Download Data Mining Association Analysis: Basic

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
O
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
Bread, Diaper, Beer, Eggs
Milk, Diaper, Beer, Coke
Bread, Milk, Diaper, Beer
5
Bread, Milk, Diaper, Coke
© Tan,Steinbach, Kumar
Introduction to Data Mining
Example of Association Rules
{Diaper} → {Beer},
{Milk, Bread} → {Eggs,Coke},
{Beer, Bread} → {Milk},
Implication means co-occurrence,
not causality!
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2
Definition: Frequent Itemset
O
Itemset
– A collection of one or more items
‹
Example: {Milk, Bread, Diaper}
– k-itemset
‹
O
An itemset that contains k items
Support count (σ)
– Frequency of occurrence of an itemset
– E.g. σ({Milk, Bread,Diaper}) = 2
O
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
O
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
3
Definition: Association Rule
O
Association Rule
– An implication expression of the form
X → Y, where X and Y are itemsets
– Example:
{Milk, Diaper} → {Beer}
O
Rule Evaluation Metrics
– Support (s)
‹
Measures how often items in Y
appear in transactions that
contain X
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=
c=
© Tan,Steinbach, Kumar
Items
1
Example:
Fraction of transactions that contain
both X and Y
– Confidence (c)
‹
TID
Introduction to Data Mining
σ (Milk , Diaper, Beer)
|T|
=
2
= 0 .4
5
σ (Milk, Diaper, Beer) 2
= = 0.67
3
σ (Milk, Diaper)
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Association Rule Mining Task
O
Given a set of transactions T, the goal of
association rule mining is to find all rules having
– support ≥ minsup threshold
– confidence ≥ minconf threshold
O
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
5
Mining Association Rules
Example of Rules:
TID
Items
1
Bread, Milk
2
Bread, Diaper, Beer, Eggs
3
4
5
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
6
Mining Association Rules
O
Two-step approach:
1. Frequent Itemset Generation
–
Generate all itemsets whose support ≥ minsup
2. Rule Generation
–
O
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
7
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
O
Brute-force approach:
– Each itemset in the lattice is a candidate frequent itemset
– Count the support of each candidate by scanning the
database
List of
Candidates
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
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
9
Computational Complexity
O
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
10
Frequent Itemset Generation Strategies
O
Reduce the number of candidates (M)
– Complete search: M=2d
– Use pruning techniques to reduce M
O
Reduce the number of transactions (N)
– Reduce size of N as the size of itemset increases
– Used by DHP and vertical-based mining algorithms
O
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
11
Reducing Number of Candidates
O
Apriori principle:
– If an itemset is frequent, then all of its subsets must also
be frequent
O
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
12
Illustrating Apriori Principle
null
A
Found to be
Infrequent
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
Pruned
supersets
© Tan,Steinbach, Kumar
ABDE
ACDE
BCDE
ABCDE
Introduction to Data Mining
4/18/2004
13
Illustrating Apriori Principle
Item
Bread
Coke
Milk
Beer
Diaper
Eggs
Count
4
2
4
3
4
1
Items (1-itemsets)
Itemset
{Bread,Milk}
{Bread,Beer}
{Bread,Diaper}
{Milk,Beer}
{Milk,Diaper}
{Beer,Diaper}
Minimum Support = 3
Pairs (2-itemsets)
(No need to generate
candidates involving Coke
or Eggs)
Triplets (3-itemsets)
If every subset is considered,
6C + 6C + 6C = 41
1
2
3
With support-based pruning,
6 + 6 + 1 = 13
