Selecting the Right Interestingness Measure for Association Patterns

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Selecting the Right Interestingness Measure for
Association Patterns

• Pang-Ning Tan, Vipin Kumar, and Jaideep
  Srivastava
• Department of Computer Science and Engineering
• University of Minnesota
• Presented by Ahmet Bulut

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Motivation

• Major Data mining problem analysis of
  relationships among variables
• Finding sets of binary variables that co-occur
  together
• Association Rule Mining Agrawal et al.
• How to find a suitable metric to capture the
  dependencies between variables (defined in terms
  of contingency tables)
• Many metrics provide conflicting information
• Goal Automated way of choosing the best metric
  for an application domain

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Justification for Conflicts

• E10 is ranked highest by I measure but lowest
  according to coefficient.
• Recognize intrinsic properties of the existing
  measures

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Analysis of a Measure

• The relationship between the gender of a student
  and the grade obtained in a course
• Number of male students X 2, Number of female
  students X 10
• One expects scale-invariance in this particular
  application
• Most measures are sensitive to scaling of rows
  and columns
• such as gini index, interest, mutual information
  etc.

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Solutions to zero-in

• Support based pruning
• Eliminate uncorrelated and poorly correlated
  patterns
• Table Standardization
• To modify contingency tables to have uniform
  margins
• Many measures provide non-conflicting information
• Expectation of domain experts
• Choose the measure that agrees with the
  expectations the most.
• Number of contingency tables, T, is high
• It is possible to extract a small set, S, of
  contingency tables
• Find the best measure for S to approximate for T

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Preliminaries

• The similarity between any two measures M1 and
  M2 the similarity between OM1(T) and OM2(T)
• The similarity metric used is correlation
  coefficient
• corr(OM1(T), OM2(T) ) gt threshold then similar

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Desired Properties of a Measure M
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Properties of a Measure M contd.

• Denote 2X2 contingency table as a contingency
  matrix
• Interestingness measure is a matrix operator, O
  such that
• OM k where k is a scalar.
• For instance for Coefficient as the
  interestingness measure
• k equals to normalized form of the determinant
  operator
• Det(M) f11f00 f01f10
• Statistical Independence
• a singular matrix M whose determinant equal to 0.

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Properties of a Measure M contd.

• Property 1 Symmetry under variable permutation
  O(MT) O(M)
• cosine (IS), interest factor(I), odds ratio ( )
• Property 2 Row/Column Scaling Invariance
  RCk1 0 0 k2
• R x M is row scaling and M x R is column scaling
• If O(RM) O(M) and O(MR) O(M), then M is
  row/scale invariant
• odds ratio ( ) satisfies this property along
  with Yules Q and Y

• Property 3 Antisymmetry under row/column
  permutation
• S 0 11 0
• If O(SM) -O(M), antisymmetric under row
  permutation
• If O(MS) -O(M), antisymmetric under column
  permutation
• Measures that are symmetric under the row and
  column permutation operations no distinction
  between positive and negative correlations of a
  table
• For example gini index

• Property 4 Inversion Invariance
• S0 11 0
• row and column permutation together
• If O(SMS)O(M), inversion invariant
• Insight flip presence with absence and vice
  versa for binary variables.
• coefficient, odds ratio, collective strength
  are symmetric binary measures
• Jaccard measure is asymmetric

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Property 4 and Property 5

• Market Basket analysis requires unequal treatment
  of binary values of a variable
• A symmetric measure like the one above is not
  suitable
• Property 5 Null Invariance If O(MC) O(M)
  where C0 00 k and k is a positive constant
• For binary variables more records added that do
  not contain the two variables under
  consideration Co-occurrence emphasized

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Effect of Support Based Pruning

• Randomly generated synthetic dataset of 10,000
  contingency tables
• Darker cells, correlation gt 0.85, and lighter
  cells indicate otherwise
• Tighter bounds on the support of the patterns
  many measures become correlated

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Elimination of poorly correlated tables using
Support-based Pruning

• Minimum support threshold to prune out the low
  support patterns
• Having a maximum support threshold equal
  elimination of uncorrelated, negatively
  correlated and positively correlated tables
• Having a lower bound of support will prune out
  the negatively correlated or uncorrelated tables.

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Table standardization

• A standardized table visual depiction of the
  disjoint distribution of two variables after
  elimination of non-uniform marginals

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Implications of standardization

• The rankings from different measures become
  identical

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Implications of standardization contd

• After standardization, a matrix has x y y x
  where
• y N/2-x and x f11
• If you consider monotonically increasing
  functions of x (nearly all of the measures are)
• Identical rankings on standardized, positively
  correlated tables
• Some measures do not satisfy this property
• Consider the values of x where N/4 lt x lt N/2
• IPF favors odds ratio measure, therefore final
  rankings agree with odds ratio rankings before
  standardization
• Leave with Different standardization techniques
  may be more appropriate for different application
  domains

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Measure Selection Based on Rankings by Experts

• Ideally, experts rank all the contingency tables,
  choose the best measure accordingly
• Laborious task if the number of tables is too
  large
• Provide a smaller set of tables to decide the
  best measure

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Table Selection via Disjoint Algorithm

• Use Disjoint algorithm to choose a subset of
  tables of cardinality k.
• Rank tables according to various measures
• Compute the similarity between different measures
• A good table selection scheme minimizes

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Experimental Results
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Conclusions

• Key properties to consider for selecting the
  right measure
• No measure is consistently better than others
• Situations where most measures provide correlated
  info
• Choosing the right measure on a non-biased small
  set of all the tables give good estimates to the
  ideal solution
• As a future work
• Extension to k-way contingency tables
• Association between mixed data types