Data Mining
sweety225Data Mining
Association Analysis: Basic Concepts
and Algorithms
Lecture Notes
Introduction to Data Mining
by
Tan, Steinbach, Kumar
© Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 *
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!
TID |
Items |
1 |
Bread, Milk |
2 |
Bread, Diaper, Beer, Eggs |
3 |
Milk, Diaper, Beer, Coke |
4 |
Bread, Milk, Diaper, Beer |
5 |
Bread, Milk, Diaper, Coke |
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
TID |
Items |
1 |
Bread, Milk |
2 |
Bread, Diaper, Beer, Eggs |
3 |
Milk, Diaper, Beer, Coke |
4 |
Bread, Milk, Diaper, Beer |
5 |
Bread, Milk, Diaper, Coke |
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 that
contain X
Example:
TID |
Items |
1 |
Bread, Milk |
2 |
Bread, Diaper, Beer, Eggs |
3 |
Milk, Diaper, Beer, Coke |
4 |
Bread, Milk, Diaper, Beer |
5 |
Bread, Milk, Diaper, Coke |
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!
Mining Association Rules
Example of Rules:
{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
TID |
Items |
1 |
Bread, Milk |
2 |
Bread, Diaper, Beer, Eggs |
3 |
Milk, Diaper, Beer, Coke |
4 |
Bread, Milk, Diaper, Beer |
5 |
Bread, Milk, Diaper, Coke |
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
Frequent Itemset Generation
Given d items, there are 2d possible candidate itemsets
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) => Expensive since M = 2d !!!
N�
w�
M�
List of Candidates�
Computational Complexity
- Given d unique items:
- Total number of itemsets = 2d
- Total number of possible association rules:
If d=6, R = 602 rules
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
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
Illustrating Apriori Principle
Found to be Infrequent
Pruned supersets
Illustrating Apriori Principle
Items (1-itemsets)
Pairs (2-itemsets)
(No need to generate
candidates involving Coke
or Eggs)
Triplets (3-itemsets)
Minimum Support = 3
If every subset is considered,
6C1 + 6C2 + 6C3 = 41
With support-based pruning,
6 + 6 + 1 = 13
Item
Count
Bread
4
Coke
2
Milk
4
Beer
3
Diaper
4
Eggs
1
Itemset
Count
{Bread,Milk}
3
{Bread,Beer}
2
{Bread,Diaper}
3
{Milk,Beer}
2
{Milk,Diaper}
3
{Beer,Diaper}
3
Itemset |
Count |
{Bread,Milk,Diaper} |
3 |
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
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
N�
k�
Buckets�
Hash Structure�
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)
2 3 4
5 6 7
1 4 5
1 3 6
1 2 4
4 5 7
1 2 5
4 5 8
1 5 9
3 4 5
3 5 6
3 5 7
6 8 9
3 6 7
3 6 8
1,4,7
2,5,8
3,6,9
Hash function
Association Rule Discovery: Hash tree
1,4,7
2,5,8
3,6,9
Hash Function
Candidate Hash Tree
Hash on 1, 4 or 7
1 5 9
1 4 5
1 3 6
3 4 5
3 6 7
3 6 8
3 5 6
3 5 7
6 8 9
2 3 4
5 6 7
1 2 4
4 5 7
1 2 5
4 5 8
Association Rule Discovery: Hash tree
1,4,7
2,5,8
3,6,9
Hash Function
Candidate Hash Tree
Hash on 2, 5 or 8
1 5 9
1 4 5
1 3 6
3 4 5
3 6 7
3 6 8
3 5 6
3 5 7
6 8 9
2 3 4
5 6 7
1 2 4
4 5 7
1 2 5
4 5 8
Association Rule Discovery: Hash tree
1,4,7
2,5,8
3,6,9
Hash Function
Candidate Hash Tree
Hash on 3, 6 or 9
1 5 9
1 4 5
1 3 6
3 4 5
3 6 7
3 6 8
3 5 6
3 5 7
6 8 9
2 3 4
5 6 7
1 2 4
4 5 7
1 2 5
4 5 8
Subset Operation
Given a transaction t, what are the possible subsets of size 3?
