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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 customer 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> + trans_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>The following metrics are defined for any given itemset "X".</p><ul> +<li>Count: The number of transactions that contain X</li> +<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> +</ul> +<p>Given any association rule "If X, then Y", the association rules function will also calculate the following metrics:</p><ul> +<li>Count: The number of transactions that contain X,Y</li> +<li>Support: The ratio of transactions that contain X,Y to all transactions, T <p class="formulaDsp"> +<img class="formulaDsp" alt="\[ S (X \Rightarrow Y) = \frac{Total(X \cup Y)}{Total transactions} \]" src="form_1.png"/> +</p> +</li> +<li>Confidence: The ratio of transactions that contain <img class="formulaInl" alt="$ X,Y $" src="form_2.png"/> to transactions that contain <img class="formulaInl" alt="$ X $" src="form_3.png"/>. One could view this metric as the conditional probability of <img class="formulaInl" alt="$ Y $" src="form_4.png"/> , given <img class="formulaInl" alt="$ X $" src="form_3.png"/> . <img class="formulaInl" alt="$ P(Y|X) $" src="form_5.png"/> <p class="formulaDsp"> +<img class="formulaDsp" alt="\[ C (X \Rightarrow Y) = \frac{s(X \cap Y )}{s(X)} \]" src="form_6.png"/> +</p> +</li> +<li>Lift: The ratio of observed support of <img class="formulaInl" alt="$ X,Y $" src="form_2.png"/> to the expected support of <img class="formulaInl" alt="$ X,Y $" src="form_2.png"/> , assuming <img class="formulaInl" alt="$ X $" src="form_3.png"/> and <img class="formulaInl" alt="$ Y $" src="form_4.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_7.png"/> +</p> +</li> +<li><p class="startli">Conviction: The ratio of expected support of <img class="formulaInl" alt="$ X $" src="form_3.png"/> occurring without <img class="formulaInl" alt="$ Y $" src="form_4.png"/> assuming <img class="formulaInl" alt="$ X $" src="form_3.png"/> and <img class="formulaInl" alt="$ \neg Y $" src="form_8.png"/> are independent, to the observed support of <img class="formulaInl" alt="$ X $" src="form_3.png"/> occuring without <img class="formulaInl" alt="$ Y $" src="form_4.png"/>. If conviction is greater than 1, then this metric shows that incorrect predictions ( <img class="formulaInl" alt="$ X \Rightarrow Y $" src="form_9.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_10.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 is called Apriori [1] which we have 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 $" src="form_11.png"/> are generated from sets of order <img class="formulaInl" alt="$ n - 1 $" src="form_12.png"/>. Using the downward closure property, all sets must have frequent subsets. There are two steps in this algorithm; generating frequent itemsets, and using these itemsets to construct the association rules. A simplified version of the algorithm is as follows, and assumes a minimum level of support and confidence is provided:</p> +<p><em>Initial</em> <em>step</em> </p><ol type="1"> +<li>Generate all itemsets of order 1.</li> +<li>Eliminate itemsets that have support less than minimum support.</li> +</ol> +<p><em>Main</em> <em>algorithm</em> </p><ol type="1"> +<li>For <img class="formulaInl" alt="$ n \ge 2 $" src="form_13.png"/>, generate itemsets of order <img class="formulaInl" alt="$ n $" src="form_11.png"/> by combining the itemsets of order <img class="formulaInl" alt="$ n - 1 $" src="form_12.png"/>. This is done by doing the union of two itemsets that have identical items except one.</li> +<li>Eliminate itemsets that have (n-1) order subsets with insufficient support.</li> +<li>Eliminate itemsets with insufficient support.</li> +<li>Repeat until itemsets cannot be generated, or maximum itemset size is exceeded.</li> +</ol> +<p><em>Association</em> <em>rule</em> <em>generation</em> </p> +<p>Given a frequent itemset <img class="formulaInl" alt="$ A $" src="form_14.png"/> generated from the Apriori algorithm, and all subsets <img class="formulaInl" alt="$ B $" src="form_15.png"/> , we generate rules such that <img class="formulaInl" alt="$ B \Rightarrow (A - B) $" src="form_16.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 has the following syntax: <pre class="syntax"> +assoc_rules( support, + confidence, + tid_col, + item_col, + input_table, + output_schema, + verbose, + max_itemset_size + );</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">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">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">Name of the column storing the transaction ids.