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Posted to commits@madlib.apache.org by es...@apache.org on 2017/08/29 19:09:59 UTC
[28/51] [partial] incubator-madlib-site git commit: Update website
for 1.21 release.
http://git-wip-us.apache.org/repos/asf/incubator-madlib-site/blob/4ddf1756/docs/v1.12/group__grp__graph.html
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diff --git a/docs/v1.12/group__grp__graph.html b/docs/v1.12/group__grp__graph.html
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+<!-- 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">
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+<title>MADlib: Graph</title>
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+ <tbody>
+ <tr style="height: 56px;">
+ <td id="projectlogo"><a href="http://madlib.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">1.12</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"
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+ onkeyup="searchBox.OnSearchFieldChange(event)"/>
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+ class="ui-resizable-handle">
+ </div>
+</div>
+<script type="text/javascript">
+$(document).ready(function(){initNavTree('group__grp__graph.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="summary">
+<a href="#groups">Modules</a> </div>
+ <div class="headertitle">
+<div class="title">Graph</div> </div>
+</div><!--header-->
+<div class="contents">
+<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
+<p>Contains graph algorithms. </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__apsp"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__apsp.html">All Pairs Shortest Path</a></td></tr>
+<tr class="memdesc:group__grp__apsp"><td class="mdescLeft"> </td><td class="mdescRight">Finds the shortest paths between every vertex pair in a given graph. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:group__grp__bfs"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__bfs.html">Breadth-First Search</a></td></tr>
+<tr class="memdesc:group__grp__bfs"><td class="mdescLeft"> </td><td class="mdescRight">Finds the nodes reachable from a given source vertex using a breadth-first approach. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:group__grp__graph__measures"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__graph__measures.html">Measures</a></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:group__grp__pagerank"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__pagerank.html">PageRank</a></td></tr>
+<tr class="memdesc:group__grp__pagerank"><td class="mdescLeft"> </td><td class="mdescRight">Find the PageRank of all vertices in a directed graph. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:group__grp__sssp"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__sssp.html">Single Source Shortest Path</a></td></tr>
+<tr class="memdesc:group__grp__sssp"><td class="mdescLeft"> </td><td class="mdescRight">Finds the shortest path from a single source vertex to every other vertex in a given graph. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:group__grp__wcc"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__wcc.html">Weakly Connected Components</a></td></tr>
+<tr class="memdesc:group__grp__wcc"><td class="mdescLeft"> </td><td class="mdescRight">Find all weakly connected components of a graph. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+</table>
+</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 Aug 29 2017 09:14:34 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/4ddf1756/docs/v1.12/group__grp__graph.js
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diff --git a/docs/v1.12/group__grp__graph.js b/docs/v1.12/group__grp__graph.js
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+var group__grp__graph =
+[
+ [ "All Pairs Shortest Path", "group__grp__apsp.html", null ],
+ [ "Breadth-First Search", "group__grp__bfs.html", null ],
+ [ "Measures", "group__grp__graph__measures.html", "group__grp__graph__measures" ],
+ [ "PageRank", "group__grp__pagerank.html", null ],
+ [ "Single Source Shortest Path", "group__grp__sssp.html", null ],
+ [ "Weakly Connected Components", "group__grp__wcc.html", null ]
+];
\ No newline at end of file
http://git-wip-us.apache.org/repos/asf/incubator-madlib-site/blob/4ddf1756/docs/v1.12/group__grp__graph__avg__path__length.html
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diff --git a/docs/v1.12/group__grp__graph__avg__path__length.html b/docs/v1.12/group__grp__graph__avg__path__length.html
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@@ -0,0 +1,238 @@
+<!-- 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">
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+<title>MADlib: Average Path Length</title>
+<link href="tabs.css" rel="stylesheet" type="text/css"/>
