Transcript Frequent Itemset

Data Mining
Association Analysis: Basic Concepts
and Algorithms
Lecture Notes for Chapter 6
Introduction to Data Mining
by
Tan, Steinbach, Kumar
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
1
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
TID
Items
1
Bread, Milk
2
3
4
5
Bread, Diaper, Beer, Eggs
Milk, Diaper, Beer, Coke
Bread, Milk, Diaper, Beer
Bread, Milk, Diaper, Coke
© Tan,Steinbach, Kumar
Introduction to Data Mining
Example of Association Rules
{Diaper}  {Beer},
{Milk, Bread}  {Eggs,Coke},
{Beer, Bread}  {Milk},
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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
TID
Items
1
Bread, Milk
2
3
4
5
Bread, Diaper, Beer, Eggs
Milk, Diaper, Beer, Coke
Bread, Milk, Diaper, Beer
Bread, Milk, Diaper, Coke
– 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
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
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
TID
Items
1
Bread, Milk
2
3
4
5
Bread, Diaper, Beer, Eggs
Milk, Diaper, Beer, Coke
Bread, Milk, Diaper, Beer
Bread, Milk, Diaper, Coke
– Support (s)

Example:
Fraction of transactions that contain
both X and Y
{Milk , Diaper }  Beer
– Confidence (c)

Measures how often items in Y
appear in transactions that
contain X
© Tan,Steinbach, Kumar
s
 (Milk, Diaper, Beer )
|T|

2
 0.4
5
 (Milk, Diaper, Beer ) 2
c
  0.67
 (Milk, Diaper )
3
Introduction to Data Mining
4/18/2004
‹#›
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!
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Mining Association Rules
Example of Rules:
TID
Items
1
Bread, Milk
2
3
4
5
Bread, Diaper, Beer, Eggs
Milk, Diaper, Beer, Coke
Bread, Milk, Diaper, Beer
Bread, Milk, Diaper, Coke
{Milk,Diaper}  {Beer} (s=0.4, c=0.67)
{Milk,Beer}  {Diaper} (s=0.4, c=1.0)
{Diaper,Beer}  {Milk} (s=0.4, c=0.67)
{Beer}  {Milk,Diaper} (s=0.4, c=0.67)
{Diaper}  {Milk,Beer} (s=0.4, c=0.5)
{Milk}  {Diaper,Beer} (s=0.4, c=0.5)
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
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Mining Association Rules

Two-step approach:
1. Frequent Itemset Generation
–
Generate all itemsets whose support  minsup
2. 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
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Frequent Itemset Generation
null
A
B
C
D
E
AB
AC
AD
AE
BC
BD
BE
CD
CE
DE
ABC
ABD
ABE
ACD
ACE
ADE
BCD
BCE
BDE
CDE
ABCD
ABCE
ABDE
ACDE
ABCDE
© Tan,Steinbach, Kumar
Introduction to Data Mining
BCDE
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
Transactions
N
TID
1
2
3
4
5
Items
Bread, Milk
Bread, Diaper, Beer, Eggs
Milk, Diaper, Beer, Coke
Bread, Milk, Diaper, Beer
Bread, Milk, Diaper, Coke
List of
Candidates
M
w
– Match each transaction against every candidate
– Complexity ~ O(NMw) => Expensive since M = 2d !!!
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Computational Complexity

Given d unique items:
– Total number of itemsets = 2d
– Total number of possible association rules:
 d   d  k 
R       

 k   j 
 3  2 1
d 1
d k
k 1
j 1
d
d 1
If d=6, R = 602 rules
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Frequent Itemset Generation Strategies

Reduce the number of candidates (M)
– Complete search: M=2d
– Use pruning techniques to reduce M

Reduce the number of transactions (N)
– Reduce size of N as the size of itemset increases

