2. Discovery of Association Rules
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1 2. Discovery of Association Rules Part I Motivation: market basket data Basic notions: association rule, frequency and confidence Problem of association rule mining (Sub)problem of frequent set mining Monotonicity Candidate set generation Algorithm Apriori Association rule generation algorithm
2 Motivation supermarkets store all customer transactions massive market basket databases examine customer behavior in terms of the purchased products i.e., find all combinations of products that are frequently purchased together e.g., beer chips or, find all associations between products beer chips
3 Example Customer 1: mustard, sausage, beer, chips Customer 2: sausage, ketchup Customer 3: beer, chips, cigarettes beer chips frequency (support): 12 % of all customers purchased chips and beer confidence (conditional probability): 70 % of the customers that purchased beer also purchased chips
4 Problem formulation: data a set a set of items a 0/1 relation of subsets of the elements of is called an itemset over the number of rows in the size of is a collection (or multiset) are called rows is denoted by is denoted by
5 Example: binary relation Row ID Row over the set Figure 1: An example 0/1 relation.
6 Problem formulation (cont.) itemset the set of rows in, i.e., the cover of ) matches a row matched by, if is denoted by (also called the (relative) frequency or support of denoted by, is in, Given a frequency threshold is frequent, if, the set
7 Example Row ID A B C D E F G H I J K a 0/1 relation over the schema
8 Association rules, confidence itemsets is an association rule over. The confidence of, is in The confidence probability that a row in it matches, denoted by is the conditional matches given that
9 Association rules, frequency is in of The frequency, and a, the rule. is Given a frequency threshold confidence threshold frequent if confident if
10 Association rule mining task given,,, and find all frequent and confident association rules in with respect to and
11 How to find association rules 1. Find all frequent itemsets frequencies. and their 2. Then test separately for all whether the rule holds with sufficient confidence. The latter is easy
12 Frequent sets given (a set), (a 0/1 relation over ), and (a frequency threshold) the collection of frequent sets in the example relation: =, the only if and nontrivial association rule is
13 Challenges Transaction databases tend to be very large and can not be stored in main memory. The search space of all itemsets contains exactly different itemsets ( rules) typically contains thousands of items, hence is usually more than the (estimated) number of atoms in the universe ( ). It is infeasible to generate and count the frequencies of all possible itemsets within a reasonable amount of time.
14 General solution 1. generate a group of candidate itemsets 2. count their frequencies 3. remove the infrequent itemsets 4. repeat until all frequent itemsets are found candidate generation can utilize monotonicity principle
15 Monotonicity principle assume then and if is frequent then, is also frequent confidence is monotonic w.r.t. the consequent of a rule if then also if
16 Search space: itemsets {} {beer} {chips} {pizza} {wine} {beer, chips} {beer, pizza} {beer, wine} {chips, pizza} {chips, wine} {pizza, wine} {beer, chips, pizza} {beer, chips, wine} {beer, pizza, wine} {chips, pizza, wine} {beer, chips, pizza, wine}
17 Candidate generation monotonicity if any of the proper subsets then 1. is not frequent, and 2. there is a non-frequent subset. itemset selected to be a candidate if all of its subsets are known to be frequent is not frequent of size
18 Example are the only possible and then members of store collections of item sets as arrays which are sorted in the lexicographical order
19 Candidate generation Input: A lexicographically sorted array sets of size Output: candidate set in lexicographical order of frequent for all such that and do // Join if for all such that, then // Prune if a subset not frequent
20 Algorithm Apriori 1. find all frequent itemsets of size 2. generate all candidate itemsets of size 3. read the database one transaction at a time increment the frequency (support) of all candidate itemsets that are contained in that transaction 4. exclude the candidate itemsets that are not frequent 5. increment by one 6. iterate lines 2 5 until no frequent sets are found
21 Apriori: drawbacks frequency counting very time- and space-consuming for each transaction in the database: for each candidate itemset we need to check whether, or we need to generate all subsets of size of, and check for each of them whether the subset is a candidate itemset number of transactions, size of transactions, and number of candidate itemsets can be large whole database must be scanned for each size of of frequents sets many optimizations and other solutions developed
22 Rule generation a simple algorithm for association rule generation Input: Output: frequent and confident association rules in // find frequent sets compute // generate rules do for all do with for all then if, and, output the rule
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