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Posted to commits@madlib.apache.org by xt...@apache.org on 2016/09/20 18:31:21 UTC
[09/51] [partial] incubator-madlib-site git commit: Update doc for
1.9.1 release
http://git-wip-us.apache.org/repos/asf/incubator-madlib-site/blob/bed9253d/docs/v1.9.1/group__grp__association__rules.js
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+var group__grp__association__rules =
+[
+ [ "Apriori Algorithm", "group__grp__assoc__rules.html", null ]
+];
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http://git-wip-us.apache.org/repos/asf/incubator-madlib-site/blob/bed9253d/docs/v1.9.1/group__grp__bayes.html
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+<title>MADlib: Naive Bayes Classification</title>
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+ <div id="projectbrief">User Documentation for MADlib</div>
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+<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> </p><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 $" src="form_18.png"/> (often referred to as attributes in this context) independently contribute to the probability that a data point belongs to a certain class <img class="formulaInl" alt="$ c $" src="form_19.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_20.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_21.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_22.png"/>
+</p>
+ where <img class="formulaInl" alt="$\mu_c$" src="form_23.png"/> and <img class="formulaInl" alt="$\sigma^{2}_c$" src="form_24.png"/> are the population mean and variance of the attribute for the class <img class="formulaInl" alt="$c$" src="form_25.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_26.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_27.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_28.png"/>
+</p>
+<p> where <img class="formulaInl" alt="$ c $" src="form_19.png"/> ranges over all classes in the training data and probabilites are estimated with relative frequencies from the training set. There are different ways to estimate the feature probabilities <img class="formulaInl" alt="$ P(A_i = a \mid C = c) $" src="form_29.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_30.png"/>
+</p>
+<p> where</p><ul>
+<li><img class="formulaInl" alt="$ \#(c,i,a) $" src="form_31.png"/> denotes the # of training samples where attribute <img class="formulaInl" alt="$ i $" src="form_32.png"/> is <img class="formulaInl" alt="$ a $" src="form_33.png"/> and class is <img class="formulaInl" alt="$ c $" src="form_19.png"/></li>
+<li><img class="formulaInl" alt="$ \#c $" src="form_34.png"/> denotes the # of training samples where class is <img class="formulaInl" alt="$ c $" src="form_19.png"/>.</li>
+</ul>
+<p>Since the maximum likelihood sometimes results in estimates of "0", you might want to use a "smoothed" estimate. To do this, you add a number of "virtual" samples and make the assumption that these samples are evenly distributed among the values assumed by attribute <img class="formulaInl" alt="$ i $" src="form_32.png"/> (that is, the set of all values observed for attribute <img class="formulaInl" alt="$ a $" src="form_33.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_35.png"/>
+</p>
+<p> where</p><ul>
+<li><img class="formulaInl" alt="$ \#i $" src="form_36.png"/> denotes the # of distinct values for attribute <img class="formulaInl" alt="$ i $" src="form_32.png"/> (for all classes)</li>
+<li><img class="formulaInl" alt="$ s \geq 0 $" src="form_37.png"/> denotes the smoothing factor.</li>
+</ul>
+<p>The case <img class="formulaInl" alt="$ s = 1 $" src="form_38.png"/> is known as "Laplace smoothing". The case <img class="formulaInl" alt="$ s = 0 $" src="form_39.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 -->
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+ <li class="footer">Generated on Tue Sep 20 2016 11:27:01 for MADlib by
+ <a href="http://www.doxygen.org/index.html">