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
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
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:
X , Y : ( X  Y )  s( X )  s(Y )
– Support of an itemset never exceeds the support of its
subsets
– This is known as the anti-monotone property of support
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Illustrating Apriori Principle
null
A
B
C
D
E
AB
AC
AD
AE
BC
BD
BE
CD
CE
DE
ABC
ABD
ABE
ACD
ACE
ADE
BCD
BCE
BDE
CDE
Found to be
Infrequent
ABCD
ABCE
Pruned
supersets
© Tan,Steinbach, Kumar
Introduction to Data Mining
ABDE
ACDE
BCDE
ABCDE
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‹#›
Illustrating Apriori Principle
Item
Bread
Coke
Milk
Beer
Diaper
Eggs
Count
4
2
4
3
4
1
Items (1-itemsets)
Minimum Support = 3
Itemset
{Bread,Milk}
{Bread,Beer}
{Bread,Diaper}
{Milk,Beer}
{Milk,Diaper}
{Beer,Diaper}
Count
3
2
3
2
3
3
Pairs (2-itemsets)
(No need to generate
candidates involving Coke
or Eggs)
Triplets (3-itemsets)
Itemset
{Bread,Milk,Diaper}
© Tan,Steinbach, Kumar
Introduction to Data Mining
Count
3
4/18/2004
‹#›
Apriori Algorithm

Method:
– Let k=1
– Generate frequent itemsets of length 1
– Repeat until no new frequent itemsets are identified
 Generate
length (k+1) 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
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
The Apriori Algorithm — Example
Database D
TID
100
200
300
400
itemset sup.
C1
{1}
2
{2}
3
Scan D
{3}
3
{4}
1
{5}
3
Items
134
235
1235
25
C2 itemset sup
L2 itemset sup
2
2
3
2
{1
{1
{1
{2
{2
{3
C3 itemset
{2 Kumar
3 5}
© Tan,Steinbach,
Scan D
{1 3}
{2 3}
{2 5}
{3 5}
2}
3}
5}
3}
5}
5}
1
2
1
2
3
2
L1 itemset sup.
{1}
{2}
{3}
{5}
2
3
3
3
C2 itemset
{1 2}
Scan D
L3 itemset sup
{2 3 5} 2
Introduction to Data Mining
{1
{1
{2
{2
{3
3}
5}
3}
5}
5}
4/18/2004
‹#›
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

Transactions
N
TID
1
2
3
4
5
Hash Structure
Items
Bread, Milk
Bread, Diaper, Beer, Eggs
Milk, Diaper, Beer, Coke
Bread, Milk, Diaper, Beer
Bread, Milk, Diaper, Coke
k
Buckets
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Reducing Number of Comparisons
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Generate Hash Tree
Suppose you have 15 candidate itemsets of length 3:
{1 4 5}, {1 2 4}, {4 5 7}, {1 2 5}, {4 5 8}, {1 5 9}, {1 3 6}, {2 3 4}, {5 6 7}, {3 4 5},
{3 5 6}, {3 5 7}, {6 8 9}, {3 6 7}, {3 6 8}
You need:
• Hash function
• Max leaf size: max number of itemsets stored in a leaf node (if number of
candidate itemsets exceeds max leaf size, split the node)
Hash function
3,6,9
1,4,7
234
567
345
136
145
2,5,8
124
457
© Tan,Steinbach, Kumar
125
458
Introduction to Data Mining
356
357
689
367
368
159
4/18/2004
‹#›
Association Rule Discovery: Hash tree
Hash Function
1,4,7
Candidate Hash Tree
3,6,9
2,5,8
234
567
145
136
345
Hash on
1, 4 or 7
124
457
© Tan,Steinbach, Kumar
125
458
159
Introduction to Data Mining
356
357
689
367
368
4/18/2004
‹#›
Association Rule Discovery: Hash tree
Hash Function
1,4,7
Candidate Hash Tree
3,6,9
2,5,8
234
567
145
136
345
Hash on
2, 5 or 8
124
457
© Tan,Steinbach, Kumar
125
458
159
Introduction to Data Mining
356
357
689
367
368
4/18/2004
‹#›
Association Rule Discovery: Hash tree
Hash Function
1,4,7
Candidate Hash Tree
3,6,9
2,5,8
234
567
145
136
345
Hash on
3, 6 or 9
124
457
© Tan,Steinbach, Kumar
125
458
159
Introduction to Data Mining
356
357
689
367
368
4/18/2004
‹#›
Subset Operation
Given a transaction t, what
are the possible subsets of
size 3?
Transaction, t
1 2 3 5 6
Level 1
1 2 3 5 6
2 3 5 6
3 5 6
Level 2
12 3 5 6
13 5 6
123
125
126
135
136
Level 3
© Tan,Steinbach, Kumar
15 6
156
23 5 6
235
236
25 6
256
35 6
356
Subsets of 3 items
Introduction to Data Mining
4/18/2004
‹#›
Subset Operation Using Hash Tree
Hash Function
1 2 3 5 6 transaction
1+ 2356
2+ 356
1,4,7
3+ 56
3,6,9
2,5,8
234
567
145
136
345
124
457
125
458
© Tan,Steinbach, Kumar
159
356
357
689
Introduction to Data Mining
367
368
4/18/2004
‹#›
Subset Operation Using Hash Tree
Hash Function
1 2 3 5 6 transaction
1+ 2356
2+ 356
12+ 356
1,4,7
3+ 56
3,6,9
2,5,8
13+ 56
234
567
15+ 6
145
136
345
124
457
© Tan,Steinbach, Kumar
125
458
159
Introduction to Data Mining
356
357
689
367
368
4/18/2004
‹#›
Subset Operation Using Hash Tree
Hash Function
1 2 3 5 6 transaction
1+ 2356
2+ 356
12+ 356
1,4,7
3+ 56
3,6,9
2,5,8
13+ 56
234
567
15+ 6
145
136
345
124
457
© Tan,Steinbach, Kumar
125
458
159
356
357
689
367
368
Match transaction against 11 out of 15 candidates
Introduction to Data Mining
4/18/2004
‹#›
Factors Affecting Complexity