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+ jax: ["input/TeX","output/HTML-CSS"],
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+</script><script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js"></script>
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+<table cellspacing="0" cellpadding="0">
+ <tbody>
+ <tr style="height: 56px;">
+ <td id="projectlogo"><a href="http://madlib.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">1.12</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()"
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+ onfocus="searchBox.OnSearchFieldFocus(true)"
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+ <a id="MSearchClose" href="javascript:searchBox.CloseResultsWindow()"><img id="MSearchCloseImg" border="0" src="search/close.png" alt=""/></a>
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+ class="ui-resizable-handle">
+ </div>
+</div>
+<script type="text/javascript">
+$(document).ready(function(){initNavTree('group__grp__graph__avg__path__length.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">Average Path Length<div class="ingroups"><a class="el" href="group__grp__graph.html">Graph</a> » <a class="el" href="group__grp__graph__measures.html">Measures</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> <ul>
+<li>
+<a href="#avg_path_length">Average Path Length</a> </li>
+<li>
+<a href="#examples">Examples</a> </li>
+</ul>
+</div><p>This function computes the average of the shortest paths between each pair of vertices. Average path length is based on "reachable target vertices", so it ignores infinite-length paths between vertices that are not connected.</p>
+<dl class="section note"><dt>Note</dt><dd>This function assumes a valid output from a prior APSP run - both the APSP table and the associated output summary table. APSP is a computationally expensive algorithm because it finds the shortest path between all nodes in the graph. The worst case run-time for this implementation is O(V^2 * E) where V is the number of vertices and E is the number of edges. In practice, run-time will be generally be much less than this, depending on the graph.</dd></dl>
+<p><a class="anchor" id="avg_path_length"></a></p><dl class="section user"><dt>Average Path Length</dt><dd><pre class="syntax">
+graph_avg_path_length( apsp_table,
+ output_table
+ )
+</pre></dd></dl>
+<p><b>Arguments</b> </p><dl class="arglist">
+<dt>apsp_table </dt>
+<dd><p class="startdd">TEXT. Name of the output table generated by a prior run of all pairs shortest path (APSP). </p>
+<p class="enddd"></p>
+</dd>
+<dt>out_table </dt>
+<dd>TEXT. Name of the table to store the average path length. It contains a row for every group, and the average path value. </dd>
+</dl>
+<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
+<ol type="1">
+<li>Create vertex and edge tables to represent the graph: <pre class="syntax">
+DROP TABLE IF EXISTS vertex, edge;
+CREATE TABLE vertex(
+ id INTEGER,
+ name TEXT
+ );
+CREATE TABLE edge(
+ src_id INTEGER,
+ dest_id INTEGER,
+ edge_weight FLOAT8
+ );
+INSERT INTO vertex VALUES
+(0, 'A'),
+(1, 'B'),
+(2, 'C'),
+(3, 'D'),
+(4, 'E'),
+(5, 'F'),
+(6, 'G'),
+(7, 'H');
+INSERT INTO edge VALUES
+(0, 1, 1.0),
+(0, 2, 1.0),
+(0, 4, 10.0),
+(1, 2, 2.0),
+(1, 3, 10.0),
+(2, 3, 1.0),
+(2, 5, 1.0),
+(2, 6, 3.0),
+(3, 0, 1.0),
+(4, 0, -2.0),
+(5, 6, 1.0),
+(6, 7, 1.0);
+</pre></li>
+<li>Calculate the all-pair shortest paths: <pre class="syntax">
+DROP TABLE IF EXISTS out_apsp, out_apsp_summary;
+SELECT madlib.graph_apsp('vertex', -- Vertex table
+ 'id', -- Vertix id column (NULL means use default naming)
+ 'edge', -- Edge table
+ 'src=src_id, dest=dest_id, weight=edge_weight',
+ -- Edge arguments (NULL means use default naming)
+ 'out_apsp'); -- Output table of shortest paths
+</pre></li>
+<li>Compute the average path length measure: <pre class="syntax">
+DROP TABLE IF EXISTS out_avg_path_length;
+SELECT madlib.graph_avg_path_length('out_apsp', 'out_avg_path_length');
+SELECT * FROM out_avg_path_length;
+</pre> <pre class="result">
+ avg_path_length
+------------------
+ 2.01785714285714
+(1 row)
+</pre></li>
+<li>Create a graph with 2 groups and find APSP for each group: <pre class="syntax">
+DROP TABLE IF EXISTS edge_gr;
+CREATE TABLE edge_gr AS
+(
+ SELECT *, 0 AS grp FROM edge
+ UNION
+ SELECT *, 1 AS grp FROM edge WHERE src_id < 6 AND dest_id < 6
+);
+INSERT INTO edge_gr VALUES
+(4,5,-20,1);
+</pre></li>
+<li>Find APSP for all groups: <pre class="syntax">
+DROP TABLE IF EXISTS out_gr, out_gr_summary;
+SELECT madlib.graph_apsp(
+ 'vertex', -- Vertex table
+ NULL, -- Vertex id column (NULL means use default naming)
+ 'edge_gr', -- Edge table
+ 'src=src_id, dest=dest_id, weight=edge_weight',
+ 'out_gr', -- Output table of shortest paths
