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<title>MADlib: Apriori Algorithm</title>
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<div class="title">Apriori Algorithm<div class="ingroups"><a class="el" href="group__grp__unsupervised.html">Unsupervised Learning</a> &raquo; <a class="el" href="group__grp__association__rules.html">Association Rules</a></div></div> </div>
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<div class="toc"><b>Contents</b> </p><ul>
<li>
<a href="#rules">Rules</a> </li>
<li>
<a href="#algorithm">Apriori Algorithm</a> </li>
<li>
<a href="#syntax">Function Syntax</a> </li>
<li>
<a href="#examples">Examples</a> </li>
<li>
<a href="#notes">Notes</a> </li>
<li>
<a href="#related">Related Topics</a> </li>
</ul>
</div><p>This module implements the association rules data mining technique on a transactional data set. Given the names of a table and the columns, minimum support and confidence values, this function generates all single and multidimensional association rules that meet the minimum thresholds.</p>
<p>Association rule mining is a widely used technique for discovering relationships between variables in a large data set (e.g items in a store that are commonly purchased together). The classic market basket analysis example using association rules is the "beer and diapers" rule. According to data mining urban legend, a study of customers' purchase behavior in a supermarket found that men often purchased beer and diapers together. After making this discovery, the managers strategically placed beer and diapers closer together on the shelves and saw a dramatic increase in sales. In addition to market basket analysis, association rules are also used in bioinformatics, web analytics, and several other fields.</p>
<p>This type of data mining algorithm uses transactional data. Every transaction event has a unique identification, and each transaction consists of a set of items (or itemset). Purchases are considered binary (either it was purchased or not), and this implementation does not take into consideration the quantity of each item. For the MADlib association rules function, it is assumed that the data is stored in two columns with one item and transaction id per row. Transactions with multiple items will span multiple rows with one row per item.</p>
<pre>
tran_id | product
---------+---------
1 | 1
1 | 2
1 | 3
1 | 4
2 | 3
2 | 4
2 | 5
3 | 1
3 | 4
3 | 6
...
</pre><p><a class="anchor" id="rules"></a></p><dl class="section user"><dt>Rules</dt><dd></dd></dl>
<p>Association rules take the form "If X, then Y", where X and Y are non-empty itemsets. X and Y are called the antecedent and consequent, or the left-hand- side and right-hand-side, of the rule respectively. Using our previous example, the association rule may state "If {diapers}, then {beer}" with .2 support and .85 confidence.</p>
<p>Given any association rule "If X, then Y", the association rules function will also calculate the following metrics:</p><ul>
<li>Support: The ratio of transactions that contain X to all transactions, T <p class="formulaDsp">
<img class="formulaDsp" alt="\[ S (X) = \frac{Total X}{Total transactions} \]" src="form_0.png"/>
</p>
</li>
<li>Confidence: The ratio of transactions that contain <img class="formulaInl" alt="$ X,Y $" src="form_1.png"/> to transactions that contain <img class="formulaInl" alt="$ X $" src="form_2.png"/>. One could view this metric as the conditional probability of <img class="formulaInl" alt="$ Y $" src="form_3.png"/> , given <img class="formulaInl" alt="$ X $" src="form_2.png"/> . <img class="formulaInl" alt="$ P(Y|X) $" src="form_4.png"/> <p class="formulaDsp">
<img class="formulaDsp" alt="\[ C (X \Rightarrow Y) = \frac{s(X \cap Y )}{s(X)} \]" src="form_5.png"/>
</p>
</li>
<li>Lift: The ratio of observed support of <img class="formulaInl" alt="$ X,Y $" src="form_1.png"/> to the expected support of <img class="formulaInl" alt="$ X,Y $" src="form_1.png"/> , assuming <img class="formulaInl" alt="$ X $" src="form_2.png"/> and <img class="formulaInl" alt="$ Y $" src="form_3.png"/> are independent. <p class="formulaDsp">
<img class="formulaDsp" alt="\[ L (X \Rightarrow Y) = \frac{s(X \cap Y )}{s(X) \cdot s(Y)} \]" src="form_6.png"/>
</p>
</li>
