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 <div class="header">
   <div class="headertitle">
 <div class="title">k-Means Clustering<div class="ingroups"><a class="el" href="group__grp__clustering.html">Clustering</a></div></div>  </div>
 </div><!--header-->
 <div class="contents">
 <dl class="section user"><dt>About</dt><dd></dd></dl>
 <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, one is given \( n \) points \( x_1, \dots, x_n \in \mathbb R^d \), and 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. 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>
 <dl class="section user"><dt>Implementation Notes</dt><dd></dd></dl>
 <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 will be skipped during analysis.</p>
 <p>The following methods are available for the centroid seeding:</p>
 <ul>
 <li><b>random selection</b>: Select \( k \) centroids randomly among the input points.</li>
 <li><b>kmeans++</b> [2]: Start with a single centroid chosen randomly among the input points. Then iteratively choose new centroids from the input points until there is a total of \( k \) centroids. The probability for picking a particular point is proportional to its minimum distance to any existing centroid. <br/>
  Intuitively, kmeans++ favors seedings where centroids are spread out over the whole range of the input points, while at the same time not being too susceptible to outliers [2].</li>
 <li><b>user-specified set of initial centroids</b>: See below for a description of the expected format of the set of initial centroids.</li>
 </ul>
 <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 [5])</li>
 <li><b>user defined function</b> with signature DOUBLE PRECISION[] x DOUBLE PRECISION[] -&gt; DOUBLE PRECISION</li>
 </ul>
 <p>The following aggregate functions for determining centroids can be used:</p>
 <ul>
 <li><b><a class="el" href="linalg_8sql__in.html#a1aa37f73fb1cd8d7d106aa518dd8c0b4">avg</a></b>: average</li>
 <li><b><a class="el" href="linalg_8sql__in.html#a0b04663ca206f03e66aed5ea2b4cc461">normalized_avg</a></b>: normalized average</li>
 </ul>
 <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 (default: 0.001).</li>
 <li>The algorithm reaches the maximum number of allowed iterations (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>
 <dl class="section user"><dt>Input</dt><dd>The <b>source relation</b> is expected to be of the following form (or to be implicitly convertible into the following form): <pre>{TABLE|VIEW} <em>rel_source</em> (
     ...
     <em>expr_points</em> FLOAT8[],
     ...
 )</pre> where:<ul>
 <li><em>expr_points</em> is the name of a column with point coordinates. Types such as <code>svec</code> or <code>INTEGER[]</code> are implicitly converted to <code>FLOAT8[]</code>.</li>
 </ul>
 </dd></dl>
 <p>If kmeans is called with a set of initial centroids, the centroid relation is expected to be of the following form: </p>
 <pre>{TABLE|VIEW} <em>rel_initial_centroids</em> (
     ...
     <em>expr_centroid</em> DOUBLE PRECISION[],
     ...
 )</pre><p> where:</p>
 <ul>
 <li><em>expr_centroid</em> is the name of a column with coordinates.</li>
 </ul>
 <dl class="section user"><dt>Usage</dt><dd>The k-means algorithm can be invoked in four possible ways:<ul>
 <li>using <em>random</em> centroid seeding method for a provided \( k \): <pre>SELECT * FROM <a class="el" href="kmeans_8sql__in.html#a66ac1cab8811c4d842de1bc221886b53">kmeans_random</a>(
   '<em>rel_source</em>', '<em>expr_point</em>', k,
   [ '<em>fn_dist</em>', '<em>agg_centroid</em>',
   <em>max_num_iterations</em>, <em>min_frac_reassigned</em> ]
 );</pre></li>
 <li>using <em>kmeans++</em> centroid seeding method for a provided \( k \): <pre>SELECT * FROM <a class="el" href="kmeans_8sql__in.html#a639178dacebca2a2114923038398d6bb">kmeanspp</a>(
   '<em>rel_source</em>', '<em>expr_point</em>', k,
   [ '<em>fn_dist</em>', '<em>agg_centroid</em>',
   <em>max_num_iterations</em>, <em>min_frac_reassigned</em> ]
 );</pre></li>
 <li>with a provided centroid set: <pre>SELECT * FROM <a class="el" href="kmeans_8sql__in.html#afdae42b563f1f8bca3937dbbbacaa1c3">kmeans</a>(
   '<em>rel_source</em>', '<em>expr_point</em>',
   '<em>rel_initial_centroids</em>', '<em>expr_centroid</em>',
   [ '<em>fn_dist</em>', '<em>agg_centroid</em>',
   <em>max_num_iterations</em>, <em>min_frac_reassigned</em> ]
 );</pre> ---------&mdash; OR ------------&mdash; <pre>SELECT * FROM <a class="el" href="kmeans_8sql__in.html#afdae42b563f1f8bca3937dbbbacaa1c3">kmeans</a>(
   '<em>rel_source</em>', '<em>expr_point</em>',
   initial_centroids,
   [ '<em>fn_dist</em>', '<em>agg_centroid</em>',
   <em>max_num_iterations</em>, <em>min_frac_reassigned</em> ]
 );</pre> where:<ul>
 <li><em>initial_centroids</em> is of type <code>DOUBLE PRECISION[][]</code>.</li>
 </ul>
 </li>
 </ul>
 </dd></dl>
 <p>The output of the k-means module is a table that includes the final centroid positions (DOUBLE PRECISION[][]), the objective function, the fraction of reassigned points in the last iteration, and the number of total iterations: </p>
 <pre>
             centroids             |   objective_fn   | frac_reassigned | num_iterations
 ----------------------------------+------------------+-----------------+----------------
              ...