</p> +<p class="enddd"></p> +</dd> +<dt>item_col </dt> +<dd><p class="startdd">Name of the column storing the products.</p> +<p class="enddd"></p> +</dd> +<dt>input_table </dt> +<dd><p class="startdd">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. The schema must be created before calling the function. Alternatively, use <code>NULL</code> to output to the current schema.</p> +<p>The results containing the rules, support, count, 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>count </th><td>integer </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 or Apache HAWQ, the table is distributed by the <code>ruleid</code> 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 described earlier. </p> +<p class="enddd"></p> +</dd> +<dt>verbose </dt> +<dd><p class="startdd">BOOLEAN, default: FALSE. Determines if details are printed for each iteration as the algorithm progresses.</p> +<p class="enddd"></p> +</dd> +<dt>max_itemset_size </dt> +<dd>INTEGER, default: generate itemsets of all sizes. Determines the maximum size of frequent itemsets that are used for generating association rules. Must be 2 or more. This parameter can be used to reduce run time for data sets where itemset size is large. </dd> +</dl> +<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl> +<p>Let's 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 $" src="form_17.png"/> and <img class="formulaInl" alt="$ min(confidence) = .5 $" src="form_18.png"/>, and the output schema is set to <code>NULL</code> indicating output to the current schema. In this example we set verbose to TRUE so that we have some insight into progress of the function. We can now generate association rules as follows: <pre class="example"> +SELECT * FROM madlib.assoc_rules( .25, -- Support + .5, -- Confidence + 'trans_id', -- Transaction id col + 'product', -- Product col + 'test_data', -- Input data + NULL, -- Output schema + TRUE -- Verbose output + ); +</pre> Result (iteration details not shown): <pre class="result"> + output_schema | output_table | total_rules | total_time +---------------+--------------+-------------+----------------- + public | assoc_rules | 7 | 00:00:00.569254 +(1 row) +</pre> The association rules are stored in the assoc_rules table: <pre class="example"> +SELECT * FROM assoc_rules +ORDER BY support DESC, confidence DESC; +</pre> Result: <pre class="result"> + ruleid | pre | post | count | support | confidence | lift | conviction +--------+-----------------+----------------+-------+-------------------+-------------------+-------------------+------------------- + 2 | {diapers} | {beer} | 5 | 0.714285714285714 | 1 | 1 | 0 + 6 | {beer} | {diapers} | 5 | 0.714285714285714 | 0.714285714285714 | 1 | 1 + 5 | {chips} | {beer} | 3 | 0.428571428571429 | 1 | 1 | 0 + 4 | {chips,diapers} | {beer} | 2 | 0.285714285714286 | 1 | 1 | 0 + 1 | {chips} | {diapers,beer} | 2 | 0.285714285714286 | 0.666666666666667 | 0.933333333333333 | 0.857142857142857 + 7 | {chips} | {diapers} | 2 | 0.285714285714286 | 0.666666666666667 | 0.933333333333333 | 0.857142857142857 + 3 | {beer,chips} | {diapers} | 2 | 0.285714285714286 | 0.666666666666667 | 0.933333333333333 | 0.857142857142857 +(7 rows) +</pre></li> +<li>Limit association rules generated from itemsets of size at most 2: <pre class="example"> +SELECT * FROM madlib.assoc_rules( .25, -- Support + .5, -- Confidence + 'trans_id', -- Transaction id col + 'product', -- Product col + 'test_data', -- Input data + NULL, -- Output schema + TRUE, -- Verbose output + 2 -- Max itemset size + ); +</pre> Result (iteration details not shown): <pre class="result"> + output_schema | output_table | total_rules | total_time +---------------+--------------+-------------+----------------- + public | assoc_rules | 4 | 00:00:00.565176 +(1 row) +</pre> The association rules are again stored in the assoc_rules table: <pre class="example"> +SELECT * FROM assoc_rules +ORDER BY support DESC, confidence DESC; +</pre> Result: <pre class="result"> + ruleid | pre | post | count | support | confidence | lift | conviction +--------+-----------+-----------+-------+-------------------+-------------------+-------------------+------------------- + 1 | {diapers} | {beer} | 5 | 0.714285714285714 | 1 | 1 | 0 + 2 | {beer} | {diapers} | 5 | 0.714285714285714 | 0.714285714285714 | 1 | 1 + 3 | {chips} | {beer} | 3 | 0.428571428571429 | 1 | 1 | 0 + 4 | {chips} | {diapers} | 2 | 0.285714285714286 | 0.666666666666667 | 0.933333333333333 | 0.857142857142857 +(4 rows) +</pre></li> +<li>Post-processing can now be done on the output table in the case that you want to filter the results. For example, if you