AMCS/CS 340: Data Mining
Association Rules
Xiangliang Zhang
King Abdullah University of Science and Technology
Outline: Mining Association Rules
• Motivation and Definition
• High computational complexity
• Frequent itemsets mining

Apriori algorithm – reduce the number of candidate

Frequent-Pattern tree (FP-tree)
• Rule Generation
• Rule Evaluation
• Mining rules with multiple minimum supports
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
2
The Market-Basket Model
• A large set of items
e.g., things sold in a supermarket
• A large set of baskets, each of which is a small
set of the items
e.g., each basket lists the things a customer buys once
• Goal: Find “interesting” connections between items
• Can be used to model any
many‐many relationship,
not just in the retail setting
TID
Items
1
Bread, Milk
2
3
4
5
Bread, Diaper, Beer, Eggs
Milk, Diaper, Beer, Coke
Bread, Milk, Diaper, Beer
Bread, Milk, Diaper, Coke
3
The Frequent Itemsets
• Simplest question: Find sets of items that
appear together “frequently” in baskets
TID
Items
1
Bread, Milk
2
3
4
5
Bread, Diaper, Beer, Eggs
Milk, Diaper, Beer, Coke
Bread, Milk, Diaper, Beer
Bread, Milk, Diaper, Coke
4
Definition: Frequent Itemset
• Itemset
A collection of one or more items
Example: {Milk, Bread, Diaper}
k-itemset
An itemset that contains k items
TID
Items
1
Bread, Milk
2
3
4
5
Bread, Diaper, Beer, Eggs
Milk, Diaper, Beer, Coke
Bread, Milk, Diaper, Beer
Bread, Milk, Diaper, Coke
• Support 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
5
Definition: Frequent Itemset
• Itemset
A collection of one or more items
Example: {Milk, Bread, Diaper}
k-itemset
An itemset that contains k items
TID
Items
1
Bread, Milk
2
3
4
5
Bread, Diaper, Beer, Eggs
Milk, Diaper, Beer, Coke
Bread, Milk, Diaper, Beer
Bread, Milk, Diaper, Coke
• Support 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
Example:
Set minsup = 0.5
The frequent 2-itemsests :
{Bread, Milk},
{Bread, Dipper}
{Milk, Diaper},
{Diaper, Beer}
6
Definition: Association Rule
Association Rule
– If-then rules, an implication
expression of the form X  Y,
where X and Y are itemsets
– Example:
{Milk, Diaper}  {Beer}
TID
Items
1
Bread, Milk
2
3
4
5
Bread, Diaper, Beer, Eggs
Milk, Diaper, Beer, Coke
Bread, Milk, Diaper, Beer
Bread, Milk, Diaper, Coke
Rule Evaluation Metrics
Example:
– Support (s)
of transactions that
contain both X and Y
{Milk, Diaper}  Beer
 Fraction
– Confidence (c)
 Measures
how often items in Y
appear in transactions that
contain X
s
c
 (Milk , Diaper, Beer )
|T|

2
 0.4
5
 (Milk, Diaper, Beer ) 2
  0.67
 (Milk , Diaper )
3
7
Application of Association Rule
1. Items = products;
Baskets = sets of products a customer bought;
many people buy beer and diapers together
 Run a sale on diapers; raise price of beer
2. Items = documents;
Baskets = documents containing a similar sentence;
 Items that appear together too often could represent
plagiarism
3. Items = words;
Baskets = Web pages;
 Co‐occurrence of relatively rare words may indicate an
interesting relationship
8
Outline: Mining Association Rules
• Motivation and Definition
• High computational complexity
• Frequent itemsets mining

