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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> |
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| <div class="contents"> |
| <div class="toc"><b>Contents</b> </p><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 <img class="formulaInl" alt="$ n $" src="form_10.png"/> points <img class="formulaInl" alt="$ x_1, \dots, x_n \in \mathbb R^d $" src="form_138.png"/>, the goal is to position <img class="formulaInl" alt="$ k $" src="form_97.png"/> centroids <img class="formulaInl" alt="$ c_1, \dots, c_k \in \mathbb R^d $" src="form_139.png"/> 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.</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. The centroid relation is expected to be of the following form: </p><pre> |
| {TABLE|VIEW} rel_initial_centroids ( |
| ... |
| expr_centroid DOUBLE PRECISION[], |
| ... |
| ) |
| </pre><p> where <em>expr_centroid</em> is the name of a column with coordinates. </p> |
| <p class="enddd"></p> |
| </dd> |
| <dt>expr_centroid </dt> |
| <dd><p class="startdd">TEXT. The name of the column 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>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> |
| <ol type="1"> |
| <li>Prepare some input data. <pre class="example"> |
| CREATE TABLE public.km_sample(pid int, points double precision[]); |
| COPY km_sample (pid, points) FROM stdin DELIMITER '|'; |
| 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"> |
| \x on; |
| SELECT * FROM madlib.kmeanspp( 'km_sample', |
| 'points', |
| 2, |
| 'madlib.squared_dist_norm2', |
| 'madlib.avg', |
| 20, |
| 0.001 |
| ); |
| </pre> Result: <pre class="result"> |
| centroids | {{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}, |
| {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}} |
| objective_fn | 151184.962672 |
| 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 |
| madlib.kmeanspp('km_sample', |
| 'points', |
| 2, |
| 'madlib.squared_dist_norm2', |
| 'madlib.avg', |
| 20, |
| 0.001)), |
| '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; |
| SELECT data.*, (madlib.closest_column(centroids, points)).column_id as cluster_id |
| FROM public.km_sample as data, |
| (SELECT centroids |
| FROM madlib.kmeanspp('km_sample', 'points', 2, |
| 'madlib.squared_dist_norm2', |
| 'madlib.avg', 20, 0.001)) as centroids |
| ORDER BY data.pid; |
| </pre> <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} | 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} | 1 |
| 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 |
| </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"> |
| <img class="formulaDsp" alt="\[ (c_1, \dots, c_k) \mapsto \sum_{i=1}^n \min_{j=1}^k \operatorname{dist}(x_i, c_j) \]" src="form_140.png"/> |
| </p> |
| <p> In the most common case, <img class="formulaInl" alt="$ \operatorname{dist} $" src="form_141.png"/> is the square of the Euclidean distance.</p> |
| <p>This problem is computationally difficult (NP-hard), yet the local-search heuristic proposed by Lloyd [4] performs reasonably well in practice. In fact, it is so ubiquitous today that it is often referred to as the <em>standard algorithm</em> or even just the <em>k-means algorithm</em> [1]. It works as follows:</p> |
| <ol type="1"> |
| <li>Seed the <img class="formulaInl" alt="$ k $" src="form_97.png"/> 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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