Data Mining

profilesweety225
association_analysis.ppt

Data 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(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

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

Sheet2

Sheet3

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

.

.

.

.

.

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

Sheet2

Sheet3

3

3

2

0

3

1

3

2

1

3

2

K

K

÷

÷

ø

ö

ç

ç

è

æ

-

-

÷

÷

ø

ö

ç

ç

è

æ

-

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

Chart1

-1
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-0.1
0
0.1
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0.5
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0.7
0.8
0.9
1
Correlation
All Itempairs
13
95
172
282
359
503
648
716
830
867
908
915
881
759
602
532
370
259
194
84
11

Chart2

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1
Correlation
Support > 0.9
0
0
4
7
9
7
25
35
42
86
92
75
54
28
14
16
3
5
0
0
0

Chart3

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0
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1
Correlation
Support > 0.7
7
29
70
95
150
182
267
296
336
383
375
370
332
272
202
161
89
49
41
17
4

Chart4

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1
Correlation
Support > 0.5
13
92
164
265
327
428
542
572
640
665
664
678
645
540
406
355
237
139
118
51
5

Chart5

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1
Correlation
Support < 0.01
10
21
22
18
28
28
46
40
33
25
21
7
2
1
0
0
0
0
0
0
0

Chart6

-1
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1
Correlation
Support < 0.03
13
71
68
66
67
84
94
93
93
77
56
30
14
5
3
0
0
0
0
0
0

Chart7

-1
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1
Correlation
Support < 0.05
13
91
104
106
107
128
141
151
161
136
99
67
45
15
8
4
2
1
0
0
0

Sheet1

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

Sheet2

Sheet3

Effect of Support-based Pruning

Support-based pruning eliminates mostly negatively correlated itemsets

Chart1

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1
Correlation
All Itempairs
13
95
172
282
359
503
648
716
830
867
908
915
881
759
602
532
370
259
194
84
11

Chart2

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Correlation
Support > 0.9
0
0
4
7
9
7
25
35
42
86
92
75
54
28
14
16
3
5
0
0
0

Chart3

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Correlation
Support > 0.7
7
29
70
95
150
182
267
296
336
383
375
370
332
272
202
161
89
49
41
17
4

Chart4

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1
Correlation
Support > 0.5
13
92
164
265
327
428
542
572
640
665
664
678
645
540
406
355
237
139
118
51
5

Chart5

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Correlation
Support < 0.01
10
21
22
18
28
28
46
40
33
25
21
7
2
1
0
0
0
0
0
0
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Chart6

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Correlation
Support < 0.03
13
71
68
66
67
84
94
93
93
77
56
30
14
5
3
0
0
0
0
0
0

Chart7

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Correlation
Support < 0.05
13
91
104
106
107
128
141
151
161
136
99
67
45
15
8
4
2
1
0
0
0

Sheet1

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

Sheet2

Sheet3

Chart1

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1
Correlation
All Itempairs
13
95
172
282
359
503
648
716
830
867
908
915
881
759
602
532
370
259
194
84
11

Chart2

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Correlation
Support > 0.9
0
0
4
7
9
7
25
35
42
86
92
75
54
28
14
16
3
5
0
0
0

Chart3

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Correlation
Support > 0.7
7
29
70
95
150
182
267
296
336
383
375
370
332
272
202
161
89
49
41
17
4

Chart4

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Correlation
Support > 0.5
13
92
164
265
327
428
542
572
640
665
664
678
645
540
406
355
237
139
118
51
5

Chart5

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Correlation
Support < 0.01
10
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18
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40
33
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Chart6

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Correlation
Support < 0.03
13
71
68
66
67
84
94
93
93
77
56
30
14
5
3
0
0
0
0
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Chart7

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Correlation
Support < 0.05
13
91
104
106
107
128
141
151
161
136
99
67
45
15
8
4
2
1
0
0
0

Sheet1

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

Sheet2

Sheet3

Chart1

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1
Correlation
All Itempairs
13
95
172
282
359
503
648
716
830
867
908
915
881
759
602
532
370
259
194
84
11

Chart2

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Correlation
Support > 0.9
0
0
4
7
9
7
25
35
42
86
92
75
54
28
14
16
3
5
0
0
0

Chart3

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Correlation
Support > 0.7
7
29
70
95
150
182
267
296
336
383
375
370
332
272
202
161
89
49
41
17
4

Chart4

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Correlation
Support > 0.5
13
92
164
265
327
428
542
572
640
665
664
678
645
540
406
355
237
139
118
51
5

Chart5

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Support < 0.01
10
21
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18
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46
40
33
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Chart6

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Correlation
Support < 0.03
13
71
68
66
67
84
94
93
93
77
56
30
14
5
3
0
0
0
0
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Chart7

-1
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Correlation
Support < 0.05
13
91
104
106
107
128
141
151
161
136
99
67
45
15
8
4
2
1
0
0
0

Sheet1

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

Sheet2

Sheet3

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

}

Diaper

,

Milk

{

Þ

4

.

0

5

2

|

T

|

)

Beer

Diaper,

,

Milk

(

=

=

=

s

s

67

.

0

3

2

)

Diaper

,

Milk

(

)

Beer

Diaper,

Milk,

(

=

=

=

s

s

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

-

´

÷

ø

ö

ç

è

æ

=

+

-

=

-

=

å

å

d

d

d

k

k

d

j

j

k

d

k

d

R

)

(

)

(

)

(

:

,

Y

s

X

s

Y

X

Y

X

³

Þ

Í

"

null

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

0000000000

90000000000

1111111111

0000000000

100000000000

1111111111

0000000000

1100000000000000000000

1111111111

1200000000000000000000

1111111111

1300000000000000000000

1111111111

1400000000000000000000

1111111111

1500000000000000000000

1111111111

å

=

÷

ø

ö

ç

è

æ

´

=

10

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

(

1

.

0

=

=

Lift

11

.

1

)

9

.

0

)(

9

.

0

(

9

.

0

=

=

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

.

0

2

.

0

=

´

´

´

´

-

=

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