# A Study of Multi-Step Sparse Vessel Trajectory Restoration Based on Feature Correlation
## 4 Trajectory Feature Extraction and Restoration Model Construction
### Input Feature Extraction Based on Trajectory Features Correlation
#### 4.2.1 Construction of a Composite Indicator of Feature Correlation  
Correlation analysis is a statistical method for assessing the degree of association between trajectory data variables, providing an important reference for the selection of model input features. The principle is to accurately measure the degree of correlation between variables by calculating the correlation coefficient index. To avoid the bias of a single calculation method, this study constructs a composite evaluation indicator of features correlation, uses multiple correlation calculation methods performs a weighted solution to comprehensively assess the degree of correlation between trajectory data variables, and provides a basis for determining the data features to be input into the model. 1. Pearson's Correlation Coefficient Pearson's correlation coefficient (PCC) is a commonly used correlation coefficient to quantitatively assess the degree of linear correlation between two variables. It is very sensitive to extreme values and accurately reflects the strength of the association between variables. The coefficient ranges from \(-1\) to \(1\), where \(-1\) means that the two variables are perfectly negatively correlated, \(1\) means that they are perfectly positively correlated, and \(0\)  
Figure 5: Sliding window method to divide the dataset. means that they are uncorrelated. For the given two random variables \(X\) and \(Y\), the specific Equation (5) for [PERSON]'s correlation coefficient is as follows:  
\[PCC_{(X,Y)}=\frac{\sum\limits_{i=1}^{n}{(X_{i}-\overline{X})(Y_{i}-\overline{Y})}} {\sqrt{\sum\limits_{i=1}^{n}{(X_{i}-\overline{X})^{2}}}\sqrt{\sum\limits_{i=1}^ {n}{(Y_{i}-\overline{Y})^{2}}}} \tag{5}\]  
where \(X\) and \(Y\) denote the values of the two variables, and \(\overline{X}\) and \(\overline{Y}\) denote the mean values of the two variables, respectively. 2. Spearman's Rank Coefficient Correlation  
Spearman's rank coefficient correlation (SRCC), also known as the rank correlation coefficient, is a statistical tool used to measure the correlation between variables. Unlike PCC, SRCC utilizes the rank order of the variables to assess their correlation, which makes SRCC more widely used as it does not require strict conditions for the data. Similar to PCC, SRCC can take values between \(-1\) and \(1\), with larger absolute values indicating a stronger correlation. The specific Equation (6) for SRCC is as follows:  
\[SRCC_{(X,Y)}=1-\frac{6\sum\limits_{i=1}^{N}{d_{i}^{2}}}{N(N^{2}-1)} \tag{6}\]  
where \(d_{i}=X_{i}-Y_{i}\); \(X_{i}\) and \(Y_{i}\) are the ranks of the two variables in order of magnitude; and \(N\) is the capacity of the sample. 3. Mutual Information Entropy Coefficient  
Information entropy is used to characterize the amount of information in a single random variable. Mutual information entropy (MIE), on the other hand, is used to evaluate the amount of information that the occurrence of one event contributes to the occurrence of another event, measuring the degree of correlation between two random variables \(X\) and \(Y\)[26; 27]. When the value of MIE is larger, the correlation between two random variables is stronger. In a special case where the value of mutual information is zero, the two random variables are independent of each other. To facilitate the subsequent calculation of the correlation index, this study adopts normalized mutual information entropy (NMIE) to calculate the mutual information between the variables, and the specific calculation Equation (7) is as follows:  
\[\left\{ \begin{array}{l}NMIE_{(X,Y)}=2\frac{I_{(X,Y)}}{H(X)+H(Y)}\\ I_{(X,Y)}=\sum\limits_{i=1}^{m}{\sum\limits_{j=1}^{n}{P(X_{i},Y_{j})\log_{2}{ \frac{P(X_{i},Y_{j})}{P(X_{i})P(Y_{j})}}}}\\ H(X)=-\sum\limits_{i}{P(X_{i})\log(P(X_{i}))}\\ \end{array} \right. \tag{7}\]  
where \(I_{(X,Y)}\) is the mutual information entropy of \(X\) and \(Y\); \(H(X)\) is the information entropy of \(X\); and \(P(X_{j})\) and \(P(Y_{i})\) are the edge probability density functions of \(X\) and \(Y\).