<li><p class="startli">Conviction: The ratio of expected support of <img class="formulaInl" alt="$ X $" src="form_2.png"/> occurring without <img class="formulaInl" alt="$ Y $" src="form_3.png"/> assuming <img class="formulaInl" alt="$ X $" src="form_2.png"/> and <img class="formulaInl" alt="$ \neg Y $" src="form_7.png"/> are independent, to the observed support of <img class="formulaInl" alt="$ X $" src="form_2.png"/> occuring without <img class="formulaInl" alt="$ Y $" src="form_3.png"/>. If conviction is greater than 1, then this metric shows that incorrect predictions ( <img class="formulaInl" alt="$ X \Rightarrow Y $" src="form_8.png"/> ) occur less often than if these two actions were independent. This metric can be viewed as the ratio that the association rule would be incorrect if the actions were independent (i.e. a conviction of 1.5 indicates that if the variables were independent, this rule would be incorrect 50% more often.)</p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ Conv (X \Rightarrow Y) = \frac{1 - S(Y)}{1 - C(X \Rightarrow Y)} \]" src="form_9.png"/>
</p>
</li>
</ul>
<p><a class="anchor" id="algorithm"></a></p><dl class="section user"><dt>Apriori Algorithm</dt><dd></dd></dl>
<p>Although there are many algorithms that generate association rules, the classic algorithm used is called Apriori (which we implemented in this module). It is a breadth-first search, as opposed to depth-first searches like eclat. Frequent itemsets of order <img class="formulaInl" alt="$ n $" src="form_10.png"/> are generated from sets of order <img class="formulaInl" alt="$ n - 1 $" src="form_11.png"/>. Using the downward closure property, all sets must have frequent subsets. There are two steps in this algorithm; generating frequent itemsets, and using these itemsets to construct the association rules. A simplified version of the algorithm is as follows, and assumes a minimum level of support and confidence is provided:</p>
<p><em>Initial</em> <em>step</em> </p><ol type="1">
<li>Generate all itemsets of order 1</li>
<li>Eliminate itemsets that have support is less than minimum support</li>
</ol>
<p><em>Main</em> <em>algorithm</em> </p><ol type="1">
<li>For <img class="formulaInl" alt="$ n \ge 2 $" src="form_12.png"/>, generate itemsets of order <img class="formulaInl" alt="$ n $" src="form_10.png"/> by combining the itemsets of order <img class="formulaInl" alt="$ n - 1 $" src="form_11.png"/>. This is done by doing the union of two itemsets that have identical items except one.</li>
<li>Eliminate itemsets that have (n-1) order subsets with insufficient support</li>
<li>Eliminate itemsets with insufficient support</li>
<li>Repeat until itemsets cannot be generated</li>
</ol>
<p><em>Association</em> <em>rule</em> <em>generation</em> </p>
<p>Given a frequent itemset <img class="formulaInl" alt="$ A $" src="form_13.png"/> generated from the Apriori algorithm, and all subsets <img class="formulaInl" alt="$ B $" src="form_14.png"/> , we generate rules such that <img class="formulaInl" alt="$ B \Rightarrow (A - B) $" src="form_15.png"/> meets minimum confidence requirements.</p>
<p><a class="anchor" id="syntax"></a></p><dl class="section user"><dt>Function Syntax</dt><dd>Association rules can be called with the following syntax. <pre class="syntax">
assoc_rules( support,
confidence,
tid_col,
item_col,
input_table,
output_schema,
verbose
);</pre> This generates all association rules that satisfy the specified minimum <em>support</em> and <em>confidence</em>.</dd></dl>
<p><b>Arguments</b> </p><dl class="arglist">
<dt>support </dt>
<dd><p class="startdd">The minimum level of support needed for each itemset to be included in result.</p>
<p class="enddd"></p>
</dd>
<dt>confidence </dt>
<dd><p class="startdd">The minimum level of confidence needed for each rule to be included in result.</p>
<p class="enddd"></p>
</dd>
<dt>tid_col </dt>
<dd><p class="startdd">The name of the column storing the transaction ids.</p>
<p class="enddd"></p>
</dd>
<dt>item_col </dt>
<dd><p class="startdd">The name of the column storing the products.</p>
<p class="enddd"></p>
</dd>
<dt>input_table </dt>
<dd><p class="startdd">The name of the table containing the input data.</p>
<p>The input data is expected to be of the following form: </p><pre>{TABLE|VIEW} <em>input_table</em> (
<em>trans_id</em> INTEGER,
<em>product</em> TEXT
)</pre><p>The algorithm maps the product names to consecutive integer ids starting at 1. If they are already structured this way, then the ids will not change. </p>
<p class="enddd"></p>
</dd>
<dt>output_schema </dt>
