## 2 Materials and Methods
### Theoretical Basics  
#### 2.3.1 GM (1, 1) Gray Prediction Model  
Gray theoretical models can be used to make fuzzy descriptions of the development pattern of things by building gray differential prediction models with a small amount of incomplete information. They use cumulative techniques to reduce the randomness of the dataset and thus increase the regularity of the system, and they are mostly used to make predictions on small data samples [31; 32; 33; 34; 35]. This is a first-order differential equation model with a single input variable, which overcomes the limitations of traditional methods and is thus suitable for the analysis of small data samples. In this study, due to the proposed buffer operator, the simulation and prediction accuracy of the GM (1, 1) model was improved, and four basic forms of GM (1, 1) model were formed, which are defined as follows:  
Set the original sequence as:  
\[X^{(0)}=\left\{x^{(0)}(1),x^{(0)}(2),\ldots,x^{(0)}(n)\right\} \tag{1}\]  
Generate 1-AGO sequences:  
\[X^{(1)}=\left\{x^{(1)}(1),x^{(1)}(2),\ldots,x^{(1)}(n)\right\} \tag{2}\]  
where \(x^{(1)}(k)=\sum_{i=1}^{k}x^{(0)}(i)\), \(k=1\), \(2\), \(\ldots\), \(n\)  
\[x^{(0)}+ax^{(1)}(k)=b \tag{3}\]  
Equation (3) is the original form of GM (1, 1), where a is called the developmental coefficient response to the developmental dynamics of the behavior series estimates, and b is the amount of gray action, which reflects the relationship of data change. The original form of GM (1, 1) is in fact a difference equation, where the a and b parameter vectors \(\hat{a}=\left(\begin{matrix}a\\ b\end{matrix}\right)\) can be obtained using the least squares method  
\[\hat{a}=\left(B^{T}B\right)^{-1}B^{T}Y \tag{4}\]  
where \(B\) and \(y_{n}\), respectively, are calculated as follows:  
\[B=\left[\begin{matrix}-x^{(1)}(2)&1\\ -x^{(1)}(3)&1\\ \vdots&\vdots\\ -x^{(1)}(n)&1\end{matrix}\right],\ Y=\left(\begin{matrix}x^{(0)}(2)\\ x^{(0)}(3)\\ \vdots\\ x^{(0)}(n)\end{matrix}\right) \tag{5}\]  
Different GM (1, 1) models are obtained by using different solutions of the difference equation as the time-response equation, and the different time-response equations are shown as follows:  
Time-response equation for the EGM model:  
\[\hat{x}^{(1)}(k)=\left(x^{(0)}(1)-\frac{b}{a}\right)e^{-a(k-1)}+\frac{b}{a}; \quad k=1,2,\ldots,n \tag{6}\]Time-response equation of the ODGM model:  
\[\hat{x}^{(1)}(k)=\bigg{(}x^{(0)}(1)-\frac{b}{a}\bigg{)}\bigg{(}\frac{1}{1+a} \bigg{)}^{k}+\frac{b}{a} \tag{7}\]  
Time-response equation of the EDGM model:  
\[\hat{x}^{(1)}(k)=\bigg{(}x^{(0)}(1)-\frac{b}{a}\bigg{)}\bigg{(}\frac{1-0.5a}{1+0. 5a}\bigg{)}^{k}+\frac{b}{a} \tag{8}\]  
Time-response equation of the DGM model:  
\[\hat{x}^{(1)}(k)=\bigg{(}x^{(0)}(1)-\frac{\beta_{2}}{1-\beta_{1}}\bigg{)}\beta _{1}^{k}+\frac{\beta_{2}}{1-\beta_{1}} \tag{9}\]  
of which \(\beta_{1}\) and \(\beta_{2}\), the parameter vectors of \(\hat{\beta}=\begin{pmatrix}\beta_{1}\\ \beta_{2}\end{pmatrix}\), are similar to the estimators of (12), where \(B\) and \(Y\) are, respectively:  
\[B=\begin{bmatrix}x^{(1)}(1)&1\\ x^{(1)}(2)&1\\ \vdots&\vdots\\ x^{(1)}(n-1)&1\end{bmatrix},\ Y=\begin{pmatrix}x^{(1)}(2)\\ x^{(1)}(3)\\ \vdots\\ x^{(1)}(n)\end{pmatrix} \tag{10}\]  
The method is applicable to small data samples and the average relative error of the prediction effect can be obtained.