want any single item on the left hand side and a particular item on the right hand side: <pre class="example"> +SELECT * FROM assoc_rules WHERE array_upper(pre,1) = 1 AND post = array['beer']; +</pre> Result: <pre class="result"> + ruleid | pre | post | count | support | confidence | lift | conviction +--------+-----------+--------+-------+-------------------+------------+------+------------ + 1 | {diapers} | {beer} | 5 | 0.714285714285714 | 1 | 1 | 0 + 3 | {chips} | {beer} | 3 | 0.428571428571429 | 1 | 1 | 0 +(2 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="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl> +<p>[1] <a href="https://en.wikipedia.org/wiki/Apriori_algorithm";>https://en.wikipedia.org/wiki/Apriori_algorithm</a></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> +</div><!-- contents --> +</div><!-- doc-content --> +<!-- start footer part --> +<div id="nav-path" class="navpath"><!-- id is needed for treeview function! --> + <ul> + <li class="footer">Generated on Tue May 16 2017 13:24:38 for MADlib by + <a href="http://www.doxygen.org/index.html";> + <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.13 </li> + </ul> +</div> +</body> +</html>
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--> +<div id="titlearea"> +<table cellspacing="0" cellpadding="0"> + <tbody> + <tr style="height: 56px;"> + <td id="projectlogo"><a href="http://madlib.incubator.apache.org";><img alt="Logo" src="madlib.png" height="50" style="padding-left:0.5em;" border="0"/ ></a></td> + <td style="padding-left: 0.5em;"> + <div id="projectname"> + <span id="projectnumber">1.11</span> + </div> + <div id="projectbrief">User Documentation for MADlib</div> + </td> + <td> <div id="MSearchBox" class="MSearchBoxInactive"> + <span class="left"> + <img id="MSearchSelect" src="search/mag_sel.png" + onmouseover="return searchBox.OnSearchSelectShow()" + onmouseout="return searchBox.OnSearchSelectHide()" + alt=""/> + <input type="text" id="MSearchField" value="Search" accesskey="S" + onfocus="searchBox.OnSearchFieldFocus(true)" + onblur="searchBox.OnSearchFieldFocus(false)" + onkeyup="searchBox.OnSearchFieldChange(event)"/> + </span><span class="right"> + <a id="MSearchClose" href="javascript:searchBox.CloseResultsWindow()"><img id="MSearchCloseImg" border="0" src="search/close.png" alt=""/></a> + </span> + </div> +</td> + </tr> + </tbody> +</table> +</div> +<!-- end header part --> +<!-- Generated by Doxygen 1.8.13 --> +<script type="text/javascript"> +var searchBox = new SearchBox("searchBox", "search",false,'Search'); +</script> +</div><!-- top --> +<div id="side-nav" class="ui-resizable side-nav-resizable"> + <div id="nav-tree"> + <div id="nav-tree-contents"> + <div id="nav-sync" class="sync"></div> + </div> + </div> + <div id="splitbar" style="-moz-user-select:none;" + class="ui-resizable-handle"> + </div> +</div> +<script type="text/javascript"> +$(document).ready(function(){initNavTree('group__grp__bayes.html','');}); +</script> +<div id="doc-content"> +<!-- window showing the filter options --> +<div id="MSearchSelectWindow" + onmouseover="return searchBox.OnSearchSelectShow()" + onmouseout="return searchBox.OnSearchSelectHide()" + onkeydown="return searchBox.OnSearchSelectKey(event)"> +</div> + +<!-- iframe showing the search results (closed by default) --> +<div id="MSearchResultsWindow"> +<iframe src="javascript:void(0)" frameborder="0" + name="MSearchResults" id="MSearchResults"> +</iframe> +</div> + +<div class="header"> + <div class="headertitle"> +<div class="title">Naive Bayes Classification<div class="ingroups"><a class="el" href="group__grp__early__stage.html">Early Stage Development</a></div></div> </div> +</div><!--header--> +<div class="contents"> +<div class="toc"><b>Contents</b> <ul> +<li> +<a href="#train">Training Function(s)</a> </li> +<li> +<a href="#classify">Classify Function(s)</a> </li> +<li> +<a href="#probabilities">Probabilities Function(s)</a> </li> +<li> +<a href="#adhoc">Ad Hoc Computation</a> </li> +<li> +<a href="#notes">Implementation Notes</a> </li> +<li> +<a href="#examples">Examples</a> </li> +<li> +<a href="#background">Technical Background</a> </li> +<li> +<a href="#related">Related Topics</a> </li> +</ul> +</div><dl class="section warning"><dt>Warning</dt><dd><em> This MADlib method is still in early stage development. There may be some issues that will be addressed in a future version. Interface and implementation is subject to change. </em></dd></dl> +<p>Naive Bayes refers to a stochastic model where all independent variables <img class="formulaInl" alt="$ a_1, \dots, a_n $" src="form_19.png"/> (often referred to as attributes in this context) independently contribute to the probability that a data point belongs to a certain class <img class="formulaInl" alt="$ c $" src="form_20.png"/>.