<dd><p class="startdd">The name of the schema where the final results will be stored. It is expected to be created before calling the function, or using <code>NULL</code> suggests the current schema will be used.</p>
<p>The results containing the rules, support, confidence, lift, and conviction are stored in the table <code>assoc_rules</code> in the schema specified by <code>output_schema</code>.</p>
<p>The table has the following columns. </p><table class="output">
<tr>
<th>ruleid </th><td>integer </td></tr>
<tr>
<th>pre </th><td>text </td></tr>
<tr>
<th>post </th><td>text </td></tr>
<tr>
<th>support </th><td>double </td></tr>
<tr>
<th>confidence </th><td>double </td></tr>
<tr>
<th>lift </th><td>double </td></tr>
<tr>
<th>conviction </th><td>double </td></tr>
</table>
<p>On Greenplum Database the table is distributed by the ruleid column.</p>
<p>The <code>pre</code> and <code>post</code> columns are the itemsets of left and right hand sides of the association rule respectively. The <code>support</code>, <code>confidence</code>, <code>lift</code>, and <code>conviction</code> columns are calculated as mentioned in the about section. </p>
<p class="enddd"></p>
</dd>
<dt>verbose </dt>
<dd>BOOLEAN, default FALSE. Determines if the output contains comments. </dd>
</dl>
<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
<p>Let us take a look at some sample transactional data and generate association rules.</p>
<ol type="1">
<li>Create an input dataset. <pre class="example">
DROP TABLE IF EXISTS test_data;
CREATE TABLE test_data (
trans_id INT,
product TEXT
);
INSERT INTO test_data VALUES (1, 'beer');
INSERT INTO test_data VALUES (1, 'diapers');
INSERT INTO test_data VALUES (1, 'chips');
INSERT INTO test_data VALUES (2, 'beer');
INSERT INTO test_data VALUES (2, 'diapers');
INSERT INTO test_data VALUES (3, 'beer');
INSERT INTO test_data VALUES (3, 'diapers');
INSERT INTO test_data VALUES (4, 'beer');
INSERT INTO test_data VALUES (4, 'chips');
INSERT INTO test_data VALUES (5, 'beer');
INSERT INTO test_data VALUES (6, 'beer');
INSERT INTO test_data VALUES (6, 'diapers');
INSERT INTO test_data VALUES (6, 'chips');
INSERT INTO test_data VALUES (7, 'beer');
INSERT INTO test_data VALUES (7, 'diapers');
</pre></li>
<li>Let <img class="formulaInl" alt="$ min(support) = .25 $" src="form_16.png"/> and <img class="formulaInl" alt="$ min(confidence) = .5 $" src="form_17.png"/>, and the output schema be 'myschema'. For this example, we set verbose to TRUE so that we have some insight into the progress of the function. We can now generate association rules as follows: <pre class="example">
SELECT * FROM madlib.assoc_rules( .25,
.5,
'trans_id',
'product',
'test_data',
'myschema',
TRUE
);
</pre> Result: <pre class="result">
output_schema | output_table | total_rules | total_time
---------------+--------------+-------------+-----------------
myschema | assoc_rules | 7 | 00:00:03.162094
(1 row)
</pre> The association rules are stored in the myschema.assoc_rules table: <pre class="example">
SELECT * FROM myschema.assoc_rules
ORDER BY support DESC;
</pre> Result: <pre class="result">
ruleid | pre | post | support | confidence | lift | conviction
--------+-----------------+----------------+-------------------+-------------------+-------------------+-------------------
4 | {diapers} | {beer} | 0.714285714285714 | 1 | 1 | 0
2 | {beer} | {diapers} | 0.714285714285714 | 0.714285714285714 | 1 | 1
1 | {chips} | {beer} | 0.428571428571429 | 1 | 1 | 0
5 | {chips} | {beer,diapers} | 0.285714285714286 | 0.666666666666667 | 0.933333333333333 | 0.857142857142857
6 | {chips,beer} | {diapers} | 0.285714285714286 | 0.666666666666667 | 0.933333333333333 | 0.857142857142857
7 | {chips,diapers} | {beer} | 0.285714285714286 | 1 | 1 | 0
3 | {chips} | {diapers} | 0.285714285714286 | 0.666666666666667 | 0.933333333333333 | 0.857142857142857
(7 rows)
</pre></li>
</ol>
<p><a class="anchor" id="notes"></a></p><dl class="section user"><dt>Notes</dt><dd></dd></dl>
<p>The association rules function always creates a table named <code>assoc_rules</code>. Make a copy of this table before running the function again if you would like to keep multiple association rule tables.</p>
<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd></dd></dl>
<p>File <a class="el" href="assoc__rules_8sql__in.html" title="The assoc_rules function computes association rules for a given set of data. The data is assumed to h...">assoc_rules.sql_in</a> documenting the SQL function. </p>
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