Data Mining Group - Association Rules

PMML 2.1 - Association Rules

The Association Rule model represents rules where some set of items is associated to another set of items. For example a rule can express that a certain product is often bought in combination with a certain set of other products.

The attribute definitions of the association rule model uses the entity ELEMENT-ID in order to express a semantical constraint that a value must be unique in a set of elements (contained in the same XML document) of the same type.

An Association Rule model consists of four major parts:

Model attributes
Items
ItemSets
AssociationRules


  <xs:element name="AssociationModel">
    <xs:complexType>
      <xs:sequence>
        <xs:element minOccurs="0" maxOccurs="unbounded" ref="Extension" />
        <xs:element ref="MiningSchema" />
        <xs:element minOccurs="0" maxOccurs="unbounded" ref="Item" />
        <xs:element minOccurs="0" maxOccurs="unbounded" ref="Itemset" />
        <xs:element minOccurs="0" maxOccurs="unbounded" ref="AssociationRule" />
        <xs:element minOccurs="0" maxOccurs="unbounded" ref="Extension" />
      </xs:sequence>
      <xs:attribute name="modelName" type="xs:string" />
      <xs:attribute name="functionName" type="MINING-FUNCTION" use="required" />
      <xs:attribute name="algorithmName" type="xs:string" />
      <xs:attribute name="numberOfTransactions" type="INT-NUMBER" use="required" />
      <xs:attribute name="maxNumberOfItemsPerTA" type="INT-NUMBER" />
      <xs:attribute name="avgNumberOfItemsPerTA" type="REAL-NUMBER" />
      <xs:attribute name="minimumSupport" type="PROB-NUMBER" use="required" />
      <xs:attribute name="minimumConfidence" type="PROB-NUMBER" use="required" />
      <xs:attribute name="lengthLimit" type="INT-NUMBER" />
      <xs:attribute name="numberOfItems" type="INT-NUMBER" use="required" />
      <xs:attribute name="numberOfItemsets" type="INT-NUMBER" use="required" />
      <xs:attribute name="numberOfRules" type="INT-NUMBER" use="required" />
    </xs:complexType>
  </xs:element>

Here is a description of the attributes:

numberOfTransactions: The number of transactions (baskets of items) contained in the input data.

maxNumberOfItemsPerTA The number of items contained in the largest transaction.

avgNumberOfItemsPerTA: The average number of items contained in a transaction.

minimumSupport: The minimum relative support value (#supporting transactions / #total transactions) satisfied by all rules.

minimumConfidence: The minimum confidence value satisfied by all rules. Confidence is calculated as (support (rule) / support(antecedent)).

lengthLimit: The maximum number of items contained in a rule which was used to limit the number of rules.

numberOfItems: The number of different items contained in the input data.

numberOfItemsets: The number of itemsets contained in the model.

numberOfRules: The number of rules contained in the model.

We consider items next:


  <xs:element name="Item">
    <xs:complexType>
      <xs:attribute name="id" type="xs:string" use="required" />
      <xs:attribute name="value" type="xs:string" use="required" />
      <xs:attribute name="mappedValue" type="xs:string" />
      <xs:attribute name="weight" type="REAL-NUMBER" />
    </xs:complexType>
  </xs:element>

Here is a description of the attributes in a item:

id: An identification to uniquely identify an item.

value: The value of the item as in the input data.

mappedValue: Optional, a value to which the original item value is mapped. For instance, this could be a product name if the original value is an EAN code.

weight : The weight of the item. For example, the price or value of an item.

We consider itemsets next:


  <xs:element name="Itemset">
    <xs:complexType>
      <xs:sequence>
        <xs:element minOccurs="0" maxOccurs="unbounded" ref="ItemRef" />
        <xs:element minOccurs="0" maxOccurs="unbounded" ref="Extension" />
      </xs:sequence>
      <xs:attribute name="id" type="xs:string" use="required" />
      <xs:attribute name="support" type="PROB-NUMBER" />
      <xs:attribute name="numberOfItems" type="INT-NUMBER" />
    </xs:complexType>
  </xs:element>

Here is a description of the attributes in a item:

id: An identification to uniquely identify an itemset

support: The relative support of the itemset

numberOfItems: The number of items contained in this itemset

ItemRef: Item references to point to elements of type item.


  <xs:element name="ItemRef">
    <xs:complexType>
      <xs:attribute name="itemRef" type="xs:string" use="required" />
    </xs:complexType>
  </xs:element>

The attribute itemRef is defined above.

We consider association rules of the form "<antecedent itemset> => <consequent itemset>" next:


  <xs:element name="AssociationRule">
    <xs:complexType>
      <xs:sequence>
        <xs:element minOccurs="0" maxOccurs="unbounded" ref="Extension" />
      </xs:sequence>
      <xs:attribute name="support" type="PROB-NUMBER" use="required" />
      <xs:attribute name="confidence" type="PROB-NUMBER" use="required" />
      <xs:attribute name="antecedent" type="xs:string" use="required" />
      <xs:attribute name="consequent" type="xs:string" use="required" />
    </xs:complexType>
  </xs:element>

Here is a description of the attributes in an AssociationRule:

support: The relative support of the rule

confidence: The confidence of the rule

antecedent: The id value of the itemset which is the antecedent of the rule

consequent: The id value of the itemset which is the consequent of the rule

Here is an example of an association model:


     <?xml version="1.0" ?>
     <PMML version="2.0" >
     <Header copyright="www.dmg.org"
          description="example model for association rules"/>
     <DataDictionary numberOfFields="2" >
     <DataField name="transaction" optype="categorical" />
     <DataField name="item" optype="categorical" />
     </DataDictionary>
     <AssociationModel
         functionName="associationRules"
         numberOfTransactions="4" numberOfItems="3"
         minimumSupport="0.6"     minimumConfidence="0.5"
         numberOfItemsets="3"     numberOfRules="2">
         <MiningSchema>
                <MiningField name="transaction"/>
                <MiningField name="item"/>
         </MiningSchema>

     <!-- We have three items in our input data -->
     <Item id="1" value="Cracker" />
     <Item id="2" value="Coke" />
     <Item id="3" value="Water" />

     <!-- and two frequent itemsets with a single item -->

     <Itemset id="1" support="1.0" numberOfItems="1">
        <ItemRef itemRef="1" />
     </Itemset>

     <Itemset id="2" support="1.0" numberOfItems="1">
        <ItemRef itemRef="3" />
     </Itemset>

     <!-- and one frequent itemset with two items. -->

     <Itemset id="3" support="1.0" numberOfItems="2">
        <ItemRef itemRef="1" />
        <ItemRef itemRef="3" />
     </Itemset>


     <!-- Two rules satisfy the requirements -->

     <AssociationRule support="1.0" confidence="1.0"
                      antecedent="1" consequent="2" />

     <AssociationRule support="1.0" confidence="1.0"
                      antecedent="2" consequent="1" />

    </AssociationModel>
    </PMML>