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Posted to commits@madlib.apache.org by ok...@apache.org on 2017/12/28 22:52:01 UTC
[27/51] [abbrv] [partial] madlib-site git commit: Additional updates
for 1.13 release
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+ <td style="padding-left: 0.5em;">
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+ <span id="projectnumber">1.13</span>
+ </div>
+ <div id="projectbrief">User Documentation for MADlib</div>
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+<a href="#groups">Modules</a> </div>
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+<div class="title">Measures<div class="ingroups"><a class="el" href="group__grp__graph.html">Graph</a></div></div> </div>
+</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__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__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__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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http://git-wip-us.apache.org/repos/asf/madlib-site/blob/6c103d3e/docs/v1.13/group__grp__graph__measures.js
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diff --git a/docs/v1.13/group__grp__graph__measures.js b/docs/v1.13/group__grp__graph__measures.js
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+var group__grp__graph__measures =
+[
+ [ "Average Path Length", "group__grp__graph__avg__path__length.html", null ],
+ [ "Closeness", "group__grp__graph__closeness.html", null ],
+ [ "Graph Diameter", "group__grp__graph__diameter.html", null ],
+ [ "In-Out Degree", "group__grp__graph__vertex__degrees.html", null ]
+];
\ No newline at end of file
http://git-wip-us.apache.org/repos/asf/madlib-site/blob/6c103d3e/docs/v1.13/group__grp__graph__vertex__degrees.html
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+ <td style="padding-left: 0.5em;">
+ <div id="projectname">
+ <span id="projectnumber">1.13</span>
+ </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 in-out degrees 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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+<!-- start footer part -->
+<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
+ <ul>
+ <li class="footer">Generated on Wed Dec 27 2017 19:05:57 for MADlib by
+ <a href="http://www.doxygen.org/index.html">
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+</html>
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+</script>
+<div id="doc-content">
+<!-- window showing the filter options -->
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+ onmouseover="return searchBox.OnSearchSelectShow()"
+ onmouseout="return searchBox.OnSearchSelectHide()"
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+
+<div class="header">
+ <div class="headertitle">
+<div class="title">HITS<div class="ingroups"><a class="el" href="group__grp__graph.html">Graph</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> <ul>
+<li>
+<a href="#hits">HITS</a> </li>
+<li>
+<a href="#notes">Notes</a> </li>
+<li>
+<a href="#examples">Examples</a> </li>
+<li>
+<a href="#literature">Literature</a> </li>
+</ul>
+</div><p>Given a graph, the HITS (Hyperlink-Induced Topic Search) algorithm outputs the authority score and hub score of every vertex, where authority estimates the value of the content of the page and hub estimates the value of its links to other pages. This algorithm was originally developed to rate web pages [1].</p>
+<p><a class="anchor" id="hits"></a></p><dl class="section user"><dt>HITS</dt><dd><pre class="syntax">
+hits( vertex_table,
+ vertex_id,
+ edge_table,
+ edge_args,
+ out_table,
+ max_iter,
+ threshold,
+ 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 and destination vertex.</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>
+</ul>
+<p class="enddd"></p>
+</dd>
+<dt>out_table </dt>
+<dd><p class="startdd">TEXT. Name of the table to store the result of HITS. It will contain a row for every vertex from 'vertex_table' with the following columns:</p><ul>
+<li>vertex_id : The id of a vertex. Will use the input parameter 'vertex_id' for column naming.</li>
+<li>authority : The vertex authority score.</li>
+<li>hub : The vertex hub score.</li>
