## 2 Materials and Methods
### Extreme Learning Machine  
Extreme learning machine (ELM) will be used for regression analysis in this paper, so the principle of ELM for the regression form is mainly introduced.  
Given \(N\) distinct samples \(\{(\mathbf{x}_{i},\mathbf{t}_{i})\,|\mathbf{x}_{i}\in\mathbf{R}^{n},\mathbf{t}_{i}\in\mathbf{R}^{m},i=1, 2,\cdots,N\},\mathbf{x}_{i}=[x_{i1},x_{i2},\cdots,x_{in}]\) represents the input vector with \(n\) inputs, and \(\mathbf{t}_{i}=[t_{i1},t_{i2},\cdots,t_{im}]\) represents the corresponding output vector with \(m\) outputs. In this paper, the input vector \(\mathbf{x}_{i}=[x_{i1},x_{i2},\cdots,x_{in}]\) is composed of \(n\) NTL indices, and the TOVAFAF is taken as the corresponding expected output and represents a numerical value, so the dimension \(m\) of the output vector \(\mathbf{t}_{i}=[t_{i1},t_{i2},\cdots,t_{im}]\) should be equal to 1 in the regression form of ELM. The mathematical model of ELM for the regression form is shown below:  
\[\underset{N\times L}{\mathbf{H}}\underset{L\times 1}{\mathbf{\beta}}=\underset{N \times 1}{\mathbf{T}}, \tag{1}\]  
where \(\mathbf{H}\) represents the hidden layer output matrix, i.e.,  
\[\underset{N\times L}{\mathbf{H}}=\begin{bmatrix}g(\mathbf{x}_{1}\cdot\mathbf{w}_{1}+b_{1} )&\cdots&g(\mathbf{x}_{1}\cdot\mathbf{w}_{L}+b_{L})\\ \vdots&\ddots&\vdots\\ g(\mathbf{x}_{N}\cdot\mathbf{w}_{1}+b_{1})&\cdots&g(\mathbf{x}_{N}\cdot\mathbf{w}_{L}+b_{L}) \end{bmatrix}, \tag{2}\]  
where \(\mathbf{w}_{i}=[w_{i1},w_{i2},\cdots,w_{in}]^{T}\) represents the input weight vector connecting the input layer and the \(i\)th hidden layer neuron, \(b_{i}\) represents the bias of the \(i\)th hidden layer neuron, \(L\) represents the number of hidden layer neurons, and \(g(\cdot)\) represents the activation function. Furthermore, \(\mathbf{\beta}\) represents the output weight vector connecting the hidden layer and output layer, i.e.,  
\[\underset{L\times 1}{\mathbf{\beta}}=[\beta_{1},\beta_{2},\cdots,\beta_{L}]^{T}, \tag{3}\]  
and \(\mathbf{T}\) represents the target output vector, i.e.,  
\[\underset{N\times 1}{\mathbf{T}}=[t_{1},t_{2},\cdots,t_{N}]^{T}. \tag{4}\]  
Since \(\mathbf{H}\) and \(\mathbf{T}\) are known in Equation (1) for the training dataset, \(\mathbf{\beta}\) can be obtained according to the generalized inverse matrix theory, i.e.,  
\[\mathbf{\beta}=\arg\underset{\mathbf{\beta}^{*}}{\min}\|\mathbf{H}\mathbf{\beta}^{*}-\mathbf{T}\| =\mathbf{H}^{+}\mathbf{T}, \tag{5}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Serial Number** & **Index Name** & **Abbreviation** & **Detail Description** \\ \hline V5 & Average luminous pixel light index & ALPLI & The ratio of TNLI to quantity of luminous pixels in the NTL image within the provincial-level administrative region \\ \hline V6 & Nighttime light standard deviation index & NLSDI & The standard deviation of light intensity for all pixels in the NTL image within the provincial-level administrative region \\ \hline V7 & Nighttime light variance index & NLVI & The variance of light intensity for all pixels in the NTL image within the provincial-level administrative region \\ \hline V8 & Nighttime light squared deviation sum index & NLSDSI & The sum of squared deviation of light intensity for all pixels in the NTL image within the provincial-level administrative region \\ \hline \hline \end{tabular}
\end{table}
Table 1: _Cont._where \(H^{+}\) represents the Moore-Penrose generalized inverse of matrix \(H\). Once \(\beta\) is determined, the test data can be predicted using Equation (1).