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Efficient Mining of Both Positive and Negative Association Rules

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Efficient Mining of Both Positive andNegative Association Rules

Xindong Wu (*), Chengqi Zhang (+), and Shichao Zhang (+)

(*) University of Vermont, USA

(+) University of Technology Sydney, Australia

Presenter: Mike Tripp

- Association Analysis
- Exceptions
- Problems
- Rules
- Examples

- Pruning Strategy
- Frequent Items of Potential Interest
- Infrequent Items of Potential Interest

- Procedure AllItemsOfInterest
- Extracting Positive and Negative Rules
- CPIR

- PositiveAndNegativeAssociations
- Effectiveness and Efficiency
- Experimental Results
- Related Work
- Exam Questions

- Known as a deviational pattern to a well-known fact, and exhibits unexpectedness.
- Also known as surprising patterns.
- Example:
- While birds(x) => flies(x), exception:
- bird(x), penguin(x) => −flies(x)

- Interesting fact: A => B as a valid rule does not imply −B => −A is a valid rule.

- How to effectively search for interesting itemsets.
- How to effectively identify negative association rules of interest.

- Generate all large itemsets: All itemsets that have a support greater than or equal to the user specified minimum support are generated.
- Generate all the rules that have a minimum confidence in the following naive way: For every large itemset Xand any B C X , let A =X − B . If the rule A => Bhas the minimum confidence (or supp(X)/supp(A) ≥ mc ), then it is a valid rule.

- The negation of an itemset A is indicated by −A . The support of−A, supp(−A) =1 − supp(A).
- In particular, for an itemset i1−i2i3, its support is supp(i1−i2i3 ) = supp(i1i3) − supp (i1i2i3).
- Positive rule: A => B
- Negative rules:
- A => −B
- −A => B
- −A => −B

- Still Difficult: exponential growth of infrequent itemsets
- TD={(A,B,D);(B,C,D);(B,D);(B,C,D,E);(A,B,D,F)}
- Such a simple database contains 49 infrequent item sets.

- Two cases:
- If both A and B are frequent, A U B is infrequent, is A=>~B a valid rule?
- If A is frequent, B is infrequent, is A => ~B a valid rule? Maybe, but not of our interest.

- Heuristic: Only if both A and B are frequent, will A => ~B be considered.

Consider supp(c) = 0.6, supp(t) = 0.4, supp(t U c) = 0.05, mc = 0.52

- Confidence of t => c is supp(t U c)/supp[t] = 0.05/0.4 = 0.125, which is < mc = 0.52 and, supp(t U c) = 0.05 is low.
- This indicates that t U c is an infrequent itemset and that t => c cannot be a valid rule.

- However...
- Supp(t U −c) = supp[t] – supp[t U c] = 0.4 – 0.05 = 0.35 which is high, and the confidence of t => −c is the ratio supp[t U −c]/supp[t] = 0.35/0.4 = 0.875, which is > mc
- Therefore, t => −c is a valid rule.

- Because of the exponential number of infrequent itemsets in a database, pruning is critical to efficient search for interesting itemsets.

- We want to find interesting itemsets when pruning.
- Let’s define an interestingness function interest(X, Y) = |supp(X U Y) – supp(X)supp(Y)| and a threshold mi
- If interest(X, Y) ≥ mi, then the rule X => Y is of potential interest, and X U Y is referred to as potentially interesting itemset

- Using this approach, we can establish an effective pruning strategy for efficiently identifying all frequent itemsets of potential interest in a database.
- The pruning strategy ensures we can use an Apriori like algorithm. Generating infrequent k itemsets from frequent k-1 itemsets.

Where f() is a constraint function concerning the support, confidence, and interestingness of X => Y

Where g() is a constraint function concerning f() and the support, confidence,

and interestingness of X => Y

- Using the fipiand iipimechanisms for both positive and negative rule discovery, our search is constrained to seeking interesting rules on certain measures, and pruning is the removal of all uninteresting branches that cannot lead to an interesting rule that would satisfy those constraints.

Input: D (a database); minsupp; mininterest

Output: PL (frequent itemsets); NL (infrequent itemsets)

- E.g. run of the algorithm (ms=0.3,mi=0.05)

TIDItems bought

T1{A,B,D}

T2{A,B,C,D}

T3{B,D}

T4{B,C,D,E}

T5{A,E}

T6 {B,D,F}

T7 {A,E,F}

T8 {C,F}

T9 {B,C,F}

T10 {A,B,C,D,F}

- Generate frequent and infrequent 2-itemset of interest.
- When ms= 0.3, L2 = {AB, AD, BC, BD, BF, CD, CF}, N2 = {AC, AE, AF, BE, CE, DE, DF, EF}

- Use interest measure to prune.