Choice of minimum support threshold
– lowering support threshold results in more frequent
itemsets

Dimensionality (number of items) of the data set
– 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
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Compact Representation of Frequent Itemsets

Some itemsets are redundant because they have
identical support as their supersets
TID A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 C1 C2 C3 C4 C5 C6 C7 C8 C9 C10
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
7
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
9
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
10
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
11
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
12
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
13
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
14
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
15
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1

10 
Number of frequent itemsets  3    
k

Need a compact representation
10
k 1
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Maximal Frequent Itemset
An itemset is maximal frequent if none of its immediate supersets
is frequent
null
Maximal
Itemsets
A
B
C
D
E
AB
AC
AD
AE
BC
BD
BE
CD
CE
DE
ABC
ABD
ABE
ACD
ACE
ADE
BCD
BCE
BDE
CDE
ABCD
ABCE
ABDE
Infrequent
Itemsets
ABCD
E
© Tan,Steinbach, Kumar
Introduction to Data Mining
ACDE
BCDE
Border
4/18/2004
‹#›
Closed Itemset

An itemset is closed if none of its immediate supersets
has the same support as the itemset
TID
1
2
3
4
5
Items
{A,B}
{B,C,D}
{A,B,C,D}
{A,B,D}
{A,B,C,D}
© Tan,Steinbach, Kumar
Itemset
{A}
{B}
{C}
{D}
{A,B}
{A,C}
{A,D}
{B,C}
{B,D}
{C,D}
Introduction to Data Mining
Support
4
5
3
4
4
2
3
3
4
3
Itemset Support
{A,B,C}
2
{A,B,D}
3
{A,C,D}
2
{B,C,D}
3
{A,B,C,D}
2
4/18/2004
‹#›
Closed Itemsets
TID
Items
1
ABC
2
ABCD
3
BCE
4
ACDE
5
DE
Closed
Not closed
124
123
A
12
124
AB
12
24
AC
AD
ABC
ABD
ABE
2
AE
2
3
BD
4
ACD
345
D
BC
BE
2
4
ACE
ADE
E
24
CD
34
CE
3
BCD
45
ABCE
ABDE
ACDE
BDE
CDE
BCDE
ABCDE
Introduction to Data Mining
DE
4
BCE
4
ABCD
Not supported by
any transactions
© Tan,Steinbach, Kumar
245
C
123
24
2
1234
B
4
Transaction Ids
null
4/18/2004
‹#›
Maximal vs Closed Frequent Itemsets
Minimum support = 2
124
123
A
12
124
AB
12
ABC
24
AC
AD
ABD
ABE
1234
B
AE
345
D
2
3
BC
BD
4
ACD
245
C
123
4
24
2
Closed but
not maximal
null
24
BE
2
4
ACE
E
ADE
CD
Closed and
maximal
34
CE
3
BCD
45
DE
4
BCE
BDE
CDE
4
2
ABCD
ABCE
ABDE
ACDE
BCDE
# Closed = 9
# Maximal = 4
ABCDE
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Maximal vs Closed Itemsets
Frequent
Itemsets
Closed
Frequent
Itemsets
Maximal
Frequent
Itemsets
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Alternative Methods for Frequent Itemset Generation