</p> +<p>Naives Bayes classification estimates feature probabilities and class priors using maximum likelihood or Laplacian smoothing. For numeric attributes, Gaussian smoothing can be used to estimate the feature probabilities.These parameters are then used to classify new data.</p> +<p><a class="anchor" id="train"></a></p><dl class="section user"><dt>Training Function(s)</dt><dd></dd></dl> +<p>For data with only categorical attributes, precompute feature probabilities and class priors using the following function:</p> +<pre class="syntax"> +create_nb_prepared_data_tables ( trainingSource, + trainingClassColumn, + trainingAttrColumn, + numAttrs, + featureProbsName, + classPriorsName + ) +</pre><p>For data containing both categorical and numeric attributes, use the following form to precompute the Gaussian parameters (mean and variance) for numeric attributes alongside the feature probabilities for categorical attributes and class priors.</p> +<pre class="syntax"> +create_nb_prepared_data_tables ( trainingSource, + trainingClassColumn, + trainingAttrColumn, + numericAttrsColumnIndices, + numAttrs, + featureProbsName, + numericAttrParamsName, + classPriorsName + ) +</pre><p>The <em>trainingSource</em> is expected to be of the following form: </p><pre>{TABLE|VIEW} <em>trainingSource</em> ( + ... + <em>trainingClassColumn</em> INTEGER, + <em>trainingAttrColumn</em> INTEGER[] OR NUMERIC[] OR FLOAT8[], + ... +)</pre><p><em>numericAttrsColumnIndices</em> should be of type TEXT, specified as an array of indices (starting from 1) in the <em>trainingAttrColumn</em> attributes-array that correspond to numeric attributes.</p> +<p>The two output tables are:</p><ul> +<li><em>featureProbsName</em> – stores feature probabilities</li> +<li><em>classPriorsName</em> – stores the class priors</li> +</ul> +<p>In addition to the above, if the function specifying numeric attributes is used, an additional table <em>numericAttrParamsName</em> is created which stores the Gaussian parameters for the numeric attributes.</p> +<p><a class="anchor" id="classify"></a></p><dl class="section user"><dt>Classify Function(s)</dt><dd></dd></dl> +<p>Perform Naive Bayes classification: </p><pre class="syntax"> +create_nb_classify_view ( featureProbsName, + classPriorsName, + classifySource, + classifyKeyColumn, + classifyAttrColumn, + numAttrs, + destName + ) +</pre><p>For data with numeric attributes, use the following version:</p> +<pre class="syntax"> +create_nb_classify_view ( featureProbsName, + classPriorsName, + classifySource, + classifyKeyColumn, + classifyAttrColumn, + numAttrs, + numericAttrParamsName, + destName + ) +</pre><p>The <b>data to classify</b> is expected to be of the following form: </p><pre>{TABLE|VIEW} <em>classifySource</em> ( + ... + <em>classifyKeyColumn</em> ANYTYPE, + <em>classifyAttrColumn</em> INTEGER[], + ... +)</pre><p>This function creates the view <code><em>destName</em></code> mapping <em>classifyKeyColumn</em> to the Naive Bayes classification. </p><pre class="result"> +key | nb_classification + ---+------------------ +... +</pre><p><a class="anchor" id="probabilities"></a></p><dl class="section user"><dt>Probabilities Function(s)</dt><dd></dd></dl> +<p>Compute Naive Bayes probabilities. </p><pre class="syntax"> +create_nb_probs_view( featureProbsName, + classPriorsName, + classifySource, + classifyKeyColumn, + classifyAttrColumn, + numAttrs, + destName + ) +</pre><p>For data with numeric attributes , use the following version:</p> +<pre class="syntax"> +create_nb_probs_view( featureProbsName, + classPriorsName, + classifySource, + classifyKeyColumn, + classifyAttrColumn, + numAttrs, + numericAttrParamsName, + destName + ) +</pre><p>This creates the view <code><em>destName</em></code> mapping <em>classifyKeyColumn</em> and every single class to the Naive Bayes probability: </p><pre class="result"> +key | class | nb_prob + ---+-------+-------- +... +</pre><p><a class="anchor" id="adhoc"></a></p><dl class="section user"><dt>Ad Hoc Computation Function</dt><dd></dd></dl> +<p>With ad hoc execution (no precomputation), the functions <a class="el" href="bayes_8sql__in.html#a798402280fc6db710957ae3ab58767e0" title="Create a view with columns (key, nb_classification) ">create_nb_classify_view()</a> and <a class="el" href="bayes_8sql__in.html#a163afffd0c845d325f060f74bcf02243" title="Create view with columns (key, class, nb_prob) ">create_nb_probs_view()</a> can be used in an ad-hoc fashion without the precomputation step. In this case, replace the function arguments</p> +<pre>'<em>featureProbsName</em>', '<em>classPriorsName</em>'</pre><p> with </p><pre>'<em>trainingSource</em>', '<em>trainingClassColumn</em>', '<em>trainingAttrColumn</em>'</pre><p> for data without any any numeric attributes and with </p><pre>'<em>trainingSource</em>', '<em>trainingClassColumn</em>', '<em>trainingAttrColumn</em>', '<em>numericAttrsColumnIndices</em>'</pre><p> for data containing numeric attributes as well.