© Tan,Steinbach, Kumar
Count
3
2
3
2
3
3
Introduction to Data Mining
Ite m s e t
{ B r e a d ,M ilk ,D ia p e r }
Count
3
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Apriori Algorithm
O
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
15
Reducing Number of Comparisons
O
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
16
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
159
Introduction to Data Mining
356
357
689
367
368
4/18/2004
17
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
18
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
356
357
689
Introduction to Data Mining
367
368
4/18/2004
19
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
20
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
15 6
156
23 5 6
235
236
25 6
35 6
256
356
Subsets of 3 items
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
21
Subset Operation Using Hash Tree
Hash Function
1 2 3 5 6 transaction
1+ 2356
2+ 356
3,6,9
1,4,7
3+ 56
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
22
Subset Operation Using Hash Tree
Hash Function
1 2 3 5 6 transaction
1+ 2356
2+ 356
12+ 356
1,4,7
3,6,9
2,5,8
3+ 56
13+ 56
234
567
15+ 6
145
136
345
124
457
159
125
458
© Tan,Steinbach, Kumar
356
357
689
367
368
Introduction to Data Mining
4/18/2004
23
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
24
Factors Affecting Complexity
O
Choice of minimum support threshold
–
–
O
Dimensionality (number of items) of the data set
–
–
O
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
–
O
lowering support threshold results in more frequent itemsets
this may increase number of candidates and max length of
frequent itemsets
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)
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
25
Compact Representation of Frequent Itemsets
O
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
1
1
1
1
1
1
1
1
1
1
2
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
1
1
1
1
1
1
1
1
1
1
4
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
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
10 
= 3× ∑  
k
O
Number of frequent itemsets
O
Need a compact representation
© Tan,Steinbach, Kumar
Introduction to Data Mining
10
k =1
4/18/2004
26
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
Infrequent
Itemsets
ABDE
ACDE
Border
ABCD
E
© Tan,Steinbach, Kumar
BCDE
Introduction to Data Mining
4/18/2004
27
Closed Itemset
O
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
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Maximal vs Closed Itemsets
TID
Items
1
ABC
2
Transaction Ids
null
124
ABCD
3
BCE
4
ACDE
5
DE
123
A
12
124
AB
12
24
AD
ABD
ABE
123
AE
2
3
BD
4
ACD
345
D
BC
24
2
245
C
4
AC
2
ABC
1234
B
E
24
BE
2
4
ACE
ADE
BCD
3
34
CD
BCE
CE
45
4
BDE
4
ABCD
ABCE
ABDE
Not supported by
any transactions
ACDE
BCDE
ABCDE
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
29
Maximal vs Closed Frequent Itemsets
Minimum support = 2
124
123
A
12
124
AB
12
ABC
24
AC
AD
ABD
ABE
2
1234
B
AE
345
D
2
BC
3
BD
4
ACD
245
C
123
4
24
2
Closed but
not maximal
null
2
4
ACE
BE
ADE
BCD
E
24
3
CD
BCE
Closed and
maximal
34
CE
BDE
45
4
DE
CDE
4
ABCD
ABCE
ABDE
ACDE
BCDE
# Closed = 9
# Maximal = 4
ABCDE
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
30
DE
CDE
Maximal vs Closed Itemsets
Frequent
Itemsets
Closed
Frequent
Itemsets
Maximal
Frequent
Itemsets
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
31
Alternative Methods for Frequent Itemset Generation
O
Traversal of Itemset Lattice
– General-to-specific vs Specific-to-general
Frequent
itemset
border
null
null
..
..
..
..
{a 1,a2,...,a n}
{a1,a 2,...,a n}
(a) General-to-specific
© Tan,Steinbach, Kumar
Frequent
itemset
border
..
..
