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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 --> +<!-- start footer part --> +<div id="nav-path" class="navpath"><!-- id is needed for treeview function! --> + <ul> + <li class="footer">Generated on Mon Oct 15 2018 11:24:30 for MADlib by + <a href="http://www.doxygen.org/index.html";> + <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.14 </li> + </ul> +</div> +</body> +</html>
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There may be some issues that will be addressed in a future version. Interface and implementation are subject to change. </em></dd></dl> +<p><a class="anchor" id="knn"></a> K-nearest neighbors is a method for finding the k closest points to a given data point in terms of a given metric. Its input consists of data points as features from testing examples, and it looks for k closest points in the training set for each of the data points in test set. The output of KNN depends on the type of task. For classification, the output is the majority vote of the classes of the k nearest data points. That is, the testing example gets assigned the most popular class from the nearest neighbors. For regression, the output is the average of the values of k nearest neighbors of the given test point.</p> +<p><a class="anchor" id="usage"></a></p><dl class="section user"><dt>Usage</dt><dd><pre class="syntax"> +knn( point_source, + point_column_name, + point_id, + label_column_name, + test_source, + test_column_name, + test_id, + output_table, + k, + output_neighbors, + fn_dist, + weighted_avg + ) +</pre></dd></dl> +<p><b>Arguments</b> </p><dl class="arglist"> +<dt>point_source </dt> +<dd><p class="startdd">TEXT. Name of the table containing the training data points. Training data points are expected to be stored row-wise in a column of type <code>DOUBLE PRECISION[]</code>. </p> +<p class="enddd"></p> +</dd> +<dt>point_column_name </dt> +<dd><p class="startdd">TEXT. Name of the column with training data points or expression that evaluates to a numeric array</p> +<p class="enddd"></p> +</dd> +<dt>point_id </dt> +<dd><p class="startdd">TEXT. Name of the column in 'point_sourceâ containing source data ids. The ids are of type INTEGER with no duplicates. They do not need to be contiguous. This parameter must be used if the list of nearest neighbors are to be output, i.e., if the parameter 'output_neighbors' below is TRUE or if 'label_column_name' is NULL.</p> +<p class="enddd"></p> +</dd> +<dt>label_column_name </dt> +<dd><p class="startdd">TEXT. Name of the column with labels/values of training data points. If this column is a Boolean, integer or text, it will run KNN classification, else if it is double precision values will run KNN regression. If you set this to NULL, it will only return the set of neighbors without actually doing classification or regression.</p> +<p class="enddd"></p> +</dd> +<dt>test_source </dt> +<dd><p class="startdd">TEXT. Name of the table containing the test data points. Testing data points are expected to be stored row-wise in a column of type <code>DOUBLE PRECISION[]</code>. </p> +<p class="enddd"></p> +</dd> +<dt>test_column_name </dt> +<dd><p class="startdd">TEXT. Name of the column with testing data points or expression that evaluates to a numeric array</p> +<p class="enddd"></p> +</dd> +<dt>test_id </dt> +<dd><p class="startdd">TEXT. Name of the column having ids of data points in test data table.</p> +<p class="enddd"></p> +</dd> +<dt>output_table </dt> +<dd><p class="startdd">TEXT. Name of the table to store final results.</p> +<p class="enddd"></p> +</dd> +<dt>k (optional) </dt> +<dd><p class="startdd">INTEGER. default: 1. Number of nearest neighbors to consider. For classification, should be an odd number to break ties, otherwise the result may depend on ordering of the input data.</p> +<p class="enddd"></p> +</dd> +<dt>output_neighbors (optional) </dt> +<dd><p class="startdd">BOOLEAN default: TRUE. Outputs the list of k-nearest neighbors that were used in the voting/averaging, sorted from closest to furthest.</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 used to calculate the distance between data points.