+ <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.10 </li>
+ </ul>
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+</html>
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+<title>MADlib: Conjugate Gradient</title>
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+ <div class="headertitle">
+<div class="title">Conjugate Gradient<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> </p><ul>
+<li>
+<a href="#syntax">Function Syntax</a> </li>
+<li>
+<a href="#examples">Examples</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>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_44.png"/>
+</p>
+<p> where <img class="formulaInl" alt="$ \boldsymbol A $" src="form_45.png"/> is a symmetric, positive definite matrix and <img class="formulaInl" alt="$x$" src="form_42.png"/> and <img class="formulaInl" alt="$ \boldsymbol b $" src="form_43.png"/> are vectors.</p>
+<p><a class="anchor" id="syntax"></a></p><dl class="section user"><dt>Function Syntax</dt><dd>Conjugate gradient returns x as an array. It has the following syntax.</dd></dl>
+<pre class="syntax">
+conjugate_gradient( table_name,
+ name_of_row_values_col,
+ name_of_row_number_col,
+ aray_of_b_values,
+ desired_precision
+ )
+</pre><p>Matrix <img class="formulaInl" alt="$ \boldsymbol A $" src="form_45.png"/> is assumed to be stored in a table where each row consists of at least two columns: array containing values of a given row, row number: </p><pre>{TABLE|VIEW} <em>matrix_A</em> (
+ <em>row_number</em> FLOAT,
+ <em>row_values</em> FLOAT[],
+)</pre><p> The number of elements in each row should be the same.</p>
+<p><img class="formulaInl" alt="$ \boldsymbol b $" src="form_43.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 -->
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+ <a href="http://www.doxygen.org/index.html">
+ <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.10 </li>
+ </ul>
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+</html>
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+<title>MADlib: Clustered Variance</title>
+<link href="tabs.css" rel="stylesheet" type="text/css"/>
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+<div id="MSearchSelectWindow"
+ onmouseover="return searchBox.OnSearchSelectShow()"
+ onmouseout="return searchBox.OnSearchSelectHide()"
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+<div class="title">Clustered Variance<div class="ingroups"><a class="el" href="group__grp__super.html">Supervised Learning</a> » <a class="el" href="group__grp__regml.html">Regression Models</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> </p><ul>
+<li>
+<a href="#train_linregr">Clustered Variance Linear Regression Training Function</a> </li>
+<li>
+<a href="#train_logregr">Clustered Variance Logistic Regression Training Function</a> </li>
+<li>
+<a href="#train_mlogregr">Clustered Variance Multinomial Logistic Regression Training Function</a> </li>
+<li>
+<a href="#train_cox">Clustered Variance for Cox Proportional Hazards model</a> </li>
+<li>
+<a href="#examples">Examples</a> </li>
+<li>
+<a href="#notes">Notes</a> </li>
+<li>
+<a href="#background">Technical Background</a> </li>
+<li>
+<a href="#related">Related Topics</a> </li>
+</ul>
+</div><p>The Clustered Variance module adjusts standard errors for clustering. For example, replicating a dataset 100 times should not increase the precision of parameter estimates, but performing this procedure with the IID assumption will actually do this. Another example is in economics of education research, it is reasonable to expect that the error terms for children in the same class are not independent. Clustering standard errors can correct for this.</p>
+<p>The MADlib Clustered Variance module includes functions to calculate linear, logistic, and multinomial logistic regression problems.</p>
+<p><a class="anchor" id="train_linregr"></a></p><dl class="section user"><dt>Clustered Variance Linear Regression Training Function</dt><dd></dd></dl>