+<li>grouping_cols : Grouping column values (if any) associated with the vertex_id. </li>
+</ul>
+<p>A summary table is also created that contains information regarding the number of iterations required for convergence. It is named by adding the suffix '_summary' to the 'out_table' parameter.</p>
+<p class="enddd"></p>
+</dd>
+<dt>max_iter (optional) </dt>
+<dd><p class="startdd">INTEGER, default: 100. The maximum number of iterations allowed. Each iteration consists of both authority and hub phases.</p>
+<p class="enddd"></p>
+</dd>
+<dt>threshold (optional) </dt>
+<dd><p class="startdd">FLOAT8, default: (1/number of vertices * 1000). Threshold must be set to a value between 0 and 1, inclusive of end points. If the difference between two consecutive iterations of authority AND two consecutive iterations of hub is smaller than 'threshold', then the computation stops. That is, both authority and hub value differences must be below the specified threshold for the algorithm to stop. If you set the threshold to 0, then you will force the algorithm to run for the full number of iterations specified in 'max_iter'. </p>
+<p class="enddd"></p>
+</dd>
+<dt>grouping_cols (optional) </dt>
+<dd>TEXT, default: NULL. A single column or a list of comma-separated columns that divides the input data into discrete groups, resulting in one distribution per group. When this value is NULL, no grouping is used and a single model is generated for all data. <dl class="section note"><dt>Note</dt><dd>Expressions are not currently supported for 'grouping_cols'. </dd></dl>
+</dd>
+</dl>
+<p><a class="anchor" id="notes"></a></p><dl class="section user"><dt>Notes</dt><dd></dd></dl>
+<p>This algorithm supports multigraph and each duplicated edge is considered for counting when calculating authority and hub scores.</p>
+<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
+ );
+CREATE TABLE edge(
+ src INTEGER,
+ dest INTEGER,
+ user_id INTEGER
+ );
+INSERT INTO vertex VALUES
+(0),
+(1),
+(2),
+(3),
+(4),
+(5),
+(6);
+INSERT INTO edge VALUES
+(0, 1, 1),
+(0, 2, 1),
+(0, 4, 1),
+(1, 2, 1),
+(1, 3, 1),
+(2, 3, 1),
+(2, 5, 1),
+(2, 6, 1),
+(3, 0, 1),
+(4, 0, 1),
+(5, 6, 1),
+(6, 3, 1);
+</pre></li>
+<li>Running HITS with default values for optional parameters: <pre class="syntax">
+DROP TABLE IF EXISTS hits_out, hits_out_summary;
+SELECT madlib.hits(
+ 'vertex', -- Vertex table
+ 'id', -- Vertex id column
+ 'edge', -- Edge table
+ 'src=src, dest=dest', -- Comma delimited string of edge arguments
+ 'hits_out'); -- Output table of HITS
+SELECT * FROM hits_out ORDER BY id;
+</pre> <pre class="result">
+ id | authority | hub
+----+----------------------+----------------------
+ 0 | 8.43871829093e-07 | 0.338306115082665
+ 1 | 0.158459587238244 | 0.527865350448059
+ 2 | 0.405627969689677 | 0.675800764727558
+ 3 | 0.721775835521825 | 3.95111934817e-07
+ 4 | 0.158459587238244 | 3.95111934817e-07
+ 5 | 0.316385413093048 | 0.189719957843216
+ 6 | 0.405199928761102 | 0.337944978189241
+(7 rows)
+</pre> <pre class="syntax">
+SELECT * FROM hits_out_summary;
+</pre> <pre class="result">
+ __iterations__
+-----------------+
+ 17
+(1 row)
+</pre></li>
+<li>Running HITS with max_iter of 3 results in different authority and hub scores: <pre class="syntax">
+DROP TABLE IF EXISTS hits_out, hits_out_summary;
+SELECT madlib.hits(
+ 'vertex', -- Vertex table
+ 'id', -- Vertex id column
+ 'edge', -- Edge table
+ 'src=src, dest=dest', -- Comma delimited string of edge arguments
+ 'hits_out', -- Output table
+ 3); -- Max iteration
+SELECT * FROM hits_out ORDER BY id;
+</pre> <pre class="result">
+ id | authority | hub
+----+-------------------+--------------------
+ 0 | 0.08653327387778 | 0.375721659592363
+ 1 | 0.18388320699029 | 0.533118571043218
+ 2 | 0.43266636938891 | 0.654974244424525
+ 3 | 0.70308285025699 | 0.040618557793769
+ 4 | 0.18388320699029 | 0.040618557793769
+ 5 | 0.30286645857224 | 0.182783510071961
+ 6 | 0.38939973245002 | 0.330025782074373
+(7 rows)
+</pre> <pre class="syntax">
+SELECT * FROM hits_out_summary;
+</pre> <pre class="result">
+ __iterations__
+-----------------+
+ 3
+(1 row)
+</pre></li>
+<li>Running HITS with a low threshold of 0.00001 results in more iterations for convergence: <pre class="syntax">
