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Association Analysis

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Title: Association Analysis


1
Association Analysis
2
Association Rule Mining Definition
  • Given a set of records each of which contain some
    number of items from a given collection
  • Produce dependency rules which will predict
    occurrence of an item based on occurrences of
    other items.

Rules Discovered Milk --gt Coke
Diaper, Milk --gt Beer
3
Association Rules
  • Marketing and Sales Promotion
  • Let the rule discovered be
  • Bagels, --gt Potato Chips
  • Potato Chips as consequent gt
  • Can be used to determine what should be done to
    boost its sales.
  • Bagels in the antecedent gt
  • Can be used to see which products would be
    affected if the store discontinues selling bagels.

4
Two key issues
  • First, discovering patterns from a large
    transaction data set can be computationally
    expensive.
  • Second, some of the discovered patterns are
    potentially spurious because they may happen
    simply by chance.

5
Items and transactions
  • Let
  • I i1, i2,,id be the set of all items in a
    market basket data and
  • T t1, t2 ,, tN be the set of all
    transactions.
  • Each transaction ti contains a subset of items
    chosen from I.
  • Itemset
  • A collection of one or more items
  • Example Milk, Bread, Diaper
  • k-itemset
  • An itemset that contains k items
  • Transaction width
  • The number of items present in a transaction.
  • A transaction tj is said to contain an itemset X,
    if X is a subset of tj.
  • E.g., the second transaction contains the itemset
    Bread, Diapers but not Bread, Milk.

6
Definition Frequent Itemset
  • Support count (?)
  • Frequency of occurrence of an itemset
  • E.g. ?(Milk, Bread,Diaper) 2
  • Support
  • Fraction of transactions that contain an itemset
  • E.g. s(Milk, Bread, Diaper) 2/5 ?/N
  • Frequent Itemset
  • An itemset whose support is greater than or equal
    to a minsup threshold

7
Definition Association Rule
  • Association Rule
  • An implication expression of the form X ? Y,
    where X and Y are itemsets
  • Example Milk, Diaper ? Beer
  • Rule Evaluation Metrics (X ? Y)
  • Support (s)
  • Fraction of transactions that contain both X and
    Y
  • Confidence (c)
  • Measures how often items in Y appear in
    transactions thatcontain X

8
Why Use Support and Confidence?
  • Support
  • A rule that has very low support may occur simply
    by chance.
  • Support is often used to eliminate uninteresting
    rules.
  • Support also has a desirable property that can be
    exploited for the efficient discovery of
    association rules.
  • Confidence
  • Measures the reliability of the inference made by
    a rule.
  • For a rule X ? Y , the higher the confidence, the
    more likely it is for Y to be present in
    transactions that contain X.
  • Confidence provides an estimate of the
    conditional probability of Y given X.

9
Association Rule Mining Task
  • Given a set of transactions T, the goal of
    association rule mining is to find all rules
    having
  • support ? minsup threshold
  • confidence ? minconf threshold
  • Brute-force approach
  • List all possible association rules
  • Compute the support and confidence for each rule
  • Prune rules that fail the minsup and minconf
    thresholds
  • ? Computationally prohibitive!

10
Brute-force approach
  • Suppose there are d items. We first choose k of
    the items to form the left hand side of the rule.
    There are Cd,k ways for doing this.
  • Now, there are Cd-k,i ways to choose the
    remaining items to form the right hand side of
    the rule, where 1 i d-k.

11
Brute-force approach
  • R3d-2d11
  • For d6,
  • 36-271602 possible rules
  • However, 80 of the rules are discarded after
    applying minsup20 and minconf50, thus making
    most of the computations become wasted.
  • So, it would be useful to prune the rules early
    without having to compute their support and
    confidence values.

An initial step toward improving the performance
decouple the support and confidence requirements.
12
Mining Association Rules
Example of Rules Milk,Diaper ? Beer (s0.4,
c0.67)Milk,Beer ? Diaper (s0.4,
c1.0) Diaper,Beer ? Milk (s0.4,
c0.67) Beer ? Milk,Diaper (s0.4, c0.67)
Diaper ? Milk,Beer (s0.4, c0.5) Milk ?
Diaper,Beer (s0.4, c0.5)
  • Observations
  • All the above rules are binary partitions of the
    same itemset Milk, Diaper, Beer
  • Rules originating from the same itemset have
    identical support but can have different
    confidence
  • Thus, we may decouple the support and confidence
    requirements
  • If the itemset is infrequent, then all six
    candidate rules can be pruned immediately without
    us having to compute their confidence values.

13
Mining Association Rules
  • Two-step approach
  • Frequent Itemset Generation
  • Generate all itemsets whose support ? minsup
    (these itemsets are called frequent itemset)
  • Rule Generation
  • Generate high confidence rules from each frequent
    itemset, where each rule is a binary partitioning
    of a frequent itemset (these rules are called
    strong rules)
  • The computational requirements for frequent
    itemset generation are more expensive than those
    of rule generation.
  • We focus first on frequent itemset generation.

14
Frequent Itemset Generation
Given d items, there are 2d possible candidate
itemsets
15
Frequent Itemset Generation
  • Brute-force approach
  • Each itemset in the lattice is a candidate
    frequent itemset
  • Count the support of each candidate by scanning
    the database
  • Match each transaction against every candidate
  • Complexity O(NMw) gt Expensive since M 2d !!!
  • w is max transaction width.