+ 'grp' -- Grouping columns
+);
+</pre></li>
+<li>Find the average path length in every group <pre class="syntax">
+DROP TABLE IF EXISTS out_gr_path;
+SELECT madlib.graph_avg_path_length('out_gr', 'out_gr_path');
+SELECT * FROM out_gr_path ORDER BY grp;
+</pre> <pre class="result">
+ grp | avg_path_length
+----—+-----------------—
+ 0 | 2.01785714285714
+ 1 | 0.466666666666667
+(2 rows)
+</pre> </li>
+</ol>
+</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 Aug 29 2017 09:14:34 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/4ddf1756/docs/v1.12/group__grp__graph__closeness.html
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diff --git a/docs/v1.12/group__grp__graph__closeness.html b/docs/v1.12/group__grp__graph__closeness.html
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+<!-- 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">
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+<title>MADlib: Closeness</title>
+<link href="tabs.css" rel="stylesheet" type="text/css"/>
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+<div class="title">Closeness<div class="ingroups"><a class="el" href="group__grp__graph.html">Graph</a> » <a class="el" href="group__grp__graph__measures.html">Measures</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> <ul>
+<li>
+<a href="#closeness">Closeness</a> </li>
+<li>
+<a href="#examples">Examples</a> </li>
+</ul>
+</div><p>The Closeness function returns various closeness centrality measures and the k-degree for given subset of vertices. The closeness measures are the inverse of the sum, the inverse of the average, and the sum of inverses of the shortest distances to all reachable target vertices (excluding the source vertex).</p>
+<dl class="section note"><dt>Note</dt><dd>The closeness measures require a valid output from a prior APSP run - both the APSP table and the associated output summary table. APSP is a computationally expensive algorithm because it finds the shortest path between all nodes in the graph. The worst case run-time for this implementation is O(V^2 E) where V is the number of vertices and E is the number of edges. In practice, run-time will be generally be much less than this, depending on the graph.</dd></dl>
+<p><a class="anchor" id="closeness"></a></p><dl class="section user"><dt>Closeness</dt><dd><pre class="syntax">
+graph_closeness( apsp_table,
+ output_table,
+ vertex_filter_expr
+ )
+</pre></dd></dl>
+<p><b>Arguments</b> </p><dl class="arglist">
+<dt>apsp_table </dt>
+<dd><p class="startdd">TEXT. Name of the output table generated by a prior run of all pairs shortest path (APSP). </p>
+<p class="enddd"></p>
+</dd>
+<dt>out_table </dt>
+<dd><p class="startdd">TEXT. Name of the table to store the closeness measures. It contains a row for every vertex of every group and have the following columns (in addition to the grouping columns):</p><ul>
+<li>inverse_sum_dist: Inverse of the sum of shortest distances to all reachable vertices.</li>
+<li>inverse_average_dist: Inverse of the average of shortest distances to all reachable vertices.</li>
+<li>sum_inverse_dist: Sum of the inverse of shortest distances to all reachable vertices.</li>
+<li>k_degree: Total number of reachable vertices. </li>
+</ul>
+<p class="enddd"></p>
+</dd>
+<dt>vertex_filter_expr (optional) </dt>
+<dd><p class="startdd">TEXT, default = NULL. Valid PostgreSQL expression that describes the vertices to generate closeness measures for. If this parameter is not specified, closeness measures are generated for all vertices in the apsp table. This input should be treated like a WHERE clause.</p>
+<p>Some example inputs:</p><ul>
+<li>If you want a short list of vertices, say 1, 2 and 3: <pre>vertex_id IN (1, 2, 3)</pre></li>
+<li>If you want a range of vertices between 1000 and 2000: <pre>vertix_id BETWEEN 1000 AND 2000</pre></li>
+<li>If you want a set of vertices from a separate table satisfying to a condition <pre>EXISTS (SELECT vertex_id FROM vertices_of_interest
+ WHERE vertex_id > 5000 AND condition = 'xyz')
+</pre></li>
+</ul>
+<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>Create vertex and edge tables to represent the graph: <pre class="syntax">
+DROP TABLE IF EXISTS vertex, edge;
+CREATE TABLE vertex(
+ id INTEGER,
+ name TEXT
+ );
+CREATE TABLE edge(
+ src_id INTEGER,
+ dest_id INTEGER,
+ edge_weight FLOAT8
+ );
+INSERT INTO vertex VALUES
+(0, 'A'),
+(1, 'B'),
+(2, 'C'),
+(3, 'D'),
+(4, 'E'),
+(5, 'F'),
+(6, 'G'),
+(7, 'H');
+INSERT INTO edge VALUES
+(0, 1, 1.0),
+(0, 2, 1.0),
+(0, 4, 10.0),
+(1, 2, 2.0),
+(1, 3, 10.0),
+(2, 3, 1.0),
+(2, 5, 1.0),
+(2, 6, 3.0),
+(3, 0, 1.0),
+(4, 0, -2.0),
+(5, 6, 1.0),
+(6, 7, 1.0);
+</pre></li>
+<li>Calculate the all-pair shortest paths: <pre class="syntax">
