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Retail shops are often interested in associations between different items that people buy Examples of such associations are: Someone who buys bread is quite likely also to buy milk A person who bought the book Database System Concepts is quite likely also to buy the book Operating System Concepts Association information can be used in several ways When a customer buys a particular book, an online shop may suggest associated books A grocery shop may decide to place bread close to milk, since they are often bought together, to help shoppers nish their task faster Or the shop may place them at opposite ends of a row, and place other associated items in between to tempt people to buy those items as well, as the shoppers walk from one end of the row to the other A shop that offers discounts on one associated item may not offer a discount on the other, since the customer will probably buy the other anyway Association Rules An example of an association rule is bread milk In the context of grocery-store purchases, the rule says that customers who buy bread also tend to buy milk with a high probability An association rule must have an associated population: the population consists of a set of instances In the grocery-store example, the population may consist of all grocery store purchases; each purchase is an instance In the case of a bookstore, the population may consist of all people who made purchases, regardless of when they made a purchase Each customer is an instance Here, the analyst has decided that when a purchase is made is not signi cant, whereas for the grocery-store example, the analyst may have decided to concentrate on single purchases, ignoring multiple visits by the same customer

The McGraw Hill Companies, 2001

Transforming to the time domain i 1 (t) = 211 cos (4t 493 ) The current I2 is given by I2 = 3I1 = 3(211) 493 = 633 493 In the time domain i 2 (t) = 633 cos (4t 493 )

.

Rules have an associated support, as well as an associated con dence These are de ned in the context of the population: Support is a measure of what fraction of the population satis es both the antecedent and the consequent of the rule For instance, suppose only 0001 percent of all purchases include milk and screwdrivers The support for the rule milk screwdrivers is low The rule may not even be statistically signi cant perhaps there was only a single purchase that included both milk and screwdrivers Businesses are usually not interested in rules that have low support, since they involve few customers, and are not worth bothering about On the other hand, if 50 percent of all purchases involve milk and bread, then support for rules involving bread and milk (and no other item) is relatively high, and such rules may be worth attention Exactly what minimum degree of support is considered desirable depends on the application Con dence is a measure of how often the consequent is true when the antecedent is true For instance, the rule bread milk has a con dence of 80 percent if 80 percent of the purchases that include bread also include milk A rule with a low con dence is not meaningful In business applications, rules usually have con dences signi cantly less than 100 percent, whereas in other domains, such as in physics, rules may have high con dences Note that the con dence of bread milk may be very different from the con dence of milk bread, although both have the same support Finding Association Rules To discover association rules of the form i1 , i2 , , in i0 we rst nd sets of items with suf cient support, called large itemsets In our example we nd sets of items that are included in a suf ciently large number of instances We will shortly see how to compute large itemsets For each large itemset, we then output all rules with suf cient con dence that involve all and only the elements of the set For each large itemset S, we output a rule S s s for every subset s S, provided S s s has suf cient con dence; the con dence of the rule is given by support of s divided by support of S We now consider how to generate all large itemsets If the number of possible sets of items is small, a single pass over the data suf ces to detect the level of support for all the sets A count, initialized to 0, is maintained for each set of items When a purchase record is fetched, the count is incremented for each set of items such that.