Representation of Database
– horizontal vs vertical data layout
Horizontal
Data Layout
TID
1
2
3
4
5
6
7
8
9
10
Items
A,B,E
B,C,D
C,E
A,C,D
A,B,C,D
A,E
A,B
A,B,C
A,C,D
B
© Tan,Steinbach, Kumar
Vertical Data Layout
A
1
4
5
6
7
8
9
Introduction to Data Mining
B
1
2
5
7
8
10
C
2
3
4
8
9
D
2
4
5
9
E
1
3
6
4/18/2004
‹#›
FP-growth Algorithm

Use a compressed representation of the
database using an FP-tree

Once an FP-tree has been constructed, it uses a
recursive divide-and-conquer approach to mine
the frequent itemsets
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
FP-tree construction
null
After reading TID=1:
TID
1
2
3
4
5
6
7
8
9
10
Items
{A,B}
{B,C,D}
{A,C,D,E}
{A,D,E}
{A,B,C}
{A,B,C,D}
{B,C}
{A,B,C}
{A,B,D}
{B,C,E}
A:1
B:1
After reading TID=2:
null
A:1
B:1
B:1
C:1
D:1
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
FP-tree construction
After reading TID=3:
TID
1
2
3
4
5
6
7
8
9
10
Items
{A,B}
{B,C,D}
{A,C,D,E}
{A,D,E}
{A,B,C}
{A,B,C,D}
{B,C}
{A,B,C}
{A,B,D}
{B,C,E}
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
FP-Tree Construction
TID
1
2
3
4
5
6
7
8
9
10
Items
{A,B}
{B,C,D}
{A,C,D,E}
{A,D,E}
{A,B,C}
{A,B,C,D}
{B,C}
{A,B,C}
{A,B,D}
{B,C,E}
Header table
Item
Pointer
A
B
C
D
E
© Tan,Steinbach, Kumar
After reading TID=10:
null
B:1
A:7
B:5
C:1
C:1
D:1
D:1
C:3
D:1
D:1
D:1
E:1
E:1
E:1
Pointers are used to assist
frequent itemset generation
Introduction to Data Mining
4/18/2004
‹#›
FP-growth Algorithm
Conditional Pattern base
for D:
P = {(A:7,B:5,C:3),
(A:7,B:5),
(A:7,C:1),
(A:7),
(B:1,C:1)}
null
A:7
B:5
B:1
C:1
C:3
D:1
D:1
C:1
Recursively apply FPgrowth on P
D:1
Frequent Itemsets found
(with sup > 1):
A, AB, ABC
D:1
D:1
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Why Is FP-Growth Fast?