+<p>The clustered variance linear regression training function has the following syntax. </p><pre class="syntax">
+clustered_variance_linregr ( source_table,
+ out_table,
+ dependent_varname,
+ independent_varname,
+ clustervar,
+ grouping_cols
+ )
+</pre><p> <b>Arguments</b> </p><dl class="arglist">
+<dt>source_table </dt>
+<dd><p class="startdd">TEXT. The name of the table containing the input data.</p>
+<p class="enddd"></p>
+</dd>
+<dt>out_table </dt>
+<dd><p class="startdd">VARCHAR. Name of the generated table containing the output model. The output table contains the following columns. </p><table class="output">
+<tr>
+<th>coef </th><td>DOUBLE PRECISION[]. Vector of the coefficients of the regression. </td></tr>
+<tr>
+<th>std_err </th><td>DOUBLE PRECISION[]. Vector of the standard error of the coefficients. </td></tr>
+<tr>
+<th>t_stats </th><td>DOUBLE PRECISION[]. Vector of the t-stats of the coefficients. </td></tr>
+<tr>
+<th>p_values </th><td>DOUBLE PRECISION[]. Vector of the p-values of the coefficients. </td></tr>
+</table>
+<p>A summary table named <out_table>_summary is also created, which is the same as the summary table created by linregr_train function. Please refer to the documentation for linear regression for details.</p>
+<p></p>
+<p class="enddd"></p>
+</dd>
+<dt>dependent_varname </dt>
+<dd>TEXT. An expression to evaluate for the dependent variable. </dd>
+<dt>independent_varname </dt>
+<dd>TEXT. An Expression to evalue for the independent variables. </dd>
+<dt>clustervar </dt>
+<dd>TEXT. A comma-separated list of the columns to use as cluster variables. </dd>
+<dt>grouping_cols (optional) </dt>
+<dd>TEXT, default: NULL. <em>Not currently implemented. Any non-NULL value is ignored.</em> An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL GROUP BY clause. When this value is null, no grouping is used and a single result model is generated. </dd>
+</dl>
+<p><a class="anchor" id="train_logregr"></a></p><dl class="section user"><dt>Clustered Variance Logistic Regression Training Function</dt><dd></dd></dl>
+<p>The clustered variance logistic regression training function has the following syntax. </p><pre class="syntax">
+clustered_variance_logregr( source_table,
+ out_table,
+ dependent_varname,
+ independent_varname,
+ clustervar,
+ grouping_cols,
+ max_iter,
+ optimizer,
+ tolerance,
+ verbose_mode
+ )
+</pre><p> <b>Arguments</b> </p><dl class="arglist">
+<dt>source_table </dt>
+<dd>TEXT. The name of the table containing the input data. </dd>
+<dt>out_table </dt>
+<dd><p class="startdd">VARCHAR. Name of the generated table containing the output model. The output table has the following columns: </p><table class="output">
+<tr>
+<th>coef </th><td>Vector of the coefficients of the regression. </td></tr>
+<tr>
+<th>std_err </th><td>Vector of the standard error of the coefficients. </td></tr>
+<tr>
+<th>z_stats </th><td>Vector of the z-stats of the coefficients. </td></tr>
+<tr>
+<th>p_values </th><td>Vector of the p-values of the coefficients. </td></tr>
+</table>
+<p>A summary table named <out_table>_summary is also created, which is the same as the summary table created by logregr_train function. Please refer to the documentation for logistic regression for details.</p>
+<p class="enddd"></p>
+</dd>
+<dt>dependent_varname </dt>
+<dd>TEXT. An expression to evaluate for the dependent variable. </dd>
+<dt>independent_varname </dt>
+<dd>TEXT. An expression to evaluate for the independent variable. </dd>
+<dt>clustervar </dt>
+<dd>TEXT. A comma-separated list of columns to use as cluster variables. </dd>
+<dt>grouping_cols (optional) </dt>
+<dd>TEXT, default: NULL. <em>Not yet implemented. Any non-NULL values are ignored.</em> An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL GROUP BY clause. When this value is NULL, no grouping is used and a single result model is generated. </dd>
+<dt>max_iter (optional) </dt>
+<dd>INTEGER, default: 20. The maximum number of iterations that are allowed. </dd>
+<dt>optimizer (optional) </dt>