</p> +<p><a class="anchor" id="notes"></a></p><dl class="section user"><dt>Implementation Notes</dt><dd><ul> +<li>The probabilities computed on the platforms of PostgreSQL and Greenplum database have a small difference due to the nature of floating point computation. Usually this is not important. However, if a data point has <p class="formulaDsp"> +<img class="formulaDsp" alt="\[ P(C=c_i \mid A) \approx P(C=c_j \mid A) \]" src="form_21.png"/> +</p> + for two classes, this data point might be classified into diferent classes on PostgreSQL and Greenplum. This leads to the differences in classifications on PostgreSQL and Greenplum for some data sets, but this should not affect the quality of the results.</li> +<li>When two classes have equal and highest probability among all classes, the classification result is an array of these two classes, but the order of the two classes is random.</li> +<li>The current implementation of Naive Bayes classification is suitable for discontinuous (categorial) attributes as well as continuous (numeric) attributes.<br /> +For continuous data, a typical assumption, usually used for small datasets, is that the continuous values associated with each class are distributed according to a Gaussian distribution, and the probabilities <img class="formulaInl" alt="$ P(A_i = a \mid C=c) $" src="form_22.png"/> are estimated using the Gaussian Distribution formula: <p class="formulaDsp"> +<img class="formulaDsp" alt="\[ P(A_i=a \mid C=c) = \frac{1}{\sqrt{2\pi\sigma^{2}_c}}exp\left(-\frac{(a-\mu_c)^{2}}{2\sigma^{2}_c}\right) \]" src="form_23.png"/> +</p> + where <img class="formulaInl" alt="$\mu_c$" src="form_24.png"/> and <img class="formulaInl" alt="$\sigma^{2}_c$" src="form_25.png"/> are the population mean and variance of the attribute for the class <img class="formulaInl" alt="$c$" src="form_26.png"/>.<br /> +Another common technique for handling continuous values, which is better for large data sets, is to use binning to discretize the values, and convert the continuous data into categorical bins. This approach is currently not implemented.</li> +<li>One can provide floating point data to the Naive Bayes classification function. If the corresponding attribute index is not specified in <em>numericAttrsColumnIndices</em>, floating point numbers will be used as symbolic substitutions for categorial data. In this case, the classification would work best if there are sufficient data points for each floating point attribute. However, if floating point numbers are used as continuous data without the attribute being marked as of type numeric in <em>numericAttrsColumnIndices</em>, no warning is raised and the result may not be as expected.</li> +</ul> +</dd></dl> +<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl> +<p>The following is an extremely simplified example of the above option #1 which can by verified by hand.</p> +<ol type="1"> +<li>The training and the classification data. <pre class="example"> +SELECT * FROM training; +</pre> Result: <pre class="result"> + id | class | attributes + ---+-------+------------ + 1 | 1 | {1,2,3} + 2 | 1 | {1,2,1} + 3 | 1 | {1,4,3} + 4 | 2 | {1,2,2} + 5 | 2 | {0,2,2} + 6 | 2 | {0,1,3} +(6 rows) +</pre> <pre class="example"> +SELECT * FROM toclassify; +</pre> Result: <pre class="result"> + id | attributes + ---+------------ + 1 | {0,2,1} + 2 | {1,2,3} +(2 rows) +</pre></li> +<li>Precompute feature probabilities and class priors. <pre class="example"> +SELECT madlib.create_nb_prepared_data_tables( 'training', + 'class', + 'attributes', + 3, + 'nb_feature_probs', + 'nb_class_priors' + ); +</pre></li> +<li>Optionally check the contents of the precomputed tables. <pre class="example"> +SELECT * FROM nb_class_priors; +</pre> Result: <pre class="result"> + class | class_cnt | all_cnt + ------+-----------+--------- + 1 | 3 | 6 + 2 | 3 | 6 +(2 rows) +</pre> <pre class="example"> +SELECT * FROM nb_feature_probs; +</pre> Result: <pre class="result"> + class | attr | value | cnt | attr_cnt + ------+------+-------+-----+---------- + 1 | 1 | 0 | 0 | 2 + 1 | 1 | 1 | 3 | 2 + 1 | 2 | 1 | 0 | 3 + 1 | 2 | 2 | 2 | 3 +... +</pre></li> +<li>Create the view with Naive Bayes classification and check the results. <pre