## 3 Research Methods and Data Sources
### Research Methods
#### 3.1.4 Gray Prediction Model  
Gray system theory was proposed and founded by Professor [PERSON] in the 1980s. It is a systems engineering discipline mainly based on mathematical theory. It is widely used in water conservancy, petroleum, transportation, and other aspects [49]. Gray systems mainly solve the uncertainty problem of \"small samples and poor information\" and have been widely used in research in various fields [50].  
The gray system can be considered as an intermediary system that bridges the gap between the black system and the white system. A system that contains both known and unknown information is referred to as a gray system. The classification of technological innovation and the development of the ecological environment as a gray system arises from the presence of both certain and intrinsically unknown components influencing these processes. The GM (1, 1) model is well recognized as the predominant gray model in use. Furthermore, when compared to alternative models, the gray GM (1, 1) model exhibits several advantageous features, including a reduced reliance on sample data, enhanced accuracy in fitting, and simplified calculation [51]. Consequently, this model is employed to forecast the coupling and coordinated advancement of technological innovation and ecological environment development in China's coastal cities over the forthcoming five-year period. By doing so, it furnishes a dependable theoretical foundation for the developmental trajectory of China's coastal cities.  
The GM (1, 1) model consists of a differential equation of the first order with a single variable. It is mainly suitable for the characteristic combination and prediction of a certain dominant factor and is used to represent the changing trend of future development and the changing form of the dominant factor. The modeling steps of gray system GM (1, 1) can be found in the literature, which is implemented through Matlab (R2023a). After constructing the prediction model, it is essential to test its accuracy against the evaluation criteria outlined in Table 3 [52]. If both the \(p\)-value and \(C\)-value fall within acceptable ranges, predictive analysis can proceed; otherwise, further analysis of residual sequences and formula revision are necessary. The posterior difference ratio \(C\) is the ratio of the standard deviation \(S_{2}\) of the residual sequence \(\delta\) and the standard deviation \(S_{1}\) of the original sequence \(x_{i}\). The calculation method is the following:  
\[\mathrm{C}=\frac{S_{2}}{S_{1}} \tag{9}\]  
\[S_{1}^{2}=\frac{1}{n}\underset{i=1}{\sum}_{i=1}^{n}[x_{i}-\overline{x}]^{2} \tag{10}\]  
\begin{table}
\begin{tabular}{c c} \hline
**Coupling Coordination Degree** & **Coupling and Coordinated Development Stage** \\ \hline
0.1–0.3 & Weak coupling coordination stage \\
0.3–0.5 & Moderate coupling coordination stage \\
0.5–0.7 & Good coupling and coordination stage \\
0.7–1.0 & High-quality coupling coordination stage \\ \hline \end{tabular}
\end{table}
Table 2: Standards for dividing the stages of coupled and coordinated development of regional scientific and technological innovation and ecological environment development.  
\[S_{2}^{2}=\frac{1}{n}{\sum_{i=1}^{n}}\big{[}\delta_{i}-\overline{ \delta}\big{]}^{2} \tag{11}\] \[\mathrm{P}=\big{\{}\big{|}\delta_{i}-\overline{\delta}\big{|}<0.6745 \mathrm{S}_{1}\big{\}} \tag{12}\]