UC Become a Millionaire Game Calculations & Monetary Value Analysis Play now?Play later? You can become a millionaire!That’s what the junk mail said.B

UC Become a Millionaire Game Calculations & Monetary Value Analysis Play now?Play later? You can become a millionaire!That’s what the junk mail said.But then there was the fine print: If you send in your entry before midnight tonight, then here are your chances: 0.1% that you win $1,000,000 75% that you win nothing Otherwise, you must PAY $1,000 But wait, there’s more!If you don’t win the million AND you don’t have to pay on your first attempt, then you can choose to play one more time.If you choose to play again, then here are your chances: 2% that you win $100,000 20% that you win $500 Otherwise, you must PAY $2,000 What is your expected outcome for attempting this venture?Solve this problem using a decision tree and clearly show all calculations and the expected monetary value at each node. Use maximization of expected value as your decision criterion. Answer these questions: 1) Should you play at all? (5%)If you play, what is your expected (net) monetary value? (15%) 2) If you play and don’t win at all on the first try (but don’t lose money), should you try again? (5%) Why? (10%) 3) Clearly show the decision tree (40%) and expected net monetary value at each node (25%) 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},
Implication means co-occurrence,
not causality!
4/18/2004
‹#›
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
4/18/2004
‹#›
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
– 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
© 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
4/18/2004
‹#›
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
Itemset
{Bread,Milk,Diaper}
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
‹#›
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
‹#›
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
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)
© 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
10
k =1
Need a compact representation
© 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
‹#›
Maximal vs Closed Itemsets
TID
Items
1
ABC
2
ABCD
3
BCE
4
ACDE
5
DE
124
123
A
12
124
AB
12
24
AC
ABE
2
245
C
123
4
AE
24
ABD
1234
B
AD
2
ABC
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
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
Traversal of Itemset Lattice
– General-to-specific vs Specific-to-general
Frequent
itemset
border
null
null
..
..
..
..
{a1,a2,…,an}
{a1,a2,…,an}
(a) General-to-specific
© Tan,Steinbach, Kumar
Frequent
itemset
border
..
..
Frequent
itemset
border
(b) Specific-to-general
Introduction to Data Mining
null
{a1,a2,…,an}
(c) Bidirectional
4/18/2004
‹#›
Alternative Methods for Frequent Itemset Generation
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
‹#›
Alternative Methods for Frequent Itemset Generation
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
‹#›
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
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
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
‹#›
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
Frequent Itemsets found
(with sup > 1):
AD, BD, CD, ACD, BCD
D:1
D:1
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
‹#›
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
‹#›
Tree Projection
Items are listed in lexicographic order
Each node P stores the following information:
–
–
–
–
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
‹#›
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
‹#›
ECLAT
For each item, store a list of transaction ids (tids)
Horizontal
Data Layout
TID
1
2
3
4
5
6
7
8
9
10
© Tan,Steinbach, Kumar
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
Vertical Data Layout
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
‹#›
ECLAT
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

B
1
2
5
7
8
10
→
AB
1
5
7
8
3 traversal approaches:
– top-down, bottom-up and hybrid
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
‹#›
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…
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