## 2 GMDH Network  
The high-order polynomial network generated by the GMDH algorithm is essentially a feed-forward and multilayer neural network, which was proposed by the Former Soviet scientist [PERSON]. It is a heuristic and self-organized modeling method. The self-organizing form minimizes the need for prior knowledge. The model does not need to specify any initial assumptions, such as the numbers of neurons and hidden layers, which reduces the subjectivity and complexity of modeling [27; 28; 29; 39; 40]. The input data of GMDH is composed of multiple sets of external factor variables and observed values. The variables describing external factors are called characteristic variables, and the characteristic variables and observed values are collectively referred to as input variables. Input data is first divided into a training set and testing set according to a certain proportion by the GMDH algorithm. The training sample data is used for the estimation of the coefficients of the Kolmogorov-Gabor (K-G) polynomial. The testing set is supplied to the GMDH network for error checking. Then, the characteristic variables of the training set are cross-recombined in pairs, and the corresponding observation hold value does not change. After recombination, each pair of characteristic variables is a group and is trained as a neuron of the network. The output of each neuron is a high-order function of both its input, which is referred to as a K-G polynomial function. Then, the output of each neuron is evaluated and tested by an external criterion. We eliminated the neurons that were predicted the worst, and retain the neurons with a good performance as the next layer. The entire process of restructuring, training, testing, and selection was performed again on this new layer. The GMDH network is not completed until the forecast error of the neurons no longer decreases. The GMDH network structure is shown in Figure 1.  
In Figure 1, the input column is cross-recombined by all the characteristic variables of the training set. Each reconstructed neuron consists of any two characteristic variables. Suppose there are \(N\) groups of data in the input data, each group of data has m characteristic variables and 1 observation value. The number of combinations is given by:  
\[k=\frac{m\times(m-1)}{2}, \tag{1}\]  
where, \(m\) denotes the number of characteristic variables. Cross-recombination of any two characteristic variables can fully express all of the combinations between characteristic variables, and it is easy to determine the effect of each characteristic variable on the observed value. There are only two characteristic variables for each neuron, which enables rapid screening of neurons at lower complexity and acquisition of a suitable model. In Figure 1, \(x\) is the input characteristic variable and \(y\) is the observed value, that is, the output variable of each neuron. The output of the previous layer is the input of the next layer. The relationship between the input and the output of each neuron can be represented by a K-G polynomial:  
\[\hat{y}_{n}=f(x_{1},x_{2},\ldots,x_{m})=\sum_{i=1}^{m}a_{i}x_{in}+\sum_{i=1}^ {m}\sum_{j=1}^{m}a_{ij}x_{in}x_{jn}+\sum_{i=1}^{m}\sum_{j=1}^{m}\sum_{k=1}^{m }a_{ijk}x_{in}x_{jn}x_{kn}+\cdots, \tag{2}\]  
where, \(a_{i}\) is the K-G polynomial coefficient and \(f\) is the output. When only two characteristic variables are the inputs for each neuron, the second-order K-G polynomial is:  
\[\hat{y}_{n}=f(x_{in},x_{jn})=a_{0}+a_{1}x_{in}+a_{2}x_{jn}+a_{3}x_{in}x_{jn}+ a_{4}x_{in}^{2}+a_{5}x_{jn}^{2}, \tag{3}\]  
Gray neurons in Figure 1 are the neurons that have been excluded after screening by external criteria. The regularity criteria can be expressed by Equation (4), where \(y_{n}\) is the observed value, \(\hat{y}_{n}\) is the corresponding estimated value of \(y_{n}\), and \(w\) is the calculation result of the external criterion. GMDH determines the correlation between neurons and the observed value by the size of the threshold \(w\) and selects neurons.