What Shall we do. With the Ratios? Sarah GIESSING, Federal Statistical Office of Germany, Division Mathematical Statistical Methods

Transcription

1 What Shall we do Sarah GIESSING, Federal Statistical Office of Germany Division Mathematical Statistical Methods With the Ratios?

2 Recall: Consistent post-tabular multiplicative noise method (Giessing, 12) Create seed variable in the micro data Compute consistent univariate random numbers (Seed) at the table cell level Apply consistent multiplicative noise (using Seed) at the table cell level Round noisy data (rounding interval confidence interval) = Consistency = Transparency = Safety Folie 2

3 Recall: Seed based multiplicative noise at the table cell level For each cell/variable Y do Determine deviation sense d (using seed) Check primary sensitivity (f.i. by p%-rule) µ=2p; for sensitive cells µ=0; for non-sensitive cells Folie 3

4 Recall: Seed based multiplicative noise at the table cell level For each cell/variable Y do Determine deviation sense d (using seed) Check primary sensitivity (f.i. by p%-rule) µ=2p; for sensitive cells µ=0; for non-sensitive cells Draw z from N(0, σ) distribution (using seed; σ p/k) Multiply largest contribution y 1 by noise y 1 post =: y 1 (1+d (µ+abs(z))) Folie 4

5 Recall: Seed based multiplicative noise at the table cell level For each cell/variable Y do Determine deviation sense d (using seed) Check primary sensitivity (f.i. by p%-rule) µ=2p; for sensitive cells µ=0; for non-sensitive cells Draw z from N(0, σ) distribution (using seed; σ p/k) Multiply largest contribution y 1 by noise y 1 post =: y 1 (1+d (µ+abs(z))) Exchange true largest contribution by perturbed largest contribution Y post = Y orig - y 1 + y 1 post Folie 5

6 Recall: Seed based multiplicative noise at the table cell level For each cell/variable Y do Determine deviation sense d (using seed) Check primary sensitivity (f.i. by p%-rule) µ=2p; for sensitive cells µ=0; for non-sensitive cells Draw z from N(0, σ) distribution (using seed; σ p/k) Multiply largest contribution y 1 by noise y 1 post =: y 1 (1+d (µ+abs(z))) Exchange true largest contribution by perturbed largest contribution Y post = Y orig - y 1 + y 1 Release Y post and its approximate (!) ζ γ -confidence interval Y post ± y 1 (µ+σζ γ ) post Folie 6

7 What happens to ratios? Example: Ratio Y/X from true data vs. Ratio Y post / X post from perturbed data Assume: X =, x 1 = 300, z y = z x = σ = 0.05, d y = Folie 7

8 What happens to ratios? Example: Ratio Y/X from true data vs. Ratio Y post / X post from perturbed data Assume: X =, x 1 = 300, z y = z x = σ = 0.05, d y = -1 Y 0 00 y

9 What happens to ratios? Example: Ratio Y/X from true data vs. Ratio Y post / X post from perturbed data Assume: X =, x 1 = 300, z y = z x = σ = 0.05, d y = -1 Y 0 00 y Y post ξ γ = 1 ll y uu y Folie 9

10 What happens to ratios? Example: Ratio Y/X from true data vs. Ratio Y post / X post from perturbed data Assume: X =, x 1 = 300, z y = z x = σ = 0.05, d y = -1 Y 0 00 y Y post ξ γ = 1 ll y uu y Folie 10

11 What happens to ratios? Example: Ratio Y/X from true data vs. Ratio Y post / X post from perturbed data Assume: X =, x 1 = 300, z y = z x = σ = 0.05, d y = -1 Y 0 00 y Y post ξ γ = 1 ll y uu y 0 00 X post ξ γ = 1 ll y uu y X post ll X uu X Folie 11

12 What happens to ratios? Example: Ratio Y/X from true data vs. Ratio Y post / X post from perturbed data Assume: X =, x 1 = 300, z y = z x = σ = 0.05, d y = -1 Y 0 00 y Y/X Y post ξ γ = 1 ll y uu y 0 00 X post ξ γ = 1 ll y uu y Y post /X post X post ll X uu X Y post /X post Folie 12

13 What happens to ratios? Example: Ratio Y/X from true data vs. Ratio Y post / X post from perturbed data Assume: X =, x 1 = 300, z y = z x = σ = 0.05, d y = -1 Y 0 00 y Y/X Y post ξ γ = 1 ll y uu y 0 00 X post ξ γ = 1 ll y uu y Y post /X post uu Y ll X ll Y uu X X post ll X uu X Y post /X post uu Y ll X ll Y uu X Folie 13

