Introduction to Association Analysis

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Introduction to Association Analysis

• Zhangxi Lin
• ISQS 3358
• Texas Tech University

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Outline

• Basic concepts
• Itemset generation - Apriori principle
• Association rule discovery and generation
• Evaluation of association patterns
• Sequential pattern analysis

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Basic Concepts
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Association Rule Mining

• Given a set of transactions, find rules that will
  predict the occurrence of an item based on the
  occurrences of other items in the transaction

Market-Basket transactions
Example of Association Rules
Diaper ? Beer,Milk, Bread ?
Eggs,Coke,Beer, Bread ? Milk,
Implication means co-occurrence, not causality!
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Definition Frequent Itemset

• Itemset
• A collection of one or more items
• Example Milk, Bread, Diaper
• k-itemset
• An itemset that contains k items
• 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
• Frequent Itemset
• An itemset whose support is greater than or equal
  to a minsup threshold

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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
• Support (s)
• Fraction of transactions that contain both X and
  Y
• Confidence (c)
• Measures how often items in Y appear in
  transactions thatcontain X

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• Count (CKG, SVG) 1
• Support 1 / 5 20
• Count (CKG) 3
• Confidence 1 / 3 0.33
• Count (CKG) 2
• Count (CKG, SVG) 2
• Confidence (CKG, SVG) 2 / 2 100

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Formal Definitions

• Support s(X ? Y)
• Confidence, c(X ? Y)

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Itemset generation - Apriori principle
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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!

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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

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Mining Association Rules

• Two-step approach
• Frequent Itemset Generation
• Generate all itemsets whose support ? minsup
• Rule Generation
• Generate high confidence rules from each frequent
  itemset, where each rule is a binary partitioning
  of a frequent itemset
• Frequent itemset generation is still
  computationally expensive

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Frequent Itemset Generation
Given d items, there are 2d possible candidate
itemsets
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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 !!!

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Frequent Itemset Generation Strategies

• Reduce the number of candidates (M)
• Complete search M2d
• Use pruning techniques to reduce M
• Reduce the number of transactions (N)
• Reduce size of N as the size of itemset increases
• Used by DHP and vertical-based mining algorithms
• Reduce the number of comparisons (NM)
• Use efficient data structures to store the
  candidates or transactions
• No need to match every candidate against every
  transaction

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Reducing Number of Candidates

• Apriori principle
• If an itemset is frequent, then all of its
  subsets must also be frequent
• Apriori principle holds due to the following
  property of the support measure
• Support of an itemset never exceeds the support
  of its subsets
• This is known as the anti-monotone property of
  support

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Illustrating Apriori Principle
Found to be Frequent
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Illustrating Apriori Principle
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Apriori Algorithm

• Method
• Let k1
• Generate frequent itemsets of length 1
• Repeat until no new frequent itemsets are
  identified
• Generate length (k1) candidate itemsets from
  length k frequent itemsets
• Prune candidate itemsets containing subsets of
  length k that are infrequent
• Count the support of each candidate by scanning
  the DB
• Eliminate candidates that are infrequent, leaving
  only those that are frequent

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Association rule discovery and generation
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Reducing Number of Comparisons

• Candidate counting
• Scan the database of transactions to determine
  the support of each candidate itemset
• To reduce the number of comparisons, store the
  candidates in a hash structure
• Instead of matching each transaction against
  every candidate, match it against candidates
  contained in the hashed buckets

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Factors Affecting Complexity

• Choice of minimum support threshold
• lowering support threshold results in more
  frequent itemsets
• this may increase number of candidates and max
  length of frequent itemsets
• Dimensionality (number of items) of the data set
• more space is needed to store support count of
  each item
• if number of frequent items also increases, both
  computation and I/O costs may also increase
• Size of database
• since Apriori makes multiple passes, run time of
  algorithm may increase with number of
  transactions
• Average transaction width
• transaction width increases with denser data
  sets
• This may increase max length of frequent itemsets
  and traversals of hash tree (number of subsets in
  a transaction increases with its width)

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Rule Generation

• Given a frequent itemset L, find all non-empty
  subsets f ? L such that f ? L f satisfies the
  minimum confidence requirement
• If A,B,C,D is a frequent itemset, candidate
  rules
• ABC ?D, ABD ?C, ACD ?B, BCD ?A, A ?BCD, B
  ?ACD, C ?ABD, D ?ABCAB ?CD, AC ? BD, AD ? BC,
  BC ?AD, BD ?AC, CD ?AB,
• If L k, then there are 2k 2 candidate
  association rules (ignoring L ? ? and ? ? L)