 </pre><dl class="section user"><dt>Examples</dt><dd><ol type="1">
 <li>Prepare some input data. <div class="fragment"><div class="line">sql&gt; <a class="code" href="robust_8sql__in.html#ac9ebd21770ba37efb90e1ccee36fc103">SELECT</a> * FROM <span class="keyword">public</span>.km_sample LIMIT 5;</div>
 <div class="line">                  points</div>
 <div class="line">-------------------------------------------</div>
 <div class="line"> {1,1}:{15.8822241332382,105.945462542586}</div>
 <div class="line"> {1,1}:{34.5065216883086,72.3126099305227}</div>
 <div class="line"> {1,1}:{22.5074400822632,95.3209559689276}</div>
 <div class="line"> {1,1}:{70.2589857042767,68.7395178806037}</div>
 <div class="line"> {1,1}:{30.9844257542863,25.3213323024102}</div>
 <div class="line">(5 rows)</div>
 </div><!-- fragment --> Note: the example <em>points</em> is type <code><a class="el" href="group__grp__svec.html">SVEC</a></code>.</li>
 <li>Run k-means clustering using kmeans++ for centroid seeding: <div class="fragment"><div class="line">sql&gt; <a class="code" href="robust_8sql__in.html#ac9ebd21770ba37efb90e1ccee36fc103">SELECT</a> * FROM madlib.kmeanspp(<span class="stringliteral">&#39;km_sample&#39;</span>, <span class="stringliteral">&#39;points&#39;</span>, 2, <span class="stringliteral">&#39;madlib.squared_dist_norm2&#39;</span>, <span class="stringliteral">&#39;madlib.avg&#39;</span>, 20, 0.001);</div>
 <div class="line">);</div>
 <div class="line">                                centroids                                |   objective_fn   | frac_reassigned | num_iterations</div>
 <div class="line">-------------------------------------------------------------------------+------------------+-----------------+----------------</div>
 <div class="line"> {{68.01668579784,48.9667382972952},{28.1452167573446,84.5992507653263}} | 586729.010675982 |           0.001 |              5</div>
 </div><!-- fragment --></li>
 <li>Calculate the simplified silhouette coefficient: <div class="fragment"><div class="line">sql&gt; <a class="code" href="robust_8sql__in.html#ac9ebd21770ba37efb90e1ccee36fc103">SELECT</a> * from madlib.simple_silhouette(<span class="stringliteral">&#39;km_test_svec&#39;</span>,<span class="stringliteral">&#39;points&#39;</span>,</div>
 <div class="line">     (select centroids from madlib.kmeanspp(<span class="stringliteral">&#39;km_test_svec&#39;</span>,<span class="stringliteral">&#39;points&#39;</span>,2,<span class="stringliteral">&#39;madlib.squared_dist_norm2&#39;</span>,<span class="stringliteral">&#39;madlib.avg&#39;</span>,20,0.001)),</div>
 <div class="line">     <span class="stringliteral">&#39;madlib.dist_norm2&#39;</span>);</div>
 <div class="line"> <a class="code" href="kmeans_8sql__in.html#a71e7675758c99acbe7785819b6a85a8f" title="Compute a simplified version of the silhouette coefficient. ">simple_silhouette</a></div>
 <div class="line">-------------------</div>
 <div class="line"> 0.611022970398174</div>
 </div><!-- fragment --></li>
 </ol>
 </dd></dl>
 <dl class="section user"><dt>Literature</dt><dd></dd></dl>
 <p>[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>[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>[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>[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>[5] Leisch, Friedrich: A Toolbox for K-Centroids Cluster Analysis. In: Computational Statistics and Data Analysis, 51(2). pp. 526-544. 2006.</p>
 <dl class="section see"><dt>See Also</dt><dd>File <a class="el" href="kmeans_8sql__in.html" title="Set of functions for k-means clustering. ">kmeans.sql_in</a> documenting the SQL functions. </dd></dl>
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	<div class="header">
	<div class="headertitle">
	<div class="title">k-Means Clustering<div class="ingroups"><a class="el" href="group__grp__clustering.html">Clustering</a></div></div> </div>
	</div><!--header-->
	<div class="contents">
	<dl class="section user"><dt>About</dt><dd></dd></dl>
	<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, one is given \( n \) points \( x_1, \dots, x_n \in \mathbb R^d \), and 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. 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>
	<dl class="section user"><dt>Implementation Notes</dt><dd></dd></dl>
	<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 will be skipped during analysis.</p>