Frequent
itemset
border
(b) Specific-to-general
Introduction to Data Mining
null
{a 1,a2 ,...,a n}
(c) Bidirectional
4/18/2004
32
Alternative Methods for Frequent Itemset Generation
O
Traversal of Itemset Lattice
– Equivalent Classes
null
A
AB
ABC
B
AC
AD
ABD
ACD
null
C
BC
BD
D
CD
A
AB
BCD
AC
ABC
B
C
BC
AD
ABD
D
BD
CD
ACD
BCD
ABCD
ABCD
(a) Prefix tree
© Tan,Steinbach, Kumar
Introduction to Data Mining
(b) Suffix tree
4/18/2004
33
Alternative Methods for Frequent Itemset Generation
O
Traversal of Itemset Lattice
– Breadth-first vs Depth-first
(a) Breadth first
© Tan,Steinbach, Kumar
(b) Depth first
Introduction to Data Mining
4/18/2004
34
Alternative Methods for Frequent Itemset Generation
O
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
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FP-growth Algorithm
O
Use a compressed representation of the
database using an FP-tree
O
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
36
FP-tree construction
null
After reading TID=1:
TID
1
2
3
4
5
6
7
8
9
10
A:1
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}
B:1
After reading TID=2:
null
B:1
A:1
B:1
C:1
D:1
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
37
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
A
B
C
D
E
Pointer
© Tan,Steinbach, Kumar
Transaction
Database
null
B:3
A:7
B:5
C:1
C:3
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
38
FP-growth
C:1
Conditional Pattern base
for D:
P = {(A:1,B:1,C:1),
(A:1,B:1),
(A:1,C:1),
(A:1),
(B:1,C:1)}
D:1
Recursively apply FPgrowth on P
null
A:7
B:5
B:1
C:1
C:3
D:1
D:1
D:1
Frequent Itemsets found
(with sup > 1):
AD, BD, CD, ACD, BCD
D:1
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
39
Tree Projection
null
Set enumeration tree:
A
Possible Extension:
E(A) = {B,C,D,E}
B
C
D
E
AB
AC
AD
AE
BC
BD
BE
CD
CE
DE
ABC
ABD
ABE
ACD
ACE
ADE
BCD
BCE
BDE
CDE
Possible Extension:
E(ABC) = {D,E}
ABCD
ABCE
ABDE
ACDE
BCDE
ABCDE
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
40
Tree Projection
Items are listed in lexicographic order
O Each node P stores the following information:
O
–
–
–
–
Itemset for node P
List of possible lexicographic extensions of P: E(P)
Pointer to projected database of its ancestor node
Bitvector containing information about which
transactions in the projected database contain the
itemset
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
41
Projected Database
Original Database:
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}
Projected Database
for node A:
TID
1
2
3
4
5
6
7
8
9
10
Items
{B}
{}
{C,D,E}
{D,E}
{B,C}
{B,C,D}
{}
{B,C}
{B,D}
{}
For each transaction T, projected transaction at node A is T ∩ E(A)
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
42
ECLAT
O
For each item, store a list of transaction ids (tids)
Horizontal
Data Layout
TID
1
2
3
4
5
6
7
8
9
10
Vertical Data Layout
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
A
1
4
5
6
7
8
9
B
1
2
5
7
8
10
C
2
3
4
8
9
D
2
4
5
9
E
1
3
6
TID-list
Introduction to Data Mining
4/18/2004
43
ECLAT
O
Determine support of any k-itemset by intersecting tid-lists
of two of its (k-1) subsets.
A
1
4
5
6
7
8
9
O
∧
B
1
2
5
7
8
10
→
AB
1
5
7
8
3 traversal approaches:
– top-down, bottom-up and hybrid
O
O
Advantage: very fast support counting
Disadvantage: intermediate tid-lists may become too
large for memory
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
44
Rule Generation
O
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,
O
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
45
Rule Generation
O
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
46
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
ABC=>D
AC=>BD
B=>ACD
AB=>CD
A=>BCD
Pruned
Rules
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
47
