3.1 Support

\[Support( \{X\} \rightarrow \{Y\}) = \frac{Number\ of\ transactions\ with\ both\ X\ and\ Y}{Total\ number\ of\ transactions} = P(X \cap Y)\] Support is basically a measure of how frequent an itemset is in all the transactions. In our case, the blouse with horse print will have a higher support then the elegant blouse (0.35 vs 0.30).

Support of \(\{blouse\_horse\} \rightarrow \{blouse\_elegant\}\) will be significantly lower, at approximately 0.01 as there are only 7 transactions containing both products.

3.2 Confidence

\[Confidence( \{X\} \rightarrow \{Y\}) = \frac{Number\ of\ transactions\ with\ both\ X\ and\ Y}{Total\ number\ of\ transactions\ with\ X} = \frac{P(X \cap Y)}{P(X)}\] Confidence answers to question: of all transactions containings {blouse_horse}, how many had also {blouse elegant}? The answer to that is, naturally, 7/246 = 0.03. Confidence for \(\{blouse\_horse\} \rightarrow \{blouse\_\}\) will be much higher, at 0.12. It is worth noticing that confidence might be misleading as high confidence value does not necessarily mean strong items association. If the consequent (the item on right hand side - {X}) is very frequent, it does not matter what the antecedent is (left hand side - {Y}) - the confidence will always be high.

3.3 Lift

\[Lift( \{X\} \rightarrow \{Y\}) = \frac{(No.\ of\ transactions\ with\ both\ X\ and\ Y)/No.\ of\ transactions\ with\ X}{Fraction\ of\ transactions\ with\ Y} = \] \[ = \frac{P(X\cap Y)}{P(X).P(Y)}\] Lift has been introduced to overcome the mentioned challenge of confidence. Lift is the rise in probability of having {Y} in transaction knowing that {X} is in the basket over the probability of having {Y} in transaction without any knowledge about presence of {X}. A value of lift greater than 1 implies that there is high association between {X} and {Y}. In other words, customer will be more likely to buy {Y} if he has already bought {X}. In our case, \(Lift(\{blouse\_horse\} \rightarrow \{blouse\_\})\) equals to (29/246)/(254/667) = 0.31, so having a blouse with horse print on the cart does not increase probabilities of occurence of a regular blouse.

4.1 Grouped matrix-based visualization

Grouped matrix-based visualization comes in very handy while mining interesting patterns in transactions.

Firstly, I will render a balloon plot with antecedent groups as columns and consequents as rows. The size and color of the balloons show the aggregated support and the aggregated lift respectively. Columns and rows in the plot are ordered such that lift value is decreasing from top down and from left to right, so the maximum value is in the top left corner.

rules <- ruleInduction(freq.items, trans1, confidence=0.1)
plot(rules, method = "grouped")

We can clearly see that having {blouse_dog, blouse_} significantly increases the chances of adding {blouse_cat} - the number of such transactions (support) is not very big though as in the analyzed period number of cat print blouse models were not that big. There is also a very strong association between {blouse_physical education, blouse_} and {shorts_physical education} as well as {shorts_physical education, blouse_} and {blouse_physical education}.

4.2 Graph-based visualizations

In graphs items (annodated with item labels) are connected with itemsets or rules using arrows. Rules are represented by vertices. The color and size of vertices show respectively lift and support values. Arrows pointing from items to rule indicate left-hand-side items and arrows pointing from a rule to items indicate right-hand-side items. Graphs are a very clear visualization of rules, however they become messy for a big set of rules. Therefore I will limit the set of rules to 15, choosing only rules with the highest lift value.

plot(head(rules, n = 15, by = "lift"), method="graph")

We can clearly see that shorts for physical education are associated with blouse for physical education. In general, having bottoms on basket increases the probability of adding another type of bottoms and having one kind of a blouse increases the probability of buying another top.

4.3 Rules inspection

Firstly, I will inspect only significant rules, using significance level 5%. To help me find significant rules, I will use is.significant function which uses Fisher’s exact test.

inspectDT(rules[is.significant(rules, trans1, alpha=0.05)])

There are just 7 significant rules. There is strong link between warm trousers and trousers, shorts for pe and blouse for pe, as well as owl print blouse and blouse with sequins. What’s interesting, having a blouse with a dog print increases the chances of buying a blouse with a cat, but not the other way around. The lack of significance of other rules does not mean that they will not be used - we need to build a recommendation system, so a rule for each combination on basket is essential, even if it is not statistically significant.

To dig deeper on relationship between all physical education clothing, let’s inspect all rules regarding it.

rules.shorts_pe <- apriori(trans1, parameter = list(support=0.003, confidence = 0.1), appearance = list(rhs = "shorts_physical education"))
rules.shorts_pe <- sort(rules.shorts_pe, by="lift", decreasing = TRUE)

rules.blouse_pe <-  apriori(trans1, parameter = list(support=0.003, confidence = 0.1), appearance = list(rhs = "blouse_physical education"))
rules.blouse_pe <- sort(rules.blouse_pe, by="lift", decreasing = TRUE)

inspectDT(rules.shorts_pe)

inspectDT(rules.blouse_pe)

There is a strong link between blouse for pe and shorts for pe, but more surprisingly if the client has also a regular blouse on the basket, the lift and confidence value is even higher.

This company sells rather a big number of blouses with horse print (almost 1/3 of total transactions), so I will inspect what customers buy having a blouse with a horse on the basket and see if the company is able to use cross-selling.

rules.horse <- apriori(trans1, parameter = list(support=0.005, confidence = 0.02), appearance = list(lhs = "blouse_horse"))
rules.horse <- sort(rules.horse, by="lift", decreasing = TRUE)

inspectDT(rules.horse)

We can see that lift value is rather big for cat print blouse and owl print blouse. This suggests developing other animals print collection as customers having a blouse with a horse on the basket, are more likely to buy a blouse with another animal. Unfortunately the confidence values are not that high.