# Data Protection and Multi-Database Data-Driven Models
## 3 Methodology
### Masking Methods  
We have considered methods for microaggregation (MDAV and Mondrian algorithms), noise addition (both Gaussian and Laplacian noise), two methods based on dimensionality reduction (SVD and PCA), and another method based on non-negative matrix factorization. We briefly describe the methods below. These masking methods are described in detail in books on data privacy [3; 4].  
* Microaggregation: in order to protect a set of records, small clusters are built and records are replaced by a cluster representative. In order to guarantee privacy, each cluster needs to have at least \(k\) records. Thus, \(k\) is a parameter of the method. The larger the \(k\), the larger the privacy level, and the lower the utility of the data. Optimal microaggregation is often defined in terms of the error between the cluster representatives and the original records. Optimal microaggregation is an NP-hard problem for multivariate data (see e.g., [22]). Because of this, several alternative heuristic algorithms have been developed. We have used two alternative algorithms: MDAV [23; 24], and Mondrian [25]. The difference is related to the method of building the clusters. We use the mean of the records in a cluster as the cluster representative. Microaggregation provides \(k\)-anonymity by definition, as each record in the protected data set is indistinguishable from at least \(k-1\) other records.
* Noise addition: in order to protect a record, noise is added to it. In other words, the original numerical value \(x\) is replaced by \(x+\epsilon\), where \(\epsilon\) follows a given distribution. We use two different alternative distributions for protection. They are a normal distribution with mean zero and standard deviation \(\sqrt{(variance*k)}\), and a Laplace distribution with mean zero and standard deviation as above. Naturally, \(k\) corresponds to the parameter of the method. As in the case of microaggregation, the larger the \(k\), the larger the distortion. Therefore, the larger the \(k\), the larger the protection and the smaller the utility of the resulting protected data. Noise addition using Laplacian noise is the standard approach to implementing differential privacy. In the case of publishing a database, as we do here, this corresponds to local differential privacy.
* Transform-based protection: these methods reduce the quality of the data by transforming the data into another space in which we can remove details. We have considered the use of singular value decomposition (SVD), principal component analysis (PCA), and non-negative matrix factorization (NMF) for this purpose. In the case of SVD and PCA, we apply the decomposition, select the principal components, and then we rebuild the matrix with only these selected components. The parameter of the method is the number of components selected. We denote this \(k\). In this case, the larger the \(k\), the better the reconstruction of the original data. Therefore, the smaller the \(k\), the larger the protection, and at the same time, the smaller the utility of the resulting protected data. This approach is similar to non-negative matrix factorization. For NMF, as above, the smaller the number of components \(k\), the larger the protection and the smaller the utility. While SVD and PCA can be applied to matrices with arbitrary real numbers, NMF can only be applied to positive data. Because of this, data are scaled into the [0,1] interval before the application of the NMF protection. The data are re-scaled back after the NMF protection.