+DROP TABLE IF EXISTS hits_out, hits_out_summary;
+SELECT madlib.hits(
+ 'vertex', -- Vertex table
+ 'id', -- Vertex id column
+ 'edge', -- Edge table
+ 'src=src, dest=dest', -- Comma delimited string of edge arguments
+ 'hits_out', -- Output table
+ NULL, -- Default max_iter
+ 0.00001); -- Threshold
+SELECT * FROM hits_out ORDER BY id;
+</pre> <pre class="result">
+ id | authority | hub
+----+----------------------+---------------------
+ 0 | 1.15243075426e-09 | 0.33800946769422
+ 1 | 0.158264459912827 | 0.527792117750177
+ 2 | 0.405384672299625 | 0.675965453766535
+ 3 | 0.72186275724613 | 5.39583282614e-10
+ 4 | 0.158264459912827 | 5.39583282614e-10
+ 5 | 0.316493740997913 | 0.189793242747412
+ 6 | 0.405356461070609 | 0.337985666133163
+(7 rows)
+</pre> <pre class="syntax">
+SELECT * FROM hits_out_summary;
+</pre> <pre class="result">
+ __iterations__
+-----------------+
+ 25
+(1 row)
+</pre></li>
+<li>Running HITS with both max_iter and threshold: <pre class="syntax">
+DROP TABLE IF EXISTS hits_out, hits_out_summary;
+SELECT madlib.hits(
+ 'vertex', -- Vertex table
+ 'id', -- Vertex id column
+ 'edge', -- Edge table
+ 'src=src, dest=dest', -- Comma delimited string of edge arguments
+ 'hits_out', -- Output table
+ 20, -- Default max_iter
+ 0.00001); -- Threshold
+SELECT * FROM hits_out ORDER BY id;
+</pre> <pre class="result">
+ id | authority | hub
+----+----------------------+---------------------
+ 0 | 7.11260011825e-08 | 0.33810307986005
+ 1 | 0.158326035587958 | 0.527815233930963
+ 2 | 0.405461453180491 | 0.675913495026452
+ 3 | 0.721835343230399 | 3.33021322089e-08
+ 4 | 0.158326035587958 | 3.33021322089e-08
+ 5 | 0.316459563893809 | 0.189770119973925
+ 6 | 0.405307074424261 | 0.337972831786458
+(7 rows)
+</pre> <pre class="syntax">
+SELECT * FROM hits_out_summary;
+</pre> <pre class="result">
+ __iterations__
+-----------------+
+ 20
+(1 row)
+</pre> The algorithm stopped at 20 iterations even though the convergence for threshold of 0.00001 is at 25 iterations. This is because max_iter was set to 20.</li>
+<li>Running HITS with grouping column and default values for max_iter and threshold. Add more rows to the edge table to create different graphs based on the user_id column. <pre class="syntax">
+INSERT INTO edge VALUES
+(0, 1, 2),
+(0, 2, 2),
+(0, 4, 2),
+(1, 2, 2),
+(1, 3, 2),
+(2, 3, 2),
+(3, 0, 2),
+(4, 0, 2),
+(5, 6, 2),
+(6, 3, 2);
+DROP TABLE IF EXISTS hits_out, hits_out_summary;
+SELECT madlib.hits(
+ 'vertex', -- Vertex table
+ 'id', -- Vertex id column
+ 'edge', -- Edge table
+ 'src=src, dest=dest', -- Comma delimited string of edge arguments
+ 'hits_out', -- Output table
+ NULL, -- Default max_iter
+ NULL, -- Threshold
+ 'user_id'); -- Grouping column
+SELECT * FROM hits_out ORDER BY user_id, id;
+</pre> <pre class="result">
+ user_id | id | authority | hub
+---------+----+----------------------+----------------------
+ 1 | 0 | 8.43871829093e-07 | 0.338306115082665
+ 1 | 1 | 0.158459587238244 | 0.527865350448059
+ 1 | 2 | 0.405627969689677 | 0.675800764727558
+ 1 | 3 | 0.721775835521825 | 3.95111934817e-07
+ 1 | 4 | 0.158459587238244 | 3.95111934817e-07
+ 1 | 5 | 0.316385413093048 | 0.189719957843216
+ 1 | 6 | 0.405199928761102 | 0.337944978189241
+ 2 | 0 | 1.60841750444e-05 | 0.632262085114062
+ 2 | 1 | 0.316079985713431 | 0.632529390899584
+ 2 | 2 | 0.632364174872359 | 0.316347297480213
+ 2 | 3 | 0.632694582987791 | 8.04208767442e-06
+ 2 | 4 | 0.316079985713431 | 8.04208767442e-06
+ 2 | 5 | 0 | 1.22712519446e-10
+ 2 | 6 | 2.45425034248e-10 | 0.316347297480213
+(14 rows)
+</pre> <pre class="syntax">
+SELECT * FROM hits_out_summary order by user_id;
+</pre> <pre class="result">
+ user_id | __iterations__
+---------+----------------
+ 1 | 17
+ 2 | 16
+(2 rows)
+</pre></li>
+</ol>
+<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
+<p>[1] Kleinerg, Jon M., "Authoritative Sources in a Hyperlinked
+Environment", Journal of the ACM, Sept. 1999. <a href="https://www.cs.cornell.edu/home/kleinber/auth.pdf">https://www.cs.cornell.edu/home/kleinber/auth.pdf</a> </p>
+</div><!-- contents -->
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+ <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 -->
