## 2 Trajectory Initialization Based on MDF-Clustering
### Fuzzy Clustering Based on Multidimensional Features  
\(\chi=(X_{1},\ldots,X_{M})\) is a set of input data after target detection,. There are \(n\) targets in the data, which in turn contain \(M\) target feature variables, where these \(M\) variables, as a kind of mixed feature variables, possess multiple types of features, such as data types, qualitative data, time series data, etc. That is, the input dataset \(\chi\) contains \(S\) types of feature variables, and for every kind of feature attribute, there are \(p_{s}\) variables, so  
\[s=1,\ldots,S;1<S\leq P;\ 1\leq p_{s}<P \tag{7}\]  
can be interpreted as follows: the feature variables of the first \(n_{1}\\(\chi_{2}\) is the set of two qualitative variables. Then, \(S=2\), \(n_{1}=n_{2}=2\), \(P=4\), \(\chi_{1}=\{X_{1},X_{2}\}\) and \(\chi_{2}=\{X_{3},X_{4}\}\). For data type variables, \(\chi_{is}=X_{is}\) is a vector of \(p_{s}\) values observed on the \(i\)th target. For a data sequence of length \(Q\), \(\chi_{is}=X_{is}\) is a \(Q\times p_{s}\) matrix, where the rows of the matrix represent the features entered in the sequence \(q(q=1,\ldots,Q)\); and the columns of the matrix represent the sequence on the \(p_{s}\) variable of the data input on the \(i\)th target [19]. The same is true for the qualitative variable types. Multidimensional data presents a unique challenge in trajectory initialization due to the varied nature of the target features. These features can include both numeric variables, such as speed and irradiance, and qualitative variables, such as target shape and contrast. To handle these diverse data types, the MDF-clustering algorithm employs a combination of Mahalanobis distance for numeric variables and a symbolic distance metric for qualitative variables. The Mahalanobis distance is particularly effective in this context as it adjusts for correlations between different features [20], ensuring that the scale of the features does not skew the clustering results. Meanwhile, the symbolic distance metric is designed to handle qualitative attributes, which do not have a natural numeric representation but are nonetheless crucial for distinguishing between different targets. By integrating these distance metrics [21], the algorithm calculates the overall weighted distance between the target's feature set and the clustering center. This weighted distance is then minimized through an iterative process, allowing the algorithm to continuously refine its cluster assignments and improve the accuracy of trajectory initialization in cluttered and noisy environments. On the other hand, traditional metrics for numeric data are not suitable for symbolic data, as symbolic data possess unique properties and structures. The expression and meaning of symbolic data can differ significantly from numerical data. Therefore, a specific metric tailored to symbolic data characteristics, such as the symbolic data difference measure, is necessary to assess similarities or differences between symbolic data accurately. Therefore, the distance between different targets, i.e., the distance between the \(i\)th unit and the \(i^{\prime}\) unit calculated according to the nature of the \(i\)th variable type, is as follows:  
\[{}_{s}d_{ii^{\prime}}=d(\chi_{is^{\prime}},\chi_{i^{\prime}s}) \tag{9}\]  
We can obtain  
\[d_{ii^{\prime}}^{2}=\sum_{s=1}^{S}\left(w_{s^{\prime}s}d_{ii^{\prime}}\right) ^{2}=\sum_{s=1}^{S}[w_{s}\cdot d(\chi_{is},\chi_{i^{\prime}s})]^{2} \tag{10}\]  
The above equation is the overall weighted squared distance considering \(S\) attribute types. Since the weights of the squared distances are in quadratic form, the distances are weighted squared. In the scenario of this paper, \({}_{1}d_{ii^{\prime}}=d(\chi_{i1},\chi_{i^{\prime}1})\), \({}_{2}d_{ii^{\prime}}=d(\chi_{i2},\chi_{i^{\prime}2})\) are the pairwise distance matrices, such as the Martian distance of \(\chi_{1}\) and the symbolic distance of \(\chi_{2}\), respectively.