Rule Generation for Apriori Algorithm
O
Candidate rule is generated by merging two rules
that share the same prefix
in the rule consequent
CD=>AB
O
join(CD=>AB,BD=>AC)
would produce the candidate
rule D => ABC
O
Prune rule D=>ABC if its
subset AD=>BC does not have
high confidence
© Tan,Steinbach, Kumar
Introduction to Data Mining
BD=>AC
D=>ABC
4/18/2004
48
Effect of Support Distribution
O
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
49
Effect of Support Distribution
O
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
O
Using a single minimum support threshold may
not be effective
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
50
Multiple Minimum Support
O
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
51
Multiple Minimum Support
Item
MS (I)
S u p (I)
A
0.10% 0.25%
B
0.20% 0.26%
C
0.30% 0.29%
D
0.50% 0.05%
E
© Tan,Steinbach, Kumar
3%
4.20%
AB
ABC
AC
ABD
AD
ABE
AE
AC D
BC
AC E
BD
AD E
BE
BC D
CD
BC E
CE
BD E
DE
CDE
A
B
C
D
E
Introduction to Data Mining
4/18/2004
52
Multiple Minimum Support
Item
MS (I)
Su p (I)
AB
ABC
AC
ABD
AD
ABE
AE
ACD
BC
ACE
BD
AD E
BE
BCD
CD
BCE
CE
BDE
DE
CDE
A
A
0.10% 0.25%
B
0.20% 0.26%
B
C
C
0.30% 0.29%
D
0.50% 0.05%
D
E
E
© Tan,Steinbach, Kumar
3%
4.20%
Introduction to Data Mining
4/18/2004
53
Multiple Minimum Support (Liu 1999)
O
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
O
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
54
Multiple Minimum Support (Liu 1999)
O
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
55
Pattern Evaluation
O
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
O
Interestingness measures can be used to
prune/rank the derived patterns
O
In the original formulation of association rules,
support & confidence are the only measures used
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
56
Application of Interestingness Measure
Knowledge
Interestingness
Measures
Patterns
Postprocessing
Preprocessed
Data
Prod
Prod
uct
Prod
uct
Prod
uct
Prod
uct
Prod
uct
Prod
uct
Prod
uct
Prod
uct
Prod
uct
uct
Featur
Featur
e
Featur
e
Featur
e
Featur
e
Featur
e
Featur
e
Featur
e
Featur
e
Featur
e
e
M ining
Selected
Data
Preprocessing
Data
Selection
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
57
Computing Interestingness Measure
O
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
X
© Tan,Steinbach, Kumar
support, confidence, lift, Gini,
J-measure, etc.
Introduction to Data Mining
4/18/2004
58
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
59
Statistical Independence
O
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) => Statistical independence
– P(S∧B) > P(S) × P(B) => Positively correlated
– P(S∧B) < P(S) × P(B) => Negatively correlated
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
60
Statistical-based Measures
O
Measures that take into account statistical
dependence
P (Y | X )
P(Y )
P( X , Y )
Interest =
P ( X ) P (Y )
PS = P( X , Y ) − P ( X ) P (Y )
Lift =
φ − coefficient =
© Tan,Steinbach, Kumar
P ( X , Y ) − P( X ) P(Y )
P( X )[1 − P ( X )]P (Y )[1 − P (Y )]
Introduction to Data Mining
4/18/2004
61
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)
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
62
Drawback of Lift & Interest
Y
Y
X
10
0
10
X
0
90
90
10
90
100
Lift =
0.1
= 10
(0.1)(0.1)
Y
Y
X
90
0
90
X
0
10
10
90
10
100
Lift =
0 .9
= 1.11
(0.9)(0.9)
Statistical independence:
If P(X,Y)=P(X)P(Y) => Lift = 1
© Tan,Steinbach, Kumar
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?
Introduction to Data Mining
4/18/2004
63
Properties of A Good Measure
O
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
65
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
66
Property under Variable Permutation
B
p
r
A
A
B
q
s
A
p
q
B
B
A
r
s
Does M(A,B) = M(B,A)?