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+<a href="#groups">Modules</a> </div>
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+<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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+<div class="title">k-Means Clustering<div class="ingroups"><a class="el" href="group__grp__unsupervised.html">Unsupervised Learning</a> » <a class="el" href="group__grp__clustering.html">Clustering</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> <ul>
+<li class="level1">
+<a href="#train">Training Function</a> </li>
+<li class="level1">
+<a href="#output">Output Format</a> </li>
+<li class="level1">
+<a href="#assignment">Cluster Assignment</a> </li>
+<li class="level1">
+<a href="#examples">Examples</a> </li>
+<li class="level1">
+<a href="#notes">Notes</a> </li>
+<li class="level1">
+<a href="#background">Technical Background</a> </li>
+<li class="level1">
+<a href="#literature">Literature</a> </li>
+<li class="level1">
+<a href="#related">Related Topics</a> </li>
+</ul>
+</div><p>Clustering refers to the problem of partitioning a set of objects according to some problem-dependent measure of <em>similarity</em>. In the k-means variant, given \( n \) points \( x_1, \dots, x_n \in \mathbb R^d \), the goal is to position \( k \) centroids \( c_1, \dots, c_k \in \mathbb R^d \) so that the sum of <em>distances</em> between each point and its closest centroid is minimized. Each centroid represents a cluster that consists of all points to which this centroid is closest.</p>
+<p><a class="anchor" id="train"></a></p><dl class="section user"><dt>Training Function</dt><dd></dd></dl>
+<p>The k-means algorithm can be invoked in four ways, depending on the source of the initial set of centroids:</p>
+<ul>
+<li>Use the random centroid seeding method. <pre class="syntax">
+kmeans_random( rel_source,
+ expr_point,
+ k,
+ fn_dist,
+ agg_centroid,
+ max_num_iterations,
+ min_frac_reassigned
+ )
+</pre></li>
+<li>Use the kmeans++ centroid seeding method. <pre class="syntax">
+kmeanspp( rel_source,
+ expr_point,
+ k,
+ fn_dist,
+ agg_centroid,
+ max_num_iterations,
+ min_frac_reassigned,
+ seeding_sample_ratio
+ )
+</pre></li>
+<li>Supply an initial centroid set in a relation identified by the <em>rel_initial_centroids</em> argument. <pre class="syntax">
+kmeans( rel_source,
+ expr_point,
+ rel_initial_centroids,
+ expr_centroid,
+ fn_dist,
+ agg_centroid,
+ max_num_iterations,
+ min_frac_reassigned
+ )
+</pre></li>
+<li>Provide an initial centroid set as an array expression in the <em>initial_centroids</em> argument. <pre class="syntax">
+kmeans( rel_source,
+ expr_point,
+ initial_centroids,
+ fn_dist,
+ agg_centroid,
+ max_num_iterations,
+ min_frac_reassigned
+ )
+</pre> <b>Arguments</b> <dl class="arglist">
+<dt>rel_source </dt>
+<dd><p class="startdd">TEXT. The name of the table containing the input data points.</p>
+<p>Data points and predefined centroids (if used) are expected to be stored row-wise, in a column of type <code><a class="el" href="group__grp__svec.html">SVEC</a></code> (or any type convertible to <code><a class="el" href="group__grp__svec.html">SVEC</a></code>, like <code>FLOAT[]</code> or <code>INTEGER[]</code>). Data points with non-finite values (NULL, NaN, infinity) in any component are skipped during analysis. </p>
+<p class="enddd"></p>
+</dd>
+<dt>expr_point </dt>
+<dd><p class="startdd">TEXT. The name of the column with point coordinates or an array expression.</p>
+<p class="enddd"></p>
+</dd>
+<dt>k </dt>
+<dd><p class="startdd">INTEGER. The number of centroids to calculate.</p>
+<p class="enddd"></p>
+</dd>
+<dt>fn_dist (optional) </dt>
+<dd><p class="startdd">TEXT, default: squared_dist_norm2'. The name of the function to use to calculate the distance from a data point to a centroid.</p>
+<p>The following distance functions can be used (computation of barycenter/mean in parentheses): </p><ul>
+<li>
+<b><a class="el" href="linalg_8sql__in.html#aad193850e79c4b9d811ca9bc53e13476">dist_norm1</a></b>: 1-norm/Manhattan (element-wise median [Note that MADlib does not provide a median aggregate function for support and performance reasons.]) </li>
+<li>
+<b><a class="el" href="linalg_8sql__in.html#aa58e51526edea6ea98db30b6f250adb4">dist_norm2</a></b>: 2-norm/Euclidean (element-wise mean) </li>