1 2 3 5 6�
Transaction, t�
2 3 5 6�
3 5 6�
2�
1�
5 6�
1 3�
3 5 6�
1 2�
6�
1 5�
5 6�
2 3�
6�
2 5�
5 6�
3�
1 2 3�1 2 5�1 2 6�
1 3 5�1 3 6�
1 5 6�
2 3 5�2 3 6�
2 5 6�
3 5 6�
Subsets of 3 items�
Level 1�
Level 2�
Level 3�
6�
3 5�
Subset Operation Using Hash Tree
transaction
1 4 5
3 6 7
3 6 8
3 5 6
3 5 7
6 8 9
2 3 4
5 6 7
1 2 4
4 5 7
1 2 5
4 5 8
1 5 9
1 3 6
3 4 5
1 2 3 5 6
1 +
2 3 5 6
3 5 6
2 +
5 6
3 +
2,5,8
3,6,9
Hash Function
1,4,7
Subset Operation Using Hash Tree
1 5 9
1 3 6
3 4 5
transaction
1 4 5
3 6 7
3 6 8
3 5 6
3 5 7
6 8 9
2 3 4
5 6 7
1 2 4
4 5 7
1 2 5
4 5 8
2,5,8
3,6,9
Hash Function
1,4,7
1 2 3 5 6
3 5 6
1 2 +
5 6
1 3 +
6
1 5 +
3 5 6
2 +
5 6
3 +
1 +
2 3 5 6
Subset Operation Using Hash Tree
1 5 9
1 3 6
3 4 5
transaction
Match transaction against 11 out of 15 candidates
1 4 5
3 6 7
3 6 8
3 5 6
3 5 7
6 8 9
2 3 4
5 6 7
1 2 4
4 5 7
1 2 5
4 5 8
2,5,8
3,6,9
Hash Function
1,4,7
1 2 3 5 6
3 5 6
1 2 +
5 6
1 3 +
6
1 5 +
3 5 6
2 +
5 6
3 +
1 +
2 3 5 6
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)
Compact Representation of Frequent Itemsets
- Some itemsets are redundant because they have identical support as their supersets
- Number of frequent itemsets
- Need a compact representation
Maximal Frequent Itemset
Border
Infrequent Itemsets
Maximal Itemsets
An itemset is maximal frequent if none of its immediate supersets is frequent
null�
AB�
AC�
AD�
AE�
BC�
BD�
BE�
CD�
CE�
DE�
ABC�
ABD�
ABE�
ACD�
ACE�
ADE�
BCD�
BCE�
BDE�
CDE�
A�
B�
C�
D�
E�
ABCD�
ABCE�
ABDE�
ACDE�
BCDE�
ABCDE�
Closed Itemset
- An itemset is closed if none of its immediate supersets has the same support as the itemset
Sheet1
TID | Items | |
1 | {A,B} | |
2 | {B,C,D} | |
3 | {A,B,C,D} | |
4 | {A,B,D} | |
5 | {A,B,C,D} |
Sheet2
Sheet3
Sheet1
Itemset | Support | |
{A} | 4 | |
{B} | 5 | |
{C} | 3 | |
{D} | 4 | |
{A,B} | 4 | |
{A,C} | 2 | |
{A,D} | 3 | |
{B,C} | 3 | |
{B,D} | 4 | |
{C,D} | 3 |
Sheet2
Sheet3
Sheet1
Itemset | Support | |
{A,B,C} | 2 | |
{A,B,D} | 3 | |
{A,C,D} | 2 | |
{B,C,D} | 3 | |
{A,B,C,D} | 2 |
Sheet2
Sheet3
Maximal vs Closed Itemsets
Transaction Ids
Not supported by any transactions
Sheet1
TID | Items |
1 | ABC |
2 | ABCD |
3 | BCE |
4 | ACDE |
5 | DE |
Maximal vs Closed Frequent Itemsets
Minimum support = 2
# Closed = 9
# Maximal = 4
Closed and maximal
Closed but not maximal
Maximal vs Closed Itemsets
Frequent Itemsets�
Closed Frequent Itemsets�
Maximal Frequent Itemsets�
Alternative Methods for Frequent Itemset Generation
- Traversal of Itemset Lattice
- General-to-specific vs Specific-to-general
....�
Frequent itemset border�
null�
{a1,a2,...,an}�
(a) General-to-specific�
....�
null�
{a1,a2,...,an}�
Frequent itemset border�
(b) Specific-to-general�
....�
Frequent itemset border�
null�
{a1,a2,...,an}�
(c) Bidirectional�
Alternative Methods for Frequent Itemset Generation
- Traversal of Itemset Lattice
- Equivalent Classes
null�
AB�
AC�
AD�
null�
BC�
BD�
AB�
CD�
AC�
AD�
ABC�
ABD�
BC�
ACD�
BD�
CD�
BCD�
A�
B�
C�
A�
B�
C�
D�
D�
ABC�
ABD�
ACD�
BCD�
ABCD�
(a) Prefix tree�
(b) Suffix tree�
ABCD�
Alternative Methods for Frequent Itemset Generation
- Traversal of Itemset Lattice
- Breadth-first vs Depth-first
(a) Breadth first�
(b) Depth first�
Alternative Methods for Frequent Itemset Generation
- Representation of Database
- horizontal vs vertical data layout
Horizontal Data Layout�
Vertical Data Layout�
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
FP-tree construction
null
A:1
B:1
null
A:1
B:1
B:1
C:1
D:1
After reading TID=1:
After reading TID=2:
Sheet1
TID | Items | |
1 | {A,B} | |
2 | {B,C,D} | |
3 | {A,C,D,E} | |
4 | {A,D,E} | |
5 | {A,B,C} | |
6 | {A,B,C,D} | |
7 | {B,C} | |
8 | {A,B,C} | |
9 | {A,B,D} | |
10 | {B,C,E} |
Sheet2
Sheet3
FP-Tree Construction
null
A:7
B:5
B:3
C:3
D:1
C:1
D:1
C:3
D:1
D:1
E:1
E:1
Pointers are used to assist frequent itemset generation
D:1
E:1
Transaction Database
Header table
Sheet1
TID | Items | |
1 | {A,B} | |
2 | {B,C,D} | |
3 | {A,C,D,E} | |
4 | {A,D,E} | |
5 | {A,B,C} | |
6 | {A,B,C,D} | |
7 | {B,C} | |
8 | {A,B,C} | |
9 | {A,B,D} | |
10 | {B,C,E} |
Sheet2
Sheet3
Sheet1
Item | Pointer | |
A | ||
B | ||
C | ||
D | ||
E |
Sheet2
Sheet3
FP-growth
null
A:7
B:5
B:1
C:1
D:1
C:1
D:1
C:3
D:1
D: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)}
Recursively apply FP-growth on P
Frequent Itemsets found (with sup > 1):
AD, BD, CD, ACD, BCD
D:1
Tree Projection
Set enumeration tree:
Possible Extension: E(A) = {B,C,D,E}
Possible Extension: E(ABC) = {D,E}
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
Projected Database
Original Database:
Projected Database for node A:
For each transaction T, projected transaction at node A is T E(A)
Sheet1
TID | Items | |
1 | {A,B} | |
2 | {B,C,D} | |
3 | {A,C,D,E} | |
4 | {A,D,E} | |
5 | {A,B,C} | |
6 | {A,B,C,D} | |
7 | {B,C} | |
8 | {A,B,C} | |
9 | {A,B,D} | |
10 | {B,C,E} |
Sheet2
Sheet3
Sheet1
TID | Items | |
1 | {B} | |
2 | {} | |
3 | {C,D,E} | |
4 | {D,E} | |
5 | {B,C} | |
6 | {B,C,D} | |
7 | {} | |
8 | {B,C} | |
9 | {B,D} | |
10 | {} |
Sheet2
Sheet3
ECLAT
- For each item, store a list of transaction ids (tids)
TID-list
ECLAT
- Determine support of any k-itemset by intersecting tid-lists of two of its (k-1) subsets.