Apriori algorithm – reduce the number of candidate

Frequent-Pattern tree (FP-tree)
• Rule Generation
• Rule Evaluation
• Mining rules with multiple minimum supports
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
9
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!
10
Mining Association Rules
TID
Items
1
Bread, Milk
2
3
4
5
Bread, Diaper, Beer, Eggs
Milk, Diaper, Beer, Coke
Bread, Milk, Diaper, Beer
Bread, Milk, Diaper, Coke
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
11
Mining Association Rules
Two-step approach:
1. Frequent Itemset Generation
– Generate all itemsets whose support  minsup
2. Rule Generation
– Generate high confidence rules from each frequent
itemset, where each rule is a binary partitioning of a
frequent itemset
Frequent itemset generation is still computationally
expensive
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
12
Computational Complexity
Given d unique items in all transactions:
• Total number of itemsets = 2d -1
• Total number of possible association rules:
 d  d  k  d  k 
R       

k 1
j

1
 j 
 k 
 3d  2d 1  1
d 1
If d=6, R = 602 rules
d (#items) can be 100K (Wal‐Mart)
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
13
Outline: Mining Association Rules
• Motivation and Definition
• High computational complexity
• Frequent itemsets mining

Apriori algorithm – reduce the number of candidate

Frequent-Pattern tree (FP-tree)
• Rule Generation
• Rule Evaluation
• Mining rules with multiple minimum supports
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
14
Reduce the number of candidates
Complete search of frequent items = 2d -1
Priory principle:
X , Y : X  Y
- If an itemset is frequent, then all of its subsets
must also be frequent
if Y is frequent, all X are frequent
- If an itemset is infrequent, then all of this
parent-sets must also be infrequent
if X is infrequent, all Y are infrequent
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
15
Illustrating Apriori Principle
Database D
TID
Items
1
Bread, Milk
2
Bread, Diaper, Beer, Eggs
3
4
Milk, Diaper, Beer, Coke
Bread, Milk, Diaper, Beer
5
Bread, Milk, Diaper, Coke
Minimum Support = 3/5
Items (1-itemsets)
Scan D
Item
Bread
Coke
Milk
Beer
Diaper
Eggs
Count
4
2
4
3
4
1
Item
Bread
Milk
Beer
Diaper
Eliminate
Count
4
4
3
4
Generate
Pairs (2-itemsets)
Itemset
Count
{Bread,Milk}
3
{Bread,Diaper}
3
{Milk,Diaper}
3
{Beer,Diaper}
3
Itemset
Count
3
Eliminate {Bread,Milk}
{Bread,Beer}
2
{Bread,Diaper}
3
{Milk,Beer}
2
{Milk,Diaper}
3
{Beer,Diaper}
3
Scan D
Generate
Itemset
{Bread,Milk}
{Bread,Beer}
{Bread,Diaper}
{Milk,Beer}
{Milk,Diaper}
{Beer,Diaper}
Triplets (3-itemsets)
Itemset
{Bread,Milk,Diaper}
{Bread,Diaper,Beer}
Milk
{Mile,Diaper,Beer}
Prune
Itemset
{Bread,Milk,Diaper}
Scan D
Itemset
Count
{Bread,Milk,Diaper} 23
Not a frequent 3-itemset
Apriori Algorithm
• Let k=1
• Generate frequent itemsets of length 1
• Repeat until no new frequent itemsets are identified
1. Generate length (k+1) candidate itemsets from length k
frequent itemsets
2. Prune candidate itemsets containing subsets of length
k that are infrequent
3. Count the support of each candidate by scanning the
DB
4. Eliminate candidates that are infrequent, leaving only
those that are frequent
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
17
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
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
18
Outline: Mining Association Rules
• Motivation and Definition
• High computational complexity
• Frequent itemsets mining