+<dd>TEXT, default: 'irls'. The name of the optimizer to use: <ul>
+<li>
+'newton' or 'irls': Iteratively reweighted least squares </li>
+<li>
+'cg': conjugate gradient </li>
+<li>
+'igd': incremental gradient descent. </li>
+</ul>
+</dd>
+<dt>tolerance (optional) </dt>
+<dd>FLOAT8, default: 0.0001 The difference between log-likelihood values in successive iterations that should indicate convergence. A zero disables the convergence criterion, so that execution stops after <em>n</em> Iterations have completed. </dd>
+<dt>verbose_mode (optional) </dt>
+<dd>BOOLEAN, default FALSE. Provides verbose_mode output of the results of training. </dd>
+</dl>
+<p><a class="anchor" id="train_mlogregr"></a></p><dl class="section user"><dt>Clustered Variance Multinomial Logistic Regression Training Function</dt><dd></dd></dl>
+<pre class="syntax">
+clustered_variance_mlogregr( source_table,
+ out_table,
+ dependent_varname,
+ independent_varname,
+ cluster_varname,
+ ref_category,
+ grouping_cols,
+ optimizer_params,
+ verbose_mode
+ )
+</pre><p> <b>Arguments</b> </p><dl class="arglist">
+<dt>source_table </dt>
+<dd>TEXT. The name of the table containing the input data. </dd>
+<dt>out_table </dt>
+<dd><p class="startdd">TEXT. The name of the table where the regression model will be stored. The output table has the following columns: </p><table class="output">
+<tr>
+<th>category </th><td>The category. </td></tr>
+<tr>
+<th>ref_category </th><td>The refererence category used for modeling. </td></tr>
+<tr>
+<th>coef </th><td>Vector of the coefficients of the regression. </td></tr>
+<tr>
+<th>std_err </th><td>Vector of the standard error of the coefficients. </td></tr>
+<tr>
+<th>z_stats </th><td>Vector of the z-stats of the coefficients. </td></tr>
+<tr>
+<th>p_values </th><td>Vector of the p-values of the coefficients. </td></tr>
+</table>
+<p class="enddd">A summary table named <out_table>_summary is also created, which is the same as the summary table created by mlogregr_train function. Please refer to the documentation for multinomial logistic regression for details. </p>
+</dd>
+<dt>dependent_varname </dt>
+<dd>TEXT. An expression to evaluate for the dependent variable. </dd>
+<dt>independent_varname </dt>
+<dd>TEXT. An expression to evaluate for the independent variable. </dd>
+<dt>cluster_varname </dt>
+<dd>TEXT. A comma-separated list of columns to use as cluster variables. </dd>
+<dt>ref_category (optional) </dt>
+<dd>INTEGER. Reference category in the range [0, num_category). </dd>
+<dt>groupingvarng_cols (optional) </dt>
+<dd>TEXT, default: NULL. <em>Not yet implemented. Any non-NULL values are ignored.</em> A comma-separated list of columns to use as grouping variables. </dd>
+<dt>optimizer_params (optional) </dt>
+<dd>TEXT, default: NULL, which uses the default values of optimizer parameters: max_iter=20, optimizer='newton', tolerance=1e-4. It should be a string that contains pairs of 'key=value' separated by commas. </dd>
+<dt>verbose_mode (optional) </dt>
+<dd>BOOLEAN, default FALSE. If TRUE, detailed information is printed when computing logistic regression. </dd>
+</dl>
+<p><a class="anchor" id="train_cox"></a></p><dl class="section user"><dt>Clustered Variance for Cox Proportional Hazards model</dt><dd></dd></dl>
+<p>The clustered robust variance estimator function for the Cox Proportional Hazards model has the following syntax. </p><pre class="syntax">
+clustered_variance_coxph(model_table, output_table, clustervar)
+</pre><p><b>Arguments</b> </p><dl class="arglist">
+<dt>model_table </dt>
+<dd>TEXT. The name of the model table, which is exactaly the same as the 'output_table' parameter of <a class="el" href="cox__prop__hazards_8sql__in.html#a737450bbfe0f10204b0074a9d45b0cef" title="Compute cox-regression coefficients and diagnostic statistics. ">coxph_train()</a> function. </dd>
+<dt>output_table </dt>
+<dd>TEXT. The name of the table where the output is saved. It has the following columns: <table class="output">
+<tr>
+<th>coef </th><td>FLOAT8[]. Vector of the coefficients. </td></tr>
