1. Association Rules Zbigniew W. Ras*,#) presented by
*) University of North Carolina, Charlotte
#) Warsaw University of Technology

2. Market Basket Analysis (MBA)
Customer buying habits by finding associations and correlations between the different items that customers place in their “shopping basket”
Milk, eggs, sugar, bread
Milk, eggs, cereal, bread
Eggs, sugar
Customer1
Customer2
Customer3

3. Market Basket Analysis
Given: a database of customer transactions, where each transaction is a set of items
Find groups of items which are frequently purchased together

4. Goal of MBA Extract information on purchasing behavior
Actionable information: can suggest
new store layouts
new product assortments
which products to put on promotion
MBA applicable whenever a customer purchases multiple things in proximity

5. Association Rules
Express how product/services relate to each other, and tend to group together
“if a customer purchases three-way calling, then will also purchase call-waiting”
Simple to understand
Actionable information: bundle three-way calling and call-waiting in a single package

6. Basic Concepts Transactions: Relational format Compact format
<Tid,item> <Tid,itemset>
<1, item1> <1, {item1,item2}>
<1, item2> <2, {item3}>
<2, item3>
Item: single element, Itemset: set of items
Support of an itemset I [denoted by sup(I)]: card(I)
Threshold for minimum support: 
Itemset I is Frequent if: sup(I)  .
Frequent Itemset represents set of items which are
positively correlated
itemset

7. Frequent Itemsets sup({dairy}) = 3 sup({fruit}) = 3
Customer 1
sup({dairy}) = 3
sup({fruit}) = 3
sup({dairy, fruit}) = 2
If  = 3, then
{dairy} and {fruit} are frequent while {dairy,fruit} is not.
Customer 2

8. Association Rules: AR(s,c)
{A,B} - partition of a set of items
r = [A  B]
Support of r: sup(r) = sup(AB)
Confidence of r: conf(r) = sup(AB)/sup(A)
Thresholds:
minimum support - s
minimum confidence – c
r  AS(s, c), if sup(r)  s and conf(r)  c

9. Association Rules - Example
Min. support – 2 [50%]
Min. confidence - 50%
For rule A  C:
sup(A  C) = 2
conf(A  C) = sup({A,C})/sup({A}) = 2/3
The Apriori principle:
Any subset of a frequent itemset must be frequent

10. The Apriori algorithm [Agrawal]
Fk : Set of frequent itemsets of size k
Ck : Set of candidate itemsets of size k
F1 := {frequent items}; k:=1;
while card(Fk)  1 do begin
Ck+1 := new candidates generated from Fk ;
for each transaction t in the database do
increment the count of all candidates in Ck+1 that
are contained in t ;
Fk+1 := candidates in Ck+1 with minimum support
k:= k+1
end
Answer := { Fk: k  1 & card(Fk)  1}

11. Apriori - Example a b c d c, d b, d b, c a, d a, c a, b a, b, d
a, c, d
a, b, c
a,b,c,d
{a,d} is not frequent, so the 3-itemsets {a,b,d}, {a,c,d} and the 4-itemset {a,b,c,d}, are not generated.

12. Algorithm Apriori: Illustration
The task of mining association rules is mainly to discover strong association rules (high confidence and strong support) in large databases.
Mining association rules is composed of two steps:
TID Items
A, B, C
A, C
A, D
B, E, F
1. discover the large items, i.e., the sets of itemsets that have
transaction support above a
predetermined minimum support s.
2. Use the large itemsets to generate
the association rules
{A}
{B}
{C}
{A,C}
Large support
items
MinSup = 2

13. Algorithm Apriori: IllustrationC1 F1
Database D
S = 2
Itemset Count
{A}
{B}
{C}
{D}
{E}
Itemset Count
{A}
{B}
{C}
{E}
TID Items
A, C, D
B, C, E
A, B, C, E
B, E 2
Scan D 3 3 1 3 C2 C2 F2
{A, B}
{A, C}
{A, E}
{B, C}
{B, E}
{C, E}
Itemset
{A,B}
{A,C}
{A,E}
{B,C}
{B,E}
{C,E}
Itemset Count
{A, C}
{B, C}
{B, E}
{C, E}
Itemset Count 1
Scan D 2 1 2 3 2 C3 C3 F3
Scan D
{B, C, E}
Itemset
{B, C, E}
{B, C, E}
Itemset Count
Itemset Count

14. Representative Association Rules
Definition 1.
Cover C of a rule X  Y is denoted by C(X  Y)
and defined as follows:
C(X  Y) = { [X  Z]  V : Z, V are disjoint subsets of Y}.
Definition 2.
Set RR(s, c) of Representative Association Rules
is defined as follows:
RR(s, c) =
{r  AR(s, c): ~(rl  AR(s, c)) [rl  r & r  C(rl)]}
s – threshold for minimum support
c – threshold for minimum confidence
Representative Rules (informal description):
[as short as possible]  [as long as possible]

15. Representative Association Rules
Transactions:
{A,B,C,D,E}
{A,B,C,D,E,F}
{A,B,C,D,E,H,I}
{A,B,E}
{B,C,D,E,H,I}
Find RR(2,80%)
Representative Rules
From (BCDEHI):
{H}  {B,C,D,E,I}
{I}  {B,C,D,E,H}
From (ABCDE):
{A,C}  {B,D,E}
{A,D}  {B,C,E}
Last set: (BCDEHI, 2)

16. Questions?
Thank You