Apriori algorithm – reduce the number of candidate

Frequent-Pattern tree (FP-tree)
• Rule Generation
• Rule Evaluation
• Mining rules with multiple minimum supports
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
19
Mining Frequent Patterns Without
Candidate Generation
• Compress a large database into a compact, FrequentPattern tree (FP-tree) structure
- highly condensed, but complete for frequent pattern
mining
- avoid costly database scans
• Develop an efficient, FP-tree-based frequent pattern mining
method
- A divide-and-conquer methodology: decompose mining
tasks into smaller ones
- Avoid candidate generation: sub-database test only!
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
20
FP-tree construction
Header Table
Item frequency
a
b
c
d
e
8
7
6
5
3
Steps:
1. Scan DB once, find frequent 1-itemset (single item pattern)
2. Scan DB again, construct FP-tree
(One transaction  one path in the FP-tree)
21
FP-tree construction: pointers
Header table
Item
a
b
c
d
e
Pointer
Pointers are used to assist
frequent itemset generation
22
Benefits of the FP-tree Structure
Completeness:
• never breaks a long pattern of any transaction
• preserves complete information for frequent pattern mining
Compactness
• One transaction  one path in the FP-tree
• Paths may overlap:
the more the paths overlap, the more compression achieved
• Never be larger than the original database (if not count node-links
and counts)
• Best case: only one single tranche of nodes
• The size of a FP-tree depends on how the items are ordered
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
23
Different FP-tree by ordering items
Header Table
Item frequency
a
8
b
7
c
6
d
5
e
3
Header Table
Item frequency
e
3
d
5
c
6
b
7
a
8
24
Generating frequent itemset
FP-Growth Algorithm:
Bottom-up fashion to derive the frequent
itemsets ending with a particular item
Paths can be accessed rapidly using
the pointers associated to e
25
Generating frequent itemset
FP-Growth Algorithm:
Decompose the frequent itemset generation
problem into multiple sub-problems
Example: find frequent itemsets including e
How to solve the sub-problem?
Start from paths containing node e
Construct conditional tree for e
(if e is frequent)
Prefix Paths ending in {de}
conditional tree for {de}
Prefix Paths ending in {ce}
conditional tree for {ce}
Prefix Paths ending in {ae}
conditional tree for {ae}
27
Example: find frequent itemsets including e
How to solve the sub-problem?
1. Check e is frequent or not
2. Convert the prefix paths into
conditional FP-tree
1) Update support counts
2) Remove node e
3) Ignore infrequent node b
Count(e) =3 > Minsup=2
Frequent Items:
{e}
28
Example: find frequent itemsets including e
How to solve the sub-problem?
(prefix paths and conditional tree with {de})
1. Check d is frequent or not in the
prefix paths ending in {de}
Count(d,e) =2
(Minsup=2)
Frequent Items:
{e} {d,e}
29
Example: find frequent itemsets including e
How to solve the sub-problem?
(prefix paths and conditional tree with {de})
2. Convert the prefix paths (ending in {de})
into conditional FP-tree for {de}
1) Update support counts
2) Remove node d
3) Ignore infrequent node c
Frequent Items:
{e} {d,e}
30
Example: find frequent itemsets including e
How to solve the sub-problem?
(prefix paths and conditional tree with {de})
1. Check a is frequent or not in the
prefix paths ending in {ade}
Count(a) =2
(Minsup=2)
Frequent Items:
{e} {d,e} {a,d,e}
31
Example: find frequent itemsets including e
How to solve the sub-problem?
(prefix paths and conditional tree with {ce})
1. Check c is frequent or not in the
prefix paths ending in {e}
Count(c,e) =2
(Minsup=2)
Frequent Items:
{e} {d,e} {a,d,e} {c,e}
32
Example: find frequent itemsets including e
How to solve the sub-problem?
(prefix paths and conditional tree with {ce})
2. Convert the prefix paths (ending in {ce})
into conditional FP-tree for {ce}
1) Update support counts
2) Remove node c
Frequent Items:
{e} {d,e} {a,d,e} {c,e}
33
Example: find frequent itemsets including e
How to solve the sub-problem?
(prefix paths and conditional tree with {ce})
1. Check a is frequent or not in the
prefix paths ending in {ace}
Count(a) =1
(Minsup=2)
Frequent Items:
{e} {d,e} {a,d,e} {c,e}
34
Example: find frequent itemsets including e
How to solve the sub-problem?
(prefix paths and conditional tree with {ae})
1. Check a is frequent or not in the
prefix paths ending in {ae}
Count(a,e) =2
(Minsup=2)
Frequent Items:
{e} {d,e} {a,d,e} {c,e}
{a,e}
35
Mining Frequent Itemset using FP-tree
General idea (divide-and-conquer)
• Recursively grow frequent pattern path using the FP-tree
• At each step, a conditional FP-tree is constructed by updating the
frequency counts along the prefix paths and removing all infrequent
items
Properties
• Sub-problems are disjoint  No duplicate itemsets
• FP-growth is an order of magnitude faster Apriori algorithm (depends
on the compaction factor of data)
- No candidate generation, no candidate test
- Use compact data structure
- Eliminate repeated database scan
- Basic operation is counting and FP-tree building
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
36
Compacting the Output of Frequent Itemsets --Maximal vs Closed Itemsets
• Maximal Frequent itemsets: no immediate superset is
frequent
• Closed itemsets: no immediate superset has the same
count (> 0).
Stores not only frequent information, but exact counts.
Frequent
Itemsets
Closed
Frequent
Itemsets
Maximal
Frequent
Itemsets
51
Example: Maximal vs Closed Itemsets
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
52
Outline: Mining Association Rules
• Motivation and Definition
• High computational complexity
• Frequent itemsets mining

