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