class="example"> +SELECT madlib.create_nb_classify_view( 'nb_feature_probs', + 'nb_class_priors', + 'toclassify', + 'id', + 'attributes', + 3, + 'nb_classify_view_fast' + ); +  +SELECT * FROM nb_classify_view_fast; +</pre> Result: <pre class="result"> + key | nb_classification + ----+------------------- + 1 | {2} + 2 | {1} +(2 rows) +</pre></li> +<li>Look at the probabilities for each class (note that we use "Laplacian smoothing"), <pre class="example"> +SELECT madlib.create_nb_probs_view( 'nb_feature_probs', + 'nb_class_priors', + 'toclassify', + 'id', + 'attributes', + 3, + 'nb_probs_view_fast' + ); +  +SELECT * FROM nb_probs_view_fast; +</pre> Result: <pre class="result"> + key | class | nb_prob + ----+-------+--------- + 1 | 1 | 0.4 + 1 | 2 | 0.6 + 2 | 1 | 0.75 + 2 | 2 | 0.25 +(4 rows) +</pre></li> +</ol> +<p>The following is an example of using a dataset with both numeric and categorical attributes</p> +<ol type="1"> +<li>The training and the classification data. Attributes {height(numeric),weight(numeric),shoe size(categorical)}, Class{sex(1=male,2=female)} <pre class="example"> +SELECT * FROM gaussian_data; +</pre> Result: <pre class="result"> + id | sex | attributes + ----+-----+--------------- + 1 | 1 | {6,180,12} + 2 | 1 | {5.92,190,12} + 3 | 1 | {5.58,170,11} + 4 | 1 | {5.92,165,11} + 5 | 2 | {5,100,6} + 6 | 2 | {5.5,150,6} + 7 | 2 | {5.42,130,7} + 8 | 2 | {5.75,150,8} +(8 rows) +</pre> <pre class="example"> +SELECT * FROM gaussian_test; +</pre> Result: <pre class="result"> + id | sex | attributes +----+-----+-------------- + 9 | 1 | {5.8,180,11} + 10 | 2 | {5,160,6} +(2 rows) +</pre></li> +<li>Precompute feature probabilities and class priors. <pre class="example"> +SELECT madlib.create_nb_prepared_data_tables( 'gaussian_data', + 'sex', + 'attributes', + 'ARRAY[1,2]', + 3, + 'categ_feature_probs', + 'numeric_attr_params', + 'class_priors' + ); +</pre></li> +<li>Optionally check the contents of the precomputed tables. <pre class="example"> +SELECT * FROM class_priors; +</pre> Result: <pre class="result"> +class | class_cnt | all_cnt + -------+-----------+--------- + 1 | 4 | 8 + 2 | 4 | 8 +(2 rows) +</pre> <pre class="example"> +SELECT * FROM categ_feature_probs; +</pre> Result: <pre class="result"> + class | attr | value | cnt | attr_cnt +-------+------+-------+-----+---------- + 2 | 3 | 6 | 2 | 5 + 1 | 3 | 12 | 2 | 5 + 2 | 3 | 7 | 1 | 5 + 1 | 3 | 11 | 2 | 5 + 2 | 3 | 8 | 1 | 5 + 2 | 3 | 12 | 0 | 5 + 1 | 3 | 6 | 0 | 5 + 2 | 3 | 11 | 0 | 5 + 1 | 3 | 8 | 0 | 5 + 1 | 3 | 7 | 0 | 5 +(10 rows) +</pre> <pre class="example"> +SELECT * FROM numeric_attr_params; +</pre> Result: <pre class="result"> +class | attr | attr_mean | attr_var +-------+------+----------------------+------------------------ + 1 | 1 | 5.8550000000000000 | 0.03503333333333333333 + 1 | 2 | 176.2500000000000000 | 122.9166666666666667 + 2 | 1 | 5.4175000000000000 | 0.09722500000000000000 + 2 | 2 | 132.5000000000000000 | 558.3333333333333333 +(4 rows) +</pre></li> +<li>Create the view with Naive Bayes classification and check the results. <pre class="example"> +SELECT madlib.create_nb_classify_view( 'categ_feature_probs', + 'class_priors', + 'gaussian_test', + 'id', + 'attributes', + 3, + 'numeric_attr_params', + 'classify_view' + ); +  +SELECT * FROM classify_view; +</pre> Result: <pre class="result"> + key | nb_classification + ----+------------------- + 9 | {1} + 10 | {2} +(2 rows) +</pre></li> +<li>Look at the probabilities for each class <pre class="example"> +SELECT madlib.create_nb_probs_view( 'categ_feature_probs', + 'class_priors', + 'gaussian_test', + 'id', + 'attributes', + 3, + 'numeric_attr_params', + 'probs_view' + ); +  +SELECT * FROM probs_view; +</pre> Result: <pre class="result"> + key | class | nb_prob +-----+-------+---------------------- + 9 | 1 | 0.993556745948775 + 9 | 2 | 0.00644325405122553 + 10 | 1 | 5.74057538627122e-05 + 10 | 2 | 0.999942594246137 +(4 rows) +</pre></li> +</ol> +<p><a class="anchor" id="background"></a></p><dl class="section user"><dt>Technical Background</dt><dd></dd></dl> +<p>In detail, <b>Bayes'</b> theorem states that </p><p class="formulaDsp"> +<img class="formulaDsp" alt="\[ \Pr(C = c \mid A_1 = a_1, \dots, A_n = a_n) = \frac{\Pr(C = c) \cdot \Pr(A_1 = a_1, \dots, A_n = a_n \mid C = c)} {\Pr(A_1 = a_1, \dots, A_n = a_n)} \,, \]" src="form_27.png"/> +</p> +<p> and the <b>naive</b> assumption is that </p><p class="formulaDsp"> +<img class="formulaDsp" alt="\[ \Pr(A_1 = a_1, \dots, A_n = a_n \mid C = c) = \prod_{i=1}^n \Pr(A_i = a_i \mid C = c) \,. \]" src="form_28.png"/> +</p> +<p> Naives Bayes classification estimates feature probabilities and class priors using maximum likelihood or Laplacian smoothing. These parameters are then used to classifying new data.