+DROP TABLE IF EXISTS out_apsp, out_apsp_summary;
+SELECT madlib.graph_apsp('vertex', -- Vertex table
+ 'id', -- Vertix id column (NULL means use default naming)
+ 'edge', -- Edge table
+ 'src=src_id, dest=dest_id, weight=edge_weight',
+ -- Edge arguments (NULL means use default naming)
+ 'out_apsp'); -- Output table of shortest paths
+</pre></li>
+<li>Compute the closeness measure for all nodes: <pre class="syntax">
+DROP TABLE IF EXISTS out_closeness;
+SELECT madlib.graph_closeness('out_apsp', 'out_closeness');
+SELECT * FROM out_closeness;
+</pre> <pre class="result">
+ src_id | inverse_sum_dist | inverse_avg_dist | sum_inverse_dist | k_degree
+--------+--------------------+-------------------+------------------+----------
+ 1 | 0.0285714285714286 | 0.2 | 1.93809523809524 | 7
+ 3 | 0.0357142857142857 | 0.25 | 2.87424242424242 | 7
+ 4 | -1 | -7 | -1 | 7
+ 0 | 0.0434782608695652 | 0.304347826086957 | 3.68333333333333 | 7
+ 6 | 1 | 1 | 1 | 1
+ 2 | 0.0416666666666667 | 0.291666666666667 | 3.75 | 7
+ 5 | 0.333333333333333 | 0.666666666666667 | 1.5 | 2
+ 7 | [NULL] | [NULL] | 0 | 0
+(8 rows)
+</pre></li>
+<li>Create a graph with 2 groups and find APSP for each group: <pre class="syntax">
+DROP TABLE IF EXISTS edge_gr;
+CREATE TABLE edge_gr AS
+(
+ SELECT *, 0 AS grp FROM edge
+ UNION
+ SELECT *, 1 AS grp FROM edge WHERE src_id < 6 AND dest_id < 6
+);
+INSERT INTO edge_gr VALUES
+(4,5,-20,1);
+</pre></li>
+<li>Find APSP for all groups: <pre class="syntax">
+DROP TABLE IF EXISTS out_gr, out_gr_summary;
+SELECT madlib.graph_apsp(
+ 'vertex', -- Vertex table
+ NULL, -- Vertex id column (NULL means use default naming)
+ 'edge_gr', -- Edge table
+ 'src=src_id, dest=dest_id, weight=edge_weight',
+ 'out_gr', -- Output table of shortest paths
+ 'grp' -- Grouping columns
+);
+</pre></li>
+<li>Compute closeness measure for vertex 0 to vertex 5 in every group <pre class="syntax">
+DROP TABLE IF EXISTS out_gr_path;
+SELECT madlib.graph_closeness('out_gr', 'out_gr_closeness', 'src_id >= 0 and src_id <=5');
+SELECT * FROM out_gr_closeness ORDER BY grp;
+</pre> <pre class="result">
+ grp | src_id | inverse_sum_dist | inverse_avg_dist | sum_inverse_dist | k_degree
+----—+-------—+--------------------—+-------------------—+------------------—+---------—
+ 0 | 0 | 0.0434782608695652 | 0.304347826086957 | 3.68333333333333 | 7
+ 0 | 5 | 0.333333333333333 | 0.666666666666667 | 1.5 | 2
+ 0 | 4 | -1 | -7 | -1 | 7
+ 0 | 3 | 0.0357142857142857 | 0.25 | 2.87424242424242 | 7
+ 0 | 1 | 0.0285714285714286 | 0.2 | 1.93809523809524 | 7
+ 0 | 2 | 0.0416666666666667 | 0.291666666666667 | 3.75 | 7
+ 1 | 3 | 0.142857142857143 | 0.714285714285714 | 1.97979797979798 | 5
+ 1 | 5 | [NULL] | [NULL] | 0 | 0
+ 1 | 0 | 0.25 | 1.25 | 2.5 | 5
+ 1 | 1 | 0.0588235294117647 | 0.294117647058824 | 0.988095238095238 | 5
+ 1 | 2 | 0.1 | 0.5 | 1.79166666666667 | 5
+ 1 | 4 | -0.0416666666666667 | -0.208333333333333 | -2.55 | 5
+(12 rows)
+</pre> </li>
+</ol>
+</div><!-- contents -->
+</div><!-- doc-content -->
+<!-- start footer part -->
+<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
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+ <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.13 </li>
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+<div class="header">
+ <div class="headertitle">
+<div class="title">Graph Diameter<div class="ingroups"><a class="el" href="group__grp__graph.html">Graph</a> » <a class="el" href="group__grp__graph__measures.html">Measures</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> <ul>
+<li>
+<a href="#diameter">Diameter</a> </li>
+<li>
+<a href="#examples">Examples</a> </li>
+</ul>
+</div><p>Diameter is defined as the longest of all shortest paths in a graph.</p>
+<dl class="section note"><dt>Note</dt><dd>This function assumes a valid output from a prior APSP run - both the APSP table and the associated output summary table. APSP is a computationally expensive algorithm because it finds the shortest path between all nodes in the graph. The worst case run-time for this implementation is O(V^2 * E) where V is the number of vertices and E is the number of edges. In practice, run-time will be generally be much less than this, depending on the graph.</dd></dl>
+<p><a class="anchor" id="diameter"></a></p><dl class="section user"><dt>Diameter</dt><dd><pre class="syntax">
+graph_diameter( apsp_table,
+ output_table
+ )
+</pre></dd></dl>
+<p><b>Arguments</b> </p><dl class="arglist">
+<dt>apsp_table </dt>
+<dd><p class="startdd">TEXT. Name of the output table generated by a prior run of all pairs shortest path (APSP). </p>
+<p class="enddd"></p>
+</dd>
+<dt>out_table </dt>
+<dd><p class="startdd">TEXT. Name of the table to store the diameter. It contains a row for every group, the diameter value and the two vertices that are the farthest apart. </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>Create vertex and edge tables to represent the graph: <pre class="syntax">