+<tr>
+<th>loglikelihood </th><td>FLOAT8. Log-likelihood value of the MLE estimate. </td></tr>
+<tr>
+<th>std_err </th><td>FLOAT8[]. Vector of the standard error of the coefficients. </td></tr>
+<tr>
+<th>clustervar </th><td>TEXT. A comma-separated list of columns to use as cluster variables. </td></tr>
+<tr>
+<th>clustered_se </th><td>FLOAT8[]. Vector of the robust standard errors of the coefficients. </td></tr>
+<tr>
+<th>clustered_z </th><td>FLOAT8[]. Vector of the robust z-stats of the coefficients. </td></tr>
+<tr>
+<th>clustered_p </th><td>FLOAT8[]. Vector of the robust p-values of the coefficients. </td></tr>
+<tr>
+<th>hessian </th><td>FLOAT8[]. The Hessian matrix. </td></tr>
+</table>
+</dd>
+<dt>clustervar </dt>
+<dd>TEXT. A comma-separated list of columns to use as cluster variables. </dd>
+</dl>
+<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
+<ol type="1">
+<li>View online help for the clustered variance linear regression function. <pre class="example">
+SELECT madlib.clustered_variance_linregr();
+</pre></li>
+<li>Run the linear regression function and view the results. <pre class="example">
+DROP TABLE IF EXISTS out_table;
+SELECT madlib.clustered_variance_linregr( 'abalone',
+ 'out_table',
+ 'rings',
+ 'ARRAY[1, diameter, length, width]',
+ 'sex',
+ NULL
+ );
+SELECT * FROM out_table;
+</pre></li>
+<li>View online help for the clustered variance logistic regression function. <pre class="example">
+SELECT madlib.clustered_variance_logregr();
+</pre></li>
+<li>Run the logistic regression function and view the results. <pre class="example">
+DROP TABLE IF EXISTS out_table;
+SELECT madlib.clustered_variance_logregr( 'abalone',
+ 'out_table',
+ 'rings < 10',
+ 'ARRAY[1, diameter, length, width]',
+ 'sex'
+ );
+SELECT * FROM out_table;
+</pre></li>
+<li>View online help for the clustered variance multinomial logistic regression function. <pre class="example">
+SELECT madlib.clustered_variance_mlogregr();
+</pre></li>
+<li>Run the multinomial logistic regression and view the results. <pre class="example">
+DROP TABLE IF EXISTS out_table;
+SELECT madlib.clustered_variance_mlogregr( 'abalone',
+ 'out_table',
+ 'CASE WHEN rings < 10 THEN 1 ELSE 0 END',
+ 'ARRAY[1, diameter, length, width]',
+ 'sex',
+ 0
+ );
+SELECT * FROM out_table;
+</pre></li>
+<li>Run the Cox Proportional Hazards regression and compute the clustered robust estimator. <pre class="example">
+DROP TABLE IF EXISTS lung_cl_out;
+DROP TABLE IF EXISTS lung_out;
+DROP TABLE IF EXISTS lung_out_summary;
+SELECT madlib.coxph_train('lung',
+ 'lung_out',
+ 'time',
+ 'array[age, "ph.ecog"]',
+ 'TRUE',
+ NULL,
+ NULL);
+SELECT madlib.clustered_variance_coxph('lung_out',
+ 'lung_cl_out',
+ '"ph.karno"');
+SELECT * FROM lung_cl_out;
+</pre></li>
+</ol>
+<p><a class="anchor" id="notes"></a></p><dl class="section user"><dt>Notes</dt><dd></dd></dl>
+<ul>
+<li>Note that we need to manually include an intercept term in the independent variable expression. The NULL value of <em>groupingvar</em> means that there is no grouping in the calculation.</li>
+</ul>
+<p><a class="anchor" id="background"></a></p><dl class="section user"><dt>Technical Background</dt><dd></dd></dl>
+<p>Assume that the data can be separated into <img class="formulaInl" alt="$m$" src="form_314.png"/> clusters. Usually this can be done by grouping the data table according to one or multiple columns.</p>
+<p>The estimator has a similar form to the usual sandwich estimator </p><p class="formulaDsp">
+<img class="formulaDsp" alt="\[ S(\vec{c}) = B(\vec{c}) M(\vec{c}) B(\vec{c}) \]" src="form_315.png"/>
+</p>
+<p>The bread part is the same as Huber-White sandwich estimator </p><p class="formulaDsp">
+<img class="formulaDsp" alt="\begin{eqnarray} B(\vec{c}) & = & \left(-\sum_{i=1}^{n} H(y_i, \vec{x}_i, \vec{c})\right)^{-1}\\ & = & \left(-\sum_{i=1}^{n}\frac{\partial^2 l(y_i, \vec{x}_i, \vec{c})}{\partial c_\alpha \partial c_\beta}\right)^{-1} \end{eqnarray}" src="form_316.png"/>
+</p>