- 3 traversal approaches:
- top-down, bottom-up and hybrid
- Advantage: very fast support counting
- Disadvantage: intermediate tid-lists may become too large for memory
Sheet1
TID | Items | A | B | C | D | E | |||
1 | A,B,E | 1 | 1 | 2 | 2 | 1 | |||
2 | B,C,D | 4 | 2 | 3 | 4 | 3 | |||
3 | C,E | 5 | 5 | 4 | 5 | 6 | |||
4 | A,C,D | 6 | 7 | 8 | 9 | ||||
5 | A,B,C,D | 7 | 8 | 9 | |||||
6 | A,E | 8 | 10 | ||||||
7 | A,B | 9 | |||||||
8 | A,B,C | ||||||||
9 | A,C,D | ||||||||
10 | B | A | B | ||||||
1 | 1 | ||||||||
4 | 2 | ||||||||
5 | 5 | ||||||||
6 | 7 | ||||||||
7 | 8 | ||||||||
8 | 10 | ||||||||
9 |
Sheet2
Sheet3
Sheet1
TID | Items | A | B | C | D | E | |||
1 | A,B,E | 1 | 1 | 2 | 2 | 1 | |||
2 | B,C,D | 4 | 2 | 3 | 4 | 3 | |||
3 | C,E | 5 | 5 | 4 | 5 | 6 | |||
4 | A,C,D | 6 | 7 | 8 | 9 | ||||
5 | A,B,C,D | 7 | 8 | 9 | |||||
6 | A,E | 8 | 10 | ||||||
7 | A,B | 9 | |||||||
8 | A,B,C | ||||||||
9 | A,C,D | ||||||||
10 | B | A | B | ||||||
1 | 1 | ||||||||
4 | 2 | ||||||||
5 | 5 | ||||||||
6 | 7 | ||||||||
7 | 8 | ||||||||
8 | 10 | ||||||||
9 |
Sheet2
Sheet3
Sheet1
TID | Items | A | B | C | D | E | |||
1 | A,B,E | 1 | 1 | 2 | 2 | 1 | |||
2 | B,C,D | 4 | 2 | 3 | 4 | 3 | |||
3 | C,E | 5 | 5 | 4 | 5 | 6 | |||
4 | A,C,D | 6 | 7 | 8 | 9 | ||||
5 | A,B,C,D | 7 | 8 | 9 | |||||
6 | A,E | 8 | 10 | ||||||
7 | A,B | 9 | |||||||
8 | A,B,C | ||||||||
9 | A,C,D | ||||||||
10 | B | A | B | AB | |||||
1 | 1 | 1 | |||||||
4 | 2 | 5 | |||||||
5 | 5 | 7 | |||||||
6 | 7 | 8 | |||||||
7 | 8 | ||||||||
8 | 10 | ||||||||
9 |
Sheet2
Sheet3
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 ABC
AB 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)
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
Rule Generation for Apriori Algorithm
Lattice of rules
Low Confidence Rule
Pruned Rules
ABCD=>{ }�
BC=>AD�
BD=>AC�
CD=>AB�
AD=>BC�
AC=>BD�
AB=>CD�
D=>ABC�
C=>ABD�
B=>ACD�
A=>BCD�
ACD=>B�
ABD=>C�
ABC=>D�
BCD=>A�
Rule Generation for Apriori Algorithm
- Candidate rule is generated by merging two rules that share the same prefix
in the rule consequent - join(CD=>AB,BD=>AC)
would produce the candidate
rule D => ABC - Prune rule D=>ABC if its
subset AD=>BC does not have
high confidence
Effect of Support Distribution
- Many real data sets have skewed support distribution
Support distribution of a retail data set
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
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
Multiple Minimum Support
Multiple Minimum Support
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
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.
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
Application of Interestingness Measure
Interestingness Measures
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
Used to define various measures
- support, confidence, lift, Gini,
J-measure, etc.
