assoc_rules.sql_in

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/* ----------------------------------------------------------------------- *//**
 *
 * @file assoc_rules.sql_in
 *
 * @brief The \ref assoc_rules function computes association rules for a given set of data. The data is assumed to have two dimensions; items (between which we are trying to discover associations), and a transaction id. This tranaction id groups the items by event and could also be a user id, date, etc. depending on the context of the data. This function assumes the data is stored in two columns with one transaction id and one item per row.

 * @date June 2011
 * @modified August 2012
 *
 * @sa For a brief introduction to the association rules implementation, see the module
 * description \ref grp_assoc_rules.
 *
 *//* ----------------------------------------------------------------------- */

m4_include(`SQLCommon.m4')


/**
@addtogroup grp_assoc_rules

@about
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.

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.

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.

<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>

\b Rules

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.

Given any association rule "If X, then Y", the association rules function will also calculate the following metrics:
- Support: The ratio of transactions that contain X to all transactions, T
\f[
S (X) = \frac{Total X}{Total transactions}
\f]

- Confidence: The ratio of transactions that contain \f$ X,Y \f$ to transactions that contain \f$ X \f$. One could view this metric as the conditional probability of \f$ Y \f$ , given \f$ X \f$ . \f$ P(Y|X) \f$

\f[
C (X \Rightarrow Y) = \frac{s(X \cap Y )}{s(X)}
\f]

- Lift: The ratio of observed support of \f$ X,Y \f$ to the expected support of \f$ X,Y \f$ , assuming \f$ X \f$ and \f$ Y \f$ are independent.
\f[
L (X \Rightarrow Y) = \frac{s(X \cap Y )}{s(X) \cdot s(Y)}
\f]

- Conviction: The ratio of expected support of \f$ X \f$ occurring without\f$ Y \f$ assuming \f$ X \f$ and \f$ \neg Y \f$ are independent, to the observed support of \f$ X \f$ occuring without \f$ Y \f$. If conviction is greater than 1, then this metric shows that incorrect predictions ( \f$ X \Rightarrow Y \f$ ) 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.)

\f[
Conv (X \Rightarrow Y) = \frac{1 - S(Y)}{1 - C(X \Rightarrow Y)}
\f]


\b Apriori \b algorithm

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 \f$ n \f$ are generated from sets of order \f$ n - 1 \f$. 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:

\e Initial \e step
-# Generate all itemsets of order 1
-# Eliminate itemsets that have support is less than minimum support

\e Main \e algorithm
-# For \f$ n \ge 2 \f$, generate itemsets of order \f$ n \f$ by combining the itemsets of order \f$ n - 1 \f$. This is done by doing the union of two itemsets that have identical items except one.
-# Eliminate itemsets that have (n-1) order subsets with insufficient support
-# Eliminate itemsets with insufficient support
-# Repeat until itemsets cannot be generated

\e Association \e rule \e generation

Given a frequent itemset \f$ A \f$ generated from the Apriori algorithm, and all subsets \f$ B \f$ , we generate rules such that \f$ B \Rightarrow (A - B) \f$ meets minimum confidence requirements.

@input

The input data is expected to be of the following form:
<pre>{TABLE|VIEW} <em>input_table</em> (
    <em>trans_id</em> INTEGER,
    <em>product</em> TEXT
)</pre>

The algorithm will map the product names to consective integer ids starting at 1. If they are already structured this way, then the ids will not change.

@usage
- Association rules can be called by:
  <pre>SELECT \ref assoc_rules(
    <em>support</em>, <em>confidence</em>,'<em>tid_col</em>','<em>item_col</em>',
    '<em>input_table</em>','<em>output_schema</em>', <em> verbose </em>
    );</pre>
  This will generate all association rules that meet a minimum support of <em>support</em> and confidence of <em>confidence</em>.

- The results containing the rules, support, confidence, lift, and conviction are stored in the table assoc_rules in the schema specified by <em>output_schema</em>.
<pre>
    Table "output_schema.assoc_rules"
      Column   |       Type       | Modifiers 
   ------------+------------------+-----------
    ruleid     | integer          | 
    pre        | text[]           | 
    post       | text[]           | 
    support    | double precision | 
    confidence | double precision | 
    lift       | double precision | 
    conviction | double precision | 
    Distributed by: (ruleid)

</pre>
    The \c pre and \c post are the itemsets of left and right hand sides of the association rule respectively. The \c support, \c confidence, \c lift, and \c conviction columns are calculated as mentioned in the about section.

@implementation

The association rules function will always create a table named assoc_rules. Please make a copy of this table before running the function again if you would like to keep multiple association rule tables.