+DROP TABLE IF EXISTS vertex, edge;
+CREATE TABLE vertex(
+ id INTEGER,
+ name TEXT
+ );
+CREATE TABLE edge(
+ src_id INTEGER,
+ dest_id INTEGER,
+ edge_weight FLOAT8
+ );
+INSERT INTO vertex VALUES
+(0, 'A'),
+(1, 'B'),
+(2, 'C'),
+(3, 'D'),
+(4, 'E'),
+(5, 'F'),
+(6, 'G'),
+(7, 'H');
+INSERT INTO edge VALUES
+(0, 1, 1.0),
+(0, 2, 1.0),
+(0, 4, 10.0),
+(1, 2, 2.0),
+(1, 3, 10.0),
+(2, 3, 1.0),
+(2, 5, 1.0),
+(2, 6, 3.0),
+(3, 0, 1.0),
+(4, 0, -2.0),
+(5, 6, 1.0),
+(6, 7, 1.0);
+</pre></li>
+<li>Calculate the all-pair shortest paths: <pre class="syntax">
+DROP TABLE IF EXISTS out_apsp, out_apsp_summary;
+SELECT madlib.graph_apsp('vertex', -- Vertex table
+ 'id', -- Vertix id column (NULL means use default naming)
+ 'edge', -- Edge table
+ 'src=src_id, dest=dest_id, weight=edge_weight',
+ -- Edge arguments (NULL means use default naming)
+ 'out_apsp'); -- Output table of shortest paths
+</pre></li>
+<li>Compute the diameter measure for the graph: <pre class="syntax">
+DROP TABLE IF EXISTS out_diameter;
+SELECT madlib.graph_diameter('out_apsp', 'out_diameter');
+SELECT * FROM out_diameter;
+</pre> <pre class="result">
+diameter | diameter_end_vertices
+---------+-----------------------
+ 14 | {{1,4}}
+(1 row)
+</pre></li>
+<li>Create a graph with 2 groups and find APSP for each group: <pre class="syntax">
+DROP TABLE IF EXISTS edge_gr;
+CREATE TABLE edge_gr AS
+(
+ SELECT *, 0 AS grp FROM edge
+ UNION
+ SELECT *, 1 AS grp FROM edge WHERE src_id < 6 AND dest_id < 6
+);
+INSERT INTO edge_gr VALUES
+(4,5,-20,1);
+</pre></li>
+<li>Find APSP for all groups: <pre class="syntax">
+DROP TABLE IF EXISTS out_gr, out_gr_summary;
+SELECT madlib.graph_apsp(
+ 'vertex', -- Vertex table
+ NULL, -- Vertex id column (NULL means use default naming)
+ 'edge_gr', -- Edge table
+ 'src=src_id, dest=dest_id, weight=edge_weight',
+ 'out_gr', -- Output table of shortest paths
+ 'grp' -- Grouping columns
+);
+</pre></li>
+<li>Find the diameter of graph in every group <pre class="syntax">
+DROP TABLE IF EXISTS out_gr_path;
+SELECT madlib.graph_diameter('out_gr', 'out_gr_diameter');
+SELECT * FROM out_gr_diameter ORDER BY grp;
+</pre> <pre class="result">
+grp | diameter | diameter_end_vertices
+---—+---------—+----------------------—
+ 0 | 14 | {{1,4}}
+ 1 | 14 | {{1,4}}
+(2 rows)
+</pre> </li>
+</ol>
+</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 Aug 29 2017 09:14:34 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>
+ <div id="projectbrief">User Documentation for MADlib</div>
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+</div><!--header-->
+<div class="contents">
+<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
+<p>Graph Measures </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__graph__closeness"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__graph__closeness.html">Closeness</a></td></tr>
+<tr class="memdesc:group__grp__graph__closeness"><td class="mdescLeft"> </td><td class="mdescRight">Computes the closeness centrality value of each node in the graph. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:group__grp__graph__diameter"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__graph__diameter.html">Graph Diameter</a></td></tr>
+<tr class="memdesc:group__grp__graph__diameter"><td class="mdescLeft"> </td><td class="mdescRight">Computes the diameter of a graph. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:group__grp__graph__avg__path__length"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__graph__avg__path__length.html">Average Path Length</a></td></tr>
+<tr class="memdesc:group__grp__graph__avg__path__length"><td class="mdescLeft"> </td><td class="mdescRight">Computes the average shortest-path length of a graph. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:group__grp__graph__vertex__degrees"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__graph__vertex__degrees.html">In-Out degree</a></td></tr>
+<tr class="memdesc:group__grp__graph__vertex__degrees"><td class="mdescLeft"> </td><td class="mdescRight">Computes the degrees for each vertex. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
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+</div><!-- contents -->
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+];
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+ </div>
+ <div id="projectbrief">User Documentation for MADlib</div>
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+
+<div class="header">
+ <div class="headertitle">
+<div class="title">In-Out degree<div class="ingroups"><a class="el" href="group__grp__graph.html">Graph</a> » <a class="el" href="group__grp__graph__measures.html">Measures</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> <ul>
