Anonymization Algorithms - Microaggregation and Clustering

1. Anonymization Algorithms - Microaggregation and Clustering Li Xiong CS573 Data Privacy and Anonymity

2. Anonymization using Microaggregation or Clustering • Practical Data-Oriented Microaggregation for Statistical Disclosure Control, Domingo-Ferrer, TKDE 2002 • Ordinal, Continuous and Heterogeneous k-anonymity through microaggregation, Domingo-Ferrer, DMKD 2005 • Achieving anonymity via clustering, Aggarwal, PODS 2006 • Efficient k-anonymization using clustering techniques, Byun, DASFAA 2007

3. Anonymization Methods • Perturbative: distort the data • statistics computed on the perturbed dataset should not differ significantly from the original • microaggregation, additive noise • Non-perturbative: don't distort the data • generalization: combine several categories to form new less specific category • suppression: remove values of a few attributes in some records, or entire records

4. Types of data • Continuous: attribute is numeric and arithmetic operations can be performed on it • Categorical: attribute takes values over a finite set and standard arithmetic operations don't make sense • Ordinal: ordered range of categories • ≤, min and max operations are meaningful • Nominal: unordered • only equality comparison operation is meaningful

5. Measure tradeoffs • k-Anonymity: a dataset satisfies k-anonymity for k > 1 if at least k records exist for each combination of quasi-identifier values • assuming k-anonymity is enough protection against disclosure risk, one can concentrate on information loss measures

6. Critique of Generalization/Suppression • Satisfying k-anonymity using generalization and suppression is NP-hard • Computational cost of finding the optimal generalization • How to determine the subset of appropriate generalizations • semantics of categories and intended use of data • e.g., ZIP code: • {08201, 08205} -> 0820* makes sense • {08201, 05201} -> 0*201 doesn't

7. Problems cont. • How to apply a generalization • globally • may generalize records that don't need it • locally • difficult to automate and analyze • number of generalizations is even larger • Generalization and suppression on continuous data are unsuitable • a numeric attribute becomes categorical and loses its numeric semantics

8. Problems cont. • How to optimally combine generalization and suppression is unknown • Use of suppression is not homogenous • suppress entire records or only some attributes of some records • blank a suppressed value or replace it with a neutral value

9. Microaggregation/Clustering • Two steps: • Partition original dataset into clusters of similar records containing at least k records • For each cluster, compute an aggregation operation and use it to replace the original records • e.g., mean for continuous data, median for categorical data

10. Advantages: • a unified approach, unlike combination of generalization and suppression • Near-optimal heuristics exist • Doesn't generate new categories • Suitable for continuous data without removing their numeric semantics

11. Advantages cont. Reduces data distortion K-anonymity requires an attribute to be generalized or suppressed, even if all but one tuple in the set have the same value. Clustering allows a cluster center to be published instead, “enabling us to release more information.”

12. Original Table

13. 2-Anonymity with Generalization Generalization allows pre-specified ranges

14. 2-Anonymity with Clustering 27=(25+27+29)/3 70=(50+60+100)/3 37=(35+39)/2 115=(110+120)/2 Cluster centers ([27,70], [37,115]) published

15. Another example: no common value among each attribute

16. Generalization vs. clustering Generalized version of the table would need to suppress all attributes. Clustered Version of the table would publish the cluster center as (1, 1, 1, 1), and the radius as 1.

17. Anonymization using Microaggregation or Clustering Practical Data-Oriented Microaggregation for Statistical Disclosure Control, Domingo-Ferrer, TKDE 2002 Ordinal, Continuous and Heterogeneous k-anonymity through microaggregation, Domingo-Ferrer, DMKD 2005 Achieving anonymity via clustering, Aggarwal, PODS 2006 Efficient k-anonymization using clustering techniques, Byun, DASFAA 2007

18. Multivariate microaggregation algorithm • MDAV-generic: Generic version of MDAV algorithm (Maximum Distance to Average Vector) from previous papers • Works with any type of data (continuous, ordinal, nominal), aggregation operator and distance calculation

19. MDAV-generic(R: dataset, k: integer) while |R| ≥ 3k 1. compute average record ~x of all records in R 2. find most distant record xr from ~x 3. find most distant record xs from xr 4. form two clusters from k-1 records closest to xr and k-1 closest to xs 5. Remove the clusters from R and run MDAV-generic on the remaining dataset end while if 3k-1 ≤ |R| ≤ 2k 1. compute average record ~x of remaining records in R 2. find the most distant record xr from ~x 3. form a cluster from k-1 records closest to ~x 4. form another cluster containing the remaining records else (fewer than 2k records in R) form a new cluster from the remaining records

