## 3 Materials and Methods
### Problem Formulation  
In the context of this study, the problem of privacy preservation when dealing with spatio-temporal databases goes one step further, and is related to the work [9]. The spatio-temporal data is the location data of a number of mobile users along with the time-stamp of each position, as shown in Table 1. Through the SMaRT system, we have in our disposal offline trajectory data that give us information about Hough-X, as well as Hough-Y of spatial data \((x,y)\). Hence, for each database record per time-stamp, that is, the mobile user trajectory point, we can consider the values of four attributes \((x,y,\theta,u)\) (as in Table 1) along with the values of an additional eight attributes' \((U_{x},a_{x},U_{y},a_{y},b_{x},w_{x},b_{y},w_{y})\) (as in Table 2).  
So, we have chosen to anonymize dual point attributes by employing the \(k\)-NN method, which enables us to form the \(k\)-anonymity set of each mobile object per time-stamp, as depicted in Table 3. The data anonymization is handled both as a clustering and a no-clustering problem. In both approaches, the anonymity set is formed again by the \(k\) nearest neighbors IDS. For each mobile user \(i\) and per time-stamp \(l\), we compute its \(k\) nearest neighbors IDS and keep them in a vector with form \(k\
us_{il}=[id_{il1}id_{il2}\ldots id_{ilk}]\) for \(l=1,2,\ldots,L\). In Table 3, an example of such sets for \(N\) mobile users' dual points is presented. For each user, we measure the number of the \(k\) nearest neighbors dual points that remained the same from one time-stamp to another.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**ObjId** & **Timestamp** & **TimeToNextPoint** & **x** & **y** & **Angle** & **Velocity** \\ \hline
1 & 2013-03-09 10:00:01 & 0 & 21,082 & 56,436 & 1.23 & 0 \\
1 & 2013-03-09 10:00:04 & 3 & 21,099 & 56,432 & 1.16 & 4.5 \\
1 & 2013-03-09 10:00:11 & 7 & 21,221 & 56,484 & 1.51 & 14.6 \\
1 & 2013-03-09 10:00:19 & 8 & 21,331 & 56,524 & 1.95 & 11.3 \\
1 & 2013-03-09 10:00:21 & 2 & 21,402 & 56,495 & 0 & 29.5 \\
2 & 2013-03-09 10:00:03 & 0 & 35,587 & 59,829 & \(-\)2.76 & 0 \\
2 & 2013-03-09 10:00:08 & 5 & 35,568 & 59,782 & 2.94 & 7.8 \\
2 & 2013-03-09 10:00:16 & 8 & 35,580 & 59,723 & \(-\)2.07 & 5.8 \\
2 & 2013-03-09 10:00:25 & 9 & 35,530 & 59,668 & \(-\)1.52 & 6.4 \\
2 & 2013-03-09 10:00:34 & 9 & 35,476 & 59,671 & \(-\)2.85 & 4.6 \\ \hline \hline \end{tabular}
\end{table}
Table 1: An overview of an original spatio-temporal database.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**ObjId** & **Timestamp** & \(\mathbf{U_{x}}\) & \(\mathbf{a_{x}}\) & \(\mathbf{U_{y}}\) & \(\mathbf{a_{y}}\) & \(\mathbf{b_{x}}\) & \(\mathbf{w_{x}}\) & \(\mathbf{b_{y}}\) & \(\mathbf{w_{y}}\) \\ \hline
1 & 2013-03-09 10:00:01 & 4.37 & 22,242,219.9 & 1.03 & 4,800,692.9 & 0.23 & \(-\)5,093,637.7  
By employing the dual transformation methods as described in Section 3.1, the \(k\)-anonymity set of mobile users is formulated based on their dual points. Hence, an alternative definition for the \(k\)-anonymity is as follows:  
Definition 10: (\(k\)DUST-anonymity). A transformed database record is \(k\)-anonymous with respect to Hough-\(X\) dual points--that is, velocity and intersection attributes \((U_{x},a_{x})\) or \((U_{y},a_{y})\), if \(k-1\) discrete records in the same specific time-stamp \(\tau\) at least have the same dual point attributes so that no record of \(k\) is distinguished from its \(k-1\) neighboring records.  
Remark 1: _As we already mentioned in [9], \(k\)-anonymization intuitively hides each individual among \(k-1\) others. This means that linking cannot be performed with confidence greater than \(\frac{1}{k}\). Nevertheless, \(k\)-anonymity may not protect users against the unveiling of dual point attributes._