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<title>MADlib: k-Means Clustering</title>
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<div class="title">k-Means Clustering<div class="ingroups"><a class="el" href="group__grp__unsupervised.html">Unsupervised Learning</a> &raquo; <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 -&gt; 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>
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