## 2 Correlation Analysis among FE of Marine Structure
### Correlation Analysis Method  
Consider a structure denoted by the set \(S\) comprising the overall elements. Each individual element within this structure can be represented as \(e_{i}\), where \(i\) denotes the index of the element, ranging from 1 to \(m\). Here, \(m\) represents the total number of elements present in the structure. For several load steps, the stress response matrix of all the elements is denoted as \(X\). \[\mathbf{X}=\left[\mathbf{x_{1}}\;\mathbf{x_{2}}\;\mathbf{x_{3}}\ldots\mathbf{x_ {i}}\ldots\mathbf{x_{m}}\right]= \begin{bmatrix}x_{1}(1)&x_{2}(1)&\ldots&x_{i}(1 )&\ldots&x_{m}(1)\\ x_{1}(2)&x_{2}(2)&\ldots&x_{i}(2)&\ldots&x_{m}(2)\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ x_{1}(k)&x_{2}(k)&\ldots&x_{i}(k)&\ldots&x_{m}(k)\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ x_{1}(n)&x_{2}(n)&\ldots&x_{i}(n)&\ldots&x_{m}(n)\end{bmatrix}  \tag{1}\]  
with  
\[\mathbf{x}_{i}=\left[\mathbf{x}_{i}(\mathbf{1})\;\mathbf{x}_{i}(\mathbf{2}) \ldots\mathbf{x}_{i}(\mathbf{n})\right] \tag{2}\]  
In Equation (1), the variable \(x_{i}(n)\) corresponds to the structural response of the finite element indexed by \(i\) under the specific load condition \(n\). In Equation (2), \(i=1,2,\ldots,m\). Due to the similarity in the responses of any two finite elements, \(e_{i}\) and \(e_{j}\), for the same load steps in a spatial structure, covariance \(\text{Cov}(x_{i},x_{j})\) can be used as a representation. \[Cov(x_{i},x_{j})=E(x_{i}x_{j})-E(x_{i})E(x_{j}) \tag{3}\]  
In Equation (3), \(E(x_{i})\) and \(E(x_{j})\) represent the expected values of the two real random variables \(x_{i}\) and \(x_{j}\), respectively. Due to the variables having different dimensions, their covariance can exhibit significant numerical differences. To address this, a dimensionless correlation analysis method such as the Pearson correlation coefficient [21], as shown in Equation (4), is introduced:  
\[P_{i,j}=\frac{cov\big{(}x_{i},x_{j}\big{)}}{\sigma_{i}\sigma_{j}} \tag{4}\]  
with  
\[cov\big{(}x_{i},x_{j}\big{)}=E(x_{i}-\bar{x}_{i})\big{(}x_{j}-\bar{x}_{j}\big{)} =\frac{1}{n-1}\sum_{k=1}^{n}(x_{i}(k)-\bar{x}_{i})\big{(}x_{j}(k)-\bar{x}_{j} \big{)} \tag{5}\]  
\[\sigma_{i}=\frac{1}{n-1}\sum_{k=1}^{n}(x_{i}(k)-\bar{x}_{i})^{2} \tag{6}\]  
\[\sigma_{j}=\frac{1}{n-1}\sum_{k=1}^{n}\big{(}x_{j}(k)-\bar{x}_{j}\big{)}^{2} \tag{7}\]  
In Equations (4)-(7), \(\sigma_{i}\) is the variance of the response value of the finite element with index \(i\) for \(n\) load steps, \(\sigma_{j}\) is the variance of the response value of the finite element with index \(j\) for \(n\) load steps, \(\bar{x}_{i}\) is the mean response value of the finite element with index \(i\) for \(n\) load steps, and \(\bar{x}_{j}\) is the mean response value of the finite element with index \(j\) for \(n\) load steps. The correlation coefficient, \(p\), has a range of values from \(-1\) to \(1\). When \(p=1\), \(x_{i}\) and \(x_{j}\) are perfectly positively correlated, whereas when \(p=-1\), \(x_{i}\) and \(x_{j}\) are perfectly negatively correlated. The strength of the correlation increases with the absolute value of \(p\) and weakens as \(p\) approaches \(0\). The correlation matrix \(P\) between any two finite elements \(x_{i}\) and \(x_{j}\) is given as follows. \[\mathbf{P}=\left[\mathbf{P}_{\bar{i}}\right]_{m\times m}= \begin{bmatrix}p_{11} &p_{12}&\ldots&p_{1j}&\ldots&p_{1m}\\ p_{21}&p_{22}&\ldots&p_{2j}&\ldots&p_{2m}\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ p_{i1}&p_{i2}&\ldots&p_{ij}&\ldots&p_{im}\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ p_{m1}&p_{m2}&\ldots&p_{mj}&\ldots&p_{mm}\end{bmatrix}  \tag{8}\]Typically, a value of \(p_{ij}>0.95\) indicates a significant correlation between \(x_{i}\) and \(x_{j}\). If the finite elements within a spatial structure exhibit a significant correlation in load feedback, they can be grouped together, resulting in the partitioning of the overall structure \(S\) into \(k\) subsets \(T\).