## 2 Materials and Methods
### Creating China Vegetation Functional Zones (CVFZ) Based on the Selected Endmembers  
After selecting the vegetation endmembers, pixels in MOD-2011 with more than 20 periods missing were filtered out to determine the category of the rest pixels, which removed 6.13% of the total number of pixels. The n-dimensional vector expression of each pixel position in the coordinate system is obtained:  
\[X=\left[x_{1},x_{2},\cdots x_{m}\right]^{T} \tag{3}\]  
The distance (\(D\)) between two multi-temporal pixels was characterized using a compound distance constructed by combining the Euclidean distance and the spectral shape angle cosine. The spectral shape angle cosine is independent of the gain coefficient of the LAI multi-temporal image and ranges between 0 and 1, reflecting the similarity of the two multi-temporal pixels. This helps to address the limitation of using only Euclidean distance, which cannot fully reflect the change in the pixel LAI time series. By combining the two spectral characteristics of time phase amplitude and time phase shape, the accuracy of similarity measurement between multi-temporal pixels is improved [70; 71]. \(D\) can be expressed as follows:  
\[D=\sqrt{\frac{\sum_{i=1}^{m}(x_{i}-x^{\prime}_{i})^{2}}{\mathfrak{m}}}*\left( 1-\frac{\sum_{i=1}^{m}x_{i}x^{\prime}_{i}}{\sqrt{\sum_{i=1}^{m}x^{\prime}_{i} }^{2}\sqrt{\sum_{i=1}^{m}x^{\prime}_{i}}^{2}}\right) \tag{4}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(P_{occurrence}\) & **Row** & **Column** & **Location** & **Vegetation Type** \\ \hline
100.0\% & 1689 & 1724 & Yunnan & Evergreen Needleleaf Forests \\
3.9\% & 1802 & 2707 & Taiwan & Savannas \\
3.7\% & 2090 & 2125 & Hainan & Savannas \\
2.2\% & 1828 & 2707 & Taiwan & Savannas \\
1.8\% & 1131 & 2117 & Shanxi & Deciduous Broadleaf Forests \\
1.6\% & 1165 & 2070 & Gansu & Deciduous Broadleaf Forests \\
1.5\% & 752 & 2754 & Liaoning & Grasslands \\
1.2\% & 1288 & 2178 & Henan & Grasslands \\
1.1\% & 1738 & 2556 & Fujian & Evergreen Broadleaf Forests \\
1.1\% & 1063 & 2198 & Shanxi & Deciduous Broadleaf Forests \\ \hline \hline \end{tabular}
\end{table}
Table 1: Extraction results of vegetation endmembers. The locations of endmembers are accurate to the province, and vegetation types are defined based on the MCD12 QI type3 band.  
where 1 minus the angular cosine of the spectral shape is used to unify the increase or decrease in the two distance indicators, further obtaining the absolute distance \(D_{A}\) and the relative distance \(D_{R}\).  
\[D_{A}=\left|D_{pixel-zeros}-D_{endmember-zeros}\right| \tag{5}\]  
\[D_{R}=D_{pixel-endmember} \tag{6}\]  
where _zeros_ is an all-one vector of \(1\times 46\) dimensions. \(D_{A}\) represents the absolute distance between the pixel to be classified and the endmember, and \(D_{R}\) represents the relative distance between them.  
The MLC (Maximum Likelihood Classification) algorithm achieves the effect of classification by substituting the image to be classified into the determined classification function on a pixel-by-pixel basis [72]. This algorithm builds upon [PERSON] et al. [73] by using the idea of equidistant loss to adjust the distance between samples with relative distance constraints. Thus, absolute distances are added to intervene in the classification process of vegetation pixels. The algorithm calculates the \(D_{R}\) of each pixel to 10 endmembers in turn. If \(D_{R}\) is the smallest and \(D_{A}\) is not greater than the Threshold, the pixel belongs to the category. The existence of a Threshold results in not all pixels being classified. The above classification process is called the VCA-MLC algorithm (Figure 2). The derivative product is called the Chinese Vegetation Functional Zones (CVFZ).