Sampling Large Databases for Association Rules

1
Sampling Large Databases for Association Rules

• Jingting Zeng
• CIS 664 Presentation
• March 13, 2007

2
Association Rules Outline

• Association Rules Problem Overview
• Association Rules Definitions
• Previous Work on Association Rules
• Toivonens Algorithm
• Experiments Result
• Conclusion

3
Overview

• Purpose
• If people tend to buy A and B together, then a
  buyer of A is a good target for an advertisement
  for B.

4
The Market-Basket Example

• Items frequently purchased together
• Bread ?PeanutButter
• Uses
• Placement
• Advertising
• Sales
• Coupons
• Objective increase sales and reduce costs

5
Other Example

• The same technology has other uses
• University course enrollment data has been
  analyzed to find combinations of courses taken by
  the same students

6
Scale of Problem

• WalMart sells 100,000 items and can store
  hundreds of millions of baskets.
• The Web has 100,000,000 words and several billion
  pages.

7
Association Rule Definitions

• Set of items II1,I2,,Im
• Transactions Dt1,t2, , tn, tj? I
• Support of an itemset Percentage of transactions
  which contain that itemset.
• Frequent itemset Itemset whose number of
  occurrences is above a threshold.

8
Association Rule Definitions

• Association Rule (AR) implication X ? Y where
  X,Y ? I and X ? Y
• Support of AR (s) X ? Y Percentage of
  transactions that contain X ?Y
• Confidence of AR (a) X ? Y Ratio of number of
  transactions that contain X ? Y to the number
  that contain X

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Example

• B1 m, c, b B2 m, p, j
• B3 m, b B4 c, j
• B5 m, p, b B6 m, c, b, j
• B7 c, b, j B8 b, c
• Association Rule
• m, b ? c
• Support 2/8 25
• Confidence 2/4 50

10
Association Rule Problem

• Given a set of items II1,I2,,Im and a
  database of transactions Dt1,t2, , tn where
  tiIi1,Ii2, , Iik and Iij ? I, the Association
  Rule Problem is to identify all association rules
  X ? Y with a minimum support and confidence
  threshold.

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Association Rule Techniques

• Find all frequent itemsets
• Generate strong association rules from the
  frequent itemsets

12
APriori Algorithm

• A two-pass approach called a-priori limits the
  need for main memory.
• Key idea monotonicity if a set of items
  appears at least s times, so does every subset.
• Converse for pairs if item i does not appear in
  s baskets, then no pair including i can appear
  in s baskets.

13
APriori Algorithm (contd.)

• Pass 1 Read baskets and count in main memory the
  occurrences of each item.
• Requires only memory proportional to items.
• Pass 2 Read baskets again and count in main
  memory only those pairs both of which were found
  in Pass 1 to have occurred at least s times.
• Requires memory proportional to square of
  frequent items only.

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Partitioning

• Divide database into partitions D1,D2,,Dp
• Apply Apriori to each partition
• Any large itemset must be large in at least one
  partition.

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Partitioning Algorithm

• Divide D into partitions D1,D2,,Dp
• For I 1 to p do
• Li Apriori(Di)
• C L1 ? ? Lp
• Count C on D to generate L

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Sampling

• Large databases
• Sample the database and apply Apriori to the
  sample.
• Potentially Frequent Itemsets (PL) Large
  itemsets from sample
• Negative Border (BD - )
• Generalization of Apriori-Gen applied to itemsets
  of varying sizes.
• Minimal set of itemsets which are not in PL, but
  whose subsets are all in PL.

17
Negative Border Example

• Let Items A,,F and there are
• itemsets
• A, B, C, F, A,B, A,C, A,F,
  C,F, A,C,F
• The whole negative border is
• B,C, B,F, D, E

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Toivonens Algorithm

• Start as in the simple algorithm, but lower the
  threshold slightly for the sample.
• Example if the sample is 1 of the baskets, use
  0.008 as the support threshold rather than 0.01
  .
• Goal is to avoid missing any itemset that is
  frequent in the full set of baskets.

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Toivonens Algorithm (contd.)

• Add to the itemsets that are frequent in the
  sample the negative border of these itemsets.
• An itemset is in the negative border if it is not
  deemed frequent in the sample, but all its
  immediate subsets are.
• Example ABCD is in the negative border if and
  only if it is not frequent, but all of ABC , BCD
  , ACD , and ABD are.

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Toivonens Algorithm (contd.)

• In a second pass, count all candidate frequent
  itemsets from the first pass, and also count the
  negative border.
• If no itemset from the negative border turns out
  to be frequent, then the candidates found to be
  frequent in the whole data are exactly the
  frequent itemsets.

21
Toivonens Algorithm (contd.)

• What if we find something in the negative border
  is actually frequent?
• We must start over again!
• But by choosing the support threshold for the
  sample wisely, we can make the probability of
  failure low, while still keeping the number of
  itemsets checked on the second pass low enough
  for main-memory.

22
Experiment

• Synthetic data set characteristics (T row
  size on average, I size of maximal frequent
  sets on average)

23
Experiment (contd.)
Lowered frequency thresholds () for probability
of missing any given frequent set is less than d
0.001
24
Number of trials with misses
25
Conclusions

• Advantages
• Reduced failure probability, while keeping
  candidate-count low enough for memory
• Disadvantages
• Potentially large number of candidates
• in second pass

26

• Thank you!