+<li>
+<b><a class="el" href="linalg_8sql__in.html#a00a08e69f27524f2096032214e15b668">squared_dist_norm2</a></b>: squared Euclidean distance (element-wise mean) </li>
+<li>
+<b><a class="el" href="linalg_8sql__in.html#a8c7b9281a72ff22caf06161701b27e84">dist_angle</a></b>: angle (element-wise mean of normalized points) </li>
+<li>
+<b><a class="el" href="linalg_8sql__in.html#afa13b4c6122b99422d666dedea136c18">dist_tanimoto</a></b>: tanimoto (element-wise mean of normalized points <a href="#kmeans-lit-5">[5]</a>) </li>
+<li>
+<b>user defined function</b> with signature <code>DOUBLE PRECISION[] x, DOUBLE PRECISION[] y -> DOUBLE PRECISION</code></li>
+</ul>
+<p class="enddd"></p>
+</dd>
+<dt>agg_centroid (optional) </dt>
+<dd><p class="startdd">TEXT, default: 'avg'. The name of the aggregate function used to determine centroids.</p>
+<p>The following aggregate functions can be used:</p><ul>
+<li>
+<b><a class="el" href="linalg_8sql__in.html#a1aa37f73fb1cd8d7d106aa518dd8c0b4">avg</a></b>: average (Default) </li>
+<li>
+<b><a class="el" href="linalg_8sql__in.html#a0b04663ca206f03e66aed5ea2b4cc461">normalized_avg</a></b>: normalized average</li>
+</ul>
+<p class="enddd"></p>
+</dd>
+<dt>max_num_iterations (optional) </dt>
+<dd><p class="startdd">INTEGER, default: 20. The maximum number of iterations to perform.</p>
+<p class="enddd"></p>
+</dd>
+<dt>min_frac_reassigned (optional) </dt>
+<dd><p class="startdd">DOUBLE PRECISION, default: 0.001. The minimum fraction of centroids reassigned to continue iterating. When fewer than this fraction of centroids are reassigned in an iteration, the calculation completes.</p>
+<p class="enddd"></p>
+</dd>
+<dt>seeding_sample_ratio (optional) </dt>
+<dd><p class="startdd">DOUBLE PRECISION, default: 1.0. The proportion of subsample of original dataset to use for kmeans++ centroid seeding method. Kmeans++ scans through the data sequentially 'k' times and can be too slow for big datasets. When 'seeding_sample_ratio' is greater than 0 (thresholded to be maximum value of 1.0), the seeding is run on an uniform random subsample of the data. Note: the final K-means algorithm is run on the complete dataset. This parameter only builds a subsample for the seeding and is only available for kmeans++.</p>
+<p class="enddd"></p>
+</dd>
+<dt>rel_initial_centroids </dt>
+<dd><p class="startdd">TEXT. The set of initial centroids. </p>
+<p class="enddd"></p>
+</dd>
+<dt>expr_centroid </dt>
+<dd><p class="startdd">TEXT. The name of the column (or the array expression) in the <em>rel_initial_centroids</em> relation that contains the centroid coordinates.</p>
+<p class="enddd"></p>
+</dd>
+<dt>initial_centroids </dt>
+<dd>TEXT. A string containing a DOUBLE PRECISION array expression with the initial centroid coordinates. </dd>
+</dl>
+</li>
+</ul>
+<p><a class="anchor" id="output"></a></p><dl class="section user"><dt>Output Format</dt><dd></dd></dl>
+<p>The output of the k-means module is a composite type with the following columns: </p><table class="output">
+<tr>
+<th>centroids </th><td>DOUBLE PRECISION[][]. The final centroid positions. </td></tr>
+<tr>
+<th>cluster_variance </th><td>DOUBLE PRECISION[]. The value of the objective function per cluster. </td></tr>
+<tr>
+<th>objective_fn </th><td>DOUBLE PRECISION. The value of the objective function. </td></tr>
+<tr>
+<th>frac_reassigned </th><td>DOUBLE PRECISION. The fraction of points reassigned in the last iteration. </td></tr>
+<tr>
+<th>num_iterations </th><td>INTEGER. The total number of iterations executed. </td></tr>
+</table>
+<p><a class="anchor" id="assignment"></a></p><dl class="section user"><dt>Cluster Assignment</dt><dd></dd></dl>
+<p>After training, the cluster assignment for each data point can be computed with the help of the following function:</p>
+<pre class="syntax">
+closest_column( m, x )
+</pre><p><b>Argument</b> </p><dl class="arglist">
+<dt>m </dt>
+<dd>DOUBLE PRECISION[][]. The learned centroids from the training function. </dd>
+<dt>x </dt>
+<dd>DOUBLE PRECISION[]. The data point. </dd>
+</dl>
+<p><b>Output format</b> </p><table class="output">
+<tr>
+<th>column_id </th><td>INTEGER. The cluster assignment (zero-based). </td></tr>