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
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
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(SB) = 420/1000 = 0.42
- P(S) P(B) = 0.6 0.7 = 0.42
- P(SB) = P(S) P(B) => Statistical independence
- P(SB) > P(S) P(B) => Positively correlated
- P(SB) < P(S) P(B) => Negatively correlated
Statistical-based Measures
- Measures that take into account statistical dependence
Example: Lift/Interest
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)
Coffee | Coffee | ||
Tea | 15 | 5 | 20 |
Tea | 75 | 5 | 80 |
90 | 10 | 100 |
Drawback of Lift & Interest
Statistical independence:
If P(X,Y)=P(X)P(Y) => Lift = 1
Y | Y | ||
X | 10 | 0 | 10 |
X | 0 | 90 | 90 |
10 | 90 | 100 |
Y | Y | ||
X | 90 | 0 | 90 |
X | 0 | 10 | 10 |
90 | 10 | 100 |
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 Apriori-style 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
Comparing Different Measures
10 examples of contingency tables:
Rankings of contingency tables using various measures:
Sheet1
Example | ||||||
E1 | 8123 | 83 | 424 | 1370 | 10000 | 1.1581671343 |
E2 | 8330 | 2 | 622 | 1046 | 10000 | 1.116800672 |
E3 | 9481 | 94 | 127 | 298 | 10000 | 1.030581565 |
E4 | 3954 | 3080 | 5 | 2961 | 10000 | 1.4198707063 |
E5 | 2886 | 1363 | 1320 | 4431 | 10000 | 1.6148802655 |
E6 | 1500 | 2000 | 500 | 6000 | 10000 | 2.1428571429 |
E7 | 4000 | 2000 | 1000 | 3000 | 10000 | 1.3333333333 |
E8 | 4000 | 2000 | 2000 | 2000 | 10000 | 1.1111111111 |
E9 | 1720 | 7121 | 5 | 1154 | 10000 | 1.127815235 |
E10 | 61 | 2483 | 4 | 7452 | 10000 | 3.6889211418 |
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Property under Variable Permutation
Does M(A,B) = M(B,A)?
Symmetric measures:
- support, lift, collective strength, cosine, Jaccard, etc
Asymmetric measures:
- confidence, conviction, Laplace, J-measure, etc
Property under Row/Column Scaling
Grade-Gender Example (Mosteller, 1968):
Mosteller:
Underlying association should be independent of
the relative number of male and female students
in the samples
2x
10x
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 |
Property under Inversion Operation
Transaction 1
Transaction N
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Example: -Coefficient
- -coefficient is analogous to correlation coefficient for continuous variables
Coefficient is the same for both tables
Y | Y | ||
X | 60 | 10 | 70 |
X | 10 | 20 | 30 |
70 | 30 | 100 |
Y | Y | ||
X | 20 | 10 | 30 |
X | 10 | 60 | 70 |
30 | 70 | 100 |
Property under Null Addition
Invariant measures:
- support, cosine, Jaccard, etc
Non-invariant measures:
- correlation, Gini, mutual information, odds ratio, etc
Different Measures have Different Properties
Sheet1
Symbol | Measure | Range | P1 | P2 | P3 | O1 | O2 | O3 | O3' | O4 |
F | Correlation | -1 … 0 … 1 | Yes | Yes | Yes | Yes | No | Yes | Yes | No |
l | Lambda | 0 … 1 | Yes | No | No | Yes | No | No* | Yes | No |
a | Odds ratio | Yes* | Yes | Yes | Yes | Yes | Yes* | Yes | No | |
Q | Yule's Q | -1 … 0 … 1 | Yes | Yes | Yes | Yes | Yes | Yes | Yes | No |
Y | Yule's Y | -1 … 0 … 1 | Yes | Yes | Yes | Yes | Yes | Yes | Yes | No |
k | Cohen's | -1 … 0 … 1 | Yes | Yes | Yes | Yes | No | No | Yes | No |
M | Mutual Information | 0 … 1 | Yes | Yes | Yes | Yes | No | No* | Yes | No |
J | J-Measure | 0 … 1 | Yes | No | No | No | No | No | No | No |
G | Gini Index | 0 … 1 | Yes | No | No | No | No | No* | Yes | No |
s | Support | 0 … 1 | No | Yes | No | Yes | No | No | No | No |
c | Confidence | 0 … 1 | No | Yes | No | Yes | No | No | No | Yes |
L | Laplace | 0 … 1 | No | Yes | No | Yes | No | No | No | No |
V | Conviction | No | Yes | No | Yes** | No | No | Yes | No | |
I | Interest | Yes* | Yes | Yes | Yes | No | No | No | No | |
IS | IS (cosine) | 0 .. 1 | No | Yes | Yes | Yes | No | No | No | Yes |
