association-rule.html

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    <h1 id="association-rule-top" class="title">Association Rule Mining</h1>

    <p>Association rule learning is a popular and well researched method for
        discovering interesting relations between variables in large databases.
        Let I = {i<sub>1</sub>, i<sub>2</sub>,..., i<sub>n</sub>} be a set of n
        binary attributes called items. Let D = {t<sub>1</sub>, t<sub>2</sub>,..., t<sub>m</sub>}
        be a set of transactions called the database. Each transaction in D has a
        unique transaction ID and contains a subset of the items in I.
        An association rule is defined as an implication of the form X &rArr; Y
        where X, Y &sube; I and X &cap; Y = &Oslash;. The item sets X and Y are called
        antecedent (left-hand-side or LHS) and consequent (right-hand-side or RHS)
        of the rule, respectively. The support supp(X) of an item set X is defined as
        the proportion of transactions in the database which contain the item set.
        Note that the support of an association rule X &rArr; Y is supp(X &cup; Y).
        The confidence of a rule is defined conf(X &rArr; Y) = supp(X &cup; Y) / supp(X).
        Confidence can be interpreted as an estimate of the probability P(Y | X),
        the probability of finding the RHS of the rule in transactions under the
        condition that these transactions also contain the LHS.</p>

    <p>For example, the rule {onions, potatoes} &rArr; {burger} found in the sales
        data of a supermarket would indicate that if a customer buys onions and
        potatoes together, he or she is likely to also buy burger. Such information
        can be used as the basis for decisions about marketing activities such as
        promotional pricing or product placements.</p>

    <p>Association rules are usually required to satisfy a user-specified minimum
        support and a user-specified minimum confidence at the same time. Association
        rule generation is usually split up into two separate steps:</p>

    <ul>
        <li>First, minimum support is applied to find all frequent item sets
            in a database (i.e. frequent item set mining).

        <li> Second, these frequent item sets and the minimum confidence constraint
        are used to form rules.
    </ul>

    <h2 id="fim">Frequent Itemset Mining</h2>

    <p>Finding all frequent item sets in a database is difficult since it involves
        searching all possible item sets (item combinations). The set of possible
        item sets is the power set over I (the set of items) and has size 2<sup>n</sup> - 1
        (excluding the empty set which is not a valid item set). Although the size
        of the power set grows exponentially in the number of items n in I, efficient
        search is possible using the downward-closure property of support
        (also called anti-monotonicity) which guarantees that for a frequent item set
        also all its subsets are frequent and thus for an infrequent item set, all
        its supersets must be infrequent.</p>

    <p>In practice, we may only consider the frequent item set that has the maximum
        number of items bypassing all the sub item sets. An item set is maximal
        frequent if none of its immediate supersets is frequent.</p>

    <p>For a maximal frequent item set, even though we know that all the sub item
        sets are frequent, we don't know the actual support of those sub item sets,
        which are very important to find the association rules within the item sets.
        If the final goal is association rule mining, we would like to discover
        closed frequent item sets. An item set is closed if none of its immediate
        supersets has the same support as the item set.</p>

    <p>Some well known algorithms of frequent item set mining are Apriori,
        Eclat and FP-Growth. Apriori is the best-known algorithm to mine association
        rules. It uses a breadth-first search strategy to counting the support of
        item sets and uses a candidate generation function which exploits the downward
        closure property of support. Eclat is a depth-first search algorithm using
        set intersection.</p>

    <p>FP-growth (frequent pattern growth) algorithm employs an extended prefix-tree (FP-tree) structure to
        store the database in a compressed form. The FP-growth algorithm is
        currently one of the fastest approaches to discover frequent item sets.
        FP-growth adopts a divide-and-conquer approach to decompose both the mining
        tasks and the databases. It uses a pattern fragment growth method to avoid
        the costly process of candidate generation and testing used by Apriori.</p>

    <p>The basic idea of the FP-growth algorithm can be described as a
        recursive elimination scheme: in a preprocessing step delete
        all items from the transactions that are not frequent individually,
        i.e., do not appear in a user-specified minimum
        number of transactions. Then select all transactions that
        contain the least frequent item (least frequent among those
        that are frequent) and delete this item from them. Recurse
        to process the obtained reduced (also known as projected)
        database, remembering that the item sets found in the recursion
        share the deleted item as a prefix. On return, remove
        the processed item from the database of all transactions
        and start over, i.e., process the second frequent item etc. In
        these processing steps the prefix tree, which is enhanced by
        links between the branches, is exploited to quickly find the
        transactions containing a given item and also to remove this
        item from the transactions after it has been processed.</p>