+<tr>
+<th>distance </th><td>DOUBLE PRECISION. The distance to the cluster centroid. </td></tr>
+</table>
+<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
+<p>Note: Your results may not be exactly the same as below due to the nature of the k-means algorithm.</p>
+<ol type="1">
+<li>Prepare some input data: <pre class="example">
+DROP TABLE IF EXISTS km_sample;
+CREATE TABLE km_sample(pid int, points double precision[]);
+INSERT INTO km_sample VALUES
+(1, '{14.23, 1.71, 2.43, 15.6, 127, 2.8, 3.0600, 0.2800, 2.29, 5.64, 1.04, 3.92, 1065}'),
+(2, '{13.2, 1.78, 2.14, 11.2, 1, 2.65, 2.76, 0.26, 1.28, 4.38, 1.05, 3.49, 1050}'),
+(3, '{13.16, 2.36, 2.67, 18.6, 101, 2.8, 3.24, 0.3, 2.81, 5.6799, 1.03, 3.17, 1185}'),
+(4, '{14.37, 1.95, 2.5, 16.8, 113, 3.85, 3.49, 0.24, 2.18, 7.8, 0.86, 3.45, 1480}'),
+(5, '{13.24, 2.59, 2.87, 21, 118, 2.8, 2.69, 0.39, 1.82, 4.32, 1.04, 2.93, 735}'),
+(6, '{14.2, 1.76, 2.45, 15.2, 112, 3.27, 3.39, 0.34, 1.97, 6.75, 1.05, 2.85, 1450}'),
+(7, '{14.39, 1.87, 2.45, 14.6, 96, 2.5, 2.52, 0.3, 1.98, 5.25, 1.02, 3.58, 1290}'),
+(8, '{14.06, 2.15, 2.61, 17.6, 121, 2.6, 2.51, 0.31, 1.25, 5.05, 1.06, 3.58, 1295}'),
+(9, '{14.83, 1.64, 2.17, 14, 97, 2.8, 2.98, 0.29, 1.98, 5.2, 1.08, 2.85, 1045}'),
+(10, '{13.86, 1.35, 2.27, 16, 98, 2.98, 3.15, 0.22, 1.8500, 7.2199, 1.01, 3.55, 1045}');
+</pre></li>
+<li>Run k-means clustering using kmeans++ for centroid seeding: <pre class="example">
+DROP TABLE IF EXISTS km_result;
+-- Run kmeans algorithm
+CREATE TABLE km_result AS
+SELECT * FROM madlib.kmeanspp('km_sample', 'points', 2,
+ 'madlib.squared_dist_norm2',
+ 'madlib.avg', 20, 0.001);
+\x on;
+SELECT * FROM km_result;
+</pre> Result: <pre class="result">
+centroids | {{14.036,2.018,2.536,16.56,108.6,3.004,3.03,0.298,2.038,6.10598,1.004,3.326,1340},{13.872,1.814,2.376,15.56,88.2,2.806,2.928,0.288,1.844,5.35198,1.044,3.348,988}}
+cluster_variance | {60672.638245208,90512.324426408}
+objective_fn | 151184.962671616
+frac_reassigned | 0
+num_iterations | 2
+</pre></li>
+<li>Calculate the simplified silhouette coefficient: <pre class="example">
+SELECT * FROM madlib.simple_silhouette( 'km_sample',
+ 'points',
+ (SELECT centroids FROM km_result),
+ 'madlib.dist_norm2'
+ );
+</pre> Result: <pre class="result">
+simple_silhouette | 0.68978804882941
+</pre></li>
+<li>Find the cluster assignment for each point: <pre class="example">
+\x off;
+-- Get point assignment
+SELECT data.*, (madlib.closest_column(centroids, points)).column_id as cluster_id
+FROM km_sample as data, km_result
+ORDER BY data.pid;
+</pre> Result: <pre class="result">
+ pid | points | cluster_id
+-----+--------------------------------------------------------------------+------------
+ 1 | {14.23,1.71,2.43,15.6,127,2.8,3.06,0.28,2.29,5.64,1.04,3.92,1065} | 1
+ 2 | {13.2,1.78,2.14,11.2,1,2.65,2.76,0.26,1.28,4.38,1.05,3.49,1050} | 1
+ 3 | {13.16,2.36,2.67,18.6,101,2.8,3.24,0.3,2.81,5.6799,1.03,3.17,1185} | 0
+ 4 | {14.37,1.95,2.5,16.8,113,3.85,3.49,0.24,2.18,7.8,0.86,3.45,1480} | 0
+ 5 | {13.24,2.59,2.87,21,118,2.8,2.69,0.39,1.82,4.32,1.04,2.93,735} | 1
+ 6 | {14.2,1.76,2.45,15.2,112,3.27,3.39,0.34,1.97,6.75,1.05,2.85,1450} | 0
+ 7 | {14.39,1.87,2.45,14.6,96,2.5,2.52,0.3,1.98,5.25,1.02,3.58,1290} | 0
+ 8 | {14.06,2.15,2.61,17.6,121,2.6,2.51,0.31,1.25,5.05,1.06,3.58,1295} | 0
+ 9 | {14.83,1.64,2.17,14,97,2.8,2.98,0.29,1.98,5.2,1.08,2.85,1045} | 1
+ 10 | {13.86,1.35,2.27,16,98,2.98,3.15,0.22,1.85,7.2199,1.01,3.55,1045} | 1
+(10 rows)
+</pre></li>
+<li>Unnest the cluster centroids 2-D array to get a set of 1-D centroid arrays: <pre class="example">
+DROP TABLE IF EXISTS km_centroids_unnest;
+-- Run unnest function
+CREATE TABLE km_centroids_unnest AS
+SELECT (madlib.array_unnest_2d_to_1d(centroids)).*
+FROM km_result;
+SELECT * FROM km_centroids_unnest ORDER BY 1;
+</pre> Result: <pre class="result">
+ unnest_row_id | unnest_result
+---------------+----------------------------------------------------------------------------------
+ 1 | {14.036,2.018,2.536,16.56,108.6,3.004,3.03,0.298,2.038,6.10598,1.004,3.326,1340}
+ 2 | {13.872,1.814,2.376,15.56,88.2,2.806,2.928,0.288,1.844,5.35198,1.044,3.348,988}
+(2 rows)