FP-growth is faster than Apriori

No candidate generation, no candidate test

Use compact data structure

Eliminate repeated database scan

Basic operation is counting and FP-tree
building
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
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,
A BCD,
AB CD,
BD AC,

ABD C,
B ACD,
AC  BD,
CD AB,
ACD B,
C ABD,
AD  BC,
BCD A,
D ABC
BC AD,
If |L| = k, then there are 2k – 2 candidate
association rules (ignoring L   and   L)
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Rule Generation

How to efficiently generate rules from frequent
itemsets?
– In general, confidence does not have an antimonotone 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

© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Rule Generation for Apriori Algorithm
Lattice of rules
Low
Confidence
Rule
CD=>AB
ABCD=>{ }
BCD=>A
ACD=>B
BD=>AC
D=>ABC
BC=>AD
C=>ABD
ABD=>C
AD=>BC
B=>ACD
ABC=>D
AC=>BD
AB=>CD
A=>BCD
Pruned
Rules
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Rule Generation for Apriori Algorithm

Candidate rule is generated by merging two rules
that share the same prefix
in the rule consequent
CD=>AB

BD=>AC
join(CD=>AB,BD=>AC)
would produce the candidate
rule D => ABC
D=>ABC

Prune rule D=>ABC if its
subset AD=>BC does not have
high confidence
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Effect of Support Distribution

Many real data sets have skewed support
distribution
Support
distribution of
a retail data set
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Effect of Support Distribution

How to set the appropriate minsup threshold?
– If minsup is set too high, we could miss itemsets
involving interesting rare items (e.g., expensive
products)
– If minsup is set too low, it is computationally
expensive and the number of itemsets is very large

Using a single minimum support threshold may
not be effective
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Multiple Minimum Support

How to apply multiple minimum supports?
– MS(i): minimum support for item i
– e.g.: MS(Milk)=5%,
MS(Coke) = 3%,
MS(Broccoli)=0.1%, MS(Salmon)=0.5%
– MS({Milk, Broccoli}) = min (MS(Milk), MS(Broccoli))
= 0.1%
– Challenge: Support is no longer anti-monotone


Suppose:
Support(Milk, Coke) = 1.5% and
Support(Milk, Coke, Broccoli) = 0.5%
{Milk,Coke} is infrequent but {Milk,Coke,Broccoli} is frequent
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Multiple Minimum Support (Liu 1999)

Order the items according to their minimum
support (in ascending order)
– e.g.:
MS(Milk)=5%,
MS(Coke) = 3%,
MS(Broccoli)=0.1%, MS(Salmon)=0.5%
– Ordering: Broccoli, Salmon, Coke, Milk

Need to modify Apriori such that:
– L1 : set of frequent items
– F1 : set of items whose support is  MS(1)
where MS(1) is mini( MS(i) )
– C2 : candidate itemsets of size 2 is generated from F1
instead of L1
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Multiple Minimum Support (Liu 1999)

Modifications to Apriori:
– In traditional Apriori,
A candidate (k+1)-itemset is generated by merging two
frequent itemsets of size k
 The candidate is pruned if it contains any infrequent subsets
of size k

– Pruning step has to be modified:
Prune only if subset contains the first item
 e.g.: Candidate={Broccoli, Coke, Milk} (ordered according to
minimum support)
 {Broccoli, Coke} and {Broccoli, Milk} are frequent but
{Coke, Milk} is infrequent

– Candidate is not pruned because {Coke,Milk} does not contain
the first item, i.e., Broccoli.
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Pattern Evaluation

Association rule algorithms tend to produce too
many rules
– many of them are uninteresting or redundant
– Redundant if {A,B,C}  {D} and {A,B}  {D}
have same support & confidence

Interestingness measures can be used to
prune/rank the derived patterns

In the original formulation of association rules,
support & confidence are the only measures used
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Application of Interestingness Measure
Interestingness
Measures
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Computing Interestingness Measure

Given a rule X  Y, information needed to compute rule
interestingness can be obtained from a contingency table
Contingency table for X  Y
Y
Y
X
f11
f10
f1+
X
f01
f00
fo+
f+1
f+0
|T|
f11: support of X and Y
f10: support of X and Y
f01: support of X and Y
f00: support of X and Y
Used to define various measures