    <p>When the input itemsets are already in memory, the below methods can be used.
        The parameter <code>itemsets</code> is the item set database,
        where each row is a item set, which may have different length.
        The item identifiers have to be in [0, n), where n is the number of items.
        Item set should NOT contain duplicated items. Note that it is reordered after the call.
        The parameter <code>minSupport</code> is the required minimum support of item sets in terms
        of frequency. The return can be the list of frequent item sets. Often the output
        is too big to be stored in memory. In that case, the user may provide
        a <code>PrintStream</code> or file path to save the output. The return number is
        the number of discovered frequent item sets.</p>

    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_1" data-toggle="tab">Scala</a></li>
        <li><a href="#java_1" data-toggle="tab">Java</a></li>
        <li><a href="#kotlin_1" data-toggle="tab">Kotlin</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_1">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    def fpgrowth(minSupport: Int, itemsets: Array[Array[Int]]): Stream[ItemSet]
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_1">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    public class FPTree {
        public static FPTree of(int minSupport, int[][] itemsets);
        public static FPTree of(double minSupport, int[][] itemsets);
    }

    public class FPGrowth {
        public static Stream&lt;ItemSet&gt; apply(FPTree tree);
    }
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="kotlin_1">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-kotlin"><code>
    fun fpgrowth(minSupport: Int, itemsets: Array&lt;IntArray&gt;): Stream&lt;ItemSet&gt;
    </code></pre>
            </div>
        </div>
    </div>

    <p>In practice, even the raw input is often too big to fit into the memory. To conquer this
        challenge, the below methods scan the input file twice.
        We first scan the database to obtains the frequency of
        single items. Then we scan the data again to construct the FP-Tree, which
        is a compressed form of data.
        In this way, we don't need load the whole database into the main memory.</p>

    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_2" data-toggle="tab">Scala</a></li>
        <li><a href="#java_2" data-toggle="tab">Java</a></li>
        <li><a href="#kotlin_2" data-toggle="tab">Kotlin</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_2">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    def fptree(minSupport: Int, supplier: Supplier[Stream[Array[Int]]]): FPTree

    def fpgrowth(tree: FPTree): Stream[ItemSet]
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_2">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    public class FPTree {
        public static FPTree of(int minSupport, Supplier&lt;Stream&lt;int[]&gt;&gt; supplier);
        public static FPTree of(double minSupport, Supplier&lt;Stream&lt;int[]&gt;&gt; supplier);
    }
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="kotlin_2">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-kotlin"><code>
    fun fptree(minSupport: Int, supplier: Supplier&lt;Stream&lt;IntArray&gt;&gt;): FPTree

    fun fpgrowth(tree: FPTree): Stream&lt;ItemSet&gt;
          </code></pre>
            </div>
        </div>
    </div>

    <p>In the below example, we apply FP-Growth to a toy data.</p>
    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_3" data-toggle="tab">Scala</a></li>
        <li><a href="#java_3" data-toggle="tab">Java</a></li>
        <li><a href="#kotlin_3" data-toggle="tab">Kotlin</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_3">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    smile> val itemsets = Array(
        Array(1, 3),
        Array(2),
        Array(4),
        Array(2, 3, 4),
        Array(2, 3),
        Array(2, 3),
        Array(1, 2, 3, 4),
        Array(1, 3),
        Array(1, 2, 3),
        Array(1, 2, 3)
    )

    smile> fpgrowth(3, itemsets).forEach { itemset => println(itemset) }
    4 (3)
    1 (5)
    2 1 (3)
    3 2 1 (3)
    3 1 (5)
    2 (7)
    3 2 (6)
    3 (8)
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_3">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    jshell> import smile.association.*

    jshell> int[][] itemsets = {
       ...>         {1, 3},
       ...>         {2},
       ...>         {4},
       ...>         {2, 3, 4},
       ...>         {2, 3},
       ...>         {2, 3},
       ...>         {1, 2, 3, 4},
       ...>         {1, 3},
       ...>         {1, 2, 3},
       ...>         {1, 2, 3}
       ...>     }
    itemsets ==> int[10][] { int[2] { 1, 3 }, int[1] { 2 }, int[1] ...  3 }, int[3] { 1, 2, 3 } }

    jshell> var tree = FPTree.of(3, itemsets)
    tree ==> smile.association.FPTree@2a7b6f69

    jshell> FPGrowth.apply(tree).forEach(itemset -> System.out.println(itemset))
    4 (3)
    1 (5)
    2 1 (3)
    3 2 1 (3)
    3 1 (5)
    2 (7)
    3 2 (6)
    3 (8)
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="kotlin_3">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-kotlin"><code>
    >>> val itemsets = arrayOf(
        intArrayOf(1, 3),
        intArrayOf(2),
        intArrayOf(4),
        intArrayOf(2, 3, 4),
        intArrayOf(2, 3),
        intArrayOf(2, 3),
        intArrayOf(1, 2, 3, 4),
        intArrayOf(1, 3),
        intArrayOf(1, 2, 3),
        intArrayOf(1, 2, 3)
    )