</p> +<p>A Naive Bayes classifier computes the following formula: </p><p class="formulaDsp"> +<img class="formulaDsp" alt="\[ \text{classify}(a_1, ..., a_n) = \arg\max_c \left\{ \Pr(C = c) \cdot \prod_{i=1}^n \Pr(A_i = a_i \mid C = c) \right\} \]" src="form_29.png"/> +</p> +<p> where <img class="formulaInl" alt="$ c $" src="form_20.png"/> ranges over all classes in the training data and probabilites are estimated with relative frequencies from the training set. There are different ways to estimate the feature probabilities <img class="formulaInl" alt="$ P(A_i = a \mid C = c) $" src="form_30.png"/>. The maximum likelihood estimate takes the relative frequencies. That is: </p><p class="formulaDsp"> +<img class="formulaDsp" alt="\[ P(A_i = a \mid C = c) = \frac{\#(c,i,a)}{\#c} \]" src="form_31.png"/> +</p> +<p> where</p><ul> +<li><img class="formulaInl" alt="$ \#(c,i,a) $" src="form_32.png"/> denotes the # of training samples where attribute <img class="formulaInl" alt="$ i $" src="form_33.png"/> is <img class="formulaInl" alt="$ a $" src="form_34.png"/> and class is <img class="formulaInl" alt="$ c $" src="form_20.png"/></li> +<li><img class="formulaInl" alt="$ \#c $" src="form_35.png"/> denotes the # of training samples where class is <img class="formulaInl" alt="$ c $" src="form_20.png"/>.</li> +</ul> +<p>Since the maximum likelihood sometimes results in estimates of "0", you might want to use a "smoothed" estimate. To do this, you add a number of "virtual" samples and make the assumption that these samples are evenly distributed among the values assumed by attribute <img class="formulaInl" alt="$ i $" src="form_33.png"/> (that is, the set of all values observed for attribute <img class="formulaInl" alt="$ a $" src="form_34.png"/> for any class):</p> +<p class="formulaDsp"> +<img class="formulaDsp" alt="\[ P(A_i = a \mid C = c) = \frac{\#(c,i,a) + s}{\#c + s \cdot \#i} \]" src="form_36.png"/> +</p> +<p> where</p><ul> +<li><img class="formulaInl" alt="$ \#i $" src="form_37.png"/> denotes the # of distinct values for attribute <img class="formulaInl" alt="$ i $" src="form_33.png"/> (for all classes)</li> +<li><img class="formulaInl" alt="$ s \geq 0 $" src="form_38.png"/> denotes the smoothing factor.</li> +</ul> +<p>The case <img class="formulaInl" alt="$ s = 1 $" src="form_39.png"/> is known as "Laplace smoothing". The case <img class="formulaInl" alt="$ s = 0 $" src="form_40.png"/> trivially reduces to maximum-likelihood estimates.</p> +<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl> +<p>[1] Tom Mitchell: Machine Learning, McGraw Hill, 1997. Book chapter <em>Generativ and Discriminative Classifiers: Naive Bayes and Logistic Regression</em> available at: <a href="http://www.cs.cmu.edu/~tom/NewChapters.html";>http://www.cs.cmu.edu/~tom/NewChapters.html</a></p> +<p>[2] Wikipedia, Naive Bayes classifier, <a href="http://en.wikipedia.org/wiki/Naive_Bayes_classifier";>http://en.wikipedia.org/wiki/Naive_Bayes_classifier</a></p> +<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd>File <a class="el" href="bayes_8sql__in.html" title="SQL functions for naive Bayes. ">bayes.sql_in</a> documenting the SQL functions.</dd></dl> +</div><!-- contents --> +</div><!-- doc-content --> +<!-- start footer part --> +<div id="nav-path" class="navpath"><!-- id is needed for treeview function! --> + <ul> + <li class="footer">Generated on Tue May 16 2017 13:24:39 for MADlib by + <a href="http://www.doxygen.org/index.html";> + <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.13 </li> + </ul> +</div> +</body> +</html> http://git-wip-us.apache.org/repos/asf/incubator-madlib-site/blob/b5b51c69/docs/v1.11/group__grp__cg.html ---------------------------------------------------------------------- diff --git a/docs/v1.11/group__grp__cg.html b/docs/v1.11/group__grp__cg.html new file mode 100644 index 0000000..0b279f1 --- /dev/null +++ b/docs/v1.11/group__grp__cg.html @@ -0,0 +1,179 @@ +<!