14 Can we do better? Y/X = n i=1 y i n i=1 x i = β, where β : OLS estimate for β in n i=1 β solution to H β = 0, wwwww H β = n i=1 y i E y i x i y i ββ i = 0, i.e Folie 14

15 Can we do better? Y/X = n i=1 y i n i=1 x i = β, where β : OLS estimate for β in n i=1 β solution to H β = 0, wwwww H β = n i=1 y i E y i x i y i ββ i = 0, i.e. =ββ i Folie 15

16 Can we do better? (Chipperfield and Yu, 11): Instead of solving H β = 0: Solve H β = E and release the solution β where E = u e = u max i Estimate β = Y/X. Then n y i ββ i H β = E (i.e. i=1 y i ββ i = E ) yields solution β : β = n i=1 y i n i=1 x i u e = Y X u e X Draw noise u from N (0, σ r ) = β u e X Release β and its approximate ζ γ confidence interval β ± σ r ξ γ e X Folie 16

17 We can do better! Example: Ratio Y/X from true data vs. Ratio Y post / X post from perturbed data Assume: X =, x 1 = 300, z y = z x = σ = σ r =0.05, d y = -1, u = 1*0.05=0.05 Y 0 00 y Y/X Y post ξ γ = 1 ll y uu y 0 00 X post ξ γ = 1 ll y uu y Y post /X post uu Y ll X ll Y uu X e β σ r e x Folie 17

18 We can do better! Example: Ratio Y/X from true data vs. Ratio Y post / X post from perturbed data Assume: X =, x 1 = 300, z y = z x = σ = σ r =0.05, d y = -1, u = 1*0.05=0.05 Y 0 00 y Y/X Y post ξ γ = 1 ll y uu y 0 00 X post ξ γ = 1 ll y uu y Y post /X post uu Y ll X ll Y uu X e β σ r e x Folie 18

19 Summary: Seed based multiplicative noise for a ratio Y/X For each cell/ratio Y/X where X and Y are non-sensitive do Determine deviation sense d (using seed) Draw u from N (0, σ r ) distribution (using seed; suitable σ r ) Folie 19

20 Summary: Seed based multiplicative noise for a ratio Y/X For each cell/ratio Y/X where X and Y are non-sensitive do Determine deviation sense d (using seed) Draw u from N (0, σ r ) distribution (using seed; suitable σ r ) Identify most influential observation e = max i Compute perturbed ratio: y i Y X x i β = Y X u e X Folie

21 Summary: Seed based multiplicative noise for a ratio Y/X For each cell/ratio Y/X where X and Y are non-sensitive do Determine deviation sense d (using seed) Draw u from N (0, σ r ) distribution (using seed; suitable σ r ) Identify most influential observation e = max i Compute perturbed ratio: y i Y X x i β = Y X u e X Compute width of the ζ γ -confidence interval 2σ r ξ γ e X Compute rounding basis B, considerung this width and the desired precision of the ratio Release the estimate β after rounding Folie 21

22 Rounding step Recall: Rounding for Y post Calculate rounding basis B: B=11 Round(lll 11(2y 1 (µ+σζ γ )) Round Y post to adjacent multiple of B Example: B=11 Round(lll 11( ) =1 Round Y post =19.65 to Folie 22

23 Rounding step for perturbed ratio Round β, considering desired precision d: Calculate transformed confidence interval rounding basis B: B=11 rrrrr(lll r e X 11d ). Transform estimate: 11 d β Round transformed estimate to adjacent multiple of B and divide by 11 d Example: B=11 rrrrr(lll ) =1 Round 11 d β = to 199 Backtransformation: 199/10 4 = For the data of the example: β d 11 d β rrrrr β * Folie 23

24 Secondary Disclosure Risk Issues Users/Intruders approximately know those intervals: I. β e σ r ξ γ Y β e +σ X X r ξ γ X II. X σ ξ γ x 1 X X + σ ξ γ x 1 III. They might Y σ ξ γ y 1 Y Y + σ ξ γ y 1 Use II to calculate new intervall for true Y from I Use III to calculate new intervall for true X from I Therefore: we should show that it is possible to chose σ r such that the new intervals for Y and X are (very likely to be) at least as wide as the intervals II and III Has been proven in the paper for non-sensitve (X,Y)

25 Conclusion, Future Work In the context of a flexible table server relying on post tabular stochastic noise as SDC method: Noise variance of ratios can be reduced when the server offers perturbed ratio data (instead of calculating ratios from the perturbed enumerator and denominator data). Future work to address the special case where X is the number of observations, i.e. the case Y/X = mean(y) Folie 25

26 Thanks for your attention