# Simulation of Cross-Correlated Random Fields for Transversely Anisotropic Soil Slope by Copulas
## 3 Copula Theory  
\(F_{1}(X_{1})\), \(F_{2}(X_{2})\),..., and \(F_{m}(X_{m})\) are marginal distributions of random variables \(\mathbf{X}=(X_{1},\,X_{2},\cdots,\,X_{m})\). Via [PERSON]'s theory, the joint CDF \(F_{X}(X_{1},\,X_{2},\cdots,\,X_{m})\) can be expressed by an unique copula function \(\mathbf{C}\)[27]:  
\[F_{X}(\mathbf{X})=\mathbf{C}[(F_{1}(X_{1}),F_{2}(X_{2}),\cdots,F_{m}(X_{m});\mathbf{\theta}] \tag{12}\]  
wherein \(\mathbf{C}\) is uncorrelated with each margin; \(\mathbf{\theta}\) represents the copula parameter.  
Specifically, while \(\mathbf{X}\) is mutually independent, Equation (12) can be rewritten as  
\[F_{X}(\mathbf{X})=\prod_{j=1}^{m}F_{j}(X_{j}) \tag{13}\]  
By differentiating Equation (12), the joint PDF \(f_{X}(X_{1},\,X_{2},\cdots,\,X_{m})\) can be derived:  
\[f_{X}(\mathbf{X})=\mathbf{D}[F_{1}(X_{1}),F_{2}(X_{2}),\cdots,F_{m}(X_{m});\mathbf{\theta} ]\prod_{j=1}^{m}f_{j}(X_{j}) \tag{14}\]  
where in \(f_{1}(x_{1}),f_{2}(x_{2}),\,\ldots\,\,\), and \(f_{m}(x_{m})\) are marginal PDFs of \(\mathbf{X}\) and \(\mathbf{D}(\cdot)\) is the copula PDF; when variables are mutually independent, \(\mathbf{D}(\cdot)=1\).  
Table 1 gives some candidate copulas commonly used in soil slope analysis. It is noted that the copula parameter \(\mathbf{\theta}\) is crucial to constructing the copula. Commonly, maximum likelihood estimation is used to estimate \(\mathbf{\theta}\), as expressed below [27].  
\[L(\mathbf{\theta})=\prod_{i=1}^{n}\mathbf{D}(u_{1},u_{2},\cdots u_{m};\mathbf{\theta})= \prod_{j=1}^{m}\frac{\partial^{m}\mathbf{C}(u_{1},u_{2},\cdots,u_{m};\mathbf{\theta}) }{\partial u_{1}\partial u_{2}\cdots\partial u_{m}} \tag{15}\]  
Figure 1: Independent standard normal and transversely anisotropic random field \(\mathbf{H}^{IAG}\). (**a**) Random field of \(u\). (**b**) Random field of \(v\).  
wherein \(u_{1}\), \(u_{2}\),..., and \(u_{m}\) can be represented by transforming \(\mathbf{X}\) from original space into rank space, as shown below.  
\[u_{j}^{(i)}=\frac{rank(X_{j}^{(i)})}{n+1},i=1,2,\cdots,n \tag{16}\]  
For simplification, the logarithm form of \(\text{In}L(\mathbf{\theta})\) is computed based on the following equation  
\[\ln L(\mathbf{\theta})=\prod_{j=1}^{m}\mathbf{D}(u_{1},u_{2},\cdots u_{m};\mathbf{\theta} )=\sum_{j=1}^{m}\ln\mathbf{D}(u_{1},u_{2},\cdots u_{m}) \tag{17}\]  
Via solving Equation (17), the maximum likelihood estimator \(\hat{\mathbf{\theta}}_{ML}\) that satisfies \(\ln L(\hat{\mathbf{\theta}}_{ML})\geq\ln L(\mathbf{\theta})\) can be obtained.  
\[\frac{\partial\ln L(\mathbf{\theta})}{\partial\mathbf{\theta}}=0 \tag{18}\]  
From Equations (12) and (14), it can be found that different copulas describe different dependent structures. Therefore, the optimal copula should be sieved out first. Generally, the Akaike Information Criterion (AIC) [36] and Bayesian Information Criterion (BIC) [37] are adopted to sieve out the proper candidate copula, as briefly expressed below,  
\[AIC=2k-2\sum_{j=1}^{m}\ln\mathbf{D}(u_{1},u_{2},\cdots,u_{m}) \tag{19}\] \[BIC=k\cdot\ln n-2\sum_{j=1}^{m}\ln\mathbf{D}(u_{1},u_{2},\cdots,u_{m}) \tag{20}\]  
where \(k\) is the number of parameters concerning the estimated model. The model associated with minimum AIC and BIC scores is considered the preferable candidate.