Sampling Large Databases for Association Rules (Toivenon

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Sampling Large Databases for Association
Rules(Toivenons Approach, 1996)

• Farzaneh Mirzazadeh
• Fall 2007

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Outline

• Introduction
• Preliminaries
• Definitions, and Problem Statement
• Two General Approaches
• Sampling Method for Mining Association Rules
• The algorithm
• Analysis
• Experimental Results

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Introduction

• Problem Discovery of Association Rules
• Domain Very Large Databases
• Bottleneck Time
• Main Memory Processes Ignorable
• Disk I/O An Influential Factor
• Suggestion Minimize the Number of Scans of the
  Database
• Only One Full Pass Over the Database

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Introduction(Cont)Overview of Toivonens Method

• Main Steps
• Pick a random sample from the database.
• Use the sample to determine all probable
  association rules.
• Verify the results with the rest of the database,
  i.e. Eliminated incorrectly detected association
  rules and add missing association rules.
• The Main Contribution
• To show that all exact frequencies can be found
  efficiently, by analyzing first a random sample
  and then the whole database with the proposed
  method.

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Preliminaries

• Items
• II1,I2,,Im
• Transactions
• rt1,t2, , tn, tj? I
• Support of an itemset
• Percentage of transactions which contain that
  itemset.
• Frequent Itemsets
• Association Rules
• Strong Association Rules

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Preliminaries

• Association Rule implication X ? Y where X,Y ? I
  and X ? Y Ø
• Support of Association Rule X ? Y Percentage of
  transactions that contain X ?Y
• Confidence of Association Rule X ? Y Ratio of
  number of transactions that contain X ? Y to the
  number that contain X
• Problem Find the strong association rules of a
  given set I with respect to threshold min_fr and
  confidence min_conf.

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Algorithms for Mining Association Rules

• Level-wise Algorithms
• Idea If a set is not frequent then its
  supersets can not be frequent.
• On level k, candidate itemsets X of size k are
  generated such that all subsets of X are
  frequent.
• Partition Algorithm
• Idea Partition the data to sections small
  enough to be handled in main memory.
• First Pass Find locally frequent Itemsets.
• Second Pass Union of the local frequent
  itemsets

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Sampling for Frequent Sets

• Major Steps
• Random sampling
• Finding the frequent itemsets of the sample
• Finding other probable candidates using the
  concept of Negative Border
• Using the rest of the database to check the
  candidates

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Negative Border

• All sets which are not in our frequent itemsets,
  but all their subsets are.
• minimal itemsets not in S, where S
  is the collection of frequent itemsets
• Example
• S A, B, C, F, A,B, A,C, A,F,
  C,F, A,C,F
• B, C, B, F, D, E

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Frequent Set Discovery

• Intuition Given a collection S of sets that are
  frequent, the negative border contains the
  closest itemsets that could be frequent too.
• After finding the collection of frequent
  itemsets, S, we check negative border of S
• If no frequent items are addedgt We can conclude
  that all frequent sets are already found. (Why?)
• Decrease minimum support to increase the chance
  of success.
• If at least one frequent itemset is found in
  negative border gt We can conclude that some of
  its supersets may be frequent.(Why?)
• In the case of failure, we can either report
  failure and stop, or scan the database again and
  check the supersets to find the exact result.

Success
Failure
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Toivonens Algorithm
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Failure Handling

• In the fraction of cases where a possible failure
  is reported, all frequent sets can be found by
  making a second pass over the database

The algorithm simply computes the collection of
all sets that could possibly be frequent.
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Analysis of Sampling

• Sample Size and Probability of Failure

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Experimental Results
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Conclusion

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

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References

• 1 H. Toivonen, Sampling Large Databases for
  Association Rules, Proc. of VLDB Conference,
  India, 1996.

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Questions

• ?

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• Thank you