## 2 Background of Soft Computing Methods
### Equilibrium Optimizer (EO)
The transformed time series data were used to predict the next day runoff value for both catchments. Further details of selected runoff and rainfall series with wavelet pre-processing are described in the model development section (Section 4). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Catchment** & & **Mean** & **Sd.** & **Median** & **Min** & **Max** & **Skewness** \\ \hline \multirow{2}{*}{Fal at Tragony} & Discharge (m\({}^{3}\)/s) & 2.03549 & 1.949909 & 1.37 & 0.208 & 48.24 & 3.69096 \\  & Rainfall (mm/day) & 3.37865 & 5.742897 & 0.6 & 0 & 55.9 & 2.693286 \\ \multirow{2}{*}{Teifi at Glanteifi} & Discharge (m\({}^{3}\)/s) & 29.6010 & 31.9316 & 18.290 & 0.7310 & 373.60 & 2.4780 \\  & Rainfall (mm/day) & 3.8631 & 6.2733 & 1 & 0 & 73.100 & 2.7719 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistics of dataset from two catchments. ### Proposed EO-ELM and DNN

This study proposes a new empirical R-R model, called EO-ELM, where EO optimizes ELM learning parameters to find an optimal configuration of ELM for better prediction of runoff values. Here, the concentrations of EO are ELM learning parameters. The root mean square error (RMSE) is considered as objective function for EO. The best equilibrium candidate found by EO is considered as optimal configuration of ELM for prediction tasks. In EO-ELM, initially all particles have no knowledge about the solution space. The collaboration of five equilibrium candidates help concentration updating process of particles. At initial periods of iteration, the equilibrium candidates are diverse in nature and exponential term (Equation (8)) produces large random numbers which helps particle to cover the entire solution space. Similarly, during end period of iterations, particles are surrounded equilibrium candidates which are in an optimal position with similar configuration. At these moments, the exponential term (Equation (8)) produces lower value of random number which helps fine tuning of candidate solutions. The algorithm of EO-ELM is shown in Algorithm 2. Figure 3: Daily discharge and rainfall of (**a**) the small basin (Fal at Tregony), (**b**) the large basin (Teifi at Glanteifi). ```
1. Training and testing dataset
2. Set hidden units and biases of ELM using initialization of EO populations
3. Assign of EO parameters value (\(a_{1}=2\), \(a_{2}=1\), \(GP=0.5\)) (optimized parameters based on several trials of proposed model) Output:
4. Optimized hidden units and biases of ELM from best fitness of a EO equilibrium candidate
5. Get output weights of ELM using Moore Penrose Inverse
6. EO optimized ELM testing using test dataset
7. Begin EO-ELM training
8. Initialize the fitness of four equilibrium candidates [\(\text{fitness}(\overset{\rightarrow}{P}_{eq[1]})\), \(\text{fitness}(\overset{\rightarrow}{P}_{eq[2]})\), \(\text{fitness}(\overset{\rightarrow}{P}_{eq[3]})\), \(\text{fitness}(\overset{\rightarrow}{P}_{eq[4]})\)]
9. for it = 1 to maximum iteration number do
10. for i = 1 to P do
11. Estimate the fitness of the \(\overset{\rightarrow}{P}_{eq}\) particle
12. if\(\text{fitness}(\overset{\rightarrow}{P}_{eq[1]})\) < \(\text{fitness}(\overset{\rightarrow}{P}_{eq[1]})\)
13. Replace \(\text{fitness}(\overset{\rightarrow}{P}_{eq[1]})\) with \(\text{fitness}(\overset{\rightarrow}{P}_{j})\) and \(\overset{\rightarrow}{P}_{eq[1]}\) with \(\overset{\rightarrow}{P}_{j}\)
14. elseif\(\text{fitness}(\overset{\rightarrow}{P}_{j})\) < \(\text{fitness}(\overset{\