PS | Piatetsky-Shapiro's | -0.25 … 0 … 0.25 | Yes | Yes | Yes | Yes | No | Yes | Yes | No |
F | Certainty factor | -1 … 0 … 1 | Yes | Yes | Yes | No | No | No | Yes | No |
AV | Added value | 0.5 … 1 … 1 | Yes | Yes | Yes | No | No | No | No | No |
S | Collective strength | No | Yes | Yes | Yes | No | Yes* | Yes | No | |
z | Jaccard | 0 .. 1 | No | Yes | Yes | Yes | No | No | No | Yes |
K | Klosgen's | Yes | Yes | Yes | No | No | No | No | No | |
where: | ||||||||||
O1: Symmetry under variable permutation | ||||||||||
O2: Marginal invariance | ||||||||||
O3: Antisymmetry under row or column permutation | ||||||||||
O3': Inversion invariance | ||||||||||
O4: Null invariance | ||||||||||
Yes*: Yes if measure is normalized | ||||||||||
Yes**: Yes if measure is symmetrized by taking max(M(A,B),M(B,A)) | ||||||||||
No*: Symmetry under row or column permutation |
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MBD01E1AF4B.unknown
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
Effect of Support-based Pruning
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Correlation | All | > 0.9 | > 0.7 | > 0.5 | < 0.01 | < 0.03 | < 0.05 | |
-1 | 13 | 0 | 7 | 13 | 10 | 13 | 13 | |
-0.9 | 95 | 0 | 29 | 92 | 21 | 71 | 91 | |
-0.8 | 172 | 4 | 70 | 164 | 22 | 68 | 104 | |
-0.7 | 282 | 7 | 95 | 265 | 18 | 66 | 106 | |
-0.6 | 359 | 9 | 150 | 327 | 28 | 67 | 107 | |
-0.5 | 503 | 7 | 182 | 428 | 28 | 84 | 128 | |
-0.4 | 648 | 25 | 267 | 542 | 46 | 94 | 141 | |
-0.3 | 716 | 35 | 296 | 572 | 40 | 93 | 151 | |
-0.2 | 830 | 42 | 336 | 640 | 33 | 93 | 161 | |
-0.1 | 867 | 86 | 383 | 665 | 25 | 77 | 136 | |
0 | 908 | 92 | 375 | 664 | 21 | 56 | 99 | |
0.1 | 915 | 75 | 370 | 678 | 7 | 30 | 67 | |
0.2 | 881 | 54 | 332 | 645 | 2 | 14 | 45 | |
0.3 | 759 | 28 | 272 | 540 | 1 | 5 | 15 | |
0.4 | 602 | 14 | 202 | 406 | 0 | 3 | 8 | |
0.5 | 532 | 16 | 161 | 355 | 0 | 0 | 4 | |
0.6 | 370 | 3 | 89 | 237 | 0 | 0 | 2 | |
0.7 | 259 | 5 | 49 | 139 | 0 | 0 | 1 | |
0.8 | 194 | 0 | 41 | 118 | 0 | 0 | 0 | |
0.9 | 84 | 0 | 17 | 51 | 0 | 0 | 0 | |
1 | 11 | 0 | 4 | 5 | 0 | 0 | 0 |
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Effect of Support-based Pruning
Support-based pruning eliminates mostly negatively correlated itemsets
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Correlation | All | > 0.9 | > 0.7 | > 0.5 | < 0.01 | < 0.03 | < 0.05 | |
-1 | 13 | 0 | 7 | 13 | 10 | 13 | 13 | |
-0.9 | 95 | 0 | 29 | 92 | 21 | 71 | 91 | |
-0.8 | 172 | 4 | 70 | 164 | 22 | 68 | 104 | |
-0.7 | 282 | 7 | 95 | 265 | 18 | 66 | 106 | |
-0.6 | 359 | 9 | 150 | 327 | 28 | 67 | 107 | |
-0.5 | 503 | 7 | 182 | 428 | 28 | 84 | 128 | |
-0.4 | 648 | 25 | 267 | 542 | 46 | 94 | 141 | |
-0.3 | 716 | 35 | 296 | 572 | 40 | 93 | 151 | |
-0.2 | 830 | 42 | 336 | 640 | 33 | 93 | 161 | |
-0.1 | 867 | 86 | 383 | 665 | 25 | 77 | 136 | |
0 | 908 | 92 | 375 | 664 | 21 | 56 | 99 | |
0.1 | 915 | 75 | 370 | 678 | 7 | 30 | 67 | |
0.2 | 881 | 54 | 332 | 645 | 2 | 14 | 45 | |
0.3 | 759 | 28 | 272 | 540 | 1 | 5 | 15 | |
0.4 | 602 | 14 | 202 | 406 | 0 | 3 | 8 | |
0.5 | 532 | 16 | 161 | 355 | 0 | 0 | 4 | |
0.6 | 370 | 3 | 89 | 237 | 0 | 0 | 2 | |
0.7 | 259 | 5 | 49 | 139 | 0 | 0 | 1 | |
0.8 | 194 | 0 | 41 | 118 | 0 | 0 | 0 | |
0.9 | 84 | 0 | 17 | 51 | 0 | 0 | 0 | |
1 | 11 | 0 | 4 | 5 | 0 | 0 | 0 |
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Correlation | All | > 0.9 | > 0.7 | > 0.5 | < 0.01 | < 0.03 | < 0.05 | |
-1 | 13 | 0 | 7 | 13 | 10 | 13 | 13 | |
-0.9 | 95 | 0 | 29 | 92 | 21 | 71 | 91 | |
-0.8 | 172 | 4 | 70 | 164 | 22 | 68 | 104 | |
-0.7 | 282 | 7 | 95 | 265 | 18 | 66 | 106 | |
-0.6 | 359 | 9 | 150 | 327 | 28 | 67 | 107 | |
-0.5 | 503 | 7 | 182 | 428 | 28 | 84 | 128 | |
-0.4 | 648 | 25 | 267 | 542 | 46 | 94 | 141 | |
-0.3 | 716 | 35 | 296 | 572 | 40 | 93 | 151 | |