+<p> where <img class="formulaInl" alt="$H$" src="form_317.png"/> is the hessian matrix, which is the second derivative of the target function </p><p class="formulaDsp">
+<img class="formulaDsp" alt="\[ L(\vec{c}) = \sum_{i=1}^n l(y_i, \vec{x}_i, \vec{c})\ . \]" src="form_318.png"/>
+</p>
+<p>The meat part is different </p><p class="formulaDsp">
+<img class="formulaDsp" alt="\[ M(\vec{c}) = \bf{A}^T\bf{A} \]" src="form_319.png"/>
+</p>
+<p> where the <img class="formulaInl" alt="$m$" src="form_314.png"/>-th row of <img class="formulaInl" alt="$\bf{A}$" src="form_320.png"/> is </p><p class="formulaDsp">
+<img class="formulaDsp" alt="\[ A_m = \sum_{i\in G_m}\frac{\partial l(y_i,\vec{x}_i,\vec{c})}{\partial \vec{c}} \]" src="form_321.png"/>
+</p>
+<p> where <img class="formulaInl" alt="$G_m$" src="form_322.png"/> is the set of rows that belong to the same cluster.</p>
+<p>We can compute the quantities of <img class="formulaInl" alt="$B$" src="form_206.png"/> and <img class="formulaInl" alt="$A$" src="form_41.png"/> for each cluster during one scan through the data table in an aggregate function. Then sum over all clusters to the full <img class="formulaInl" alt="$B$" src="form_206.png"/> and <img class="formulaInl" alt="$A$" src="form_41.png"/> in the outside of the aggregate function. At last, the matrix mulplitications are done in a separate function on the master node.</p>
+<p>When multinomial logistic regression is computed before the multinomial clustered variance calculation, it uses a default reference category of zero and the regression coefficients are included in the output table. The regression coefficients in the output are in the same order as multinomial logistic regression function, which is described below. For a problem with <img class="formulaInl" alt="$ K $" src="form_117.png"/> dependent variables <img class="formulaInl" alt="$ (1, ..., K) $" src="form_118.png"/> and <img class="formulaInl" alt="$ J $" src="form_119.png"/> categories <img class="formulaInl" alt="$ (0, ..., J-1) $" src="form_120.png"/>, let <img class="formulaInl" alt="$ {m_{k,j}} $" src="form_121.png"/> denote the coefficient for dependent variable <img class="formulaInl" alt="$ k $" src="form_97.png"/> and category <img class="formulaInl" alt="$ j $" src="form_122.png"/>. The output is <img class="formulaInl" alt="$ {m_{k_1, j_0}, m_{k_1, j_1} \ldots m_{k_1, j_{J-1}},
m_{k_2, j_0}, m_{k_2, j_1} \ldots m_{k_K, j_{J-1}}} $" src="form_323.png"/>. The order is NOT CONSISTENT with the multinomial regression marginal effect calculation with function <em>marginal_mlogregr</em>. This is deliberate because the interfaces of all multinomial regressions (robust, clustered, ...) will be moved to match that used in marginal.</p>
+<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
+<p>[1] Standard, Robust, and Clustered Standard Errors Computed in R, <a href="http://diffuseprior.wordpress.com/2012/06/15/standard-robust-and-clustered-standard-errors-computed-in-r/">http://diffuseprior.wordpress.com/2012/06/15/standard-robust-and-clustered-standard-errors-computed-in-r/</a></p>
+<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd>File <a class="el" href="clustered__variance_8sql__in.html">clustered_variance.sql_in</a> documenting the clustered variance SQL functions.</dd></dl>
+<p>File <a class="el" href="clustered__variance__coxph_8sql__in.html" title="SQL functions for clustered robust cox proportional hazards regression. ">clustered_variance_coxph.sql_in</a> documenting the clustered variance for Cox proportional hazards SQL functions.</p>
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+<div class="title">Clustering<div class="ingroups"><a class="el" href="group__grp__unsupervised.html">Unsupervised Learning</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
+<p>A collection of methods for clustering data </p>
+<table class="memberdecls">
+<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="groups"></a>
+Modules</h2></td></tr>