+</pre> Note that the ID column returned by <a class="el" href="array__ops_8sql__in.html#af057b589f2a2cb1095caa99feaeb3d70" title="This function takes a 2-D array as the input and unnests it by one level. It returns a set of 1-D arr...">array_unnest_2d_to_1d()</a> is not guaranteed to be the same as the cluster ID assigned by k-means. See below to create the correct cluster IDs.</li>
+<li>Create cluster IDs for 1-D centroid arrays so that cluster ID for any centroid can be matched to the cluster assignment for the data points: <pre class="example">
+SELECT cent.*, (madlib.closest_column(centroids, unnest_result)).column_id as cluster_id
+FROM km_centroids_unnest as cent, km_result
+ORDER BY cent.unnest_row_id;
+</pre> Result: <pre class="result">
+ unnest_row_id | unnest_result | cluster_id
+---------------+----------------------------------------------------------------------------------+------------
+ 1 | {14.036,2.018,2.536,16.56,108.6,3.004,3.03,0.298,2.038,6.10598,1.004,3.326,1340} | 0
+ 2 | {13.872,1.814,2.376,15.56,88.2,2.806,2.928,0.288,1.844,5.35198,1.044,3.348,988} | 1
+(2 rows)
+</pre></li>
+<li>Run the same example as above, but using array input. Create the input table: <pre class="example">
+DROP TABLE IF EXISTS km_arrayin CASCADE;
+CREATE TABLE km_arrayin(pid int,
+ p1 float,
+ p2 float,
+ p3 float,
+ p4 float,
+ p5 float,
+ p6 float,
+ p7 float,
+ p8 float,
+ p9 float,
+ p10 float,
+ p11 float,
+ p12 float,
+ p13 float);
+INSERT INTO km_arrayin VALUES
+(1, 14.23, 1.71, 2.43, 15.6, 127, 2.8, 3.0600, 0.2800, 2.29, 5.64, 1.04, 3.92, 1065),
+(2, 13.2, 1.78, 2.14, 11.2, 1, 2.65, 2.76, 0.26, 1.28, 4.38, 1.05, 3.49, 1050),
+(3, 13.16, 2.36, 2.67, 18.6, 101, 2.8, 3.24, 0.3, 2.81, 5.6799, 1.03, 3.17, 1185),
+(4, 14.37, 1.95, 2.5, 16.8, 113, 3.85, 3.49, 0.24, 2.18, 7.8, 0.86, 3.45, 1480),
+(5, 13.24, 2.59, 2.87, 21, 118, 2.8, 2.69, 0.39, 1.82, 4.32, 1.04, 2.93, 735),
+(6, 14.2, 1.76, 2.45, 15.2, 112, 3.27, 3.39, 0.34, 1.97, 6.75, 1.05, 2.85, 1450),
+(7, 14.39, 1.87, 2.45, 14.6, 96, 2.5, 2.52, 0.3, 1.98, 5.25, 1.02, 3.58, 1290),
+(8, 14.06, 2.15, 2.61, 17.6, 121, 2.6, 2.51, 0.31, 1.25, 5.05, 1.06, 3.58, 1295),
+(9, 14.83, 1.64, 2.17, 14, 97, 2.8, 2.98, 0.29, 1.98, 5.2, 1.08, 2.85, 1045),
+(10, 13.86, 1.35, 2.27, 16, 98, 2.98, 3.15, 0.22, 1.8500, 7.2199, 1.01, 3.55, 1045);
+</pre> Now find the cluster assignment for each point: <pre class="example">
+DROP TABLE IF EXISTS km_result;
+-- Run kmeans algorithm
+CREATE TABLE km_result AS
+SELECT * FROM madlib.kmeans_random('km_arrayin',
+ 'ARRAY[p1, p2, p3, p4, p5, p6,
+ p7, p8, p9, p10, p11, p12, p13]',
+ 2,
+ 'madlib.squared_dist_norm2',
+ 'madlib.avg',
+ 20,
+ 0.001);
+-- Get point assignment
+SELECT data.*, (madlib.closest_column(centroids,
+ ARRAY[p1, p2, p3, p4, p5, p6,
+ p7, p8, p9, p10, p11, p12, p13])).column_id as cluster_id
+FROM km_arrayin as data, km_result
+ORDER BY data.pid;
+</pre> This produces the result in column format: <pre class="result">
+ pid | p1 | p2 | p3 | p4 | p5 | p6 | p7 | p8 | p9 | p10 | p11 | p12 | p13 | cluster_id
+-----+-------+------+------+------+-----+------+------+------+------+--------+------+------+------+------------
+ 1 | 14.23 | 1.71 | 2.43 | 15.6 | 127 | 2.8 | 3.06 | 0.28 | 2.29 | 5.64 | 1.04 | 3.92 | 1065 | 0
+ 2 | 13.2 | 1.78 | 2.14 | 11.2 | 1 | 2.65 | 2.76 | 0.26 | 1.28 | 4.38 | 1.05 | 3.49 | 1050 | 0
+ 3 | 13.16 | 2.36 | 2.67 | 18.6 | 101 | 2.8 | 3.24 | 0.3 | 2.81 | 5.6799 | 1.03 | 3.17 | 1185 | 0
+ 4 | 14.37 | 1.95 | 2.5 | 16.8 | 113 | 3.85 | 3.49 | 0.24 | 2.18 | 7.8 | 0.86 | 3.45 | 1480 | 1
+ 5 | 13.24 | 2.59 | 2.87 | 21 | 118 | 2.8 | 2.69 | 0.39 | 1.82 | 4.32 | 1.04 | 2.93 | 735 | 0
+ 6 | 14.2 | 1.76 | 2.45 | 15.2 | 112 | 3.27 | 3.39 | 0.34 | 1.97 | 6.75 | 1.05 | 2.85 | 1450 | 1
+ 7 | 14.39 | 1.87 | 2.45 | 14.6 | 96 | 2.5 | 2.52 | 0.3 | 1.98 | 5.25 | 1.02 | 3.58 | 1290 | 1
+ 8 | 14.06 | 2.15 | 2.61 | 17.6 | 121 | 2.6 | 2.51 | 0.31 | 1.25 | 5.05 | 1.06 | 3.58 | 1295 | 1
+ 9 | 14.83 | 1.64 | 2.17 | 14 | 97 | 2.8 | 2.98 | 0.29 | 1.98 | 5.2 | 1.08 | 2.85 | 1045 | 0
+ 10 | 13.86 | 1.35 | 2.27 | 16 | 98 | 2.98 | 3.15 | 0.22 | 1.85 | 7.2199 | 1.01 | 3.55 | 1045 | 0
+(10 rows)
+</pre></li>
+</ol>
+<p><a class="anchor" id="notes"></a></p><dl class="section user"><dt>Notes</dt><dd></dd></dl>
