Information preserving regression-based tools for statistical disclosure control

Abstract

This paper presents a unified framework for regression-based statistical disclosure control for microdata. A basic method, known as information preserving statistical obfuscation (IPSO), produces synthetic data that preserve variances, covariances and fitted values. The data are then generated conditionally according to the multivariate normal distribution. Generalizations of the IPSO method are described in the literature, and these methods aim to generate data more similar to the original data. This paper describes these methods in a concise and interpretable way, which is close to efficient implementation. Decomposing the residual data into orthogonal scores and corresponding loadings is an essential part of the framework. Both QR decomposition (Gram–Schmidt orthogonalization) and singular value decomposition (principal components) may be used. Within this framework, new and generalized methods are presented. In particular, a method is described by means of which the correlations to the original principal component scores can be controlled exactly. It is shown that a suggested method of random orthogonal matrix masking can be implemented without generating an orthogonal matrix. Generalized methodology for hierarchical categories is presented within the context of microaggregation. Some information can then be preserved at the lowest level and more information at higher levels. The presented methodology is also applicable to tabular data. One possibility is to replace the content of primary and secondary suppressed cells with generated values. It is proposed replacing suppressed cell frequencies with decimal numbers, and it is argued that this can be a useful method.

Appendices

Appendix 1: Generalized QR decomposition

The QR decomposition of a $n \times m$ matrix $\varvec{A}$ with rank r can be written as

$$\begin{aligned} \varvec{A} = \varvec{Q} \varvec{R} \end{aligned}$$

(24)

where $\varvec{Q}$ is a $n \times r$ matrix whose columns form an orthonormal basis for the column space of $\varvec{A}$. This decomposition can be viewed as the matrix formulation of the Gram–Schmidt orthogonalization process. The Cholesky decomposition of $\varvec{A}^T\!\varvec{A}$ can be read from the QR decomposition of $\varvec{A}$ as $\varvec{R}^T\!\varvec{R}$.

In this paper, in order to allow linearly dependent columns of $\varvec{A}$ ($r < m$), we refer to a generalized variant of QR decomposition. In such cases, a usual decomposition (Chan 1987) is

$$\begin{aligned} \varvec{A} \varvec{\tilde{P}} = \varvec{Q} \varvec{\tilde{R}} \end{aligned}$$

(25)

where $\varvec{\tilde{P}}$ is a permutation matrix that reorders the columns (pivoting) in order to make a decomposition so that $\varvec{\tilde{R}}$ is upper triangular.

To make the decomposition unique, we require the diagonal entries of $\varvec{\tilde{R}}$ to be positive. Furthermore, we require $\varvec{\tilde{P}}$ to keep the order of the columns as close to the original order as possible (minimal pivoting). We now have $\varvec{A} = \varvec{Q} \varvec{\tilde{R}} \varvec{\tilde{P}}^{T}$ and in generalized QR decomposition (24) we use

$$\begin{aligned} \varvec{R} = \varvec{\tilde{R}} \varvec{\tilde{P}}^{T} \end{aligned}$$

(26)

The QR decomposition of a composite matrix can be written as

$$\begin{aligned} \left[ \varvec{A}_{1}\; \varvec{A}_{2} \right] = \left[ \varvec{Q}_{1}\; \varvec{Q}_{2} \right] \left[ \varvec{R}_{1}^{T}\; \varvec{R}_{2}^{T} \right] ^{T} \end{aligned}$$

(27)

Now $\varvec{Q}_{1}$ can be computed by QR decomposition of $\varvec{A}_{1}$. The matrix $\varvec{Q}_{2}$ can be computed by QR decomposition of $\varvec{A}_{2} - \varvec{Q}_{1}\varvec{Q}_{1}^{T}\varvec{A}_{2}$, which is the residual part after regressing $\varvec{A}_{2}$ onto $\varvec{A}_{1}$.

Appendix 2: The singular value decomposition

The singular value decomposition (SVD) of a $n \times m$ matrix $\varvec{A}$ with rank r can be written as

$$\begin{aligned} \varvec{A} = \varvec{U} \varvec{\varLambda } \varvec{V}^{T} \end{aligned}$$

(28)

where $\varvec{\varLambda }$ is a $r \times r$ diagonal matrix of strictly positive singular values in descending order. This is the rank-revealing version of the decomposition (Demmel et al. 1999). Other variants of SVD allow some singular values to be zero, but these can be omitted. The columns of $\varvec{U}$ form an orthonormal basis for the column space of $\varvec{A}$ and the columns of $\varvec{V}$ form an orthonormal basis for the row space.

The singular values are the square root of the eigenvalues of $\varvec{A}^T\!\varvec{A}$ and $\varvec{A}\varvec{A}^T$. The eigen decompositions of these two symmetric matrices can be read directly from the SVD of $\varvec{A}$ as $\varvec{V} \varvec{\varLambda }^2 \varvec{V}^{T}$ and $\varvec{U} \varvec{\varLambda }^2 \varvec{U}^{T}$. It is also worth mentioning that an alternative to the ordinary Cholesky decomposition, $\varvec{A}^T\!\varvec{A}=\varvec{R}^T\!\varvec{R}$, is to let $\varvec{\varLambda } \varvec{V}^{T}$ play the role of $\varvec{R}$.

To make the SVD unique, we can require all column sums of $\varvec{V}$ to be positive. In cases with equal singular values, the decomposition is not unique regardless.

There is a close relationship between SVD and PCA. In PCA, the variables are usually centered to zero means and in many cases standardized to equal variances prior to decomposition. If $\varvec{A}$ is such a centered/standardized matrix, then $\varvec{U} \varvec{\varLambda }$ is the matrix of PCA scores and $\varvec{V}$ is the matrix of PCA loadings.

The Moore–Penrose generalized inverse of $\varvec{A}$ can be written as

$$\begin{aligned} \varvec{A}^{\dagger } = \varvec{V} \varvec{\varLambda }^{-1} \varvec{U}^{T} \end{aligned}$$

(29)

We have

$$\begin{aligned} \varvec{A}^{\dagger } = (\varvec{A}^{T}\varvec{A})^{\dagger }\varvec{A}^{T} = \varvec{A}^{T} (\varvec{A}\varvec{A}^{T})^{\dagger } \end{aligned}$$

(30)

When $\varvec{A}$ is invertible, $\varvec{A}^{\dagger }= \varvec{A}^{-1}$. When $\varvec{A}^{T}\varvec{A}$ or $\varvec{A}\varvec{A}^{T}$ is invertible, this means, respectively, that $\varvec{A}^{\dagger } = (\varvec{A}^{T}\varvec{A})^{-1}\varvec{A}^{T}$ or $\varvec{A}^{\dagger } = \varvec{A}^{T} (\varvec{A}\varvec{A}^{T})^{-1}$.