+<tr class="memitem:group__grp__kmeans"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__kmeans.html">k-Means Clustering</a></td></tr>
+<tr class="memdesc:group__grp__kmeans"><td class="mdescLeft"> </td><td class="mdescRight">Partitions a set of observations into clusters by finding centroids that minimize the sum of observations' distances from their closest centroid. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
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+<div class="title">Pearson's Correlation<div class="ingroups"><a class="el" href="group__grp__stats.html">Statistics</a> » <a class="el" href="group__grp__desc__stats.html">Descriptive Statistics</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> </p><ul>
+<li>
+<a href="#usage">Correlation Function</a> </li>
+<li>
+<a href="#examples">Examples</a> </li>
+<li>
+<a href="#seealso">See Also</a> </li>
+</ul>
+</div><p>A correlation function is the degree and direction of association of two variables—how well one random variable can be predicted from the other. The coefficient of correlation varies from -1 to 1. A coefficient of 1 implies perfect correlation, 0 means no correlation, and -1 means perfect anti-correlation.</p>
+<p>This function provides a cross-correlation matrix for all pairs of numeric columns in a <em>source_table</em>. A correlation matrix describes correlation among <img class="formulaInl" alt="$ M $" src="form_174.png"/> variables. It is a square symmetrical <img class="formulaInl" alt="$ M $" src="form_174.png"/>x <img class="formulaInl" alt="$M $" src="form_380.png"/> matrix with the <img class="formulaInl" alt="$ (ij) $" src="form_381.png"/>th element equal to the correlation coefficient between the <img class="formulaInl" alt="$i$" src="form_128.png"/>th and the <img class="formulaInl" alt="$j$" src="form_129.png"/>th variable. The diagonal elements (correlations of variables with themselves) are always equal to 1.0.</p>
+<p><a class="anchor" id="usage"></a></p><dl class="section user"><dt>Correlation Function</dt><dd></dd></dl>
+<p>The correlation function has the following syntax: </p><pre class="syntax">
+correlation( source_table,
+ output_table,
+ target_cols,
+ verbose
+ )
+</pre><p>The covariance function, with a similar syntax, can be used to compute the covariance between features. </p><pre class="syntax">
+covariance( source_table,
+ output_table,
+ target_cols,
+ verbose
+ )
+</pre><dl class="arglist">
+<dt>source_table </dt>
+<dd><p class="startdd">TEXT. The name of the data containing the input data.</p>
+<p class="enddd"></p>
+</dd>
+<dt>output_table </dt>
+<dd><p class="startdd">TEXT. The name of the table where the cross-correlation matrix will be saved. The output is a table with N+2 columns and N rows, where N is the number of target columns. It contains the following columns. </p><table class="output">
+<tr>
+<th>column_position </th><td>The first column is a sequential counter indicating the position of the variable in the '<em>output_table</em>'. </td></tr>
+<tr>
+<th>variable </th><td>The second column contains the row-header for the variables. </td></tr>
+<tr>
+<th><...> </th><td>The remainder of the table is the NxN correlation matrix for the pairs of numeric 'source_table' columns. </td></tr>
+</table>
+<p>The output table is arranged as a lower-triangular matrix with the upper triangle set to NULL and the diagonal elements set to 1.0. To obtain the result from the '<em>output_table</em>' in this matrix format ensure to order the elements using the '<em>column_position</em>', as shown in the example below. </p><pre class="example">
+SELECT * FROM output_table ORDER BY column_position;
+</pre><p>In addition to output table, a summary table named <output_table>_summary is also created at the same time, which has the following columns: </p><table class="output">
+<tr>
+<th>method</th><td>'correlation' </td></tr>
+<tr>
+<th>source_table</th><td>VARCHAR. The data source table name. </td></tr>
+<tr>
+<th>output_table</th><td>VARCHAR. The output table name. </td></tr>