	<p>The following methods are available for the centroid seeding:</p>
	<ul>
	<li><b>random selection</b>: Select \( k \) centroids randomly among the input points.</li>
	<li><b>kmeans++</b> [2]: Start with a single centroid chosen randomly among the input points. Then iteratively choose new centroids from the input points until there is a total of \( k \) centroids. The probability for picking a particular point is proportional to its minimum distance to any existing centroid. <br/>
	Intuitively, kmeans++ favors seedings where centroids are spread out over the whole range of the input points, while at the same time not being too susceptible to outliers [2].</li>
	<li><b>user-specified set of initial centroids</b>: See below for a description of the expected format of the set of initial centroids.</li>
	</ul>
	<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 [5])</li>
	<li><b>user defined function</b> with signature DOUBLE PRECISION[] x DOUBLE PRECISION[] -> DOUBLE PRECISION</li>
	</ul>
	<p>The following aggregate functions for determining centroids can be used:</p>
	<ul>
	<li><b><a class="el" href="linalg_8sql__in.html#a1aa37f73fb1cd8d7d106aa518dd8c0b4">avg</a></b>: average</li>
	<li><b><a class="el" href="linalg_8sql__in.html#a0b04663ca206f03e66aed5ea2b4cc461">normalized_avg</a></b>: normalized average</li>
	</ul>
	<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 (default: 0.001).</li>
	<li>The algorithm reaches the maximum number of allowed iterations (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>
	<dl class="section user"><dt>Input</dt><dd>The <b>source relation</b> is expected to be of the following form (or to be implicitly convertible into the following form): <pre>{TABLE\|VIEW} <em>rel_source</em> (
	...
	<em>expr_points</em> FLOAT8[],
	...
	)</pre> where:<ul>
	<li><em>expr_points</em> is the name of a column with point coordinates. Types such as <code>svec</code> or <code>INTEGER[]</code> are implicitly converted to <code>FLOAT8[]</code>.</li>
	</ul>
	</dd></dl>
	<p>If kmeans is called with a set of initial centroids, the centroid relation is expected to be of the following form: </p>
	<pre>{TABLE\|VIEW} <em>rel_initial_centroids</em> (
	...
	<em>expr_centroid</em> DOUBLE PRECISION[],
	...
	)</pre><p> where:</p>
	<ul>
	<li><em>expr_centroid</em> is the name of a column with coordinates.</li>
	</ul>
	<dl class="section user"><dt>Usage</dt><dd>The k-means algorithm can be invoked in four possible ways:<ul>
	<li>using <em>random</em> centroid seeding method for a provided \( k \): <pre>SELECT * FROM <a class="el" href="kmeans_8sql__in.html#a66ac1cab8811c4d842de1bc221886b53">kmeans_random</a>(
	'<em>rel_source</em>', '<em>expr_point</em>', k,
	[ '<em>fn_dist</em>', '<em>agg_centroid</em>',
	<em>max_num_iterations</em>, <em>min_frac_reassigned</em> ]
	);</pre></li>
	<li>using <em>kmeans++</em> centroid seeding method for a provided \( k \): <pre>SELECT * FROM <a class="el" href="kmeans_8sql__in.html#a639178dacebca2a2114923038398d6bb">kmeanspp</a>(
	'<em>rel_source</em>', '<em>expr_point</em>', k,
	[ '<em>fn_dist</em>', '<em>agg_centroid</em>',
	<em>max_num_iterations</em>, <em>min_frac_reassigned</em> ]
	);</pre></li>
	<li>with a provided centroid set: <pre>SELECT * FROM <a class="el" href="kmeans_8sql__in.html#afdae42b563f1f8bca3937dbbbacaa1c3">kmeans</a>(
	'<em>rel_source</em>', '<em>expr_point</em>',
	'<em>rel_initial_centroids</em>', '<em>expr_centroid</em>',
	[ '<em>fn_dist</em>', '<em>agg_centroid</em>',
	<em>max_num_iterations</em>, <em>min_frac_reassigned</em> ]
	);</pre> ---------— OR ------------— <pre>SELECT * FROM <a class="el" href="kmeans_8sql__in.html#afdae42b563f1f8bca3937dbbbacaa1c3">kmeans</a>(
	'<em>rel_source</em>', '<em>expr_point</em>',
	initial_centroids,
	[ '<em>fn_dist</em>', '<em>agg_centroid</em>',
	<em>max_num_iterations</em>, <em>min_frac_reassigned</em> ]
	);</pre> where:<ul>
	<li><em>initial_centroids</em> is of type <code>DOUBLE PRECISION[][]</code>.</li>
	</ul>
	</li>
	</ul>
	</dd></dl>
	<p>The output of the k-means module is a table that includes the final centroid positions (DOUBLE PRECISION[][]), the objective function, the fraction of reassigned points in the last iteration, and the number of total iterations: </p>
	<pre>
	centroids \| objective_fn \| frac_reassigned \| num_iterations
	----------------------------------+------------------+-----------------+----------------
	...