© Tan,Steinbach, Kumar
support, confidence, lift, Gini,
J-measure, etc.
Introduction to Data Mining
4/18/2004
‹#›
Drawback of Confidence
Coffee
Coffee
Tea
15
5
20
Tea
75
5
80
90
10
100
Association Rule: Tea  Coffee
Confidence= P(Coffee|Tea) = 0.75
but P(Coffee) = 0.9
 Although confidence is high, rule is misleading
 P(Coffee|Tea) = 0.9375
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Correlation and statistical dependence

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



If corrS,B = 1: independence
If corrS,B > 1: positive correlation
If corrS,B < 1: negative correlation
© Tan,Steinbach, Kumar
Introduction to Data Mining
P( S  B)
corrS , B 
P( S ) P( B)
4/18/2004
‹#›
Statistical-based Measures

Measures that take into account statistical
dependence
P(Y | X )
Lift 
P(Y )
P( X , Y )
Interest 
P( X ) P(Y )
© Tan,Steinbach, Kumar
Introduction to Data Mining
Confidence (XY) / Support (Y)
Support (X, Y) / support(X)* support(Y)
4/18/2004
‹#›
Example: Lift/Interest
Coffee
Coffee
Tea
15
5
20
Tea
75
5
80
90
10
100
Association Rule: Tea  Coffee
Confidence= P(Coffee|Tea) = 0.75
but P(Coffee) = 0.9
 Lift = 0.75/0.9= 0.8333 (< 1, therefore is negatively associated)
 Interest = 0.15 / (0.9 * 0.2) = 0.8333 (< 1, therefore is negatively associated)
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Drawback of Lift & Interest
Y
Y
X
10
0
10
X
0
90
90
10
90
100
0.1
Lift 
 10
(0.1)(0.1)
Y
Y
X
90
0
90
X
0
10
10
90
10
100
0.9
Lift 
 1.11
(0.9)(0.9)
Statistical independence:
If P(X,Y)=P(X)P(Y) => Lift = 1
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
There are lots of
measures proposed
in the literature
Some measures are
good for certain
applications, but not
for others
What criteria should
we use to determine
whether a measure
is good or bad?
What about Aprioristyle support based
pruning? How does
it affect these
measures?
Properties of A Good Measure

Piatetsky-Shapiro:
3 properties a good measure M must satisfy:
– M(A,B) = 0 if A and B are statistically independent
– M(A,B) increase monotonically with P(A,B) when P(A)
and P(B) remain unchanged
– M(A,B) decreases monotonically with P(A) [or P(B)]
when P(A,B) and P(B) [or P(A)] remain unchanged
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Comparing Different Measures
10 examples of
contingency tables:
Example
f11
E1
E2
E3
E4
E5
E6
E7
E8
E9
E10
8123
8330
9481
3954
2886
1500
4000
4000
1720
61
Rankings of contingency tables
using various measures:
© Tan,Steinbach, Kumar
Introduction to Data Mining
f10
f01
f00
83
424 1370
2
622 1046
94
127 298
3080
5
2961
1363 1320 4431
2000 500 6000
2000 1000 3000
2000 2000 2000
7121
5
1154
2483
4
7452
4/18/2004
‹#›
Property under Variable Permutation
B
p
r
A
A
B
q
s
B
B
A
p
q
A
r
s
Does M(A,B) = M(B,A)?
Symmetric measures:

support, lift, collective strength, cosine, Jaccard, etc
Asymmetric measures:

confidence, conviction, Laplace, J-measure, etc
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Property under Row/Column Scaling
Grade-Gender Example (Mosteller, 1968):
Male
Female
High
2
3
5
Low
1
4
5
3
7
10
Male
Female
High
4
30
34
Low
2
40
42
6
70
76
2x
10x
Mosteller:
Underlying association should be independent of
the relative number of male and female students
in the samples
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Property under Inversion Operation
Transaction 1
.
.
.
.
.
Transaction N
A
B
C
D
E
F
1
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
1
1
1
1
1
1
1
1
0
1
1
1
1
0
1
1
1
1
1
0
1
1
1
1
1
1
1
1
0
0
0
0
0
1
0
0
0
0
0
(a)
© Tan,Steinbach, Kumar
(b)
Introduction to Data Mining
(c)
4/18/2004
‹#›
Example: -Coefficient

