The IF component of an association rule is known as the antecedent. The THEN component is known as the consequent. The antecedent and the consequent are disjoint; they have no items in common.

Oracle Data Mining supports association rules that have one or more items in the antecedent and a single item in the consequent.

Rules have an associated confidence, which is the conditional probability that the consequent occurs given the occurrence of the antecedent. You can specify the minimum confidence for rules.

Learn about storage format of transactional data.

Transactional data may be stored in native transactional format, with a non-unique case ID column and a values column, or it may be stored in some other configuration, such as a star schema. If the data is not stored in native transactional format, it must be transformed to a nested column for processing by the Apriori algorithm.

In transactional data, a collection of items is associated with each case. The collection theoretically includes all possible members of the collection. For example, all products can theoretically be purchased in a single market-basket transaction. However, in actuality, only a tiny subset of all possible items are present in a given transaction; the items in the market-basket represent only a small fraction of the items available for sale in the store.

Learn about missing items through sparsity.

Missing items in a collection indicate sparsity. Missing items may be present with a null value, or they may simply be missing.

Nulls in transactional data are assumed to represent values that are known but not present in the transaction. For example, three items out of hundreds of possible items might be purchased in a single transaction. The items that were not purchased are known but not present in the transaction.

Oracle Data Mining assumes sparsity in transactional data. The Apriori algorithm is optimized for processing sparse data.

Association Rules (AR) can use a good sample size with performance guarantee, based on the work of Riondato and Upfal.

The AR algorithm computes the sample size by the following inputs:

d-index is defined as the maximum integer d such that the dataset contains at least d transactions of length d at the minimum. It is the upper bound of Vapnik-Chervonenkis (VC) dimension. The AR algorithm computes d-index of the dataset by scanning the length of all transactions in the dataset.

Users specify absolute error ε and confidence level γ parameters. A large d-index, small AR support, small ε or large γ can cause a large sample size. The sample size theoretically guarantees that the absolute error of both the support and confidence of the approximated AR (from sampling) is less than ε compared to the exact AR with probability (or confidence level) at least γ. In this document this sample size is called AR-specific sample size.

Learn about itemsets.

The maximum number of items in an itemset is user-specified. If the maximum is two, then all the item pairs are counted. If the maximum is greater than two, then all the item pairs, all the item triples, and all the item combinations up to the specified maximum are counted.

The following table shows the itemsets derived from the transactions shown in the following example, assuming that maximum number of items in an itemset is set to 3.

TRANS_ID   ITEM_ID
---------  -------------------
11         B
11         D
11         E
12         A
12         B
12         C
12         E
13         B
13         C
13         D
13         E

Learn about Frequent Itemsets and Support.

Association rules are calculated from itemsets. If rules are generated from all possible itemsets, there can be a very high number of rules and the rules may not be very meaningful. Also, the model can take a long time to build. Typically it is desirable to only generate rules from itemsets that are well-represented in the data. Frequent itemsets are those that occur with a minimum frequency specified by the user.

The minimum frequent itemset Support is a user-specified percentage that limits the number of itemsets used for association rules. An itemset must appear in at least this percentage of all the transactions if it is to be used as a basis for rules.

The following table shows the itemsets from Table 10-1 that are frequent itemsets with support > 66%.

Example to calculating rules from Frequent itemsets.

The following tables show the itemsets and frequent itemsets that were calculated in "Association". The frequent itemsets are the itemsets that occur with a minimum support of 67%; at least 2 of the 3 transactions must include the itemset.

A rule expresses a conditional probability. Confidence in a rule is calculated by dividing the probability of the items occurring together by the probability of the occurrence of the antecedent.

For example, if B (antecedent) is present, what is the chance that C (consequent) is also present? What is the confidence for the rule "IF B, THEN C"?

As shown in Table 10-3:

• All 3 transactions include B (3/3 or 100%)

• Only 2 transactions include both B and C (2/3 or 67%)

• Therefore, the confidence of the rule "IF B, THEN C" is 67/100 or 67%.

The following table the rules that can be derived from the frequent itemsets in Table 10-4.

If the minimum confidence is 70%, ten rules are generated for these frequent itemsets. If the minimum confidence is 60%, sixteen rules are generated.

The support of a rule indicates how frequently the items in the rule occur together. For example, cereal and milk might appear together in 40% of the transactions. If so, the following rules each have a support of 40%:

cereal implies milk
milk implies cereal

Support is the ratio of transactions that include all the items in the antecedent and consequent to the number of total transactions.

Support can be expressed in probability notation as follows:

support(A implies B) = P(A, B)

The confidence of a rule indicates the probability of both the antecedent and the consequent appearing in the same transaction.

Confidence is the conditional probability of the consequent given the antecedent. For example, cereal appears in 50 transactions; 40 of the 50 might also include milk. The rule confidence is:

cereal implies milk with 80% confidence

Confidence is the ratio of the rule support to the number of transactions that include the antecedent.

Confidence can be expressed in probability notation as follows.

confidence (A implies B) = P (B/A), which is equal to P(A, B) / P(A)

Both support and confidence must be used to determine if a rule is valid. However, there are times when both of these measures may be high, and yet still produce a rule that is not useful. For example:

Convenience store customers who buy orange juice also buy milk with 
a 75% confidence. 
The combination of milk and orange juice has a support of 30%.

This at first sounds like an excellent rule, and in most cases, it would be. It has high confidence and high support. However, what if convenience store customers in general buy milk 90% of the time? In that case, orange juice customers are actually less likely to buy milk than customers in general.

A third measure is needed to evaluate the quality of the rule. Lift indicates the strength of a rule over the random co-occurrence of the antecedent and the consequent, given their individual support. It provides information about the improvement, the increase in probability of the consequent given the antecedent. Lift is defined as follows.

(Rule Support) /(Support(Antecedent) * Support(Consequent))

This can also be defined as the confidence of the combination of items divided by the support of the consequent. So in our milk example, assuming that 40% of the customers buy orange juice, the improvement would be:

30% / (40% * 90%)

which is 0.83 – an improvement of less than 1.

Any rule with an improvement of less than 1 does not indicate a real cross-selling opportunity, no matter how high its support and confidence, because it actually offers less ability to predict a purchase than does random chance.