# Early Evidence That Soil Dryness Causes Widespread Decline in Grassland Productivity in China
## 2 Materials and Methods
### Methods  
#### 2.2.1 Interannual Correlation Measures  
For extreme value theory, the correlation coefficient is a good indicator and is commonly used to measure the effect of dependence on the likelihood of binary extreme values [24]. We calculated the interannual correlation between the SM and GPP model dataset per pixel. To remove the influence of climate change signals on long-term trends, we performed a linear detrending of the bivariate prior to calculating correlations. Commonly used correlations include Pearson and Spearman [25]. Pearson correlation coefficients are only applicable to correlation analyses where the two variables are linear; however, the effect of SM on GPP is often non-linear. Therefore, we chose the Spearman's rank correlation coefficient to analyze the correlation between SM and GPP, with a stronger negative correlation between the two variables indicating more significant negative feedback. 2.2 Bivariate Linkage to Calculate the Probability of Conditions under Soil (or Atmospheric) Drought Conditions  
Based on the copula function and Bayesian equation, we drew on the novel probabilistic assessment framework constructed by [PERSON] et al. (2021) and [PERSON] et al. (2022) to calculate the conditional probability of a simultaneous soil drought (or atmospheric drought) scenario. This consists of the following three steps:  
(1) Fitting of marginal distributions  
Bivariate frequency analysis requires that the distributions of the random variables U and V be determined, so determining the marginal distribution of the bivariate is a prerequisite for constructing the joint probability distribution. We used a non-parametric method for fitting the marginal distributions, because non-parametric estimation methods do not require prior estimation or assumptions about all parameters of the copula function of the dependence structure between the bivariate variables, and can be directly estimated to obtain the fitted values at any point [26]. Non-parametric kernel density estimation is therefore more widely used in practice than conventional parameter estimation (e.g., normal, gamma), and can effectively eliminate errors in the fitted joint probability distribution due to partial singular values. Kernel density estimation is the most widely used test in the field of non-parametric estimation [27], where the kernel distribution produces a non-parametric probability density estimate that adapts itself to the data, rather than selecting a probability density estimate with a particular parameter. Here, we used kernel density estimation to derive marginal distribution fits for the bivariate. The kernel function density estimation method is described by the following equation:  
\[f_{n}(x)=\frac{1}{nh}{\sum_{i=1}^{n}K\left(\frac{x-x_{i}}{h}\right)}\]  
\(f_{n}(x)\) represents the kernel density value; \(n\) represents the number of samples in the bandwidth range; \(h\) is the window or bandwidth, representing a reasonable smoothing parameter and \(h>0\); \(K\) is the kernel smoothing function; \((x-x_{i})\) is the distance between points to \(x_{i}\); and \(x_{1}\), \(x_{2}\),... \(x_{n}\) are random samples from an unknown distribution. The kernel density estimate of the cumulative distribution function is described as follows:  
\[F_{n}(x)=\int_{-\infty}^{x}f_{n}(t)dt=\frac{1}{n}{\sum_{i=1}^{n}G\left(\frac{x -x_{i}}{h}\right)},G(x)=\int_{-\infty}^{x}K(t)dt\]  
(2) Fitting and optimization of joint probability distributions  
Copulas are multivariate distribution functions defined in the domain of [0, 1] and are used to describe correlations between multiple variables. The bivariate copula function is commonly used to describe the dependence structure between two sets of random variables and to count the joint probability of an event (such as a compound drought) occurring.