-0.2 | 830 | 42 | 336 | 640 | 33 | 93 | 161 | |
-0.1 | 867 | 86 | 383 | 665 | 25 | 77 | 136 | |
0 | 908 | 92 | 375 | 664 | 21 | 56 | 99 | |
0.1 | 915 | 75 | 370 | 678 | 7 | 30 | 67 | |
0.2 | 881 | 54 | 332 | 645 | 2 | 14 | 45 | |
0.3 | 759 | 28 | 272 | 540 | 1 | 5 | 15 | |
0.4 | 602 | 14 | 202 | 406 | 0 | 3 | 8 | |
0.5 | 532 | 16 | 161 | 355 | 0 | 0 | 4 | |
0.6 | 370 | 3 | 89 | 237 | 0 | 0 | 2 | |
0.7 | 259 | 5 | 49 | 139 | 0 | 0 | 1 | |
0.8 | 194 | 0 | 41 | 118 | 0 | 0 | 0 | |
0.9 | 84 | 0 | 17 | 51 | 0 | 0 | 0 | |
1 | 11 | 0 | 4 | 5 | 0 | 0 | 0 |
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Correlation | All | > 0.9 | > 0.7 | > 0.5 | < 0.01 | < 0.03 | < 0.05 | |
-1 | 13 | 0 | 7 | 13 | 10 | 13 | 13 | |
-0.9 | 95 | 0 | 29 | 92 | 21 | 71 | 91 | |
-0.8 | 172 | 4 | 70 | 164 | 22 | 68 | 104 | |
-0.7 | 282 | 7 | 95 | 265 | 18 | 66 | 106 | |
-0.6 | 359 | 9 | 150 | 327 | 28 | 67 | 107 | |
-0.5 | 503 | 7 | 182 | 428 | 28 | 84 | 128 | |
-0.4 | 648 | 25 | 267 | 542 | 46 | 94 | 141 | |
-0.3 | 716 | 35 | 296 | 572 | 40 | 93 | 151 | |
-0.2 | 830 | 42 | 336 | 640 | 33 | 93 | 161 | |
-0.1 | 867 | 86 | 383 | 665 | 25 | 77 | 136 | |
0 | 908 | 92 | 375 | 664 | 21 | 56 | 99 | |
0.1 | 915 | 75 | 370 | 678 | 7 | 30 | 67 | |
0.2 | 881 | 54 | 332 | 645 | 2 | 14 | 45 | |
0.3 | 759 | 28 | 272 | 540 | 1 | 5 | 15 | |
0.4 | 602 | 14 | 202 | 406 | 0 | 3 | 8 | |
0.5 | 532 | 16 | 161 | 355 | 0 | 0 | 4 | |
0.6 | 370 | 3 | 89 | 237 | 0 | 0 | 2 | |
0.7 | 259 | 5 | 49 | 139 | 0 | 0 | 1 | |
0.8 | 194 | 0 | 41 | 118 | 0 | 0 | 0 | |
0.9 | 84 | 0 | 17 | 51 | 0 | 0 | 0 | |
1 | 11 | 0 | 4 | 5 | 0 | 0 | 0 |
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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
Effect of Support-based Pruning
- Without Support Pruning (All Pairs)
- Red cells indicate correlation between
the pair of measures > 0.85 - 40.14% pairs have correlation > 0.85
Scatter Plot between Correlation & Jaccard Measure
Effect of Support-based Pruning
- 0.5% support 50%
- 61.45% pairs have correlation > 0.85
Scatter Plot between Correlation & Jaccard Measure:
Effect of Support-based Pruning
- 0.5% support 30%
- 76.42% pairs have correlation > 0.85
Scatter Plot between Correlation & Jaccard Measure
Subjective Interestingness Measure
- Objective measure:
- Rank patterns based on statistics computed from data
- e.g., 21 measures of association (support, confidence, Laplace, Gini, mutual information, Jaccard, etc).
- 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)
Interestingness via Unexpectedness
- Need to model expectation of users (domain knowledge)
- Need to combine expectation of users with evidence from data (i.e., extracted patterns)
+
Pattern expected to be frequent
-
Pattern expected to be infrequent
Pattern found to be frequent
Pattern found to be infrequent
+
-
Expected Patterns
-
+
Unexpected Patterns
Interestingness via Unexpectedness
- 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
- Use Dempster-Shafer theory to combine domain knowledge and evidence from data
TID Items
1 Bread, Milk
2 Bread, Diaper, Beer, Eggs
3 Milk, Diaper, Beer, Coke
4 Bread, Milk, Diaper, Beer
5 Bread, Milk, Diaper, Coke
TID Items
1 Bread, Milk
2 Bread, Diaper, Beer, Eggs
3 Milk, Diaper, Beer, Coke
4 Bread, Milk, Diaper, Beer
5 Bread, Milk, Diaper, Coke
Beer
}
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(
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c
null
AB
AC
AD
AE
BC
BD
BE
CD
CE
DE
A
B
C
D
E
ABC
ABD
ABE
ACD
ACE
ADE
BCD
BCE
BDE
CDE
ABCD
ABCE
ABDE
ACDE
BCDE
ABCDE
TID Items
1 Bread, Milk
2 Bread, Diaper, Beer, Eggs
3 Milk, Diaper, Beer, Coke
4 Bread, Milk, Diaper, Beer
5 Bread, Milk, Diaper, Coke
N
Transactions
List of
Candidates
M
w
1
2
3
1
1
1
1
+
-
=
ú
û
ù
ê
ë
é
÷
ø
ö
ç
è
æ
-
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ø