16
Reducing Number of Candidates
  • Apriori principle
  • If an itemset is frequent, then all of its
    subsets must also be frequent
  • This is due to the anti-monotone property of
    support
  • Apriori principle conversely said
  • If an itemset such as a, b is infrequent, then
    all of its supersets must be infrequent too.

17
Illustrating Apriori Principle
18
Illustrating Apriori Principle
Items (1-itemsets)
Minimum Support 3
If every subset is considered, 6C1 6C2 6C3
41 With support-based pruning, 6 6 1 13
19
Apriori Algorithm
  • Method
  • Let k1
  • Generate frequent itemsets of length 1
  • Repeat until no new frequent itemsets are
    identified
  • kk1
  • Generate length-k candidate itemsets from
    length-k-1 frequent itemsets
  • Prune candidate itemsets containing subsets of
    length-k-1 that are infrequent
  • Count the support of each candidate by scanning
    the DB and eliminate candidates that are
    infrequent, leaving only those that are frequent

20
Candidate generation and prunning
  • Many ways to generate candidate itemsets.
  • An effective candidate generation procedure
  • Should avoid generating too many unnecessary
    candidates.
  • A candidate itemset is unnecessary if at least
    one of its subsets is infrequent.
  • Must ensure that the candidate set is complete,
  • i.e., no frequent itemsets are left out by the
    candidate generation procedure.
  • Should not generate the same candidate itemset
    more than once.
  • E.g., the candidate itemset a, b, c, d can be
    generated in many ways---
  • by merging a, b, c with d,
  • c with a, b, d, etc.

21
Brute force
  • A bruteforce method considers every kitemset as
    a potential candidate and then applies the
    candidate pruning step to remove any unnecessary
    candidates.

22
Fk-1?F1 Method
  • Extend each frequent (k - 1)itemset with a
    frequent 1-itemset.
  • Is it complete?
  • The procedure is complete because every frequent
    kitemset is composed of a frequent (k -
    1)itemset and a frequent 1itemset.
  • However, it doesnt prevent the same candidate
    itemset from being generated more than once.
  • E.g., Bread, Diapers, Milk can be generated by
    merging
  • Bread, Diapers with Milk,
  • Bread, Milk with Diapers, or
  • Diapers, Milk with Bread.

23
Lexicographic Order
  • Avoid generating duplicate candidates by ensuring
    that the items in each frequent itemset are kept
    sorted in their lexicographic order.
  • Each frequent (k-1)itemset X is then extended
    with frequent items that are lexicographically
    larger than the items in X.
  • For example, the itemset Bread, Diapers can be
    augmented with Milk since Milk is
    lexicographically larger than Bread and Diapers.
  • However, we dont augment Diapers, Milk with
    Bread nor Bread, Milk with Diapers because
    they violate the lexicographic ordering
    condition.
  • Is it complete?

24
Lexicographic Order - Completeness
  • Is it complete?
  • Yes. Let (i1,, ik-1, ik) be a frequent k-itemset
    sorted in lexicographic order.
  • Since it is frequent, by the Apriori principle,
    (i1,, ik-1) and (ik) are frequent as well.
  • I.e. (i1,, ik-1) ?Fk-1 and (ik) ?F1.
  • Since, (ik) is lexicographically bigger than
    i1,, ik-1, we have that (i1,, ik-1) would be
    joined with (ik) for giving (i1,, ik-1, ik) as a
    candidate k-itemset.

25
Still too many candidates
  • E.g. merging Beer, Diapers with Milk is
    unnecessary because one of its subsets, Beer,
    Milk, is infrequent.
  • Heuristics available to reduce (prune) the number
    of unnecessary candidates.
  • E.g., for a candidate kitemset to be worthy,
  • every item in the candidate must be contained in
    at least k-1 of the frequent (k-1)itemsets.
  • Beer, Diapers, Milk is a viable candidate
    3itemset only if every item in the candidate,
    including Beer, is contained in at least 2
    frequent 2itemsets.
  • Since there is only one frequent 2itemset
    containing Beer, all candidate itemsets involving
    Beer must be infrequent.
  • Why?
  • Because each of k-1subsets containing an item
    must be frequent.

26
Fk-1?F1
27
Fk-1?Fk-1 Method
  • Merge a pair of frequent (k-1)itemsets only if
    their first k-2 items are identical.
  • E.g. frequent itemsets Bread, Diapers and
    Bread, Milk are merged to form a candidate
    3itemset Bread, Diapers, Milk.
  • We dont merge Beer, Diapers with Diapers,
    Milk because the first item in both itemsets is
    different.
  • Indeed, if Beer, Diapers, Milk is a viable
    candidate, it would have been obtained by merging
    Beer, Diapers with Beer, Milk instead.
  • This illustrates both the completeness of the
    candidate generation procedure and the advantages
    of using lexicographic ordering to prevent
    duplicate candidates.
  • Pruning?
  • Because each candidate is obtained by merging a
    pair of frequent (k-1) itemsets, an additional
    candidate pruning step is needed to ensure that
    the remaining k-2 subsets of k-1 elements are
    frequent.

28
Fk-1?Fk-1
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