+<li>
+<a href="#degrees">In-out degrees</a> </li>
+<li>
+<a href="#examples">Examples</a> </li>
+</ul>
+</div><p>This function computes the degree of each node. The node degree is the number of edges adjacent to that node. The node in-degree is the number of edges pointing in to the node and node out-degree is the number of edges pointing out of the node.</p>
+<p><a class="anchor" id="degrees"></a></p><dl class="section user"><dt>In-out degrees</dt><dd><pre class="syntax">
+graph_vertex_degrees(
+ vertex_table,
+ vertex_id,
+ edge_table,
+ edge_args,
+ out_table,
+ grouping_cols
+)
+</pre></dd></dl>
+<p><b>Arguments</b> </p><dl class="arglist">
+<dt>vertex_table </dt>
+<dd><p class="startdd">TEXT. Name of the table containing the vertex data for the graph. Must contain the column specified in the 'vertex_id' parameter below.</p>
+<p class="enddd"></p>
+</dd>
+<dt>vertex_id </dt>
+<dd><p class="startdd">TEXT, default = 'id'. Name of the column in 'vertex_table' containing vertex ids. The vertex ids are of type INTEGER with no duplicates. They do not need to be contiguous.</p>
+<p class="enddd"></p>
+</dd>
+<dt>edge_table </dt>
+<dd><p class="startdd">TEXT. Name of the table containing the edge data. The edge table must contain columns for source vertex, destination vertex and edge weight. Column naming convention is described below in the 'edge_args' parameter.</p>
+<p class="enddd"></p>
+</dd>
+<dt>edge_args </dt>
+<dd><p class="startdd">TEXT. A comma-delimited string containing multiple named arguments of the form "name=value". The following parameters are supported for this string argument:</p><ul>
+<li>src (INTEGER): Name of the column containing the source vertex ids in the edge table. Default column name is 'src'.</li>
+<li>dest (INTEGER): Name of the column containing the destination vertex ids in the edge table. Default column name is 'dest'.</li>
+<li>weight (FLOAT8): Name of the column containing the edge weights in the edge table. Default column name is 'weight'.</li>
+</ul>
+<p class="enddd"></p>
+</dd>
+<dt>out_table </dt>
+<dd><p class="startdd">TEXT. Name of the table to store the result. It contains a row for every vertex of every group and has the following columns (in addition to the grouping columns):</p><ul>
+<li>vertex: The id for the source vertex. Will use the input vertex column 'id' for column naming.</li>
+<li>indegree: Number of incoming edges to the vertex.</li>
+<li>outdegree: Number of outgoing edges from the vertex.</li>
+</ul>
+<p class="enddd"></p>
+</dd>
+<dt>grouping_cols </dt>
+<dd>TEXT, default = NULL. List of columns used to group the input into discrete subgraphs. These columns must exist in the edge table. When this value is null, no grouping is used and a single result is generated. </dd>
+</dl>
+<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
+<ol type="1">
+<li>Create vertex and edge tables to represent the graph: <pre class="syntax">
+DROP TABLE IF EXISTS vertex, edge;
+CREATE TABLE vertex(
+ id INTEGER,
+ name TEXT
+ );
+CREATE TABLE edge(
+ src_id INTEGER,
+ dest_id INTEGER,
+ edge_weight FLOAT8
+ );
+INSERT INTO vertex VALUES
+(0, 'A'),
+(1, 'B'),
+(2, 'C'),
+(3, 'D'),
+(4, 'E'),
+(5, 'F'),
+(6, 'G'),
+(7, 'H');
+INSERT INTO edge VALUES
+(0, 1, 1.0),
+(0, 2, 1.0),
+(0, 4, 10.0),
+(1, 2, 2.0),
+(1, 3, 10.0),
+(2, 3, 1.0),
+(2, 5, 1.0),
+(2, 6, 3.0),
+(3, 0, 1.0),
+(4, 0, -2.0),
+(5, 6, 1.0),
+(6, 7, 1.0);
+</pre></li>
+<li>Calculate the in-out degrees for each node: <pre class="syntax">
+DROP TABLE IF EXISTS degrees;
+SELECT madlib.graph_vertex_degrees(
+ 'vertex', -- Vertex table
+ 'id', -- Vertix id column (NULL means use default naming)
+ 'edge', -- Edge table
+ 'src=src_id, dest=dest_id, weight=edge_weight',
+ 'degrees'); -- Output table of shortest paths
+SELECT * FROM degrees ORDER BY id;
+</pre> <pre class="result">
+ id | indegree | outdegree
+----+----------+-----------
+ 0 | 2 | 3
+ 1 | 1 | 2
+ 2 | 2 | 3
+ 3 | 2 | 1
+ 4 | 1 | 1
+ 5 | 1 | 1
+ 6 | 2 | 1
+ 7 | 1 | 0
+</pre></li>
+<li>Create a graph with 2 groups and find degrees for each group: <pre class="syntax">
+DROP TABLE IF EXISTS edge_gr;
+CREATE TABLE edge_gr AS
+(
+ SELECT *, 0 AS grp FROM edge
+ UNION
+ SELECT *, 1 AS grp FROM edge WHERE src_id < 6 AND dest_id < 6
+);
+INSERT INTO edge_gr VALUES
+(4,5,-20,1);
+</pre></li>
+<li>Find APSP for all groups: <pre class="syntax">
+DROP TABLE IF EXISTS out_gr;
+SELECT madlib.graph_vertex_degrees(
+ 'vertex', -- Vertex table
+ NULL, -- Vertex id column (NULL means use default naming)
+ 'edge_gr', -- Edge table
+ 'src=src_id, dest=dest_id, weight=edge_weight',
+ 'out_gr', -- Output table of shortest paths
+ 'grp' -- Grouping columns
+);
+SELECT * FROM out_gr WHERE id < 2 ORDER BY grp, id;
