflow, while lines represent the predicted streamflow. It is worth noting that the model accurately captures the characteristics of the streamflow hydrograph, although some timing errors are present. This indicates that there may be a need for further improvement in the model structure.

Xinanjiang model is first proposed by Professor Zhao Renjun in China. This model is widely used in streamflow prediction in moist and semi-humid basins. The most widely used framework is the three source Xinanjiang model. And the parameter descriptions are shown in Table 5. The basic frame and the description of the model parameters can be referred to the references (Zhao, 1992; Li et al., 2009; Song et al., 2012; Rahman and Lu, 2015).

In this research, the streamflow prediction of Zishui watershed is taken as the second case study. The Zishui watershed area is about 22,640 km², and this watershed plays an important role in power generation and flood control in Hunan province. The location of the Zishui watershed is shown as Figure 5.

The Zishui watershed has 32 rainfall stations, 6 water level stations. The history of water and rainfall observation data of river basin is very complete. The observed data from the 32 rainfall stations and 6 water level stations between 2004-1-1 and 2014-12-31 is used for model calibration and validation. The observation data for the first 6 years is used for model calibration, while the rest used for model validation.

Three algorithms SCE-UA, SCE-UA-MJ and FASCE are employed for model calibration. Similar with the above case study, the Root Mean Square Error (RMSE) is selected as the training objective function. From several forwards testing of the model, we note that the best RMSE value found by FASCE algorithm is about 662 m³/s. Therefore, this value 662 m³/s is selected as a threshold to determine a trial is success or not. To reduce the influence of the random factor existed in the above mentioned three algorithms, each algorithm runs 20 times independently. The results of the experiment are summarized in Tables 6, 7.

The model parameter calibration results of the FASCE are shown in Table 8.

The results obtained by FASCE algorithm in calibration and validation period are given in Figure 6, where dots denote the observed streamflow while lines denote the predicted streamflow. It can also be indicated that the model can fit the characteristics of the streamflow hydrograph very well.

From the above results, it can be noted that three algorithms can all get very good performance, and the algorithm FASCE are more efficient to arrive at the same performance than the other two algorithms. Same with the above case study, the FASCE mainly enhances the evolving speed, so the FASCE has faster converge speed but nearly same search accuracy than SCE-MJ.