## 2 Methodology
### Improved Weighted Extreme Learning Machine (IWELM)  
(1) Basic ELM: ELM is a single layer feed-forward network which is widely accepted for its simple training process and excellent performance in generalization. An origin ELM model on one of \(N\) samples \(\left(\mathbf{x}_{n},r_{n}\right),n=1,2,\cdots\cdots N\) can be described by  
\[\sum_{l=1}^{L}g\left(\mathbf{x}_{n}\cdot\mathbf{w}_{l}+b_{l}\right)\beta_{l}= o_{n}, \tag{8}\]  
where \(\mathbf{x}_{n}\)= \(\left[x_{n1},x_{n2}\cdot\ldots\cdot x_{nK}\right]\) is the input vector, \(r_{n}\) and \(o_{n}\) are desired and actual output, respectively. \(\boldsymbol{w}_{l}\)=\(\left[w_{1l},w_{2l}\cdot\ldots\cdot w_{Kl}\right]^{T}\), \(l=1,2,\cdots\cdots L\) is the input weight, \(b_{l}\) is the bias of hidden layer, and \(\beta_{l}\) is the output weight. For all training samples, the whole model can be written as  
\[\mathbf{H}\boldsymbol{\beta}=g\left(\mathbf{X}\cdot\mathbf{W}+\boldsymbol{u} \cdot\boldsymbol{b}\right)\boldsymbol{\beta}=\boldsymbol{o}, \tag{9}\]  
where \(\mathbf{X}\)=\(\left[\mathbf{x}_{1}^{T},\mathbf{x}_{2}^{T}\cdot\ldots\cdot\mathbf{x}_{n}^{T} \right]^{T}\) is the input matrix, \(\mathbf{W}\)=\(\left[\boldsymbol{w}_{1}^{T},\boldsymbol{w}_{2}^{T}\cdot\ldots\ldots\cdot \boldsymbol{w}_{L}^{T}\right]^{T}\) is the input weight matrix, \(\boldsymbol{u}\) is a N-dimensional column vector in which all the elements are \(1,\boldsymbol{b}=\left[b_{1},b_{2}\cdot\ldots\cdot b_{L}\right]\) is the bias vector, \(\boldsymbol{r}\)=\(\left[r_{1},r_{2}\cdot\ldots\cdot r_{N}\right]^{T}\) is the output vector, and \(\mathbf{H}\) is the hidden layer output matrix, which is obtained by giving \(\mathbf{W}\), \(\boldsymbol{b}\) randomly. Then, the output weight vector can be directly calculated by \(\boldsymbol{\beta}=\mathbf{H}^{\dagger}\boldsymbol{r}\), where \(\mathbf{H}^{\dagger}\) is the Moore-Penrose generalized inverse of \(\mathbf{H}\). (2) Weighted ELM: Considering the importance degree of features, a branch of ELM called weighted extreme learning machine (WELM) inserts a coefficient matrix \(\mathbf{Y}\)= diag \(\left[\gamma_{1},\gamma_{2}\cdot\ldots\cdot\gamma_{K}\right]\) between \(\mathbf{X}\) and \(\mathbf{W}\). WELM associates each input feature with the hidden layer through a controllable parameter \(\gamma_{k}\in\mathbf{Y}\left(k=1,2\cdot\cdots\cdot K\right)\). The whole model is expressed by rewriting Equation (9) as  
\[\mathbf{H}_{\mathbf{Y}}\boldsymbol{\beta}=g\left(\mathbf{X}\cdot\mathbf{Y} \cdot\mathbf{W}+\boldsymbol{u}\cdot\boldsymbol{b}\right)\boldsymbol{\beta}= \boldsymbol{o}. \tag{10}\]  
For all the training data, the effect of each input feature on hidden layer nodes can be adjusted flexibly as is shown in Figure 4. Overall, \(\mathbf{Y}\) and \(\boldsymbol{\beta}\) are respectively regarded as the input and output structural parameters of the WELM model. However, there are two problems with the application of WELM: on the one hand, compared with the increasing computation brought by extra parameters \(\gamma_{k}\), the significance of these extra parameters to LF is ambiguous. On the other hand, the results of this model are vari  
Figure 4: The structure of the weighted extreme learning machine (WELM) model. weights and biases. In this paper, an improved WELM model with an optimization algorithm is provided to resolve the above problems. (3) IWELM: Firstly, the coefficient matrix \(\mathbf{Y}\) is divided into blocks so as to tackle the redundancies brought by the similarity of different features. The blocks of the input features decomposed by a certain wavelet family is proved to be capable of forecasting future load demand as an individual predictor [43]. To take advantage of all the individual predictors, the structure of these predictors is specified by \(\mathbf{Y}\). For example, for all the features \(x_{nk}\) (\(k\in Z,k\in[1,K]\)) decomposed by db2, it is defined that \(\gamma_{k}=\gamma_{db2}\). \(\gamma_{db2}\) is the weight of the above features in the hidden layer nodes and the output.