</p> +<p>The following distance functions can be used: </p><ul> +<li> +<b><a class="el" href="linalg_8sql__in.html#aad193850e79c4b9d811ca9bc53e13476">dist_norm1</a></b>: 1-norm/Manhattan </li> +<li> +<b><a class="el" href="linalg_8sql__in.html#aa58e51526edea6ea98db30b6f250adb4">dist_norm2</a></b>: 2-norm/Euclidean </li> +<li> +<b><a class="el" href="linalg_8sql__in.html#a00a08e69f27524f2096032214e15b668">squared_dist_norm2</a></b>: squared Euclidean distance </li> +<li> +<b><a class="el" href="linalg_8sql__in.html#a8c7b9281a72ff22caf06161701b27e84">dist_angle</a></b>: angle </li> +<li> +<b><a class="el" href="linalg_8sql__in.html#afa13b4c6122b99422d666dedea136c18">dist_tanimoto</a></b>: tanimoto </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>weighted_avg (optional) </dt> +<dd><p class="startdd">BOOLEAN, default: FALSE. Calculates classification or regression values using a weighted average. The idea is to weigh the contribution of each of the k neighbors according to their distance to the test point, giving greater influence to closer neighbors. The distance function 'fn_dist' specified above is used.</p> +<p>For classification, majority voting weighs a neighbor according to inverse distance.</p> +<p class="enddd">For regression, the inverse distance weighting approach is used from Shepard [4]. </p> +</dd> +</dl> +<p><a class="anchor" id="output"></a></p><dl class="section user"><dt>Output Format</dt><dd></dd></dl> +<p>The output of the KNN module is a table with the following columns: </p><table class="output"> +<tr> +<th>id </th><td>INTEGER. The ids of test data points. </td></tr> +<tr> +<th>test_column_name </th><td>DOUBLE PRECISION[]. The test data points. </td></tr> +<tr> +<th>prediction </th><td>INTEGER. Label in case of classification, average value in case of regression. </td></tr> +<tr> +<th>k_nearest_neighbours </th><td>INTEGER[]. List of nearest neighbors, sorted closest to furthest from the corresponding test point. </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 training data for classification: <pre class="example"> +DROP TABLE IF EXISTS knn_train_data; +CREATE TABLE knn_train_data ( + id integer, + data integer[], + label integer -- Integer label means for classification + ); +INSERT INTO knn_train_data VALUES +(1, '{1,1}', 1), +(2, '{2,2}', 1), +(3, '{3,3}', 1), +(4, '{4,4}', 1), +(5, '{4,5}', 1), +(6, '{20,50}', 0), +(7, '{10,31}', 0), +(8, '{81,13}', 0), +(9, '{1,111}', 0); +</pre></li> +<li>Prepare some training data for regression: <pre class="example"> +DROP TABLE IF EXISTS knn_train_data_reg; +CREATE TABLE knn_train_data_reg ( + id integer, + data integer[], + label float -- Float label means for regression + ); +INSERT INTO knn_train_data_reg VALUES +(1, '{1,1}', 1.0), +(2, '{2,2}', 1.0), +(3, '{3,3}', 1.0), +(4, '{4,4}', 1.0), +(5, '{4,5}', 1.0), +(6, '{20,50}', 0.0), +(7, '{10,31}', 0.0), +(8, '{81,13}', 0.0), +(9, '{1,111}', 0.0); +</pre></li> +<li>Prepare some testing data: <pre class="example"> +DROP TABLE IF EXISTS knn_test_data; +CREATE TABLE knn_test_data ( + id integer, + data integer[] + ); +INSERT INTO knn_test_data VALUES +(1, '{2,1}'), +(2, '{2,6}'), +(3, '{15,40}'), +(4, '{12,1}'), +(5, '{2,90}'), +(6, '{50,45}'); +</pre></li> +<li>Run KNN for classification: <pre class="example"> +DROP TABLE IF EXISTS knn_result_classification; +SELECT * FROM madlib.knn( + 'knn_train_data', -- Table of training data + 'data', -- Col name of training data + 'id', -- Col name of id in train data + 'label', -- Training labels + 'knn_test_data', -- Table of test data + 'data', -- Col name of test data + 'id', -- Col name of id in test data + 'knn_result_classification', -- Output table + 3, -- Number of nearest neighbors + True, -- True to list nearest-neighbors by id + 'madlib.squared_dist_norm2' -- Distance function + ); +SELECT * from knn_result_classification ORDER BY id; +</pre> Result: <pre class="result"> + id | data | prediction | k_nearest_neighbours +----+---------+------------+---------------------- + 1 | {2,1} | 1 | {2,1,3} + 2 | {2,6} | 1 | {5,4,3} + 3 | {15,40} | 0 | {7,6,5} + 4 | {12,1} | 1 | {4,5,3} + 5 | {2,90} | 0 | {9,6,7} + 6 | {50,45} | 0 | {6,7,8} +(6 rows) +</pre> Note that the nearest neighbors are sorted from closest to furthest from the corresponding test point.