## 2 Materials and Methods
### Dimensionality Reduction  
When the similarity between trajectories was measured using the statistical distance measurement method for ship trajectories obtained through the data purification, n \(\times\) n matrix data were extracted according to the number (n) of data points (ship trajectories). The similarity measurement data between ship trajectories were dimensionally reduced to convert the trajectory similarity into a 2 \(\times\) n format, which could be used for cluster analysis. Dimension reduction generally refers to reducing the volume of a dataset or converting high-dimensional data into low-dimensional data. It simplifies or summarizes the data to remove unnecessary data or create subsets. This technique is mainly used when processing large datasets and can save storage space for datasets or shorten calculation and processing times. It has the effect of improving the learning and prediction performance of the model. The representative methods for data reduction include sampling, dimension reduction, and clustering. [PERSON] and [PERSON] [30] described the application of PCA, an unsupervised learning technique, to reduce the dimensionality of large datasets; this highlights the importance of PCA in machine learning and intelligent systems, particularly in the development of algorithms that can process large amounts of data to solve various real-world problems. [PERSON] et al. [31] provided a comprehensive analysis and theoretical framework for classical multi-dimensional scaling (CMDS), providing an important theoretical contribution to understanding the performance of CMDS in dimensionality reduction, clustering, and data. Practical guidelines for applying this analysis are provided. PCA is a multivariate statistical technique that reduces data dimensionality [32]. Not only can it easily process large amounts of data, but it also improves calculation speed. The basic idea of PCA is to reorganize certain correlated variables into a new integrated set of uncorrelated variables. Assuming samples have individual variables, n-by-k data can be constructed, called a matrix X [33]. The data were standardized using Equation (7). This process set the mean of each characteristic to zero and the variance to one. \[x_{\mathrm{ij}}^{\prime}=\frac{x_{ij}-\overline{x_{j}}}{\sqrt{\frac{1}{n-1} \sum_{i=1}^{n}\left(x_{ij}-\overline{x_{j}}\right)^{2}}} \tag{7}\]  
\(x\): original data  
We can calculate the correlation matrix and calculate all eigenvalues \(\lambda_{\mathrm{j}}\). Subsequently, all related eigenvectors \(\mathrm{E_{j}}\) obtain their corresponding eigenvalues. Determine the number of Principal Components (\(\alpha\)) using Equation (8). This method calculates the proportion of the variance explained by each principal component. \[\alpha_{\mathrm{i}}=\frac{\lambda_{\mathrm{i}}}{\sum_{i=1}^{k}\lambda_{ \mathrm{i}}} \tag{8}\]  
The standardized matrix is converted into a Principal Component, as shown in Equation (9). The standardized matrix is converted into a Principal Component, as shown in Equation (9). \(\mathrm{a_{ij}}\) is the weight of the jth feature for the ith principal component and is an element of the eigenvector. Through this conversion, the original data were expressed as new principal components that were not correlated. \[ \begin{array}{l}Z_{1}=a_{11}x_{1}+a_{12}x_{2}+\cdots+a_{1k}x_{k}\\ Z_{2}=a_{21}x_{1}+a_{22}x_{2}+\cdots+a_{2k}x_{k}\\ \vdots\\ Z_{\mathrm{m}}=a_{\mathrm{m1}}x_{1}+a_{\mathrm{m2}}x_{2}+\cdots+a_{mk}x_{k}\;1 \leq\;m\;\leq k\end{array}  \tag{9}\]  
CMDS uses data as a matrix that expresses the similarity based on the distance between objects, such as the distance measured on a map. To measure the distance between objects,Euclidean Distance is used for the x and y coordinates of the object, and the application method is defined as Equation (10) [34].