## 3 Methods
### Driving Mechanisms of NPP Changes
#### 3.4.2 The Role of Climate and Human Activities in NPP Changes  
Overall, temperature, precipitation, and solar radiation were used as climate-influencing factors for NPP, and differential equations were used to assess the contributions of climate change and human activities to the vegetation NPP of the terrestrial ecosystems of China for 2001-2020. On each stage, the drivers of vegetation NPP change were discussed using a partial least squares-structural equation (PLS-SEM) approach with a five-year period, integrating four types of influencing factors--meteorological, topographic, soil, and socio-economic--to reveal how vegetation NPP responded to climatic factors across terrestrial ecosystems of the country. (1) Differential equation method  
Differential equation methods can quantitate the response for climate change and human activities to changes in vegetation NPP for a function \(y=f\) (\(x_{1}\), \(x_{2}\),...). The variability of the dependent factor \(y\) may be represented as [37]:  
\[\frac{dy}{dt}=\sum\frac{\partial f}{\partial x_{i}}\frac{dx_{i}}{dt}=\sum{f^{ \prime}}_{i}^{t}\frac{dx_{i}}{dt} \tag{13}\]  
It then follows that:  
\[\frac{d\text{NPP}}{dt}=\frac{\partial\text{NPP}}{\partial tem}\frac{dtem}{dt} -\frac{\partial\text{NPP}}{\partial pre}\frac{dpre}{dt}-\frac{\partial\text{ NPP}}{\partial radi}\frac{dradi}{dt}+\varepsilon \tag{14}\]  
Alternatively,  
\[TR_{\text{NPP}}=\frac{d\text{NPP}}{dt}=C_{tem}+C_{pre}+C_{radi}+\varepsilon=C _{cc}+C_{ha} \tag{15}\]  
where \(d_{\text{NPP}}/d_{t}\) is the trend in the long time series NPP and \(d_{\text{NPP}}/d_{t}\), \(d_{tem}/d_{t}\), \(d_{pre}/d_{t}\), and \(d_{radi}/d_{t}\) represent the slope of the linear regression of NPP, temperature, precipitation, and solar radiation at time \(t\), respectively. Moreover, \(C_{cc}=C_{tem}+C_{pre}+C_{radi}\), \(C_{ha}=\varepsilon\), \(C_{tem}\), \(C_{pre}\), and \(C_{radi}\) are the response of climate change, human activity, temperature, precipitation, and radiation to the long-term trend in NPP, respectively, and \(\varepsilon\) is the systematic error. Although there are many drivers other than climate change that affect NPP changes, human activities constitute the dominant part. Finally, the individual proportional contributions of climate change (Equation (16)) and human activity (Equation (17)) to NPP trends, \(p(x)\), can be estimated as follows:  
\[\rho_{\text{cc}}=\frac{C_{cc}}{C_{cc}+C_{ha}}\times 100\% \tag{16}\]  
\[\rho_{ha}=\frac{C_{ha}}{C_{cc}+C_{ha}}\times 100\% \tag{17}\]  
(2) PLS-SEM method  
The advantages of PLS-SEM are that it fully exploits the information in the data, minimizes the error term, and does not require a particular (high) sample size, model identification problems, or distribution state of the data. Therefore, the problem of covariance between variables can be dealt with effectively. The model is composed of two major components; one is a measuring model characterizing the relationship between latent and explicit variables, and the other is a structuring model characterizing the relationship of latent and explicit variables [38]. Its methodology for estimating the parameterization is separated into two procedures: (1) iteration to obtain the estimates of the latent variables, and (2) linear regression using partial least squares to obtain the estimates of the parameters for structural and measurement models. In this study, using a stratified random sampling method, 50,000 sample points were selected to analyze the effect of each influencing factor on the changes in NPP. Considering \(k\) latent variables and each of \(k\) sets of dominant variables which each contain \(m\) variables, each dominant variable set may be denoted as  
\[X_{i}=\{x_{i1},x_{i2},x_{i3},\ldots,x_{im}\}i=\{1,2,3,\ldots,k\} \tag{18}\]  
It is assumed that the latent variables are all linearly combined with the latent variables and the latent variables are linearly combined with the dominant variables.