## Appendix A List of abbreviations: \begin{tabular}{l l} MM5 & Mesoscale Model 5 \\ HW & Holt-Winters \\ CS & Cuckoo Search algorithm \\ SVM & Support Vector Machine \\ AE & Absolute Error \\ MAE & Mean Absolute Error \\ MSE & Mean Square Error \\ PS & Pareto-optimal Set \\ PF & Pareto Front \\ DE & Differential Evolution \\ MOP & Multiple Objective Optimization Problem \\ ARIMA & Auto-Regressive Integrate Moving Average \\ ARMA & Autoregressive Moving Average \\ BPNN & Back-Propagation Neural Network \\ NSGA-III & Non-dominated Sorting Genetic Algorithm \\ SOP & Single Objective Optimization Problem \\ OSELM & Online Extreme Learning Machine \\ MAPE & Mean Absolute Percentage Error \\ RMSE & Root Mean Squared Error \\ ANN & Artificial Neural Networks \\ \end{tabular}   
The Introduction of NSGA-III  
NSGA-III performs the following operations, performed at a generation \(t\). Primarily, different nondomination levels are classified from whole population \(P_{t}\) in the same way in NSGA-II, following the principle of nondominated sorting. Then, \(P_{t}\) creates an offspring population \(Q_{t}\) by way of the usual recombination and mutation operators. For every reference point, only one individual of the population is anticipated to be found, so there is no necessity for any selection operation in NSGA-III. Then, there is a new population \(R_{t}\) combined with \(P_{t}\) and \(Q_{t}\) (\(R_{t}\)= \(P_{t}\bigcup Q_{t}\)). Subsequently, points starting from the first nondominated front are selected for \(P_{t\,+\,1}\), one at a time, until all solutions from a complete front cannot be included. We can denote the final front as \(F_{L}\). In general, only a few solutions from \(F_{L}\) need to be selected for \(P_{t\,+\,1}\) using a niche-preserving operator, which could be described in the next one. Primarily, each population member of \(P_{t\,+\,1}\) and \(F_{L}\) is normalized by using the current population spread so that all objective vectors and reference points have commensurate values. Subsequently, each member of \(P_{t+1}\) and \(F_{L}\) is associated with a specific reference point by using the shortest perpendicular distance (d(0))of each population member, with a reference line created by joining the origin with a supplied reference point. Then, a careful niching strategy is employed to choose those \(F_{L}\) members that are associated with the least-represented reference points in \(P_{t\,+\,1}\). The niching strategy puts an emphasis on selecting a population member for as many supplied reference points as possible. A population member associated with an under-represented or unrepresented reference point is immediately preferred. With continuous stress for emphasizing nondominatedindividuals, the whole process is then expected to find one population member corresponding to each supplied reference point close to the Pareto-optimal front, provided that the genetic variation operators (recombination and mutation) are capable of producing respective solutions. The use of well-spread reference points ensures a well-distributed set of trade-off points at the end. The pseudocode for the NSGA-III is provided in **Algorithm 2**. ```
Input:\(Y=\{y_{1},\;y_{2},\;\cdots,y_{N}\}\) a sequence of original time series Output:\(\widetilde{Y}=\{\widetilde{y}_{1},\;\widetilde{y}_{2},\cdots,\;\widetilde{y}_{N}\}\)--a sequence of decomposed time series Parameters:\(N\)--the length of a time series. \(X\)--the trajectory matrix. \(L\)--the windows length. \(K\)--the number of lagged vectors. \(L^{*}\)--the minimum between \(L\) and \(K\). \(K^{*}\)--the maximum between \(L\) and \(K\). \(M\)--the number of repetitions of each trial. ```  
**Algorithm 1** Denoising  
```
1/*Obtain the trajectory matrix. */
2 FOREACH\(i\)=1:\(L\)DO
3\(X(i,\cdot)\)=(\(y_{i},\ldots,y_{K+i\cdot 1}\))
4 ENDFOR
5\(X=\left[ \begin{array}{cccc}y_{1}&y_{2}&\cdots&y_{K}\\ y_{2}&y_{3}&\cdots&y_{K+1}\\ \vdots&\vdots&\ddots&\vdots\\ y_{L}&y_{L+1}&\cdots&y_{N}\end{array} \right]\). 6\(S=XX^{T}\)/*Singular value decomposition.