Bounded Small Cell Adjustments for Flexible Frequency Table Generators

Abstract

Statistics Korea has disseminated census data through the Statistical Geographic Information Service (SGIS) system. Users can easily access the system on a web-site and obtain frequencies on the map for diverse size-of-area units according to their selection of variables. In order to control the disclosure risk for frequency tables, we thoroughly examined the Small Cell Adjustments (SCA) method to find the reasons for disclosures: we then suggested the Bounded Small Cell Adjustments (BSCA) procedure in this paper. From the analysis on the census data of a Korean city of approximately 1.5 million people, we demonstrated the efficiency of BSCA, which reduces information loss under B in most cells while maintaining B-anonymity in all cells as intended in the SCA idea. The B denotes the criterion value defining a small cell. Furthermore, we have discussed the relationship between disclosure risk and information loss by BSCA.

Appendices

Appendix A

Multi-dimensional tables can be represented in a flattened form. Table 11 shows a three-dimensional flattened frequency tables in our data. The categories of each variable are listed from lexicographic ordering. We can represent any p-dimensional frequency table of an area unit in this kind of flattened form. In general, we have ${}_P\text {C}_P = 1$ table of $T^{vP}$, ${}_P\text {C}_{P-1}$ tables of $T^{v(P-1)}$, ..., ${}_P\text {C}_1 = P$ tables of $T^{v1}$ and totally $2^P -1$ tables at each area unit for P variables.

Table 11. Three dimensional flattened frequency tables, $T^{v3}$

Appendix B

Here, we would like to discuss how disclosures can occur by applying FGR only to hierarchical tables. According to Eq. (1), frequencies having hierarchy is denoted as

$$\begin{aligned} f_j= & {} a_j B + b_j + 1, \qquad j = 1, \ldots , K \\ f= & {} aB + b + 1 = \sum _{j=1}^K a_j B + \sum _{j=1}^K b_j + K. \end{aligned}$$

Note that $a = [(f-1)/B]$ and $b \in \{0, 1, \ldots , B-1\}$. We omit subscript i for convenience. If we apply FGR to the frequencies at each hierarchical level, the masked values are as follows with $Q(\{b_1, \ldots , b_K\}) = [(\sum _{j=1}^K b_j + K - 1)/B]$:

From the masked frequencies and $\tilde{f}$ provided by the system, users can directly obtain the information of

$$\begin{aligned} f_j\in & {} C_j = [a_j B + 1, a_j B + B] \\ f\in & {} C = [aB + 1, aB + B] \end{aligned}$$

On the other hand, from , users can also infer that the true f should be in $D = [\sum _{j=1}^K a_j B + K, \sum _{j=1}^K a_j B + KB]$.

Therefore, disclosures can happen (or anonymity can decrease) if $C \not \subset D$:

$$\begin{aligned} (1)&~~&\sum _{j=1}^K a_j B + K > aB + 1 = \sum _{j=1}^K a_j B + Q(\{b_1, \ldots , b_K\}) \cdot B + 1 \quad \text{ or } \\ (2)&~~&\sum _{j=1}^K a_j B + KB < aB + B = \sum _{j=1}^K a_j B + Q(\{b_1, \ldots , b_K\}) \cdot B + B. \end{aligned}$$

An example can be found when $\{f_1, f_2\} = \{4, 4\}$, $f = 8$ and $B=3$. Note that (1) and (2) do not hold at the same time.

If we use FGR at each hierarchical level, we may not be able to avoid disclosures. However, we use FGR once when we aggregate the small cells masked by SCA in the full table and then have a moving grid step to avoid disclosures in our algorithm.