# A New Approach to Privacy-Preserving Multiple Independent Data Publishing
## 4 Merging Method
### Formulation
#### 4.1.2 Attribute Separation  
For attribute separation, related attributes are arranged in a subset, such that each attribute belongs to one subset. Each grouped subset is called a column. Specifically, in a microdata table \(T\) there will be \(c\) columns \(C_{1},C_{2},...,C_{c}\) satisfying \(\bigcup_{i=1}C_{i}=AT\) and for any \(1\leq i_{1}\
eq i_{2}\leq c\), \(C_{i_{1}}\cap C_{i_{2}}=\emptyset\). The sensitive attribute \(S\) forms the last column \(C_{c}\), called the sensitive column. The remaining columns \(\{C_{1},C_{2},...,C_{c-1}\}\) contain \(QI\) attributes.  
Related attributes are grouped by measuring the association between attributes. The widely used method to measure the association between categorical attributes is the mean square contingency coefficient [25; 42]. Given two categorical attributes \(AT_{1}\) and \(AT_{2}\) with value domains \(\{v_{11},v_{12},...,v_{1d_{1}}\}\) and \(\{v_{21},v_{22},...,v_{2d_{2}}\}\) respectively and domain sizes \(d_{1}\) and \(d_{2}\), the mean square contingency coefficient between attributes \(AT_{1}\) and \(AT_{2}\) is  
\[\phi^{2}(AT_{1},AT_{2})=\frac{1}{min\{d_{1},d_{2}\}-1}\sum_{i=1}^{d_{1}}\sum_{ j=1}^{d_{2}}\frac{(f_{ij}-f_{i}.f_{\cdot j})^{2}}{f_{i}.f_{\cdot j}}\]  
where, \(f_{i}\). and \(f_{\cdot j}\) are the fractions of occurrence of \(v_{1i}\) and \(v_{2j}\) in the data, respectively. \(f_{ij}\) is the fraction of cooccurrence of \(v_{1i}\) and \(v_{2j}\) in the data. Therefore, \(f_{i}\). and \(f_{\cdot j}\) are the marginal totals of \(f_{ij}:f_{\cdoti}=\sum_{j=1}^{d_{2}}f_{ij}\) and \(f_{\cdot j}=\sum_{i=1}^{d_{1}}f_{ij}\). It can be shown that \(0\leq\phi^{2}(AT_{1},AT_{2})\leq 1\).  
For continuous attributes, we have applied discretization to partition the domain of continuous attributes. Equal-width discretization is used for partitioning the domain into some \(k\) equal-sized intervals.  
After computation of association between attributes, the widely employed \(k\)-_medoid_ clustering algorithm Partition Around Medoids (PAM) [43] is used for partitioning attributes into the columns. In the anonymization algorithm, each attribute is determined as a point in the cluster space. The dissimilarity of two attributes is defined as \(d(AT_{1},AT_{2})=1-\phi^{2}(AT_{1},AT_{2})\), which lies in 0 and 1. In the cluster space, two correlated attributes have a lower dissimilarity between the corresponding data points.  
We arrange attributes to bring the highly correlated attributes in the same column. For the data utility and privacy, this approach performs well. In terms of data utility, grouping highly correlated attributes protect the relationships among those attributes. With regard to privacy, the association between correlated attributes reduce the identification risk. Since the association between uncorrelated attribute values are not common among the dataset and hence more distinctive, it is suitable to break the relation between uncorrelated attributes to preserve the user privacy. That way, when an adversary examines the intersection of two or more published datasets, the resulting intersection data will lead to more false matches, reducing the adversary's confidence in breaching data privacy.