Symmetric measures:
X
support, lift, collective strength, cosine, Jaccard, etc
Asymmetric measures:
X
confidence, conviction, Laplace, J-measure, etc
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
67
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
68
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)
(c)
Introduction to Data Mining
4/18/2004
69
Example: φ-Coefficient
O
φ-coefficient is analogous to correlation coefficient
for continuous variables
X
X
Y
Y
60
10
Y
70
X
20
10
X
10
60
70
30
70
100
10
20
30
70
30
100
0.6 − 0.7 × 0.7
0.7 × 0.3 × 0.7 × 0.3
= 0.5238
φ=
Y
30
0.2 − 0.3 × 0.3
0.7 × 0.3 × 0.7 × 0.3
= 0.5238
φ=
φ Coefficient is the same for both tables
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
70
Property under Null Addition
A
A
B
p
r
B
q
s
B
p
r
A
A
B
q
s+k
Invariant measures:
X
support, cosine, Jaccard, etc
Non-invariant measures:
X
correlation, Gini, mutual information, odds ratio, etc
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
71
Different Measures have Different Properties
Sym bol
Measure
Range
P1
P2
P3
O1
O2
O3
O3'
O4
Φ
λ
α
Q
Y
κ
M
J
G
s
c
L
V
I
IS
PS
F
AV
S
ζ
Correlation
-1 … 0 … 1
Yes
Yes
Yes
Yes
No
Yes
Yes
No
Lambda
0…1
Yes
No
No
Yes
No
No*
Yes
No
Odds ratio
0…1…∞
Yes*
Yes
Yes
Yes
Yes
Yes*
Yes
No
Yule's Q
-1 … 0 … 1
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
Yule's Y
-1 … 0 … 1
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
No
Cohen's
-1 … 0 … 1
Yes
Yes
Yes
Yes
No
No
Yes
Mutual Information
0…1
Yes
Yes
Yes
Yes
No
No*
Yes
No
J-Measure
0…1
Yes
No
No
No
No
No
No
No
Gini Index
0…1
Yes
No
No
No
No
No*
Yes
No
Support
0…1
No
Yes
No
Yes
No
No
No
No
Confidence
0…1
No
Yes
No
Yes
No
No
No
Yes
Laplace
0…1
No
Yes
No
Yes
No
No
No
No
Conviction
0.5 … 1 … ∞
No
Yes
No
Yes**
No
No
Yes
No
Interest
0…1…∞
Yes*
Yes
Yes
Yes
No
No
No
No
IS (cosine)
0 .. 1
No
Yes
Yes
Yes
No
No
No
Yes
Piatetsky-Shapiro's
-0.25 … 0 … 0.25
Yes
Yes
Yes
Yes
No
Yes
Yes
No
Certainty factor
-1 … 0 … 1
Yes
Yes
Yes
No
No
No
Yes
No
Added value
0.5 … 1 … 1
Yes
Yes
Yes
No
No
No
No
No
Collective strength
0…1…∞
No
Yes
Yes
Yes
No
Yes*
Yes
No
0 .. 1
No
Yes
Yes
Yes
No
No
No
Yes
Yes
Yes
No
No
No
4/18/2004
No
Jaccard
Klosgen's
K
© Tan,Steinbach, Kumar





1 
2
− 1  2 − 3 −
Yes
 K 0K
3
3  to Data
3 3 Mining
 Introduction
2
72
No
Support-based Pruning
O
Most of the association rule mining algorithms
use support measure to prune rules and itemsets
O
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
73
Effect of Support-based Pruning
All Itempairs
1000
900
800
700
600
500
400
300
200
100
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
0
1
0.
.1
.2
-0
-0
.3
.4
-0
-0
.5
.6
-0
-0
.7
.8
-0
-0
-1
-0
.9
0
Correlation
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
74
Effect of Support-based Pruning
Support < 0.01
1
7
0.
1
4
5
6
0.
3
0.
0.
0.
2
0
1
0.
.1
0.
0.
8
0.