+</pre> <pre class="result">
+ grp | id | indegree | outdegree
+----—+---—+---------—+----------—
+ 0 | 0 | 2 | 3
+ 0 | 1 | 1 | 2
+ 1 | 0 | 2 | 3
+ 1 | 1 | 1 | 2
+(4 rows)
+</pre> </li>
+</ol>
+</div><!-- contents -->
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+<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
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+</html>
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+ <td style="padding-left: 0.5em;">
+ <div id="projectname">
+ <span id="projectnumber">1.12</span>
+ </div>
+ <div id="projectbrief">User Documentation for MADlib</div>
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+<div class="header">
+ <div class="headertitle">
+<div class="title">Create Indicator Variables<div class="ingroups"><a class="el" href="group__grp__deprecated.html">Deprecated Modules</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<dl class="section warning"><dt>Warning</dt><dd><em> This version of encoding categorical variables has been deprecated. The new module with more capability can be found here <a class="el" href="group__grp__encode__categorical.html">Encoding Categorical Variables</a></em></dd></dl>
+<div class="toc"><b>Contents</b> <ul>
+<li>
+<a href="#categorical">Coding systems for categorical variables</a> </li>
+<li>
+<a href="#examples">Examples</a> </li>
+</ul>
+</div><p><a class="anchor" id="categorical"></a></p><dl class="section user"><dt>Coding systems for categorical variables</dt><dd>Categorical variables require special attention in regression analysis because, unlike dichotomous or continuous variables, they cannot be entered into the regression equation just as they are. For example, if you have a variable called race that is coded 1 = Hispanic, 2 = Asian, 3 = Black, 4 = White, then entering race in your regression will look at the linear effect of race, which is probably not what you intended. Instead, categorical variables like this need to be recoded into a series of indicator variables which can then be entered into the regression model. There are a variety of coding systems (also called as contrasts) that can be used when coding categorical variables. including dummy, effects, orthogonal, and helmert coding.</dd></dl>
+<p>We currently only support the dummy coding technique. Dummy coding is used when a researcher wants to compare other groups of the predictor variable with one specific group of the predictor variable. Often, the specific group to compare with is called the reference group.</p>
+<pre class="syntax">
+create_indicator_variables(
+ source_table,
+ output_table,
+ categorical_cols,
+ keep_null,
+ distributed_by
+ )
+</pre><p> <b>Arguments</b> </p><dl class="arglist">
+<dt>source_table </dt>
+<dd>VARCHAR. Name of the source table, containing data for categorical variables. </dd>
+<dt>output_table </dt>
+<dd>VARCHAR. Name of result table. The output table has the same columns as the original table, adding new indicator variable columns for each categorical column. The column name for the indicator variable is <em>'categorical column name'</em>_<em>'categorical value'</em>. </dd>
+<dt>categorical_cols </dt>
+<dd>VARCHAR. Comma-separated string of column names of categorical variables that need to be dummy-coded. </dd>
+<dt>keep_null (optional) </dt>
+<dd>BOOLEAN. default: FALSE. Whether 'NULL' should be treated as one of the categories of the categorical variable. If True, then an indicator variable is created corresponding to the NULL value. If False, then all indicator variables for that record will be set to NULL. </dd>
+<dt>distributed_by (optional) </dt>
+<dd>VARCHAR. default: NULL. Columns to use for the distribution policy of the output table. When NULL, the distribution policy of 'source_table' will be used. This argument is not available for POSTGRESQL platforms. </dd>
+</dl>
+<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
+<ol type="1">
+<li>Use a subset of the abalone dataset. <pre class="example">
+DROP TABLE IF EXISTS abalone;
+CREATE TABLE abalone (
+ sex character varying,
+ length double precision,
+ diameter double precision,
+ height double precision
+);
+COPY abalone (sex, length, diameter, height) FROM stdin WITH DELIMITER '|' NULL as '@';
+M| 0.455 | 0.365 | 0.095
+F| 0.53 | 0.42 | 0.135
+M| 0.35 | 0.265 | 0.09
+F| 0.53 | 0.415 | 0.15
+M| 0.44 | 0.365 | 0.125
+F| 0.545 | 0.425 | 0.125
+I| 0.33 | 0.255 | 0.08
+F| 0.55 | 0.44 | 0.15
+I| 0.425 | 0.30 | 0.095
+F| 0.525 | 0.38 | 0.140
+M| 0.475 | 0.37 | 0.125
+F| 0.535 | 0.405 | 0.145
+M| 0.43 | 0.358 | 0.11
+F| 0.47 | 0.355 | 0.100
+M| 0.49 | 0.38 | 0.135
+F| 0.44 | 0.340 | 0.100
+M| 0.5 | 0.400 | 0.13
+F| 0.565 | 0.44 | 0.155
+I| 0.355 | 0.280 | 0.085
+F| 0.550 | 0.415 | 0.135
+| 0.475 | 0.37 | 0.125
+\.
