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Data Mining
Association Analysis: Basic Concepts
and Algorithms

Lecture Notes for Chapter 6

Introduction to Data Mining

Tan, Steinbach, Kumar

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 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

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

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

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

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

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

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 (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

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

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

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

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

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

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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Diaper

Milk

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Beer

Diaper,

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Diaper

Milk

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Beer

Diaper,

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ABCDE

ABCABDABEACDACEADEBCDBCEBDECDE

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ABCDE

null

ABACADAEBCBDBECDCEDE

ABCDE

ABCABDABEACDACEADEBCDBCEBDECDE

ABCDABCEABDEACDEBCDE

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

Transactions

Hash Structure

Buckets

TIDA1A2A3A4A5A6A7A8A9A10B1B2B3B4B5B6B7B8B9B10C1C2C3C4C5C6C7C8C9C10

1111111111

00000000000000000000

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60000000000

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70000000000

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80000000000

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90000000000

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0000000000

100000000000

1111111111

0000000000

1100000000000000000000

1111111111

1200000000000000000000

1111111111

1300000000000000000000

1111111111

1400000000000000000000

1111111111

1500000000000000000000

1111111111

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}

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{}

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