9
Correlation
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
.2
-0
.4
.3
-0
-0
.6
.7
.5
-0
-0
-0
-0
-1
.9
-0
-0
1
.1
.3
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
-0
-0
-0
-0
-0
-0
-0
-0
-0
.2
0
.4
50
0
.5
100
50
.7
150
100
.6
200
150
-1
250
200
.9
250
.8
300
.8
Support < 0.03
300
Correlation
Support < 0.05
300
250
200
150
100
50
.2
.1
-0
.3
-0
-0
.5
.4
-0
.7
.6
-0
-0
-0
-1
.9
-0
.8
0
-0
Support-based pruning
eliminates mostly
negatively correlated
itemsets
Correlation
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
75
Effect of Support-based Pruning
O
Investigate how support-based pruning affects
other measures
O
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
76
Effect of Support-based Pruning
Without Support Pruning (All Pairs)
X
All P a irs (4 0 .1 4 % )
1
C onviction
Odds ratio
0.9
C ol S treng th
0.8
C orrelation
Interes t
0.7
PS
CF
0.6
J accard
Yule Y
R eliability
Kappa
0.5
0.4
Klos g en
Yule Q
0.3
C onfidence
Laplace
0.2
IS
0.1
Support
J accard
0
-1
Lambda
-0.8
-0.6
-0.4
-0.2
Gini
0
0.2
Corre la tion
0.4
0.6
0.8
1
J -meas ure
Mutual Info
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
X
Red cells indicate correlation between
the pair of measures > 0.85
X
40.14% pairs have correlation > 0.85
© Tan,Steinbach, Kumar
Scatter Plot between Correlation
& Jaccard Measure
Introduction to Data Mining
4/18/2004
77
Effect of Support-based Pruning
0.5% ≤ support ≤ 50%
X
0 .0 0 5 <= s u p p o rt <= 0 .5 0 0 (6 1 .4 5 % )
1
Interes t
C onviction
0.9
Odds ratio
C ol S treng th
0.8
Laplace
0.7
C onfidence
C orrelation
0.6
J accard
Klos g en
R eliability
PS
0.5
0.4
Yule Q
CF
0.3
Yule Y
Kappa
0.2
IS
0.1
J accard
Support
0
-1
Lambda
Gini
-0.8
-0.6
-0.4
-0.2
0
0.2
Corre la tion
0.4
0.6
0.8
J -meas ure
Mutual Info
1
X
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Scatter Plot between Correlation
& Jaccard Measure:
61.45% pairs have correlation > 0.85
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
78
1
Effect of Support-based Pruning
X
0.5% ≤ support ≤ 30%
0 .0 0 5 <= s u p p o rt <= 0 .3 0 0 (7 6 .4 2 % )
1
Support
Interes t
0.9
R eliability
C onviction
0.8
Yule Q
0.7
Odds ratio
C onfidence
0.6
J accard
CF
Yule Y
Kappa
0.5
0.4
C orrelation
C ol S treng th
0.3
IS
J accard
0.2
Laplace
PS
0.1
Klos g en
0
-0.4
Lambda
Mutual Info
-0.2
0
0.2
0.4
Corre la tion
0.6
0.8
Gini
J -meas ure
1
X
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Scatter Plot between Correlation
& Jaccard Measure
76.42% pairs have correlation > 0.85
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
79
Subjective Interestingness Measure
O
Objective measure:
– Rank patterns based on statistics computed from data
– e.g., 21 measures of association (support, confidence,
Laplace, Gini, mutual information, Jaccard, etc).
O
Subjective measure:
– Rank patterns according to user’s interpretation
‹
A pattern is subjectively interesting if it contradicts the
expectation of a user (Silberschatz & Tuzhilin)
‹
A pattern is subjectively interesting if it is actionable
(Silberschatz & Tuzhilin)
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
80
1
Interestingness via Unexpectedness
O
Need to model expectation of users (domain knowledge)
+
-
Pattern expected to be frequent
Pattern expected to be infrequent
Pattern found to be frequent
Pattern found to be infrequent
+ - +
O
Expected Patterns
Unexpected Patterns
Need to combine expectation of users with evidence from
data (i.e., extracted patterns)
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
81
Interestingness via Unexpectedness
O
Web Data (Cooley et al 2001)
– Domain knowledge in the form of site structure
– Given an itemset F = {X1, X2, …, Xk} (Xi : Web pages)
‹
L: number of links connecting the pages
‹
lfactor = L / (k × k-1)
‹
cfactor = 1 (if graph is connected), 0 (disconnected graph)
– Structure evidence = cfactor × lfactor
– Usage evidence =
P( X I X I ... I X )
P ( X ∪ X ∪ ... ∪ X )
1
1
2
2
k
k
– Use Dempster-Shafer theory to combine domain
knowledge and evidence from data
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
82