Mesures en analyse de paniers d'achats

Analyse des paniers d'achats en R

Christopher Bruffaerts

Statistician

Mesures utilisées pour extraire des règles

TID	Transaction
1	{Bread, Butter, Cheese, Wine}
2	{Bread, Butter, Wine}
3	{Bread, Butter}
4	{Butter, Cheese, Wine}
5	{Butter, Cheese}
6	{Cheese, Wine}
7	{Butter, Wine}

Objectif : extraire des règles d'association

Exemples :

{Bread} $\rightarrow$ {Butter}
- Bread = « antécédent »
- Butter = « conséquent »
{Butter, Cheese} $\rightarrow$ {Wine}

Mesures : support, confiance, lift, …

Mesure de support

TID	Transaction
1	{Bread, Butter, Cheese, Wine}
2	{Bread, Butter, Wine}
3	{Bread, Butter}
4	{Butter, Cheese, Wine}
5	{Butter, Cheese}
6	{Cheese, Wine}
7	{Butter, Wine}

Support : « popularité d'un itemset »

supp(X) = fraction des transactions qui contiennent l'itemset X.
supp(X $\cup$ Y) = fraction des transactions contenant X et Y.

Exemples :

supp({Bread}) = 3/7 = 42 %
supp({Bread} $\cup$ {Butter}) = 3/7 = 42 %

Mesure de confiance

TID	Transaction
1	{Bread, Butter, Cheese, Wine}
2	{Bread, Butter, Wine}
3	{Bread, Butter}
4	{Butter, Cheese, Wine}
5	{Butter, Cheese}
6	{Cheese, Wine}
7	{Butter, Wine}

Confiance : « fréquence à laquelle la règle est vraie »

conf(X $\rightarrow$ Y) = supp(X $\cup$ Y) / supp(X)

La confiance indique le pourcentage de cas où Y est acheté avec X.

Exemple :

X = {Bread}

Y = {Butter}

conf(X $\rightarrow$ Y) = $\frac{3/7}{3/7}$ = 100 %

Mesure de lift

TID	Transaction
1	{Bread, Butter, Cheese, Wine}
2	{Bread, Butter, Wine}
3	{Bread, Butter}
4	{Butter, Cheese, Wine}
5	{Butter, Cheese}
6	{Cheese, Wine}
7	{Butter, Wine}

Lift : « force de l'association »

lift(X $\rightarrow$ Y) = $\dfrac{supp(X \cup Y)}{supp(X) \times supp(Y)}$

Lift > 1 : Y a plus de chances d'être acheté avec X
Lift < 1 : Y a moins de chances d'être acheté si X est acheté

Exemple :

X = {Bread}; Y = {Butter}

lift(X $\rightarrow$ Y) = $\frac{3/7}{(3/7)*(6/7)} = \frac{7}{6}$ ~ 1,16

La fonction apriori pour les itemsets fréquents

library(arules)
# Frequent itemsets
supp.cw = apriori(trans, # the transactional dataset
                  # Parameter list
                  parameter=list(
                    # Minimum Support
                    supp=0.2,
                    # Minimum Confidence
                    conf=0.4,
                    # Minimum length
                    minlen=2,
                    # Target
                    target="frequent itemsets"),
                  # Appearence argument
                  appearance = list(
                    items = c("Cheese","Wine"))
                 )

La fonction apriori pour les règles

library(arules)
# Rules
rules.b.rhs = apriori(trans, # the transactional dataset
                  # Parameter list
                  parameter=list(
                    # Minimum Support
                    supp=0.2,
                    # Minimum Confidence
                    conf=0.4,
                    # Minimum length
                    minlen=2,
                    # Target
                    target="rules"),
                  # Appearence argument
                   appearance = list(
                     rhs = "Butter",
                    default = "lhs")
                 )

Itemsets fréquents avec apriori

TID	Transaction
1	{Bread, Butter, Cheese, Wine}
2	{Bread, Butter, Wine}
3	{Bread, Butter}
4	{Butter, Cheese, Wine}
5	{Butter, Cheese}
6	{Cheese, Wine}
7	{Butter, Wine}

Récupérer les itemsets fréquents

supp.all = apriori(trans, 
           parameter=list(supp=3/7,
           target="frequent itemsets"))        
inspect(head(sort(supp.all,by="support"),3))

    items          support   count
[1] {Butter}       0.8571429 6    
[2] {Wine}         0.7142857 5    
[3] {Cheese}       0.5714286 4

Examiner les mesures de confiance et de lift

TID	Transaction
1	{Bread, Butter, Cheese, Wine}
2	{Bread, Butter, Wine}
3	{Bread, Butter}
4	{Butter, Cheese, Wine}
5	{Butter, Cheese}
6	{Cheese, Wine}
7	{Butter, Wine}

Récupérer les règles

# Rules with "Butter" on rhs
rules.b.rhs = apriori(trans, 
                 parameter=list(
                    minlen=2,
                    target="rules"), 
                 appearance = list(
                     rhs="Butter",
                    default = "lhs")
           )
inspect(head(sort(rules.b.rhs,by="lift")), 5)

Examiner les mesures de confiance et de lift

TID	Transaction
1	{Bread, Butter, Cheese, Wine}
2	{Bread, Butter, Wine}
3	{Bread, Butter}
4	{Butter, Cheese, Wine}
5	{Butter, Cheese}
6	{Cheese, Wine}
7	{Butter, Wine}

Récupérer les règles

    lhs                    rhs      support  confidence lift  count
[1] {Bread}             => {Butter} 0.42      1.0        1.16     3    
[2] {Bread,Cheese}      => {Butter} 0.14      1.0        1.16     1    
[3] {Bread,Wine}        => {Butter} 0.28      1.0        1.16     2    
[4] {Bread,Cheese,Wine} => {Butter} 0.14      1.0        1.16     1    
[5] {Wine}              => {Butter} 0.57      0.8        0.93     4

Passons à la pratique !

Analyse des paniers d'achats en R