Association Rule: Numerical Problem

Question: 

Suppose half of all the documents in a computer contain the word sports, and one third of all the

documents contain the word football. Again, half of the documents that contain the word sports, also

contain the word football. Write down all the non-trivial association rules that can be deduced from

the above information, giving the support, confidence and lift of each rule.

Answer: 

Given Information: Support(sports)=1/2, Support(football)=1/3, Support(sports, football)=(1/2*1/2)​=1/4

Non-Trivial Association Rules: We can deduce two non-trivial association rules-

sports →football  & football →sports.

Calculating Support, Confidence, and Lift:

Confidence(sports →football)= Support(sports, football) / Support(sports) ​=(1/4) /1/2​​ =1​/2=0.5

Confidence(football →sports)= Support(sports, football) ​/ Support(football) = (1​​/4)/1/3 =3/4​=0.75

Lift: The lift of a rule $X \rightarrow Y$ is defined as:

$$\text{Lift}(X \rightarrow Y) = \frac{\text{Confidence}(X \rightarrow Y)}{\text{Support}(Y)}$$$\text{Lift}(\text{sports} \rightarrow \text{football}) = \frac{\text{Confidence}(\text{sports} \rightarrow \text{football})}{\text{Support}(\text{football})} = \frac{0.5}{\frac{1}{3}} = 0.5 \times 3 = 1.5$

$\text{Lift}(\text{football} \rightarrow \text{sports}) = \frac{\text{Confidence}(\text{football} \rightarrow \text{sports})}{\text{Support}(\text{sports})} = \frac{0.75}{\frac{1}{2}} = 0.75 \times 2 = 1.5$

Summary of Association Rules:

1. Rule 1: $\text{sports} \rightarrow \text{football}$

   • Support: 0.25
   • Confidence: 0.5
   • Lift: 1.5

2. Rule 2: $\text{football} \rightarrow \text{sports}$

   • Support: 0.25
   • Confidence: 0.75
   • Lift: 1.5