## 4 Discussion
### Reasons for Selecting Indicators and Testing for Collinearity and Correlation
#### 4.2.2 Correlation and Collinearity Diagnosis between Indicators  
To assess the collinearity and correlation among various indicators, we randomly extracted 27,000, 30,000, and 8000 sample points from BTH, CYRD, and PRD, respectively. This extraction was based on the principle of maintaining a distance longer than 1 km between points to avoid autocorrelation while preserving the region's area proportion. We utilized SPSS Statistics 25.0 to calculate the Variance Inflation Factor (VIF) and tolerance as indicators for diagnosing multicollinearity. A higher VIF suggests stronger collinearity. Specifically, a VIF greater than 5 indicates strong collinearity, while a VIF lower than 5 but higher than 1 implies moderate collinearity. The tolerance is an inverse measure, where a smaller value indicates stronger collinearity, and a limit of 0.2 is considered significant. The results of the multicollinearity test for various indicators across the three major urban agglomerations, after averaging, are presented in Table 7. Notably, LAI and GPP exhibited strong collinearity, and Pre, LC, and NDVI showed more collinearity than other variables. Unfortunately, HD did not pass the significance test. It is essential to highlight that, upon separate sampling and collinearity calculation for PRD, most indicators exhibited increased collinearity, with LAI, GPP, and NDVI reaching values of 11.1, 18.6, and 7.3, respectively. The distinct relationships between various indicators across different urban agglomerations may obscure a certain degree of collinearity. Additional details regarding collinearity results for each urban agglomeration can be found in Table S5. Therefore,introducing principal component analysis (PCA) to reduce data dimension and collinearity between variables is deemed feasible. As for exploring the influencing relationship between variables and EQI, we conducted a correlation diagnosis in the three urban agglomerations for the years 2001 and 2020. Actually, there are more studies introducing panel regression analysis due to its outstanding performance in controlling time dimension and solving the problem of heterogeneity to analyze the impact of various independent variables on the dependent variable [108; 109]. However, we did not choose this method for several reasons. The first reason is that we collected various datasets in 2001 and 2020 to practice our method, lacking sufficient time information to construct a proper panel dataset. There would be more problems waiting to be solved, such as cross-sectional dependence, if we used a short panel database [110]. The second reason is that our research situation may not fit well the assumptions of many panel regression models. For example, the fixed effect model assumes that all unobserved heterogeneity is time-invariant and constant across time [111], but we are not sure to what extent the indicators have temporal variability, especially those indicators with strong correlation with other variables and geographical conditions [112]. The correlation results are presented in Figure 10. The test outcomes validate the rationale behind our framework and EQI. Combining the spatial patterns of ecological quality (see Figure 3) with correlation analysis (see Figure 10) yields important findings. It aligns with common sense that vegetation parameters and land cover in ecological function (EF) play a crucial role in influencing EQI (Figure 10). However, the negative effects from human activities, such as PM2.5 and GDP, cannot be overlooked (especially in Figure 10e,f) [44]. Furthermore, comparing the correlation between NDVI and LAI in different urban agglomerations, we observe a gradual decrease in correlation as latitude increases, with values of 0.78 in BTH and 0.88 in PRD (Figure 10d,f). This decline may be attributed to the saturation effect of NDVI in areas with high vegetation cover, underscoring the significance of selecting both NDVI and LAI as evaluation indicators [104].