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Rule Generation

• How to efficiently generate rules from frequent
  itemsets?
• In general, confidence does not have an
  anti-monotone property
• c(ABC ?D) can be larger or smaller than c(AB ?D)
• But confidence of rules generated from the same
  itemset has an anti-monotone property
• e.g., L A,B,C,D c(ABC ? D) ? c(AB ? CD)
  ? c(A ? BCD)
• Confidence is anti-monotone w.r.t. number of
  items on the RHS of the rule

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Rule Generation for Apriori Algorithm
Lattice of rules
Low Confidence Rule
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Rule Generation for Apriori Algorithm

• Candidate rule is generated by merging two rules
  that share the same prefixin the rule consequent
• join(CDgtAB,BDgtAC)would produce the
  candidaterule D gt ABC
• Prune rule DgtABC if itssubset ADgtBC does not
  havehigh confidence

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Demonstration

• A bank wants to examine its customer base and
  understand which of its products individual
  customers own in combination with one another. It
  has chosen to conduct a market-basket analysis of
  a sample of its customer base. The bank has a
  data set that lists the banking products/services
  used by 7,991 customers.
• Data set BANK
• Variables
• ACCT ID, Nominal, Account Number
• SERVICE Target, Nominal, Type of Service
• VISIT Sequence, Ordinal, Order of Product
  Purchase

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Evaluation of association patterns
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Contingency Table
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Statistical Independence

• Population of 1000 students
• 600 students know how to swim (S)
• 700 students know how to bike (B)
• 420 students know how to swim and bike (S,B)
• P(S?B) 420/1000 0.42
• P(S) ? P(B) 0.6 ? 0.7 0.42
• P(S?B) P(S) ? P(B) gt Statistical independence
• P(S?B) gt P(S) ? P(B) gt Positively correlated
• P(S?B) lt P(S) ? P(B) gt Negatively correlated

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Statistical-based Measures

• Measures that take into account statistical
  dependence

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Example Lift/Interest Contingency Table
Coffee Coffee
Tea 15 5 20
Tea 75 5 80
90 10 100

• Association Rule Tea ? Coffee
• Confidence P(CoffeeTea) 0.75
• but P(Coffee) 0.9
• Lift 0.75/0.9 0.8333 (lt 1, therefore is
  negatively associated)

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Compared to Confusion Matrix
Computed Yes Computed No Total
Actual Yes 15 5 20
Actual No 75 5 80
Total 90 10 100
In classification, we are interested in P(Actual
YesComputed Yes) i.e. P(Row Column) In
associate analysis, we are interested in
P(ColumnRow)
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Sequential pattern analysis
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Examples of Sequence Data
Sequence Database Sequence Element (Transaction) Event(Item)
Customer Purchase history of a given customer A set of items bought by a customer at time t Books, diary products, CDs, etc
Web Data Browsing activity of a particular Web visitor A collection of files viewed by a Web visitor after a single mouse click Home page, index page, contact info, etc
Event data History of events generated by a given sensor Events triggered by a sensor at time t Types of alarms generated by sensors
Genome sequences DNA sequence of a particular species An element of the DNA sequence Bases A,T,G,C
Element (Transaction)
Event (Item)
E1E2
E1E3
E2
E3E4
E2
Sequence
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Examples of Sequence

• Web sequence
• lt Homepage Electronics Digital Cameras
  Canon Digital Camera Shopping Cart Order
  Confirmation Return to Shopping gt
• Sequence of initiating events causing the nuclear
  accident at 3-mile Island(http//stellar-one.com
  /nuclear/staff_reports/summary_SOE_the_initiating_
  event.htm)
• lt clogged resin outlet valve closure loss
  of feedwater condenser polisher outlet valve
  shut booster pumps trip main waterpump
  trips main turbine trips reactor pressure
  increasesgt
• Sequence of books checked out at a library
• ltFellowship of the Ring The Two Towers
  Return of the Kinggt

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Sequential Pattern Mining Definition

• Given
• a database of sequences
• a user-specified minimum support threshold,
  minsup
• Task
• Find all subsequences with support minsup

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Sequential Pattern Mining Example
Minsup 50 Examples of Frequent
Subsequences lt 1,2 gt s60 lt 2,3 gt
s60 lt 2,4gt s80 lt 3 5gt s80 lt 1
2 gt s80 lt 2 2 gt s60 lt 1 2,3
gt s60 lt 2 2,3 gt s60 lt 1,2 2,3 gt s60