@examp

Let us take a look at some sample transactional data and generate association rules:

\code
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');

\endcode

Let \f$ min(support) = .25 \f$ and \f$ min(confidence) = .5 \f$, and the output schema be 'myschema'. For this example, we will set verbose to 'true' so that we have some insight into the progress of the function. We can now generate association rules as follows:

\code

SELECT * FROM MADLIB_SCHEMA.assoc_rules (.25, .5, 'trans_id', 'product', 'test_data','myschema', false);

\endcode

This should generate this output:

\code

 output_schema | output_table | total_rules | total_time      
---------------+--------------+-------------+-----------------
 myschema      | assoc_rules  |           7 | 00:00:03.162094
(1 row)


\endcode

The association rules are stored in the myschema.assoc_rules:

\code
select * from myschema.assoc_rules order by support desc;
 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)

\endcode

@sa File assoc_rules.sql_in documenting the SQL function.

*/

/*
 * @brief The result data type for the association rule API
 *
 * output_schema the name of the output schema.
 * output_table the name of the output table.
 * total_rules the number of total rules.
 * total_time the running time.
 */
CREATE TYPE MADLIB_SCHEMA.assoc_rules_results AS
    (
    output_schema TEXT,
    output_table TEXT,
    total_rules INT,
    total_time INTERVAL
);


/*
 * @brief Given the text form of a closed frequent pattern (cfp), this function
 * generates the association rules for that pattern. We use text format
 * because text values are hash joinable. The output is a set of text
 * array. For example, assuming the input pattern is '1,2,3'.
 * The result rules:
 * array['1', '2,3']
 * array['2', '1,3']
 * array['3', '1,2']
 * array['1,2', '3']
 * array['1,3', '2']
 * array['2,3', '1']
 * Note that two meaningless rules will be excluded:
 * array['1,2,3', NULL]
 * array[NULL, '1,2,3']
 *
 * @param arg 1 The text form of a closed frequent pattern.
 * @param arg 2 The number of items in the pattern.
 *
 * @return A set of text array. Each array has two elements, corresponding to
 * the left and right parts of an association rule.
 *
 */
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.gen_rules_from_cfp
    (
    TEXT,
    INT
    )
RETURNS SETOF TEXT[] AS 'MODULE_PATHNAME'
LANGUAGE C STRICT IMMUTABLE;


/**
 *
 * @param support minimum level of support needed for each itemset to
 * be included in result
 * @param confidence minimum level of confidence needed for each rule to
 * be included in result
 * @param tid_col name of the column storing the transaction ids
 * @param item_col name of the column storing the products
 * @param input_table name of the table where the data is stored
 * @param output_schema name of the schema where the final results will be stored
 * @param verbose determining if output contains comments
 *
 * @returns The schema and table name containing association rules,
 * and total number of rules found.
 *
 * This function computes the association rules between products in a data set.
 * It reads the name of the table, the column names of the product and ids, and
 * computes ssociation rules using the Apriori algorithm, and subject to the
 * support and confidence constraints as input by the user. This version of
 * association rules has verbose functionality. When verbose is true, output of
 * function includes iteration steps and comments on Apriori algorithm steps.
 *
 */
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.assoc_rules
    (
    support FLOAT8,
    confidence FLOAT8,
    tid_col TEXT,
    item_col TEXT,
    input_table TEXT,
    output_schema TEXT,
    verbose BOOLEAN
   )
RETURNS MADLIB_SCHEMA.assoc_rules_results
AS $$

    PythonFunctionBodyOnly(`assoc_rules', `assoc_rules')

    plpy.execute("SET client_min_messages = error;")

    # schema_madlib comes from PythonFunctionBodyOnly
    return assoc_rules.assoc_rules(
        schema_madlib,
        support,
        confidence,
        tid_col,
        item_col,
        input_table,
        output_schema,
        verbose
        );

$$ LANGUAGE plpythonu;


/**
 *
 * @brief The short form of the above function with vobose removed.
 *
 */
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.assoc_rules
    (
    support FLOAT8,
    confidence FLOAT8,
    tid_col TEXT,
    item_col TEXT,
    input_table TEXT,
    output_schema TEXT
    )
RETURNS MADLIB_SCHEMA.assoc_rules_results
AS $$

    PythonFunctionBodyOnly(`assoc_rules', `assoc_rules')

    plpy.execute("SET client_min_messages = error;")

    # schema_madlib comes from PythonFunctionBodyOnly
    return assoc_rules.assoc_rules(
        schema_madlib,
        support,
        confidence,
        tid_col,
        item_col,
        input_table,
        output_schema,
        False
        );

$$ LANGUAGE plpythonu;