Apriori algorithm – reduce the number of candidate

Frequent-Pattern tree (FP-tree)
• Rule Generation
• Rule Evaluation
• Mining rules with multiple minimum supports
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
53
Rule Generation
• Given a frequent itemset L, find all non-empty
subsets f  L such that f  L – f satisfies the
minimum confidence requirement
If {A,B,C,D} is a frequent itemset, candidate rules:
ABC D,
A BCD,
AB CD,
BD AC,
ABD C,
B ACD,
AC  BD,
CD AB,
ACD B,
C ABD,
AD  BC,
BCD A,
D ABC
BC AD,
• If |L| = k, then there are 2k – 2 candidate
association rules (ignoring L   and   L)
54
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)
s(ABCD)
c(ABCD) = ------------s(ABC)
s(ABD)
c(ABD) =-----------s(AB)
only s(ABC) < s(AB), s(ABCD) < s(ABD) (support has monotone property)
 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)
Computing the confidence of a rule does not require additional scans of
data
55
Rule Generation for Apriori Algorithm
Apriori algorithm:
• level-wise approach for generating association rules
• each level: same number of items in rule consequent
• rules with t consequent items are used to generate rules with
t+1 consequent items
ABCD=>{ }
BCD=>A
CD=>AB
BD=>AC
D=>ABC
ACD=>B
BC=>AD
C=>ABD
ABD=>C
AD=>BC
B=>ACD
ABC=>D
AC=>BD
A=>BCD
AB=>CD
56
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
CD=>AB
BD=>AC
D=>ABC
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
57
Rule Generation for Apriori Algorithm
Low-confidence Rules
BCD=>A
CD=>AB
BD=>AC
D=>ABC
ABCD=>{ }
ACD=>B
BC=>AD
C=>ABD
Pruned Rules (all rules containing
item A as consequent)
ABD=>C
AD=>BC
B=>ACD
ABC=>D
AC=>BD
AB=>CD
A=>BCD
58
Outline: Mining Association Rules
• Motivation and Definition
• High computational complexity
• Frequent itemsets mining

Apriori algorithm – reduce the number of candidate

Frequent-Pattern tree (FP-tree)
• Rule Generation
• Rule Evaluation
• Mining rules with multiple minimum supports
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
59
Interesting rules ?
• Rules with high support and confidence may be useful, but
not “interesting”
The “then” part is a frequent action in “if-them” rules
Example:
{Milk,Beer}  {Diaper} (s=0.4, c=1.0)
Prob(Diaper) = 4/5 = 0.8
• high interest suggests a cause
that might be worth investigating
TID
Items
1
Bread, Milk
2
3
4
5
Bread, Diaper, Beer, Eggs
Milk, Diaper, Beer, Coke
Bread, Milk, Diaper, Beer
Bread, Milk, Diaper, Coke
60
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
Y
Y
X
f11
f10
f1+
X
f01
f00
fo+
f+0
|T|
f+1
f11:
f10:
f01:
f00:
the # of transactions containing both X and Y
the # of transactions containing only X, no Y
the # of transactions containing only Y, no X
the # of transactions containing no X, no Y
Used to define various measures
 support (f11/|T|), confidence(f11/f1+), lift, Gini, J-measure, etc
62
Drawback of Confidence
Coffee
Coffee
Tea
15
5
20
Tea
75
5
80
90
10
100
Association Rule: Tea  Coffee
Confidence= P(Coffee|Tea) = 15/20 = 0.75
but P(Coffee) = 90/100 = 0.9
 Although confidence is high, rule is misleading
 P(Coffee|Tea) =75/80= 0.9375
63
Example: Lift/Interest
Coffee
Coffee
Tea
15
5
20
Tea
75
5
80
90
10
100
Association Rule: Tea  Coffee
Confidence= P(Coffee|Tea) = 15/20 = 0.75
but P(Coffee) = 90/100 = 0.9
 Lift = 0.75/0.9= 0.8333 (< 1, therefore is negatively associated)
66
There are lots of
measures proposed
in the literature
Tan,Steinbach, Kumar
Introduction to Data Mining
Outline: Mining Association Rules
• Motivation and Definition
• High computational complexity
• Frequent itemsets mining

