# Quantitative Analysis of Influencing Factors on Changzhou Ship Lock Capacity
## 3 Methods
### Analyze Factors Based on Principal Component Analysis (PCA)  
PCA is a data analysis method that transforms multiple variables into a smaller number of comprehensive variables through variable transformation. Its essence is to seek comprehensive substitutes for correlated variables based on the correlation of the original variables [24; 28; 29; 30; 31; 32]. Each principal component is a linear combination of the original variables, mutually independent and retaining most of the information of the original variables. For a single ship passage event, the factors that affect ship passing time are independent of each other, and multiple factors collectively affect ship passing time. PCA can identify factors with a significant impact from numerous factors and conduct a quantitative analysis, which is crucial for analyzing the factors.  
A total of 11 variables were collected from the ship blockage data: ship lock sill, operation direction, ship length, ship width, ship draft, ship weight, approved load, actual load, goods type, starting place, and arrival place. Among these, the operation direction has an independent and dominant impact on ship lock capacity. The goods type has a relatively small impact on ship lock capacity. The impact of starting and arrival places on the single-level ship lock capacity can be ignored.  
PCA is aiming to screen out important variables where highly correlated variables are orthogonally transformed into new variables, thereby losing their original meaning [33]. To make the extracted variables more concise and clear, a correlation analysis was first conducted on a large amount of ship passing data to filter out significantly correlated variables, from which representative variables were selected.  
For the selected variables, the steps for quantitative analysis based on PCA include the following:  
1. Assume there are m data samples and n variables, the corresponding data matrix is R, where each column represents a ship passing record and each row represents the variables to be analyzed. \[R=\begin{pmatrix}r_{11}&\cdots&r_{1m}\\ \vdots&\ddots&\vdots\\ r_{n1}&\cdots&r_{mm}\end{pmatrix}\] (5)
2. Normalize R to matrix X, where \(\overline{r}_{i}\) and \(s_{i}\) are the mean and standard deviation of the i-th row, respectively. \[x_{ij}=(r_{ij}-\overline{r})/s_{i}\] (6) \[X=\left(x_{ij}\right)_{ij}\] (7)  
Figure 4: The process of eliminating navigability data redundancy.  
3. Calculate the covariance matrix \(D(X)\) of matrix X. \[D(X)=\begin{pmatrix}Cov(X_{1},X_{1})&\cdots&Cov(X_{1},X_{n})\\ \vdots&\ddots&\vdots\\ Cov(X_{n},X_{1})&\cdots&Cov(X_{n},X_{n})\end{pmatrix}\] (8)
4. Calculate the eigenvalues \((\lambda_{1},\lambda_{2},\cdots\lambda_{n})\) and eigenvectors \((x_{1},x_{2},\cdots x_{n})\) of \(D(X)\), arrange the eigenvalues in descending order, and calculate the variance contribution rate \(s_{i}\) corresponding to each eigenvalue \(\lambda_{i}\) \[S_{i}=\frac{\lambda_{i}}{\sum\limits_{i=1}^{n}\lambda_{i}}\] (9)
5. Assume the k-th eigenvalue is \(\lambda_{k}\), and calculate the cumulative variance contribution rate \(\lambda_{i}\) corresponding to each eigenvalue \(Y_{k}\). \[Y_{i}=\sum\limits_{i=1}^{k}s_{i}\] (10)
6. PCA represents new variables using linear combinations of n original variables. The eigenvalue \(\lambda_{k}\) serves as the weight coefficient for each variable. By discarding eigenvalues with low variance contribution rates and selecting k principal components that can significantly represent the original variables, the dimensionality of the data is reduced. Typically, k variables with a cumulative variance contribution rate reaching 85% are chosen as the principal component variables.