# Research on the Reliability of Bridge Structure Construction Process System Based on Copula Theory
## 2 Basic Theory of the Copula Function
### Metrics of Copula Function Correlation
#### 2.2.2 Kendall Rank Correlation Coefficient  
In contrast to the Pearson linear correlation coefficient, the Kendall rank correlation coefficient not only describes the linear correlation between failure modes, but also describes the nonlinear correlation between them well and has been widely used in engineering. The Kendall rank correlation coefficient is a measure of the degree of consistency between variables. Let \((x_{11},x_{21})\) and \((x_{12},x_{22})\) be two sets of observations of a two-dimensional random vector \((X_{1},X_{2})\). If \((x_{11}-x_{21})(x_{12}-x_{22})>0\), then \((x_{11},x_{21})\) and \((x_{12},x_{22})\) are said to be consistent. If \((x_{11}-x_{21})(x_{12}-x_{22})<0\), then \((x_{11},x_{21})\) and \((x_{12},x_{22})\) are said to be inconsistent. Establish the independent identically distributed vectors \(\big{(}X_{1}^{\prime},X_{2}^{\prime}\big{)}\) of the two-dimensional random vector \((X_{1},X_{2})\). Using \(P\big{[}\big{(}X_{1}-X_{1}^{\prime}\big{)}\big{(}X_{2}-X_{2}^{\prime}\big{)}> 0\big{]}\) to denote their probability of being consistent, and \(P\big{[}\big{(}X_{1}-X_{1}^{\prime}\big{)}(X_{2}-X_{2}^{\prime})<0\big{]}\) to denote their probability of being inconsistent, the Kendall rank correlation coefficient \(\tau\) is defined as the difference between the probability of being consistent and the probability of being inconsistent, with the following equation [28]:  
\[\tau=P\big{[}\big{(}X_{1}-X_{1}^{\prime}\big{)}\big{(}X_{2}-X_{2}^{\prime} \big{)}>0\big{]}-P\big{[}\big{(}X_{1}-X_{1}^{\prime}\big{)}\big{(}X_{2}-X_{2} ^{\prime}\big{)}<0\big{]} \tag{6}\]  
Similar to the range of the Pearson linear correlation coefficient \(\rho\), the range of the Kendall rank correlation coefficient \(\tau\) is also \([-1,1]\). The larger the \(|\tau|\), the stronger the correlation between the two failure modes. \(|\tau|=1\) indicates that the two failure modes are perfectly correlated, while \(|\tau|=0\) indicates that there is no correlation between the two failure modes. The Kendall rank correlation coefficient \(\tau\) can also be calculated from the observations of the two-dimensional random vectors \(X_{1}\) and \(X_{2}\)[28]:  
\[\tau=\frac{\sum\limits_{i<j}\operatorname{sign}\big{[}\big{(}x_{1i}-x_{1j} \big{)}\big{(}x_{2i}-x_{2j}\big{)}\big{]}}{0.5N(N-1)} \tag{7}\]  
where \(i,j=1,2,\cdots,N\); \(\operatorname{sign}[\cdot]\) is the symbolic function, when \(\big{(}x_{1i}-x_{1j}\big{)}\big{(}x_{2i}-x_{2j}\big{)}>0\), \(\operatorname{sign}=1\), and vice versa, \(\operatorname{sign}=-1\); \(N\) is the total number of combinations of observations. The Kendall rank correlation coefficient \(\tau\) has the following relationship with the two-dimensional Copula function \(C(u,v;\ \theta)\)[28]:  
\[\tau=4\int_{0}^{1}\int_{0}^{1}C(u,v;\ \theta)\text{d}C(u,v;\ \theta)-1 \tag{8}\]  
The Kendall rank correlation coefficient mainly reflects the correlation between two groups of random vectors and his independent identically distributed vectors. However, what this paper needs to reflect is the correlation between the two failure modes. In order to overcome the above drawbacks, this paper proposes to use the Latin hypercube sampling method to sample the uncertainty parameters in the two failure modes and then bring them into the failure mode function so as to calculate the response values of the function. The two sets of response values are the two-dimensional random vectors \(X_{1}\) and \(X_{2}\). Then the Kendall rank correlation coefficient \(\tau\) between the two functions can then be calculated using the Equation (6). The details of the Latin hypercube sampling method are described below. The Latin hypercube sampling method to draw \(N\) samples \(x=(x_{1,i},x_{2,i}\cdots x_{N,i})^{T}\)\((i=1,2\cdots N)\) from the random variable \(X=(X_{1},X_{2}\cdots X_{m})^{T}\) is as follows [31]:  
1.