ö
ç
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-
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å
d
d
d
k
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ABACADAEBCBDBECDCEDE
ABCDE
ABCABDABEACDACEADEBCDBCEBDECDE
ABCDABCEABDEACDEBCDE
ABCDE
null
ABACADAEBCBDBECDCEDE
ABCDE
ABCABDABEACDACEADEBCDBCEBDECDE
ABCDABCEABDEACDEBCDE
ABCDE
Item
Count
Bread
4
Coke
2
Milk
4
Beer
3
Diaper
4
Eggs
1
Itemset
Count
{
Bread,Milk}
3
{
Bread,Beer}
2
{
Bread,Diaper}
3
{
Milk,Beer}
2
{
Milk,Diaper}
3
{
Beer,Diaper}
3
Itemset
Count
{Bread,Milk,Diaper}
3
TID Items
1 Bread, Milk
2 Bread, Diaper, Beer, Eggs
3 Milk, Diaper, Beer, Coke
4 Bread, Milk, Diaper, Beer
5 Bread, Milk, Diaper, Coke
N
Transactions
Hash Structure
k
Buckets
1 2 3 5 6
Transaction, t
2 3 5 613 5 62
5 61 33 5 61 261 55 62 362 5
5 63
1 2 3
1 2 5
1 2 6
1 3 5
1 3 6
1 5 6
2 3 5
2 3 6
2 5 63 5 6
Subsets of 3 items
Level 1
Level 2
Level 3
63 5
TIDA1A2A3A4A5A6A7A8A9A10B1B2B3B4B5B6B7B8B9B10C1C2C3C4C5C6C7C8C9C10
1
1111111111
00000000000000000000
2
1111111111
00000000000000000000
3
1111111111
00000000000000000000
4
1111111111
00000000000000000000
5
1111111111
00000000000000000000
60000000000
1111111111
0000000000
70000000000
1111111111
0000000000
80000000000
1111111111
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90000000000
1111111111
0000000000
100000000000
1111111111
0000000000
1100000000000000000000
1111111111
1200000000000000000000
1111111111
1300000000000000000000
1111111111
1400000000000000000000
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1500000000000000000000
1111111111
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1
10
3
k
k
null
ABACADAEBCBDBECDCEDE
ABCDE
ABCABDABEACDACEADEBCDBCEBDECDE
ABCDABCEABDEACDEBCDE
ABCD
E
TIDItems
1{A,B}
2{B,C,D}
3{A,B,C,D}
4{A,B,D}
5{A,B,C,D}
ItemsetSupport
{A}4
{B}5
{C}3
{D}4
{A,B}4
{A,C}2
{A,D}3
{B,C}3
{B,D}4
{C,D}3
ItemsetSupport
{A,B,C}2
{A,B,D}3
{A,C,D}2
{B,C,D}3
{A,B,C,D}2
TIDItems
1ABC
2ABCD
3BCE
4ACDE
5DE
null
AB
AC
AD
AE
BC
BD
BE
CD
CE
DE
A
B
C
D
E
ABC
ABD
ABE
ACD
ACE
ADE
BCD
BCE
BDE
CDE
ABCD
ABCE
ABDE
ACDE
BCDE
ABCDE
124
123
1234
245
345
12
124
24
4
123
2
3
24
34
45
12
2
24
4
4
2
3
4
2
4
null
AB
AC
AD
AE
BC
BD
BE
CD
CE
DE
A
B
C
D
E
ABC
ABD
ABE
ACD
ACE
ADE
BCD
BCE
BDE
CDE
ABCD
ABCE
ABDE
ACDE
BCDE
ABCDE
124
123
1234
245
345
12
124
24
4
123
2
3
24
34
45
12
2
24
4
4
2
3
4
2
4
Frequent
Itemsets
Closed
Frequent
Itemsets
Maximal
Frequent
Itemsets
Frequent
itemset
border
null
{a
1
,a
2
,...,a
n
}
(a) General-to-specific
null
{a
1
,a
2
,...,a
n
}
Frequent
itemset
border
(b) Specific-to-general
..
..
..
..
Frequent
itemset
border
null
{a
1
,a
2
,...,a
n
}
(c) Bidirectional
..
..
null
ABACADBCBD
CD
AB
C
D
ABCABDACD
BCD
ABCD
null
ABAC
ADBCBDCD
AB
C
D
ABC
ABDACDBCD
ABCD
(a) Prefix tree(b) Suffix tree
(a) Breadth first(b) Depth first
TIDItems
1A,B,E
2B,C,D
3C,E
4A,C,D
5A,B,C,D
6A,E
7A,B
8A,B,C
9A,C,D
10B
Horizontal
Data Layout
ABCDE
11221
42343
55456
6789
789
810
9
Vertical Data Layout
TIDItems
1{A,B}
2{B,C,D}
3{A,C,D,E}
4{A,D,E}
5{A,B,C}
6{A,B,C,D}
7{B,C}
8{A,B,C}
9{A,B,D}
10{B,C,E}
ItemPointer
A
B
C
D
E
null
AB
AC
AD
AE
BC
BD
BE
CD
CE
DE
A
B
C
D
E
ABC
ABD
ABE
ACD
ACE
ADE
BCD
BCE
BDE
CDE
ABCD
ABCE
ABDE
ACDE
BCDE
ABCDE
TIDItems
1{B}
2{}
3{C,D,E}
4{D,E}
5{B,C}
6{B,C,D}
7{}
8{B,C}
9{B,D}
10{}
TID
Items
1
A,B,E
2
B,C,D
3
C,E
4
A,C,D
5
A,B,C,D
6
A,E
7
A,B
8
A,B,C
9
A,C,D
10
B
Horizontal
Data Layout
A
B
C
D
E
1
1
2
2
1
4
2
3
4
3
5
5
4
5
6
6
7
8
9
7
8
9
8
10
9
Vertical Data Layout
A
1
4
5
6
7
8
9
B
1
2
5
7
8
10
AB
1
5
7
8
ABCD=>{ }
BCD=>AACD=>BABD=>CABC=>D
BC=>ADBD=>ACCD=>ABAD=>BCAC=>BDAB=>CD
D=>ABCC=>ABDB=>ACDA=>BCD
ABCD=>{ }
BCD=>AACD=>BABD=>CABC=>D
BC=>ADBD=>ACCD=>ABAD=>BCAC=>BDAB=>CD
D=>ABCC=>ABDB=>ACDA=>BCD
BD=>
AC
CD=>
AB
D=>
ABC
A
Item
MS(I)
Sup(I)
A
0.10%
0.25%
B
0.20%
0.26%
C
0.30%
0.29%
D
0.50%
0.05%
E
3%
4.20%
B
C
D
