Linear Transformation of post-microaggregated data

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Linear Transformation of post-microaggregated
data

• Mi-Ja Woo
• National Institute of Statistical Sciences

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Motivation Example
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• Different distributions, but the same moments
  and estimates of regression coefficients.
• How about making D3 have the same mean and
  covariance?

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1. Linear Transformation

• Let D1 be the original p-dimensional data with
  mean, E1 and covariance matrix S1.
• Let D2 be the post-microaggregated p-dimensional
  data with mean, E2 and covariance matrix, S2.
• Transform D2 into T(D2) such that ET(D2)E1
  ST(D2)S1.

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How to compute A and b?

• Mathematically, A and b are obtained as
  .
• Use SVD decomposition to calculate

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NOTES

• Linear transformed masked data yields the same
  analysis based on mean and covariance.
• How about higher moments? There is no clear
  answer, but higher moments rely on distributions
  other than A, b, mean and covariance. We
  need data utility measures.
• Linear transformation does not preserve
  positivity.
• Can we improve data utility of other SDLs through
  linear transformation?

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Question Other masked data?
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Linear transformation with constraint of
positivity.

• Partition X into
• Transform X2 but not X1.
• Replace final negative values with minimum of
  original data or zero after transforming X2.
• It is the middle of non-transformed
  microaggregated and transformed microaggregated
  data.
• The utility of this method depends on how many
  negative values are in transformed
  microaggregated data.

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How to partition X?

• The way of partitioning X1. Initially,
  transform X in YAXb.2. Sort Y according to
  descending order.3. Count how many records are
  negative, n. 4. Partition Y into Y1 and Y2,
  where Y1 has 1st to (n np)-th observations of
  Y and Y2 contains the rest of them.5. Partition
  X in X1 and X2 corresponding to Y1 and Y2.
• More observations are added to Y1 in order to
  reduce the possibility of getting negative values
  after transforming X2.

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Example

• Here are eight different types of data.
• For most of data violating signs, the procedure
  above improves utilities.
• Since it is the middle of non-transformed and
  transformed microaggregated data, it does not
  always improve three data utilities comparing to
  transformed microaggregated data.
• Improvement of Non-symmetric Low Positive is the
  largest, that of Non-symmetric High Positive is
  the next, and the last one is Non-symmetric Low
  Negative.

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END