    >>> fpgrowth(3, itemsets).forEach { itemset -> println(itemset) }
    4 (3)
    1 (5)
    2 1 (3)
    3 2 1 (3)
    3 1 (5)
    2 (7)
    3 2 (6)
    3 (8)
    </code></pre>
            </div>
        </div>
    </div>

    <p>Each row is a frequent item set, ending with the frequency in parenthesis.
        The sample data directory <code>data/transaction</code> contains a couple of
        large benchmark datasets.</p>

    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_4" data-toggle="tab">Scala</a></li>
        <li><a href="#java_4" data-toggle="tab">Java</a></li>
        <li><a href="#kotlin_4" data-toggle="tab">Kotlin</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_4">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    smile> val tree = fptree(1000, () => {
      smile.util.Paths.getTestDataLines("transaction/kosarak.dat").map { line =>
          line.split(" ").map(_.toInt).toArray
      }
    })

    smile> fpgrowth(tree).limit(10).forEach { itemset => println(itemset) }
    5634 (1000)
    3805 (1001)
    3376 (1001)
    2279 (1001)
    6333 (1002)
    243 (1002)
    808 (1003)
    3875 (1004)
    2265 (1004)
    996 (1004)
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_4">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    jshell> var tree = FPTree.of(1000, () -> {
       ...>         try {
       ...>             return smile.util.Paths.getTestDataLines("transaction/kosarak.dat").
       ...>                 map(line -> Arrays.stream(line.split(" ")).
       ...>                     mapToInt(Integer::valueOf).
       ...>                     toArray()
       ...>                 );
       ...>         } catch (IOException ex) {
       ...>             ex.printStackTrace();
       ...>         }
       ...>
       ...>         return Stream.empty();
       ...>     })
    tree ==> smile.association.FPTree@3dddbe65

    jshell> FPGrowth.apply(tree).limit(10).forEach(itemset -> System.out.println(itemset))
    5634 (1000)
    3805 (1001)
    3376 (1001)
    2279 (1001)
    6333 (1002)
    243 (1002)
    808 (1003)
    3875 (1004)
    2265 (1004)
    996 (1004)
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="kotlin_4">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-kotlin"><code>
    >>> class Parser : Supplier&lt;Stream&lt;IntArray&gt;&gt; {
            override fun get(): Stream&lt;IntArray&gt; {  
               return smile.util.Paths.getTestDataLines("transaction/kosarak.dat").map { line ->
                   line.split(" ").map({ w -> w.toInt() }).toIntArray()
               }
            }
        }
    >>> val tree = fptree(1000, Parser())

    >>> fpgrowth(tree).limit(10).forEach { itemset -> println(itemset) }
    5634 (1000)
    3805 (1001)
    3376 (1001)
    2279 (1001)
    6333 (1002)
    243 (1002)
    808 (1003)
    3875 (1004)
    2265 (1004)
    996 (1004)
    </code></pre>
            </div>
        </div>
    </div>

    <p>The data <code>kosarak.data</code> has 990,002 transactions.
        With minimum support 1000, our implementation generates 711,424
        frequent item sets in about 30 seconds on a modern laptop.</p>

    <h2 id="arm">Association Rules</h2>

    <p>After mining frequent itemsets, it is straightforward to
        form association rules that meet minimum confidence constraint.
        Our implementation generates association rules in a storage efficient way
        by employing the total support tree that is a kind of compressed set enumeration tree.</p>

    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_5" data-toggle="tab">Scala</a></li>
        <li><a href="#java_5" data-toggle="tab">Java</a></li>
        <li><a href="#kotlin_5" data-toggle="tab">Kotlin</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_5">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    def arm(minSupport: Int, confidence: Double, itemsets: Array[Array[Int]]): Stream[AssociationRule]

    def arm(confidence: Double, tree: FPTree): Stream[AssociationRule]
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_5">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    public class ARM {
        public static Stream&lt;AssociationRule&gt; apply(double confidence, FPTree tree);
    }
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="kotlin_5">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-kotlin"><code>
    def arm(minSupport: Int, confidence: Double, itemsets: Array&lt;IntArray&gt;): Stream&lt;AssociationRule&gt;

    def arm(confidence: Double, tree: FPTree): Stream&lt;AssociationRule&gt;
    </code></pre>
            </div>
        </div>
    </div>

