Apriori algorithm

In computer science and data mining, Apriori is a classic algorithm for learning association rules. Apriori is designed to operate on databases containing transactions (for example, collections of items bought by customers, or details of a website frequentation). Other algorithms are designed for finding association rules in data having no transactions (Winepi and Minepi), or having no timestamps (DNA sequencing).

As is common in association rule mining, given a set of itemsets (for instance, sets of retail transactions, each listing individual items purchased), the algorithm attempts to find subsets which are common to at least a minimum number C of the itemsets. Apriori uses a "bottom up" approach, where frequent subsets are extended one item at a time (a step known as candidate generation), and groups of candidates are tested against the data. The algorithm terminates when no further successful extensions are found.

The purpose of the Apriori Algorithm is to find associations between different sets of data. It is sometimes referred to as "Market Basket Analysis". Each set of data has a number of items and is called a transaction. The output of Apriori is sets of rules that tell us how often items are contained in sets of data. Here is an example:

each line is a set of items

alpha	beta	gamma
alpha	beta	theta
alpha	beta	epsilon
alpha	beta	theta

1. 100% of sets with alpha also contain beta
2. 25% of sets with alpha, beta also have gamma
3. 50% of sets with alpha, beta also have theta

Apriori uses breadth-first search and a Hash tree structure to count candidate item sets efficiently. It generates candidate item sets of length $k$ from item sets of length $k-1$. Then it prunes the candidates which have an infrequent sub pattern. According to the downward closure lemma, the candidate set contains all frequent $k$-length item sets. After that, it scans the transaction database to determine frequent item sets among the candidates.

Apriori, while historically significant, suffers from a number of inefficiencies or trade-offs, which have spawned other algorithms. Candidate generation generates large numbers of subsets (the algorithm attempts to load up the candidate set with as many as possible before each scan). Bottom-up subset exploration (essentially a breadth-first traversal of the subset lattice) finds any maximal subset S only after all $2^{|S|}-1$ of its proper subsets.

Algorithm Pseudocode

The pseudocode for the algorithm is given below for a transaction database $T$, and a support threshold of $\epsilon$. Usual set theoretic notation is employed, though note that $T$ is a multiset. $C_{k}$ is the candidate set for level $k$. Generate() algorithm is assumed to generate the candidate sets from the large itemsets of the preceding level, heeding the downward closure lemma. $count[c]$ accesses a field of the data structure that represents candidate set $c$, which is initially assumed to be zero. Many details are omitted below, usually the most important part of the implementation is the data structure used for storing the candidate sets, and counting their frequencies.

Apriori$(T,\epsilon )$

   $L_{1}\gets \{$ large 1-itemsets $\}$
   $k\gets 2$
   while $L_{k-1}\neq \emptyset$
       $C_{k}\gets$Generate$(L_{k-1})$
       for transactions $t\in T$
           $C_{t}\gets \{c|c\in C_{k}\land c\subseteq t\}$
           for candidates $c\in C_{t}$
               $count[c]\gets count[c]+1$
       $L_{k}\gets \{c\in C_{k}|~count[c]\geq \epsilon \}$
       $k\gets k+1$
   return $\bigcup _{k}L_{k}$

Example

A large supermarket tracks sales data by stock-keeping unit (SKU) for each item, and thus is able to know what items are typically purchased together. Apriori is a moderately efficient way to build a list of frequent purchased item pairs from this data. Let the database of transactions consist of the sets {1,2,3,4}, {1,2}, {2,3,4}, {2,3}, {1,2,4}, {3,4}, and {2,4}. Each number corresponds to a product such as "butter" or "bread". The first step of Apriori is to count up the frequencies, called the supports, of each member item separately:

This table explains the working of apriori algorithm.

Item	Support
1	3
2	6
3	4
4	5

We can define a minimum support level to qualify as "frequent," which depends on the context. For this case, let min support = 3. Therefore, all are frequent. The next step is to generate a list of all 2-pairs of the frequent items. Had any of the above items not been frequent, they wouldn't have been included as a possible member of possible 2-item pairs. In this way, Apriori prunes the tree of all possible sets. In next step we again select only these items (now 2-pairs are items) which are frequent:

Item	Support
{1,2}	3
{2,3}	3
{2,4}	4
{3,4}	3

And generate a list of all 3-triples of the frequent items (by connecting frequent pairs with frequent single items). In the example, there are no frequent 3-triples. Most common 3-triples are {1,2,4} and {2,3,4}, but their support is equal to 2 which is smaller than our min support.

See also

• Dynamic itemset counting

External links

• "Implementation of the Apriori algorithm in C#"
• ARtool, GPL Java association rule mining application with GUI, offering implementations of multiple algorithms for discovery of frequent patterns and extraction of association rules (includes Apriori)