E
AB
AC
AD
AE
BC
BD
BE
CD
CE
DE
ABC
ABD
ABE
ACD
ACE
ADE
BCD
BCE
BDE
CDE
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
Item
MS(I)
Sup(I)
A
0.10%
0.25%
B
0.20%
0.26%
C
0.30%
0.29%
D
0.50%
0.05%
E
3%
4.20%
Featur
e
Prod
uct
Prod
uct
Prod
uct
Prod
uct
Prod
uct
Prod
uct
Prod
uct
Prod
uct
Prod
uct
Prod
uct
Featur
e
Featur
e
Featur
e
Featur
e
Featur
e
Featur
e
Featur
e
Featur
e
Featur
e
Selection
Preprocessing
Mining
Postprocessing
Data
Selected
Data
Preprocessed
Data
Patterns
Knowledge
)]
(
1
)[
(
)]
(
1
)[
(
)
(
)
(
)
,
(
)
(
)
(
)
,
(
)
(
)
(
)
,
(
)
(
)
|
(
Y
P
Y
P
X
P
X
P
Y
P
X
P
Y
X
P
t
coefficien
Y
P
X
P
Y
X
P
PS
Y
P
X
P
Y
X
P
Interest
Y
P
X
Y
P
Lift
-
-
-
=
-
-
=
=
=
f
10
)
1
.
0
)(
1
.
0
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=
Lift
11
.
1
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9
.
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.
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9
.
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Lift
Example
f
11
f
10
f
01
f
00
E18123834241370
E2833026221046
E3948194127298
E43954308052961
E52886136313204431
E6150020005006000
E74000200010003000
E84000200020002000
E91720712151154
E1061248347452
B
B
A
p
q
A
r
s
A
A
B
p
r
B
q
s
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
A
B
C
D
(a)
(b)
0
1
1
1
1
1
1
1
1
0
0
0
0
0
1
0
0
0
0
0
(c)
E
F
5238
.
0
3
.
0
7
.
0
3
.
0
7
.
0
7
.
0
7
.
0
6
.
0
=
´
´
´
´
-
=
f
5238
.
0
3
.
0
7
.
0
3
.
0
7
.
0
3
.
0
3
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0
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=
f
B
B
A
p
q
A
r
s
B
B
A
p
q
A
r
s + k
SymbolMeasureRangeP1P2P3O1O2O3O3'O4
Correlation-1 … 0 … 1YesYesYesYesNoYesYesNo
Lambda0 … 1YesNoNoYesNoNo*YesNo
Odds ratio0 … 1 … Yes*YesYesYesYesYes*YesNo
Q
Yule's Q-1 … 0 … 1YesYesYesYesYesYesYesNo
Y
Yule's Y-1 … 0 … 1YesYesYesYesYesYesYesNo
Cohen's-1 … 0 … 1YesYesYesYesNoNoYesNo
M
Mutual Information0 … 1YesYesYesYesNoNo*YesNo
J
J-Measure0 … 1YesNoNoNoNoNoNoNo
G
Gini Index0 … 1YesNoNoNoNoNo*YesNo
s
Support0 … 1NoYesNoYesNoNoNoNo
c
Confidence0 … 1NoYesNoYesNoNoNoYes
L
Laplace0 … 1NoYesNoYesNoNoNoNo
V
Conviction0.5 … 1 … NoYesNoYes**NoNoYesNo
I
Interest0 … 1 … Yes*YesYesYesNoNoNoNo
IS
IS (cosine)0 .. 1NoYesYesYesNoNoNoYes
PS
Piatetsky-Shapiro's-0.25 … 0 … 0.25YesYesYesYesNoYesYesNo
F
Certainty factor-1 … 0 … 1YesYesYesNoNoNoYesNo
AV
Added value0.5 … 1 … 1YesYesYesNoNoNoNoNo
S
Collective strength0 … 1 … NoYesYesYesNoYes*YesNo
Jaccard0 .. 1NoYesYesYesNoNoNoYes
K
Klosgen'sYesYesYesNoNoNoNoNo
33
2
0
3
1
321
3
2
All Itempairs
0
100
200
300
400
500
600
700
800
900
1000
-1
-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1
0
0.10.20.30.40.50.60.70.80.9
1
Correlation
Support < 0.01
0
50
100
150
200
250
300
-1
-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1
0
0.10.20.30.40.50.60.70.80.9
1
Correlation
Support < 0.03
0
50
100
150
200
250
300
-1
-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1
0
0.10.20.30.40.50.60.70.80.9
1
Correlation
Support < 0.05
0
50
100
150
200
250
300
-1
-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1
0
0.10.20.30.40.50.60.70.80.9
1
Correlation
All Pairs (40.14%)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
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
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Correlation
Jaccard
0.005 <= support <= 0.500 (61.45%)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
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
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Correlation
Jaccard
0.005 <= support <= 0.300 (76.42%)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
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
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Correlation
Jaccard
)
...
(
)
...
(
2
1
2
1
k
k
X
X
X
P
X
X
X
P
È
È
È
=
I
I
I