+<p>The algorithm stops when one of the following conditions is met:</p><ul>
+<li>The fraction of updated points is smaller than the convergence threshold (<em>min_frac_reassigned</em> argument). (Default: 0.001).</li>
+<li>The algorithm reaches the maximum number of allowed iterations (<em>max_num_iterations</em> argument). (Default: 20).</li>
+</ul>
+<p>A popular method to assess the quality of the clustering is the <em>silhouette coefficient</em>, a simplified version of which is provided as part of the k-means module. Note that for large data sets, this computation is expensive.</p>
+<p>The silhouette function has the following syntax: </p><pre class="syntax">
+simple_silhouette( rel_source,
+ expr_point,
+ centroids,
+ fn_dist
+ )
+</pre><p> <b>Arguments</b> </p><dl class="arglist">
+<dt>rel_source </dt>
+<dd>TEXT. The name of the relation containing the input point. </dd>
+<dt>expr_point </dt>
+<dd>TEXT. An expression evaluating to point coordinates for each row in the relation. </dd>
+<dt>centroids </dt>
+<dd>TEXT. An expression evaluating to an array of centroids. </dd>
+<dt>fn_dist (optional) </dt>
+<dd>TEXT, default 'dist_norm2', The name of a function to calculate the distance of a point from a centroid. See the <em>fn_dist</em> argument of the k-means training function. </dd>
+</dl>
+<p><a class="anchor" id="background"></a></p><dl class="section user"><dt>Technical Background</dt><dd></dd></dl>
+<p>Formally, we wish to minimize the following objective function: </p><p class="formulaDsp">
+\[ (c_1, \dots, c_k) \mapsto \sum_{i=1}^n \min_{j=1}^k \operatorname{dist}(x_i, c_j) \]
+</p>
+<p> In the most common case, \( \operatorname{dist} \) is the square of the Euclidean distance.</p>
+<p>This problem is computationally difficult (NP-hard), yet the local-search heuristic proposed by Lloyd [4] performs reasonably well in practice. In fact, it is so ubiquitous today that it is often referred to as the <em>standard algorithm</em> or even just the <em>k-means algorithm</em> [1]. It works as follows:</p>
+<ol type="1">
+<li>Seed the \( k \) centroids (see below)</li>
+<li>Repeat until convergence:<ol type="a">
+<li>Assign each point to its closest centroid</li>
+<li>Move each centroid to a position that minimizes the sum of distances in this cluster</li>
+</ol>
+</li>
+<li>Convergence is achieved when no points change their assignments during step 2a.</li>
+</ol>
+<p>Since the objective function decreases in every step, this algorithm is guaranteed to converge to a local optimum.</p>
+<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
+<p><a class="anchor" id="kmeans-lit-1"></a>[1] Wikipedia, K-means Clustering, <a href="http://en.wikipedia.org/wiki/K-means_clustering">http://en.wikipedia.org/wiki/K-means_clustering</a></p>
+<p><a class="anchor" id="kmeans-lit-2"></a>[2] David Arthur, Sergei Vassilvitskii: k-means++: the advantages of careful seeding, Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'07), pp. 1027-1035, <a href="http://www.stanford.edu/~darthur/kMeansPlusPlus.pdf">http://www.stanford.edu/~darthur/kMeansPlusPlus.pdf</a></p>
+<p><a class="anchor" id="kmeans-lit-3"></a>[3] E. R. Hruschka, L. N. C. Silva, R. J. G. B. Campello: Clustering Gene-Expression Data: A Hybrid Approach that Iterates Between k-Means and Evolutionary Search. In: Studies in Computational Intelligence - Hybrid Evolutionary Algorithms. pp. 313-335. Springer. 2007.</p>
+<p><a class="anchor" id="kmeans-lit-4"></a>[4] Lloyd, Stuart: Least squares quantization in PCM. Technical Note, Bell Laboratories. Published much later in: IEEE Transactions on Information Theory 28(2), pp. 128-137. 1982.</p>
+<p><a class="anchor" id="kmeans-lit-5"></a>[5] Leisch, Friedrich: A Toolbox for K-Centroids Cluster Analysis. In: Computational Statistics and Data Analysis, 51(2). pp. 526-544. 2006.</p>
+<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd></dd></dl>
+<p>File <a class="el" href="kmeans_8sql__in.html" title="Set of functions for k-means clustering. ">kmeans.sql_in</a> documenting the k-Means SQL functions</p>
+<p><a class="el" href="group__grp__svec.html">Sparse Vectors</a></p>
+<p>simple_silhouette()</p>
+</div><!-- contents -->
+</div><!-- doc-content -->
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