- So AD and CD are not of interest, they are removed from L2.

- So the resulting frequent 2-itemsets are as follows:

- Generate infrequent 2-itemsets useing the iipi measure.
- Very similar to frequent 2-itemsets.

- Continue like this to get all the itemsets.

Algorithm iteration

Frequent 1-itemsetA,B,C,D,E,F

Frequent 2-itemsetAB,BC,BD,BF,CF

Infrequent 2-itemsetAC,AE,AF,BE,

CE,CF,DE,EF

Frequent 3-itemsetBCD

Infrequent 3-itemset BCF,BDF

TIDItems bought

T1{A,B,D}

T2{A,B,C,D}

T3{B,D}

T4{B,C,D,E}

T5{A,E}

T6 {B,D,F}

T7 {A,E,F}

T8 {C,F}

T9 {B,C,F}

T10 {A,B,C,D,F}

- Pruning strategy for rule generation: Piatetsky-Shapiro’s argument.
- If Dependence(X,Y) = 1, X and Y are independent.
- If Dependence(X,Y) > 1, Y is positively dependent on X.
- The bigger the ratio (p(Y | X) – p(Y))/(1 – p(Y)), the higher the positive dependence.

- If Dependence(X,Y) < 1, Y is negatively dependent on X (~Y is positively dependent on X).
- The bigger the ratio (p(Y | X) – p(Y))/(−p(Y)), the higher the negative dependence.

- Conditional probability increment ratio.
- Used to measure the correlation between X and Y.
- When CPIR(X|Y)=0, X and Y are dependent.
- When it is 1, they are perfectly correlated.
- When it is -1, they are perfectly negatively correlated.

- Because p(~A)=1-p(A), we only need the first half of the previous equation.
or

- This value is used as confidence value.

- Let I be the set of items in a database D, i = A UB ⊆I be an itemset, A ∩B = Ø, supp(A) ≠ 0, supp(B) ≠ 0, andms, mc and mi > 0 be given by the user. Then,
- If supp(A U B) ≥ ms, interest(A, B) ≥ mi, and CIPR(B|A) ≥ mc, then A => B is a positive rule of interest
- If supp(A U −B) ≥ ms, supp(A) ≥ ms, supp(B) ≥ ms, interest(A, −B) ≥mi, and CPIR(−B|A) ≥ mc, then A => −B is a negative rule of interest
- If supp(−A U B) ≥ ms, supp(A) ≥ ms, supp(B) ≥ ms, interest(−A, B) ≥ mi, and CPIR(B|−A) ≥ mc, then −A => B is a negative rule of interest
- If supp(−A U −B) ≥ ms, supp(A) ≥ ms, supp(B) ≥ ms, interest(−A, −B) ≥ mi, and CPIR(−B|−A) ≥ mc, then −A => −B is a negative rule of interest

- For itemset B U D in PL
- B => D can be a valid positive rule of interest

- One snapshot of an iteration in the algorithm
- The result B => −Eis a valid rule.

- Generate the set PL of frequent itemset and the set NL of infrequent itemsets
- Extract positive rules of the form A => B in PL, and negative rules of the forms A => −B, −A => B, −A => −B in NL

- Aggregated Test Data: used for KDD Cup 2000 Data and Questions
- Can be found: http://www.ecn.purdue.edu/KDDCUP/
- Implemented on: Dell Workstation PWS650 w/ 2G of CPU and 2G memory
- Language: C++

- A comparison with Apriori like algorithm without pruning
- MBP = Mining By Pruning
- MNP = Mining with No-Pruning

- A comparison with no-pruning

- Effectiveness of pruning

PII = Positive Items of Interest

NII = Negative Items of Interest

- Negative relationships between frequent itemsets, but not how to find negative rules (Brin, Motwani and Silverstein 1997)
- Strong negative association mining using domain knowledge (Savasere, Ommiecinski and Navathe 1998)

- Negative rules are useful
- Pruning is essential to find frequent and infrequent itemsets.
- Pruning is important to find negative association rules.
- There could be more negative association rules if you have different conditions.

- What are the three types dependencies when dependence(X, Y) are equal, greater than, and less than 1?
- If Dependence(X,Y) = 1, X and Y are independent.
- If Dependence(X,Y) > 1, Y is positively dependent on X.
- If Dependence(X,Y) < 1, Y is negatively dependent on X(−Yis positively dependent on X).

- Give an example of a rule exception, or surprising pattern.
- While birds(x) => flies(x), exception:
- bird(x), penguin(x) => −flies(x)

- What does CPIR(X|Y) tell us?
- When CPIR(X|Y)=0, X and Y are dependent.
- When it is 1, they are perfectly correlated.
- When it is -1, they are perfectly negatively correlated.