## 4 Results
### Vulnerability Evaluation in Hough Space  
In the context of proposed work, we focused on \(k\)-anonymity from a different perspective as we employed Hough-X transformation of \((x,y)\) spatial data to formulate the anonymity set. The number of clusters is denoted by \(K\), while \(k\) refers to the number of nearest neighbors in terms of Euclidean distance. In Figures 4 and 5, both approaches achieve similar performance that can be improved as the number of \(k\) increases. Although for low values of \(k\), the vulnerability remains relatively high, the more the nearest neighbors are utilized to form the anonymity set, the lower the vulnerability becomes. Actually, Figure 4b,d depicts that the use of Hough-X transformation of y attribute considerably improved the performance in both cases. To ameliorate the classification accuracy, we considered the Hough-X attributes of y as well.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{K}\) & \(\mathbf{k}\) & **Clustering Attributes** & \(\mathbf{k}\)**-NN Attributes** \\ \hline
3 & 8 & \((x,y)\) & \((U_{x},a_{x})\) \\
3 & 8 & \((x,y)\) & \((U_{x},a_{x},U_{y},a_{y})\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: Parameters for the experiment using \((x,y)\) for clustering and Hough-X for \(k\)-NN for \(N=87\) trajectories and \(L=100\) time-stamps (Figure 7c,d).  
We should note that the information \((U_{x},a_{x})\) in combination with \((U_{y},a_{y})\) increased the robustness of both methods for the same number of nearest neighbors. This entails that the performance of the \(k\)-NN classifier improved, and thus the anonymity set shows less time variation from one time-stamp to another. In the following, we employ the suppressing \(k\)-anonymity method for the composition of the \(k\) anonymity set. In particular, we applied \(K\)-Means clustering that takes advantage of \((U_{x},a_{x},U_{y},a_{y})\), while \(k\)-NN is applied either on \((U_{x},a_{x})\) or \((U_{y},a_{y})\). Here, the clustering method presents much better performance than the non-clustering one. However, for the same number of \(k\), the performance of both methods is worse than in the first case, where we based it on attributes \((U_{x},a_{x},U_{y},a_{y})\) for both clustering and \(k\)-NN computation. The experimental results in Figure 5 present the performance of attribute suppression in terms of \(k\) anonymity set computation.  
Subsequently, a scenario with synthetic data from the real trajectory dataset was considered. More specifically, for each dual point, a number of copies \(M\) were generated, which shows that these dual points correspond to the same linear sub-trajectory of the trajectory. Figure 6a,b shows that DukNN (non-clustering) and DuCLkNN (clustering) methods have identical performance for \(M=5\), \(k=10\) and \(K=5\), and we verify that vulnerability is piece-wise preserved, except for the characteristic points.  
In Figure 6c, we compare vulnerability in the Hough-X space of \(x\) and \(y\) attributes with the one in native dimensional space of \(x\) and \(y\). We observe that the results in native space are better by almost 5% than the ones in Hough-X space. This may relate with the linear dependency of dual space of the native one.