    <p>The API is similar to <code>fpgrowth</code> except the additional parameter <code>confidence</code>.</p>

    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_6" data-toggle="tab">Scala</a></li>
        <li><a href="#java_6" data-toggle="tab">Java</a></li>
        <li><a href="#kotlin_6" data-toggle="tab">Kotlin</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_6">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    smile> arm(0.6, tree).limit(10).forEach { rule => println(rule) }
    (11) => (6) support = 32.73% confidence = 89.00% lift = 1.47 leverage = 0.1039
    (3, 11) => (6) support = 14.51% confidence = 89.09% lift = 1.47 leverage = 0.0462
    (1) => (6) support = 13.34% confidence = 66.89% lift = 1.10 leverage = 0.0123
    (3, 1) => (6) support = 5.84% confidence = 68.28% lift = 1.12 leverage = 0.0064
    (6, 1) => (11) support = 8.70% confidence = 65.17% lift = 1.77 leverage = 0.0379
    (11, 1) => (6) support = 8.70% confidence = 93.70% lift = 1.54 leverage = 0.0306
    (6, 3, 1) => (11) support = 3.81% confidence = 65.30% lift = 1.78 leverage = 0.0167
    (3, 11, 1) => (6) support = 3.81% confidence = 93.73% lift = 1.54 leverage = 0.0134
    (218) => (6) support = 7.85% confidence = 87.67% lift = 1.44 leverage = 0.0241
    (3, 218) => (6) support = 3.43% confidence = 87.86% lift = 1.45 leverage = 0.0106
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_6">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    jshell> ARM.apply(0.6, tree).limit(10).forEach(rule -> System.out.println(rule))
    (11) => (6) support = 32.73% confidence = 89.00% lift = 1.47 leverage = 0.1039
    (3, 11) => (6) support = 14.51% confidence = 89.09% lift = 1.47 leverage = 0.0462
    (1) => (6) support = 13.34% confidence = 66.89% lift = 1.10 leverage = 0.0123
    (3, 1) => (6) support = 5.84% confidence = 68.28% lift = 1.12 leverage = 0.0064
    (6, 1) => (11) support = 8.70% confidence = 65.17% lift = 1.77 leverage = 0.0379
    (11, 1) => (6) support = 8.70% confidence = 93.70% lift = 1.54 leverage = 0.0306
    (6, 3, 1) => (11) support = 3.81% confidence = 65.30% lift = 1.78 leverage = 0.0167
    (3, 11, 1) => (6) support = 3.81% confidence = 93.73% lift = 1.54 leverage = 0.0134
    (218) => (6) support = 7.85% confidence = 87.67% lift = 1.44 leverage = 0.0241
    (3, 218) => (6) support = 3.43% confidence = 87.86% lift = 1.45 leverage = 0.0106
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="kotlin_6">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-kotlin"><code>
    >>> arm(0.6, tree).limit(10).forEach { rule -> println(rule) }
    (11) => (6) support = 32.73% confidence = 89.00% lift = 1.47 leverage = 0.1039
    (3, 11) => (6) support = 14.51% confidence = 89.09% lift = 1.47 leverage = 0.0462
    (1) => (6) support = 13.34% confidence = 66.89% lift = 1.10 leverage = 0.0123
    (3, 1) => (6) support = 5.84% confidence = 68.28% lift = 1.12 leverage = 0.0064
    (6, 1) => (11) support = 8.70% confidence = 65.17% lift = 1.77 leverage = 0.0379
    (11, 1) => (6) support = 8.70% confidence = 93.70% lift = 1.54 leverage = 0.0306
    (6, 3, 1) => (11) support = 3.81% confidence = 65.30% lift = 1.78 leverage = 0.0167
    (3, 11, 1) => (6) support = 3.81% confidence = 93.73% lift = 1.54 leverage = 0.0134
    (218) => (6) support = 7.85% confidence = 87.67% lift = 1.44 leverage = 0.0241
    (3, 218) => (6) support = 3.43% confidence = 87.86% lift = 1.45 leverage = 0.0106
    </code></pre>
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        </div>
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