+</pre></li>
+<li>Create new table with dummy-coded indicator variables <pre class="example">
+drop table if exists abalone_out;
+select madlib.create_indicator_variables ('abalone', 'abalone_out', 'sex');
+select * from abalone_out;
+</pre> <pre class="result">
+ sex | length | diameter | height | sex_F | sex_I | sex_M
+  -----+--------+----------+--------+--------+--------+-------
+ F | 0.53 | 0.42 | 0.135 | 1 | 0 | 0
+ F | 0.53 | 0.415 | 0.15 | 1 | 0 | 0
+ F | 0.545 | 0.425 | 0.125 | 1 | 0 | 0
+ F | 0.55 | 0.44 | 0.15 | 1 | 0 | 0
+ F | 0.525 | 0.38 | 0.14 | 1 | 0 | 0
+ F | 0.535 | 0.405 | 0.145 | 1 | 0 | 0
+ F | 0.47 | 0.355 | 0.1 | 1 | 0 | 0
+ F | 0.44 | 0.34 | 0.1 | 1 | 0 | 0
+ F | 0.565 | 0.44 | 0.155 | 1 | 0 | 0
+ F | 0.55 | 0.415 | 0.135 | 1 | 0 | 0
+ M | 0.455 | 0.365 | 0.095 | 0 | 0 | 1
+ M | 0.35 | 0.265 | 0.09 | 0 | 0 | 0
+ M | 0.44 | 0.365 | 0.125 | 0 | 0 | 0
+ I | 0.33 | 0.255 | 0.08 | 0 | 1 | 0
+ I | 0.425 | 0.3 | 0.095 | 0 | 1 | 0
+ M | 0.475 | 0.37 | 0.125 | 0 | 0 | 0
+ M | 0.43 | 0.358 | 0.11 | 0 | 0 | 0
+ M | 0.49 | 0.38 | 0.135 | 0 | 0 | 0
+ M | 0.5 | 0.4 | 0.13 | 0 | 0 | 0
+ I | 0.355 | 0.28 | 0.085 | 0 | 1 | 0
+ NULL | 0.55 | 0.415 | 0.135 | NULL | NULL | NULL
+</pre></li>
+<li>Create indicator variable for 'NULL' value (note the additional column '"sex_NULL"') <pre class="example">
+drop table if exists abalone_out;
+select madlib.create_indicator_variables'abalone', 'abalone_out', 'sex', True);
+select * from abalone_out;
+</pre> <pre class="result">
+ sex | length | diameter | height | sex_F | sex_I | sex_M | sex_NULL
+  ---—+-----—+-------—+-----—+-----—+-----—+----—+----—
+ F | 0.53 | 0.42 | 0.135 | 1 | 0 | 0 | 0
+ F | 0.53 | 0.415 | 0.15 | 1 | 0 | 0 | 0
+ F | 0.545 | 0.425 | 0.125 | 1 | 0 | 0 | 0
+ F | 0.55 | 0.44 | 0.15 | 1 | 0 | 0 | 0
+ F | 0.525 | 0.38 | 0.14 | 1 | 0 | 0 | 0
+ F | 0.535 | 0.405 | 0.145 | 1 | 0 | 0 | 0
+ F | 0.47 | 0.355 | 0.1 | 1 | 0 | 0 | 0
+ F | 0.44 | 0.34 | 0.1 | 1 | 0 | 0 | 0
+ F | 0.565 | 0.44 | 0.155 | 1 | 0 | 0 | 0
+ F | 0.55 | 0.415 | 0.135 | 1 | 0 | 0 | 0
+ M | 0.455 | 0.365 | 0.095 | 0 | 0 | 1 | 0
+ M | 0.35 | 0.265 | 0.09 | 0 | 0 | 0 | 0
+ M | 0.44 | 0.365 | 0.125 | 0 | 0 | 0 | 0
+ I | 0.33 | 0.255 | 0.08 | 0 | 1 | 0 | 0
+ I | 0.425 | 0.3 | 0.095 | 0 | 1 | 0 | 0
+ M | 0.475 | 0.37 | 0.125 | 0 | 0 | 0 | 0
+ M | 0.43 | 0.358 | 0.11 | 0 | 0 | 0 | 0
+ M | 0.49 | 0.38 | 0.135 | 0 | 0 | 0 | 0
+ M | 0.5 | 0.4 | 0.13 | 0 | 0 | 0 | 0
+ I | 0.355 | 0.28 | 0.085 | 0 | 1 | 0 | 0
+ NULL | 0.55 | 0.415 | 0.135 | 0 | 0 | 0 | 1
+</pre> </li>
+</ol>
+</div><!-- contents -->
+</div><!-- doc-content -->
+<!-- start footer part -->
+<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
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+ <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.13 </li>
+ </ul>
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+</html>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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+ </div>
+ <div id="projectbrief">User Documentation for MADlib</div>
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+<div class="header">
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+<a href="#groups">Modules</a> </div>
+ <div class="headertitle">
+<div class="title">Inferential Statistics<div class="ingroups"><a class="el" href="group__grp__stats.html">Statistics</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 to compute inferential statistics on a dataset. </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__stats__tests"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__stats__tests.html">Hypothesis Tests</a></td></tr>
+<tr class="memdesc:group__grp__stats__tests"><td class="mdescLeft"> </td><td class="mdescRight">Provides functions to perform statistical hypothesis tests. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+</table>
+</div><!-- contents -->
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