20. MDAV-generic for continuous attributes • use arithmetic mean and Euclidean distance • standardize attributes (subtract mean and divide by standard deviation) to give them equal weight for computing distances • After MDAV-generic, destandardize attributes • xij is value of k-anonymized jth attribute for the ith record • m10(j) and m2(j) are mean and variance of the k-anonymized jth attribute • u10(j) and u2(j)are mean and variance of the original jth attribute

21. MDAV-generic for ordinal attributes • The distance between two categories a and b in an attribute Vi: • dord(a,b) = (|{i| ≤ i < b}|) / |D(Vi)| • i.e., the number of categories separating a and b divided by the number of categories in the attribute • Nominal attributes • The distance between two values is defined according to equality: 0 if they're equal, else 1

22. Empirical Results • Continuous attributes • From the U.S. Current Population Survey (1995) • 1080 records described by 13 continuous attributes • Computed k-anonymity for k = 3, ..., 9 and quasi-identifiers with 6 and 13 attributes • Categorical attributes • From the U.S. Housing Survey (1993) • Three ordinal and eight nominal attributes • Computed k-anonymity for k = 2, ..., 9 and quasi-identifiers with 3, 4, 8 and 11 attributes

23. IL measures for continuous attributes • IL1 = mean variation of individual attributes in original and k-anonymous datasets • IL2 = mean variation of attribute means in both datasets • IL3 = mean variation of attribute variances • IL4 = mean variation of attribute covariances • IL5 = mean variation of attribute Pearson's correlations • IL6 = 100 times the average of IL1-6

24. MDAV-generic preserves means and variances • The impact on the non-preserved statistics grows with the quasi-identifier length, as one would expect • For a fixed-quasi-identifier length, the impact on the non-preserved statistics grows with k

25. IL measures for categorical attributes • Dist: direct comparison of original and protected values using a categorical distance • CTBIL': mean variation of frequencies in contingency tables for original and protected data (based on another paper by Domingo-Ferrer and Torra) • ACTBIL': CTBIL' divided by the total number of cells in all considered tables • EBIL: Entropy-based information loss (based on another paper by Domingo-Ferrer and Torra)

26. Ordinal attribute protection using median

27. Ordinal attribute protection using convex median

28. Anonymization using Microaggregation or Clustering • Practical Data-Oriented Microaggregation for Statistical Disclosure Control, Domingo-Ferrer, TKDE 2002 • Ordinal, Continuous and Heterogeneous k-anonymity through microaggregation, Domingo-Ferrer, DMKD 2005 • Achieving anonymity via clustering, Aggarwal, PODS 2006 • Efficient k-anonymization using clustering techniques, Byun, DASFAA 2007

29. r-Clustering • Attributes from a table are first redefined as points in metric space. • These points are clustered, and then the cluster centers are published, rather than the original quasi-identifiers. • r is the lower bound on the number of members in each cluster. • r is used instead of k to denote the minimum degree of anonymity because k is typically used in clustering to denote the number of clusters.

30. Data published for clusters • Three features are published for the clustered data • the quasi-identifying attributes of the cluster center • the number of points within the cluster • the set of sensitive values for the cluster (which remain unchanged, as with k-anonymity) • A measure of the quality of the clusters will also be published.

31. Defining the records in metric space • Some attributes, such as age and height, are easily mapped to metric space. • Others, such as zip may first need to be converted, for example to longitude and latitude. • Some attributes may need to be scaled, such as location, which may differ by thousands of miles. • Some attributes such as race or nationality may not convert to points in metric space easily.

32. How to measure the quality of the cluster • Measures how much it distorts the original data. • Maximum radius (r-GATHER problem) • Maximum radius of all clusters • Cellular cost (r-CELLULAR CLUSTERING problem) • Each cluster incurs a “facility cost” to set up the cluster center. • Each cluster incurs a “service cost” which is equal to the radius times the number of points in the cluster • Sum of the facility and services costs for each of the clusters.