</li> +<li>Run KNN for regression: <pre class="example"> +DROP TABLE IF EXISTS knn_result_regression; +SELECT * FROM madlib.knn( + 'knn_train_data_reg', -- Table of training data + 'data', -- Col name of training data + 'id', -- Col Name of id in train data + 'label', -- Training labels + 'knn_test_data', -- Table of test data + 'data', -- Col name of test data + 'id', -- Col name of id in test data + 'knn_result_regression', -- Output table + 3, -- Number of nearest neighbors + True, -- True to list nearest-neighbors by id + 'madlib.dist_norm2' -- Distance function + ); +SELECT * FROM knn_result_regression ORDER BY id; +</pre> Result: <pre class="result"> + id | data | prediction | k_nearest_neighbours +----+---------+-------------------+---------------------- + 1 | {2,1} | 1 | {2,1,3} + 2 | {2,6} | 1 | {5,4,3} + 3 | {15,40} | 0.333333333333333 | {7,6,5} + 4 | {12,1} | 1 | {4,5,3} + 5 | {2,90} | 0 | {9,6,7} + 6 | {50,45} | 0 | {6,7,8} +(6 rows) +</pre></li> +<li>List nearest neighbors only, without doing classification or regression: <pre class="example"> +DROP TABLE IF EXISTS knn_result_list_neighbors; +SELECT * FROM madlib.knn( + 'knn_train_data_reg', -- Table of training data + 'data', -- Col name of training data + 'id', -- Col Name of id in train data + NULL, -- NULL training labels means just list neighbors + 'knn_test_data', -- Table of test data + 'data', -- Col name of test data + 'id', -- Col name of id in test data + 'knn_result_list_neighbors', -- Output table + 3 -- Number of nearest neighbors + ); +SELECT * FROM knn_result_list_neighbors ORDER BY id; +</pre> Result, with neighbors sorted from closest to furthest: <pre class="result"> + id | data | k_nearest_neighbours +----+---------+---------------------- + 1 | {2,1} | {2,1,3} + 2 | {2,6} | {5,4,3} + 3 | {15,40} | {7,6,5} + 4 | {12,1} | {4,5,3} + 5 | {2,90} | {9,6,7} + 6 | {50,45} | {6,7,8} +(6 rows) +</pre></li> +<li>Run KNN for classification using the weighted average: <pre class="example"> +DROP TABLE IF EXISTS knn_result_classification; +SELECT * FROM madlib.knn( + 'knn_train_data', -- Table of training data + 'data', -- Col name of training data + 'id', -- Col name of id in train data + 'label', -- Training labels + 'knn_test_data', -- Table of test data + 'data', -- Col name of test data + 'id', -- Col name of id in test data + 'knn_result_classification', -- Output table + 3, -- Number of nearest neighbors + True, -- True to list nearest-neighbors by id + 'madlib.squared_dist_norm2', -- Distance function + True -- For weighted average + ); +SELECT * FROM knn_result_classification ORDER BY id; +</pre> <pre class="result"> + id | data | prediction | k_nearest_neighbours +----+---------+---------------------+---------------------- + 1 | {2,1} | 1 | {2,1,3} + 2 | {2,6} | 1 | {5,4,3} + 3 | {15,40} | 0 | {7,6,5} + 4 | {12,1} | 1 | {4,5,3} + 5 | {2,90} | 0 | {9,6,7} + 6 | {50,45} | 0 | {6,7,8} +(6 rows) +</pre></li> +</ol> +<p><a class="anchor" id="background"></a></p><dl class="section user"><dt>Technical Background</dt><dd></dd></dl> +<p>The training data points are vectors in a multidimensional feature space, each with a class label. The training phase of the algorithm consists only of storing the feature vectors and class labels of the training points.</p> +<p>In the classification phase, k is a user-defined constant, and an unlabeled vector (a test point) is classified by assigning the label which is most frequent among the k training samples nearest to that test point. In case of regression, average of the values of these k training samples is assigned to the test point.</p> +<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl> +<p><a class="anchor" id="knn-lit-1"></a>[1] Wikipedia, k-nearest neighbors algorithm, <a href="https://en.wikipedia.org/wiki/K-nearest_neighbors_algorithm";>https://en.wikipedia.org/wiki/K-nearest_neighbors_algorithm</a></p> +<p><a class="anchor" id="knn-lit-2"></a>[2] N. S. Altman: An Introduction to Kernel and Nearest-Neighbor Nonparametric Regression <a href="http://www.stat.washington.edu/courses/stat527/s13/readings/Altman_AmStat_1992.pdf";>http://www.stat.washington.edu/courses/stat527/s13/readings/Altman_AmStat_1992.pdf</a></p> +<p><a class="anchor" id="knn-lit-3"></a>[3] Gongde Guo1, Hui Wang, David Bell, Yaxin Bi, Kieran Greer: KNN Model-Based Approach in Classification, <a href="https://ai2-s2-pdfs.s3.amazonaws.com/a7e2/814ec5db800d2f8c4313fd436e9cf8273821.pdf";>https://ai2-s2-pdfs.s3.amazonaws.com/a7e2/814ec5db800d2f8c4313fd436e9cf8273821.pdf</a></p> +<p><a class="anchor" id="knn-lit-4"></a>[4] Shepard, Donald (1968). "A two-dimensional interpolation function for +irregularly-spaced data". Proceedings of the 1968 ACM National Conference. pp. 517â524.</p> +</div><!-- contents --> +</div><!-- doc-content --> +<!-- start footer part --> +<div id="nav-path" class="navpath"><!-- id is needed for treeview function! --> + <ul> + <li class="footer">Generated on Mon Oct 15 2018 11:24:30 for MADlib by + <a href="http://www.doxygen.org/index.html";> + <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.14 </li> + </ul> +</div> +</body> +</html>