+<tr>
+<th>column_names</th><td>VARCHAR. Column names used for correlation computation, comma-separated string. </td></tr>
+<tr>
+<th>mean_vector</th><td>FLOAT8[]. Vector where each is the mean of a column. </td></tr>
+<tr>
+<th>total_rows_processed </th><td>BIGINT. Total numbers of rows processed. </td></tr>
+<tr>
+<th>total_rows_skipped </th><td>BIGINT. Total numbers of rows skipped due to missing values. </td></tr>
+</table>
+<p class="enddd"></p>
+</dd>
+<dt>target_cols (optional) </dt>
+<dd><p class="startdd">TEXT, default: '*'. A comma-separated list of the columns to correlate. If NULL or <code>'*'</code>, results are produced for all numeric columns.</p>
+<p class="enddd"></p>
+</dd>
+<dt>verbose (optional) </dt>
+<dd><p class="startdd">BOOLEAN, default: FALSE. Print verbose debugging information if TRUE.</p>
+<p class="enddd"></p>
+</dd>
+</dl>
+<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
+<ol type="1">
+<li>View online help for the correlation function. <pre class="example">
+SELECT madlib.correlation();
+</pre></li>
+<li>Create an input data set. <pre class="example">
+DROP TABLE IF EXISTS example_data;
+CREATE TABLE example_data(
+ id SERIAL, outlook TEXT,
+ temperature FLOAT8, humidity FLOAT8,
+ windy TEXT, class TEXT);
+INSERT INTO example_data VALUES
+(1, 'sunny', 85, 85, 'false', 'Dont Play'),
+(2, 'sunny', 80, 90, 'true', 'Dont Play'),
+(3, 'overcast', 83, 78, 'false', 'Play'),
+(4, 'rain', 70, 96, 'false', 'Play'),
+(5, 'rain', 68, 80, 'false', 'Play'),
+(6, 'rain', 65, 70, 'true', 'Dont Play'),
+(7, 'overcast', 64, 65, 'true', 'Play'),
+(8, 'sunny', 72, 95, 'false', 'Dont Play'),
+(9, 'sunny', 69, 70, 'false', 'Play'),
+(10, 'rain', 75, 80, 'false', 'Play'),
+(11, 'sunny', 75, 70, 'true', 'Play'),
+(12, 'overcast', 72, 90, 'true', 'Play'),
+(13, 'overcast', 81, 75, 'false', 'Play'),
+(14, 'rain', 71, 80, 'true', 'Dont Play'),
+(15, NULL, 100, 100, 'true', NULL),
+(16, NULL, 110, 100, 'true', NULL);
+</pre></li>
+<li>Run the <a class="el" href="correlation_8sql__in.html#ada17a10ea8a6c4580e7413c86ae5345e">correlation()</a> function on the data set. <pre class="example">
+-- Correlate all numeric columns
+SELECT madlib.correlation( 'example_data',
+ 'example_data_output'
+ );
+-- Setting target_cols to NULL or '*' also correlates all numeric columns
+SELECT madlib.correlation( 'example_data',
+ 'example_data_output',
+ '*'
+ );
+-- Correlate only the temperature and humidity columns
+SELECT madlib.correlation( 'example_data',
+ 'example_data_output',
+ 'temperature, humidity'
+ );
+</pre></li>
+<li>View the correlation matrix. <pre class="example">
+SELECT * FROM example_data_output ORDER BY column_position;
+</pre> Result: <pre class="result">
+ column_position | variable | temperature | humidity
+-----------------+-------------+-------------------+----------
+ 1 | temperature | 1.0 |
+ 2 | humidity | 0.616876934548786 | 1.0
+(2 rows)
+</pre></li>
+<li>Compute the covariance of features in the data set. <pre class="example">
+SELECT madlib.covariance( 'example_data',
+ 'cov_output'
+ );
+</pre></li>
+<li>View the covariance matrix. <pre class="example">
+SELECT * FROM cov_output ORDER BY column_position;
+</pre> Result: <pre class="result">
+ column_position | variable | temperature | humidity
+-----------------+-------------+-------------------+----------
+ 1 | temperature | 146.25 |
+ 2 | humidity | 82.125 | 121.1875
+(2 rows)
+</pre></li>
+</ol>
+<dl class="section user"><dt>Notes</dt><dd>Current implementation ignores a row that contains NULL entirely. This means any correlation in such a row (with NULLs) does not contribute to the final answer.</dd></dl>
+<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="correlation_8sql__in.html" title="SQL functions for correlation computation. ">correlation.sql_in</a> documenting the SQL functions</p>
+<p><a class="el" href="group__grp__summary.html">Summary</a> for general descriptive statistics for a table </p>
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