	</pre><dl class="section user"><dt>Examples</dt><dd><ol type="1">
	<li>Prepare some input data. <div class="fragment"><div class="line">sql> <a class="code" href="robust_8sql__in.html#ac9ebd21770ba37efb90e1ccee36fc103">SELECT</a> * FROM <span class="keyword">public</span>.km_sample LIMIT 5;</div>
	<div class="line"> points</div>
	<div class="line">-------------------------------------------</div>
	<div class="line"> {1,1}:{15.8822241332382,105.945462542586}</div>
	<div class="line"> {1,1}:{34.5065216883086,72.3126099305227}</div>
	<div class="line"> {1,1}:{22.5074400822632,95.3209559689276}</div>
	<div class="line"> {1,1}:{70.2589857042767,68.7395178806037}</div>
	<div class="line"> {1,1}:{30.9844257542863,25.3213323024102}</div>
	<div class="line">(5 rows)</div>
	</div><!-- fragment --> Note: the example <em>points</em> is type <code><a class="el" href="group__grp__svec.html">SVEC</a></code>.</li>
	<li>Run k-means clustering using kmeans++ for centroid seeding: <div class="fragment"><div class="line">sql> <a class="code" href="robust_8sql__in.html#ac9ebd21770ba37efb90e1ccee36fc103">SELECT</a> * FROM madlib.kmeanspp(<span class="stringliteral">'km_sample'</span>, <span class="stringliteral">'points'</span>, 2, <span class="stringliteral">'madlib.squared_dist_norm2'</span>, <span class="stringliteral">'madlib.avg'</span>, 20, 0.001);</div>
	<div class="line">);</div>
	<div class="line"> centroids \| objective_fn \| frac_reassigned \| num_iterations</div>
	<div class="line">-------------------------------------------------------------------------+------------------+-----------------+----------------</div>
	<div class="line"> {{68.01668579784,48.9667382972952},{28.1452167573446,84.5992507653263}} \| 586729.010675982 \| 0.001 \| 5</div>
	</div><!-- fragment --></li>
	<li>Calculate the simplified silhouette coefficient: <div class="fragment"><div class="line">sql> <a class="code" href="robust_8sql__in.html#ac9ebd21770ba37efb90e1ccee36fc103">SELECT</a> * from madlib.simple_silhouette(<span class="stringliteral">'km_test_svec'</span>,<span class="stringliteral">'points'</span>,</div>
	<div class="line"> (select centroids from madlib.kmeanspp(<span class="stringliteral">'km_test_svec'</span>,<span class="stringliteral">'points'</span>,2,<span class="stringliteral">'madlib.squared_dist_norm2'</span>,<span class="stringliteral">'madlib.avg'</span>,20,0.001)),</div>
	<div class="line"> <span class="stringliteral">'madlib.dist_norm2'</span>);</div>
	<div class="line"> <a class="code" href="kmeans_8sql__in.html#a71e7675758c99acbe7785819b6a85a8f" title="Compute a simplified version of the silhouette coefficient. ">simple_silhouette</a></div>
	<div class="line">-------------------</div>
	<div class="line"> 0.611022970398174</div>
	</div><!-- fragment --></li>
	</ol>
	</dd></dl>
	<dl class="section user"><dt>Literature</dt><dd></dd></dl>
	<p>[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>[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>[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>[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>[5] Leisch, Friedrich: A Toolbox for K-Centroids Cluster Analysis. In: Computational Statistics and Data Analysis, 51(2). pp. 526-544. 2006.</p>
	<dl class="section see"><dt>See Also</dt><dd>File <a class="el" href="kmeans_8sql__in.html" title="Set of functions for k-means clustering. ">kmeans.sql_in</a> documenting the SQL functions. </dd></dl>
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