Apriori algorithm – reduce the number of candidate

Frequent-Pattern tree (FP-tree)
• Rule Generation
• Rule Evaluation
• Mining rules with multiple minimum supports
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
71
Threshold for rules ?
• 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,
jewelry)
 If minsup is set too low,
- it is computationally expensive
- the number of itemsets is very large
- extract spurious patterns (cross-support patterns having
weak correlations, e.g. milk (s=0.7) and caviar (s=0.0004) )
• Using a single minimum support threshold may not
be effective
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
72
Problems with the single minsup
• Single minsup: It assumes that all items in the
data are of the same nature and/or have similar
frequencies.
• Not true: In many applications, some items
appear very frequently in the data, while others
rarely appear.
E.g., in a supermarket, people buy food processor and
cooking pan much less frequently than they buy bread and
milk.
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
73
Multiple minsups model
• Each item i can have a minimum item support MIS(i)
• The minimum support of a rule R is expressed as
the lowest MIS value of the items that appear in the
rule.
i.e., a rule R: a1, a2, …, ak  ak+1, …, ar satisfies its minimum support if
its actual support is 
min(MIS(a1), MIS(a2), …, MIS(ar)).
• By providing different MIS values for different items,
the user effectively expresses different support
requirements for different rules.
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
74
Multiple Minimum Support
How to apply multiple minimum supports?
MS(i): minimum support for item i
e.g.: MIS(Milk)=5%,
MIS(Coke) = 3%,
MIS(Broccoli)=0.1%,
MIS(Salmon)=0.5%
MIS({Milk, Broccoli}) = min (MIS(Milk), MIS(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
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
75
Summary
• Association rule mining has been extensively
studied in the data mining community.
• There are many efficient algorithms and model
variations
• Other related work includes
-
Multi-level or generalized rule mining
Constrained rule mining
Incremental rule mining
Maximal frequent itemset mining
Numeric association rule mining
Rule interestingness and visualization
Parallel algorithms
…
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
79
Related Resources
• Tools, Softwares
 A set of software for Frequent pattern Mining: Apriori, Eclat,
FPgrowth, RElim, SaM etc. http://www.borgelt.net/fpm.html
 Frequent Itemset Mining Implementations Repository
http://fimi.cs.helsinki.fi/src/
 Arules: Mining association rules and frequent itemsets
http://cran.r-project.org/web/packages/arules/index.html
• Annotated Bibliography on Association Rule Mining
http://michael.hahsler.net/research/bib/association_rules/
• Apriori Demo in Silverlight, http://codeding.com/?article=13
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
80
References: Frequent-pattern Mining
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large databases. SIGMOD'93, 207-216, Washington, D.C.
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Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
What you should know
• What is the motivation of association rule mining?
• What are the basic steps for mining association
rules?
• How does Apriori algorithm work?
• What is the issue of Apriori algorithm? How to solve
it?
• How does Frequent-Pattern tree work?
• How to generate rules from frequent itemsets?
• How to mine rules with multiple minimum supports?
• How to evaluate the Rules?
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
83
Demo
1. Apriori on “census” data
http://www.borgelt.net/apriori.html
2. Apriori Demo in Silverlight,
http://codeding.com/?article=13
Xiangliang Zhang, KAUST AMCS/CS 340: Data Mining
84