# Association Rules¶

Association rules mining is the method for uncovering the most important relationships between variables. Its main application is a store basket analysis, which aims at discovery of a relationship between groups of products with some level of confidence.

## Details¶

The library provides Apriori algorithm for association rule mining [Agrawal94].

Let $$I = \{i_1, i_2, \ldots, i_m\}$$ be a set of items (products) and subset $$T \subset I$$ is a transaction associated with item set I. The association rule has the form: $$X \Rightarrow Y$$, where $$X \subset I$$, $$Y \subset I$$, and intersection of $$X$$ and $$Y$$ is empty: $$X \cap Y = \emptyset$$. The left-hand-side set of items (itemset) $$X$$ is called antecedent, while the right-hand-side itemset Y is called consequent of the rule.

Let $$D = \{T_1, T_2, \ldots, T_n\}$$ be a set of transactions, each associated with item set I. Item subset $$X \subset I$$ has support $$s$$ in the transaction set $$D$$ if $$s$$ percent of transactions in $$D$$ contains $$X$$.

The association rule $$X \Rightarrow Y$$ in the transaction set $$D$$ holds with confidence $$c$$ if $$c$$ percent of transactions in $$D$$ that contain $$X$$ also contains $$Y$$. Confidence of the rule can be represented as conditional probability:

$$confidence(X \Rightarrow Y) = support (X \cup Y)/support(X)$$

For a given set of transactions $$D = \{T_1, T_2, \ldots, T_n\}$$, the minimum support s and minimum confidence c discover all item sets $$X$$ with support greater than $$s$$ and generate all association rules $$X \Rightarrow Y$$ with confidence greater than $$c$$.

Therefore, the association rule discovery is decomposed into two stages: mining (training) and discovery (prediction). The mining stage involves generation of large item sets, that is, the sets that have support greater than the given parameters. At the discovery stage, the algorithm generates association rules using the large item sets identified at the mining stage.

## Batch Processing¶

### Algorithm Input¶

The association rules algorithm accepts the input described below. Pass the Input ID as a parameter to the methods that provide input for your algorithm.

Input ID

Input

data

Pointer to the $$n \times 2$$ numeric table t with the mining data. Each row consists of two integers:

• Transaction ID, the number between 0 and $$nTransactions - 1$$.

• Item ID, the number between 0 and $$nUniqueItems - 1$$.

The input can be an object of any class derived from NumericTable except PackedTriangularMatrix and PackedSymmetricMatrix.

### Algorithm Parameters¶

The association rules algorithm has the following parameters:

Parameter

Default Value

Description

algorithmFPType

float

The floating-point type that the algorithm uses for intermediate computations. Can be float or double.

method

defaultDense

The computation method used by the algorithm. The only method supported so far is Apriori.

minSupport

$$0.01$$

Minimal support, a number in the [0,1) interval.

minConfidence

$$0.6$$

Minimal confidence, a number in the [0,1) interval.

nUniqueItems

$$0$$

The total number of unique items. If set to zero, the library automatically determines the number of unique items from the input data.

nTransactions

$$0$$

The total number of transactions. If set to zero, the library automatically determines the number transactions from the input data.

discoverRules

true

A flag that enables generation of the rules from large item sets.

itemsetsOrder

itemsetsUnsorted

The sort order of returned item sets:

• itemsetsUnsorted - not sorted

• itemsetsSortedBySupport - sorted by support in a descending order

rulesOrder

rulesUnsorted

The sort order of returned rules:

• rulesUnsorted - not sorted

• rulesSortedByConfidence - sorted by support in a descending order

minItemsetSize

$$0$$

A parameter that defines the minimal size of item sets to be included into the array of results. The value of zero imposes no limitations on the minimal size of item sets.

maxItemsetSize

$$0$$

A parameter that defines the maximal size of item sets to be included into the array of results. The value of zero imposes no limitations on the maximal size of item sets.

### Algorithm Output¶

The association rules algorithm calculates the result described below. Pass the Result ID as a parameter to the methods that access the results of your algorithm.

Result ID

Result

largeItemsets

Pointer to the numeric table with large item sets. The number of rows in the table equals the number of items in the large item sets. Each row contains two integers:

• ID of the large item set, the number between 0 and nLargeItemsets -1.

• ID of the item, the number between 0 and $$nUniqueItems-1$$.

largeItemsetsSupport

Pointer to the $$nLargeItemsets \times 2$$ numeric table of support values. Each row contains two integers:

• ID of the large item set, the number between 0 and nLargeItemsets-1.

• The support value, the number of times the item set is met in the array of transactions.

antecedentItemsets

Pointer to the $$nAntecedentItems \times 2$$ numeric table that contains the left-hand-side (X) part of the association rules. Each row contains two integers:

• Rule ID, the number between 0 and $$nAntecedentItems-1$$.

• Item ID, the number between 0 and $$nUniqueItems-1$$.

conseqentItemsets

Pointer to the $$nConsequentItems \times 2$$ numeric table that contains the right-hand-side (Y) part of the association rules. Each row contains two integers:

• Rule ID, the number between 0 and $$nConsequentItems-1$$.

• Item ID, the number between 0 and $$nUniqueItems-1$$.

confidence

Pointer to the $$nRules \times 1$$ numeric table that contains confidence values of rules, floating-point numbers between 0 and 1. Confidence value in the i-th position corresponds to the rule with the index i.

By default, the result is an object of the HomogenNumericTable class, but you can define the result as an object of any class derived from NumericTable except PackedSymmetricMatrix, PackedTriangularMatrix, and СSRNumericTable.

Note

• The library requires transactions and items for each transaction to be passed in the ascending order.

• Numbering of rules starts at 0.

• The library calculates the sizes of numeric tables intended for results in a call to the algorithm. Avoid allocating the memory in numeric tables intended for results because, in general, it is impossible to accurately estimate the required memory size. If the memory interfaced by the numeric tables is allocated and its amount is insufficient to store the results, the algorithm returns an error.

### Examples¶

Batch Processing:

Note

There is no support for Java on GPU.

Batch Processing:

Batch Processing:

## Performance Considerations¶

To get the best overall performance of the association rules algorithm, whenever possible use the following numeric tables and data types:

• A SOA numeric table of type int to store features.

• A homogenous numeric table of type int to store large item sets, support values, and left-hand-side and right-hand-side parts of association rules.

• A numeric table with the confidence values of the same data type as specified in the algorithmFPType template parameter of the class.

Optimization Notice

Intel’s compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804