-- HTML header for doxygen 1.8.4--> +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd";> +<html xmlns="http://www.w3.org/1999/xhtml";> +<head> +<meta http-equiv="Content-Type" content="text/xhtml;charset=UTF-8"/> +<meta http-equiv="X-UA-Compatible" content="IE=9"/> +<meta name="generator" content="Doxygen 1.8.13"/> +<meta name="keywords" content="madlib,postgres,greenplum,machine learning,data mining,deep learning,ensemble methods,data science,market basket analysis,affinity analysis,pca,lda,regression,elastic net,huber white,proportional hazards,k-means,latent dirichlet allocation,bayes,support vector machines,svm"/> +<title>MADlib: Conjugate Gradient</title> +<link href="tabs.css" rel="stylesheet" type="text/css"/> +<script type="text/javascript" src="jquery.js"></script> +<script type="text/javascript" src="dynsections.js"></script> +<link href="navtree.css" rel="stylesheet" type="text/css"/> +<script type="text/javascript" src="resize.js"></script> +<script type="text/javascript" src="navtreedata.js"></script> +<script type="text/javascript" src="navtree.js"></script> +<script type="text/javascript"> + $(document).ready(initResizable); +</script> +<link href="search/search.css" rel="stylesheet" type="text/css"/> +<script type="text/javascript" src="search/searchdata.js"></script> +<script type="text/javascript" src="search/search.js"></script> +<script type="text/javascript"> + $(document).ready(function() { init_search(); }); +</script> +<!-- hack in the navigation tree --> +<script type="text/javascript" src="eigen_navtree_hacks.js"></script> +<link href="doxygen.css" rel="stylesheet" type="text/css" /> +<link href="madlib_extra.css" rel="stylesheet" type="text/css"/> +<!-- google analytics --> +<script> + (function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){ + (i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o), + m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m) + })(window,document,'script','//www.google-analytics.com/analytics.js','ga'); + ga('create', 'UA-45382226-1', 'madlib.incubator.apache.org'); + ga('send', 'pageview'); +</script> +</head> +<body> +<div id="top"><!-- do not remove this div, it is closed by doxygen! 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There may be some issues that will be addressed in a future version. Interface and implementation is subject to change. </em></dd></dl> +<p>This function uses the iterative conjugate gradient method [1] to find a solution to the function: </p><p class="formulaDsp"> +<img class="formulaDsp" alt="\[ \boldsymbol Ax = \boldsymbol b \]" src="form_45.png"/> +</p> +<p> where <img class="formulaInl" alt="$ \boldsymbol A $" src="form_46.png"/> is a symmetric, positive definite matrix and <img class="formulaInl" alt="$x$" src="form_43.png"/> and <img class="formulaInl" alt="$ \boldsymbol b $" src="form_44.png"/> are vectors.</p> +<p><a class="anchor" id="syntax"></a></p><dl class="section user"><dt>Function Syntax</dt><dd>Conjugate gradient returns x as an array. It has the following syntax.</dd></dl> +<pre class="syntax"> +conjugate_gradient( table_name, + name_of_row_values_col, + name_of_row_number_col, + aray_of_b_values, + desired_precision + ) +</pre><p>Matrix <img class="formulaInl" alt="$ \boldsymbol A $" src="form_46.png"/> is assumed to be stored in a table where each row consists of at least two columns: array containing values of a given row, row number: </p><pre>{TABLE|VIEW} <em>matrix_A</em> ( + <em>row_number</em> FLOAT, + <em>row_values</em> FLOAT[], +)</pre><p> The number of elements in each row should be the same.</p> +<p><img class="formulaInl" alt="$ \boldsymbol b $" src="form_44.png"/> is passed as a FLOAT[] to the function.</p> +<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd><ol type="1"> +<li>Construct matrix A according to structure. <pre class="example"> +SELECT * FROM data; +</pre> Result: <pre class="result"> + row_num | row_val + --------+--------- + 1 | {2,1} + 2 | {1,4} +(2 rows) +</pre></li> +<li>Call the conjugate gradient function. <pre class="example"> +SELECT conjugate_gradient( 'data', + 'row_val', + 'row_num', + '{2,1}', + 1E-6,1 + ); +</pre> <pre class="result"> +INFO: COMPUTE RESIDUAL ERROR 14.5655661859659 +INFO: ERROR 0.144934004246004 +INFO: ERROR 3.12963615962926e-31 +INFO: TEST FINAL ERROR 2.90029642185163e-29 + conjugate_gradient + -------------------------- + {1,-1.31838984174237e-15} +(1 row) +</pre></li> +</ol> +</dd></dl> +<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd>[1] "Conjugate gradient method" Wikipedia - <a href="http://en.wikipedia.org/wiki/Conjugate_gradient_method";>http://en.wikipedia.org/wiki/Conjugate_gradient_method</a></dd></dl> +<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd>File <a class="el" href="conjugate__gradient_8sql__in.html" title="SQL function computing Conjugate Gradient. ">conjugate_gradient.sql_in</a> documenting the SQL function. </dd></dl> +</div><!-- contents --> +</div><!-- doc-content --> +<!-- start footer part --> +<div id="nav-path" class="navpath"><!-- id is needed for treeview function! --> + <ul> + <li class="footer">Generated on Tue May 16 2017 13:24:39 for MADlib by + <a href="http://www.doxygen.org/index.html";> + <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.13 </li> + </ul> +</div> +</body> +</html>
