How Association Rules Are Discovered: Concepts, Scale, Measures, and the Apriori Approach#

Stage 3 · 🛒 Market Basket & Association Rules · Lesson 19 of 56 · intermediate

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Important

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Three measures of a rule#

A rule like \(X \rightarrow Y\) is only worth keeping if it is both common and reliable. Three numbers, all computed from how often itemsets appear in the transactions, make “worth keeping” precise: support, confidence and lift.

Support, confidence, lift#

Each captures a different facet:

Support — how common the combination is, the fraction of all transactions containing the whole itemset:

\[\operatorname{support}(X \rightarrow Y) = \frac{\text{transactions containing } X \cup Y}{\text{all transactions}}.\]
Confidence — how reliable the rule is, the share of \(X\)-baskets that also contain \(Y\), estimating \(P(Y \mid X)\):

\[\operatorname{confidence}(X \rightarrow Y) = \frac{\operatorname{support}(X \cup Y)}{\operatorname{support}(X)}.\]
Lift — how surprising it is, confidence compared with \(Y\)’s baseline frequency:

\[\operatorname{lift}(X \rightarrow Y) = \frac{\operatorname{confidence}(X \rightarrow Y)}{\operatorname{support}(Y)}.\]

Lift \(> 1\) means \(X\) and \(Y\) occur together more than chance (positive association); \(= 1\) means independent; \(< 1\) means they repel.

The scale problem#

Why not simply score every possible rule? Because there are too many. With \(d\) distinct items there are \(2^d\) possible itemsets and more possible rules still — for a supermarket with thousands of products, an astronomically large number. Counting the support of every candidate by brute force is hopeless.

The Apriori idea#

The escape is one simple observation, the Apriori principle (also called downward closure): if an itemset is frequent, all of its subsets must be frequent too — and, turned around, if an itemset is infrequent, every superset of it is infrequent as well. That lets an algorithm prune vast regions of the search: once the pair {milk, caviar} is rare, nothing containing it can be common, so none of its extensions need be checked. The next lesson builds an algorithm around exactly this.

Hint