-coefficient is analogous to
correlation coefficient for
continuous variables
Y
Y
X
60
10
70
X
10
20
30
70
30
100
60 * 20  10 *10

70  30  70  30
 0.5238
 
f11  f 00  f 01  f10
f1  f 1  f 0   f  0
Y
Y
X
20
10
30
X
10
60
70
30
70
100
20  60  10 10

70  30  70  30
 0.5238
 Coefficient is the same for both tables
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Property under Null Addition
A
A
B
p
r
B
q
s
A
A
B
p
r
B
q
s+k
Invariant measures:

support, cosine, Jaccard, etc
Non-invariant measures:

correlation, Gini, mutual information, odds ratio, etc
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Different Measures have Different Properties
Sym bol
Measure
Range
P1
P2
P3
O1
O2
O3
O3'
O4



Q
Y

M
J
G
s
c
L
V
I
IS
PS
F
AV
S

Correlation
Lambda
Odds ratio
Yule's Q
Yule's Y
Cohen's
Mutual Information
J-Measure
Gini Index
Support
Confidence
Laplace
Conviction
Interest
IS (cosine)
Piatetsky-Shapiro's
Certainty factor
Added value
Collective strength
Jaccard
-1 … 0 … 1
0…1
0 … 1 … 
-1 … 0 … 1
-1 … 0 … 1
-1 … 0 … 1
0…1
0…1
0…1
0…1
0…1
0…1
0.5 … 1 … 
0 … 1 … 
0 .. 1
-0.25 … 0 … 0.25
-1 … 0 … 1
0.5 … 1 … 1
0 … 1 … 
0 .. 1
Yes
Yes
Yes*
Yes
Yes
Yes
Yes
Yes
Yes
No
No
No
No
Yes*
No
Yes
Yes
Yes
No
No
Yes
No
Yes
Yes
Yes
Yes
Yes
No
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
Yes
Yes
Yes
Yes
Yes
No
No
No
No
No
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
No
Yes
Yes
Yes
Yes**
Yes
Yes
Yes
No
No
Yes
Yes
No
No
Yes
Yes
Yes
No
No
No
No
No
No
No
No
No
No
No
No
No
No
No
Yes
No*
Yes*
Yes
Yes
No
No*
No
No*
No
No
No
No
No
No
Yes
No
No
Yes*
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
Yes
No
No
No
Yes
No
No
Yes
Yes
No
Yes
No
No
No
No
No
No
No
No
No
No
No
Yes
No
No
No
Yes
No
No
No
No
Yes

2
1 
2
 1  2  3 
  0 
Yes
3  Introduction
3  to Data
3 3 Mining
Yes
Yes
No
No
No
No
Klosgen's
K
© Tan,Steinbach, Kumar




4/18/2004
‹#›
No
Support-based Pruning

Most of the association rule mining algorithms
use support measure to prune rules and itemsets

Study effect of support pruning on correlation of
itemsets
– Generate 10000 random contingency tables
– Compute support and pairwise correlation for each
table
– Apply support-based pruning and examine the tables
that are removed
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Effect of Support-based Pruning
All Itempairs
1000
900
800
700
600
500
400
300
200
100
2
3
4
5
6
7
8
9
0.
0.
0.
0.
0.
0.
0.
0.
1
1
0.
0
-1
-0
.9
-0
.8
-0
.7
-0
.6
-0
.5
-0
.4
-0
.3
-0
.2
-0
.1
0
Correlation
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Effect of Support-based Pruning
Support < 0.01
1
8
9
7
0.
0.
6
0.
-1
-0
.9
-0
.8
-0
.7
-0
.6
-0
.5
-0
.4
-0
.3
-0
.2
-0
.1
Correlation
5
0
0.
0
4
50
0.
50
3
100
0.
100
2
150
0.
150
1
200
0.
200
0
250
1
250
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
300
-1
-0
.9
-0
.8
-0
.7
-0
.6
-0
.5
-0
.4
-0
.3
-0
.2
-0
.1
300
0.
Support < 0.03
Correlation
Support < 0.05
300
250
Small Support values
decrease negatively
correlated itemsets
200
150
100
50
Correlation
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
1
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
-1
-0
.9
-0
.8
-0
.7
-0
.6
-0
.5
-0
.4
-0
.3
-0
.2
-0
.1
0
Effect of Support-based Pruning

