Package smile.association

Frequent item set mining and association rule mining.

Association rule learning is a popular and well researched method for discovering interesting relations between variables in large databases. Let I = {i₁, i₂,..., i_n} be a set of n binary attributes called items. Let D = {t₁, t₂,..., t_m} be a set of transactions called the database. Each transaction in D has a unique transaction ID and contains a subset of the items in I. An association rule is defined as an implication of the form X ⇒ Y where X, Y ⊆ I and X ∩ Y = Ø. The item sets X and Y are called antecedent (left-hand-side or LHS) and consequent (right-hand-side or RHS) of the rule, respectively. The support supp(X) of an item set X is defined as the proportion of transactions in the database which contain the item set. Note that the support of an association rule X ⇒ Y is supp(X ∪ Y). The confidence of a rule is defined conf(X ⇒ Y) = supp(X ∪ Y) / supp(X). Confidence can be interpreted as an estimate of the probability P(Y | X), the probability of finding the RHS of the rule in transactions under the condition that these transactions also contain the LHS.

For example, the rule {onions, potatoes} ⇒ {burger} found in the sales data of a supermarket would indicate that if a customer buys onions and potatoes together, he or she is likely to also buy burger. Such information can be used as the basis for decisions about marketing activities such as promotional pricing or product placements.

Association rules are usually required to satisfy a user-specified minimum support and a user-specified minimum confidence at the same time. Association rule generation is usually split up into two separate steps:

• First, minimum support is applied to find all frequent item sets in a database (i.e. frequent item set mining).

• Second, these frequent item sets and the minimum confidence constraint are used to form rules.

Finding all frequent item sets in a database is difficult since it involves searching all possible item sets (item combinations). The set of possible item sets is the power set over I (the set of items) and has size 2ⁿ - 1 (excluding the empty set which is not a valid item set). Although the size of the power set grows exponentially in the number of items n in I, efficient search is possible using the downward-closure property of support (also called anti-monotonicity) which guarantees that for a frequent item set also all its subsets are frequent and thus for an infrequent item set, all its supersets must be infrequent.

In practice, we may only consider the frequent item set that has the maximum number of items bypassing all the sub item sets. An item set is maximal frequent if none of its immediate supersets is frequent.

For a maximal frequent item set, even though we know that all the sub item sets are frequent, we don't know the actual support of those sub item sets, which are very important to find the association rules within the item sets. If the final goal is association rule mining, we would like to discover closed frequent item sets. An item set is closed if none of its immediate supersets has the same support as the item set.

Some well known algorithms of frequent item set mining are Apriori, Eclat and FP-Growth. Apriori is the best-known algorithm to mine association rules. It uses a breadth-first search strategy to counting the support of item sets and uses a candidate generation function which exploits the downward closure property of support. Eclat is a depth-first search algorithm using set intersection.

FP-growth (frequent pattern growth) uses an extended prefix-tree (FP-tree) structure to store the database in a compressed form. FP-growth adopts a divide-and-conquer approach to decompose both the mining tasks and the databases. It uses a pattern fragment growth method to avoid the costly process of candidate generation and testing used by Apriori.