33. Points arranged in clusters 17 points, radius 7 25 points, radius 10 14 points, radius 8

34. Cluster quality measurements • Maximum radius = 10 • Facility cost plus service cost: • Facility cost = f(c) • Service cost = (17 x 7) + (14 x 8) + (25 x 10) = 481

35. r-GATHER problem • “The r-Gather problem is to cluster n points in a metric space into a set of clusters, such that each cluster has at least r points. The objective is to minimize the maximum radius among the clusters.”

36. “Outlier” points • r-GATHER and r-CELLULAR CLUSTERING, like k-anonymity, are sensitive to outlier points (i.e., points which are far removed from the rest of the data). • The clustering solutions in this paper are generalized to allow an e fraction of outliers to be removed from the data, that is, e fraction of the tuples can be suppressed.

37. (r,e)-GATHER Clustering • The (r, e)-GATHER clustering formulation of the problem allows an e fraction of the outlier points to be unclustered (i.e., these tuples are suppressed). • The paper finds that there is a polynomial time algorithm that provides a 4-approximation for the (r,e)-GATHER problem.

38. r-CELLULAR CLUSTERING defined • The CELLULAR CLUSTERING problem is to arrange n points into clusters with each cluster has at least r points and with the minimum total cellular cost.

39. (r,e)-CELLULAR CLUSTERING • There is also a (r,e)-CELLULAR CLUSTERING problem in which an e fraction of the points can be excluded. • The details of the constant-factor approximation of this problem are deferred to the full version of this paper.

40. Anonymization using Microaggregation or Clustering • Practical Data-Oriented Microaggregation for Statistical Disclosure Control, Domingo-Ferrer, TKDE 2002 • Ordinal, Continuous and Heterogeneous k-anonymity through microaggregation, Domingo-Ferrer, DMKD 2005 • Achieving anonymity via clustering, Aggarwal, PODS 2006 • Efficient k-anonymization using clustering techniques, Byun, DASFAA 2007

41. Anonymization And Clustering • k-Member Clustering Problem • From a given set of n records, find a set of clusters such that • Each cluster contains at least k records, and • The total intra-cluster distance is minimized. • The problem is NP-complete

42. Distance Metrics • Distance metric for records • Measure the dissimilarities between two data points • Sum of all dissimilarities between corresponding attributes. • Numerical values • Categorical values

43. Distance between two numerical values • Definition Let D be a finite numeric domain. Then the normalized distance between two values vi, vj D is defined as: where |D| is the domain size measured by the difference between the maximum and minimum values in D. • Example 1 • Distance between r1 and r2 with respect to Age attribute is |57-41|/|57-24| = 16/33 = 0.4848.. • Example 2 • Distance between r5 and r6 with respect to Age attribute is |24-45|/|57-24| = 21/33 = 0.6363..

44. Distance between two categorical values • Equally different to each other. • 0 if they are the same • 1 if they are different • Relationships can be easily captured in a taxonomy tree. Taxonomy tree of Country Taxonomy tree of Occupation

45. Definition Let D be a categorical domain and TD be a taxonomy tree defined for D. The normalized distance between two values vi, vj D is defined as: where  (x, y) is the subtree rooted at the lowest common ancestor of x and y, and H(T) represents the height of tree T. Distance between two categorical values Taxonomy tree of Country • Example: • The distance between India and USA is 3/3 = 1. • The distance between India and Iran is 2/3 = 0.66.

46. Distance between two records • Definition Let QT = {N1, . . . ,Nm, C1, . . . ,Cn} be the quasi-identifier of table T , where Ni(i = 1, . . . ,m) is an attribute with a numeric domain and Ci(i = 1, . . . , n) is an attribute with a categorical domain. The distance of two records r1, r2 T is defined as: where δN is the distance function for numeric attribute, and δC is the distance function for categorical attribute.

47. Distance between two records Continued… Taxonomy tree of Country Taxonomy tree of Occupation

48. Cost Function - Information loss (IL) • The amount of distortion (i.e., information loss) caused by the generalization process. Note: Records in each cluster are generalized to share the same quasi-identifier value that represents every original quasi-identifier value in the cluster. • Definition: Let e = {r1, . . . , rk} be a cluster (i.e., equivalence class). Then the amount of information loss in e, denoted by IL(e), is defined as: where |e| is the number of records in e, |N| represents the size of numeric domain N, (Cj) is the subtree rooted at the lowest common ancestor of every value in Cj, and H(T) is the height of tree T.

49. Cost Function - Information loss (IL) Example Taxonomy tree of Country Cluster e1 Cluster e2

50. Greedy k-member clustering algorithm