Investigate how support-based pruning affects
other measures

Steps:
– Generate 10000 contingency tables
– Rank each table according to the different measures
– Compute the pair-wise correlation between the
measures
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Effect of Support-based Pruning

Without Support Pruning (All Pairs)
All Pairs (40.14%)
Conviction
Odds ratio
Col Strength
Correlation
Interest
PS
CF
Yule Y
Reliability
Kappa
Klosgen
Yule Q
Confidence
Laplace
IS
Support
Jaccard
Lambda
Gini
J-measure
Mutual Info
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19

Red cells indicate correlation between pairs of measures > 0.85

40.14% pairs have correlation > 0.85
© Tan,Steinbach, Kumar
Introduction to Data Mining
20
4/18/2004
21
‹#›
Effect of Support-based Pruning

0.5%  support  50%
0.005 <= s upport <= 0.500 (61.45%)
Interest
Conviction
Odds ratio
Col Strength
Laplace
Confidence
Correlation
Klosgen
Reliability
PS
Yule Q
CF
Yule Y
Kappa
IS
Jaccard
Support
Lambda
Gini
J-measure
Mutual Info
1

2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
61.45% pairs have correlation > 0.85
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Effect of Support-based Pruning

0.5%  support  30%
0.005 <= s upport <= 0.300 (76.42%)
Support
Interest
Reliability
Conviction
Yule Q
Odds ratio
Confidence
CF
Yule Y
Kappa
Correlation
Col Strength
IS
Jaccard
Laplace
PS
Klosgen
Lambda
Mutual Info
Gini
J-measure
1

2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
76.42% pairs have correlation > 0.85
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
?
?
?
SIMPSON’S PARADOX
How does it happen?
© Tan,Steinbach, Kumar
Introduction to Data Mining
Pamela
Leutwyler
4/18/2004
‹#›
A perfume company is testing two new scents
Citrus
And
Orange blossom
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
28 single women volunteer to test
these products
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
15 are Eagles cheerleaders
13 are members of
Granny’s bingo club
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
13 women choose CITRUS
15 women choose
ORANGE
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
80% success rate
70% success rate
4 OF THE 5 CHEERLEADERS
WHO USED CITRUS FOUND LOVE!
7 OF THE 10 CHEERLEADERS
WHO USED ORANGE FOUND LOVE!
CITRUS appears to be more effective for cheerleaders
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
2 OF THE 8 GRANNIES
WHO USED CITRUS FOUND LOVE!
1 OF THE 5 GRANNIES
WHO USED ORANGE FOUND LOVE!
80% success rate
success rate
CITRUS appears to be more effective for 70%
grannies
25% success rate
© Tan,Steinbach, Kumar
Introduction to Data Mining
20% success rate
4/18/2004
‹#›
CITRUS is better for cheerleaders
80% success rate
70% success rate
CITRUS is better for grannies
25% success rate
© Tan,Steinbach, Kumar
Introduction to Data Mining
20% success rate
4/18/2004
‹#›
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
6 of the 13 women who used
Citrus found love. 46%
8 of the 15 women who used
Orange found love. 53%
Overall Orange has
a higher success
rate
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
How can it happen that
CITRUS works better for cheerleaders and
CITRUS works better for grannies
While
ORANGE works better overall???
Where are most of the
grannies?
© Tan,Steinbach, Kumar
Introduction to Data Mining
Where are most of the
cheerleaders?
4/18/2004
‹#›
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
Simpson’s paradox
Contingency matrix for women
Love
Not-Love
Citrus
6
7
13
Orange
8
7
15
14
14
28
Confidence (CitrusLove)=6/13= 46%
Confidence (OrangeLove)=8/15= 53%
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›