Association Rules

Association Rules presented by Zbigniew W. Ras*,#) *) University of North Carolina – Charlotte #) ICS, Polish Academy of Sciences

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

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

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

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

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 • Supportof 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

Frequent Itemsets 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

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

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

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}

a,b,c,d a, b, c a, b, d a, c, d b, c, d a, b a, c a, d b, c b, d c, d a b c d Apriori - Example {a,d}is not frequent, so the 3-itemsets{a,b,d},{a,c,d}and the4-itemset {a,b,c,d},are not generated.

Algorithm Apriori: Illustration Large support items {A} 3 {B} 2 {C} 2 {A,C} 2 • 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 1000 A, B, C 2000 A, C 3000 A, D 4000 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 MinSup = 2

Algorithm Apriori: Illustration C1 F1 Database D S = 2 Itemset Count {A} {B} {C} {D} {E} Itemset Count {A} 2 {B} 3 {C} 3 {E} 3 TID Items 100 A, C, D 200 B, C, E 300 A, B, C, E 400 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} 2 {B, C} 2 {B, E} 3 {C, E} 2 Itemset Count 1 Scan D 2 1 2 3 2 C3 C3 F3 Scan D {B, C, E} Itemset {B, C, E} 2 {B, C, E} 2 Itemset Count Itemset Count

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]

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)

Beyond Support and Confidence • Example 1: (Aggarwal & Yu) • {tea} => {coffee} has high support (20%) and confidence (80%) • However, a priori probability that a customer buys coffee is 90% • A customer who is known to buy tea is less likely to buy coffee (by 10%) • There is a negative correlation between buying tea and buying coffee • {~tea} => {coffee} has higher confidence (93%)

Correlation and Interest • Two events are independent if P(A  B) = P(A)*P(B), otherwise are correlated. • Interest = P(A  B)/P(B)*P(A) • Interest expresses measure of correlation. If: • equal to 1 A and B are independent events • less than 1 A and B negatively correlated, • greater than 1 A and B positively correlated. • In our example, I(drink tea  drink coffee ) = 0.89 i.e. they are negatively correlated.

Questions? Thank You

Association Rules