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[05/51] [partial] incubator-madlib-site git commit: Update doc for
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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> » <a class="el" href="group__grp__clustering.html">Clustering</a></div></div> </div>
+</div><!--header-->
+<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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+<title>MADlib: Latent Dirichlet Allocation</title>
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+<div class="title">Latent Dirichlet Allocation<div class="ingroups"><a class="el" href="group__grp__unsupervised.html">Unsupervised Learning</a> » <a class="el" href="group__grp__topic__modelling.html">Topic Modelling</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> </p><ul>
+<li>
+<a href="#vocabulary">Vocabulary Format</a> </li>
+<li>
+<a href="#train">Training Function</a> </li>
+<li>
+<a href="#predict">Prediction Function</a> </li>
+<li>
+<a href="#examples">Examples</a> </li>
+<li>
+<a href="#notes">Notes</a> </li>
+<li>
+<a href="#literature">Literature</a> </li>
+<li>
+<a href="#related">Related Topics</a></li>
+<li>
+</li>
+</ul>
+</div><p>Latent Dirichlet Allocation (LDA) is an interesting generative probabilistic model for natural texts and has received a lot of attention in recent years. The model is quite versatile, having found uses in problems like automated topic discovery, collaborative filtering, and document classification.</p>
+<p>The LDA model posits that each document is associated with a mixture of various topics (e.g. a document is related to Topic 1 with probability 0.7, and Topic 2 with probability 0.3), and that each word in the document is attributable to one of the document's topics. There is a (symmetric) Dirichlet prior with parameter <img class="formulaInl" alt="$ \alpha $" src="form_142.png"/> on each document's topic mixture. In addition, there is another (symmetric) Dirichlet prior with parameter <img class="formulaInl" alt="$ \beta $" src="form_143.png"/> on the distribution of words for each topic.</p>
+<p>The following generative process then defines a distribution over a corpus of documents.</p>
+<ul>
+<li>Sample for each topic <img class="formulaInl" alt="$ i $" src="form_32.png"/>, a per-topic word distribution <img class="formulaInl" alt="$ \phi_i $" src="form_144.png"/> from the Dirichlet( <img class="formulaInl" alt="$\beta$" src="form_135.png"/>) prior.</li>
+<li>For each document:<ul>
+<li>Sample a document length N from a suitable distribution, say, Poisson.</li>
+<li>Sample a topic mixture <img class="formulaInl" alt="$ \theta $" src="form_145.png"/> for the document from the Dirichlet( <img class="formulaInl" alt="$\alpha$" src="form_146.png"/>) distribution.</li>
+<li>For each of the N words:<ul>
+<li>Sample a topic <img class="formulaInl" alt="$ z_n $" src="form_147.png"/> from the multinomial topic distribution <img class="formulaInl" alt="$ \theta $" src="form_145.png"/>.</li>
+<li>Sample a word <img class="formulaInl" alt="$ w_n $" src="form_148.png"/> from the multinomial word distribution <img class="formulaInl" alt="$ \phi_{z_n} $" src="form_149.png"/> associated with topic <img class="formulaInl" alt="$ z_n $" src="form_147.png"/>.</li>
+</ul>
+</li>
+</ul>
+</li>
+</ul>
+<p>In practice, only the words in each document are observable. The topic mixture of each document and the topic for each word in each document are latent unobservable variables that need to be inferred from the observables, and this is the problem people refer to when they talk about the inference problem for LDA. Exact inference is intractable, but several approximate inference algorithms for LDA have been developed. The simple and effective Gibbs sampling algorithm described in Griffiths and Steyvers [2] appears to be the current algorithm of choice.</p>
+<p>This implementation provides a parallel and scalable in-database solution for LDA based on Gibbs sampling. Different with the implementations based on MPI or Hadoop Map/Reduce, this implementation builds upon the shared-nothing MPP databases and enables high-performance in-database analytics.</p>
+<p><a class="anchor" id="vocabulary"></a></p><dl class="section user"><dt>Vocabulary Format</dt><dd></dd></dl>
+<p>The vocabulary, or dictionary, indexes all the words found in the corpus and has the following format: </p><pre>{TABLE|VIEW} <em>vocab_table</em> (
+ <em>wordid</em> INTEGER,
+ <em>word</em> TEXT
+)</pre><p> where <code>wordid</code> refers the word ID (the index of a word in the vocabulary) and <code>word</code> is the actual word.</p>
+<dl class="section user"><dt>Usage</dt><dd><ul>
+<li><p class="startli">The training (i.e. topic inference) can be done with the following function: </p><pre>
+ SELECT <a class="el" href="lda_8sql__in.html#aeb7593251a4dedb695494f65dc2d1f80">lda_train</a>(
+ <em>'data_table'</em>,
+ <em>'model_table'</em>,
+ <em>'output_data_table'</em>,
+ <em>voc_size</em>,
+ <em>topic_num</em>,
+ <em>iter_num</em>,
+ <em>alpha</em>,
+ <em>beta</em>)
+ </pre><p class="startli">This function stores the resulting model in <code><em>model_table</em></code>. The table has only 1 row and is in the following form: </p><pre>{TABLE} <em>model_table</em> (
+ <em>voc_size</em> INTEGER,
+ <em>topic_num</em> INTEGER,
+ <em>alpha</em> FLOAT,
+ <em>beta</em> FLOAT,
+ <em>model</em> BIGINT[])
+ </pre><p class="startli">This function also stores the topic counts and the topic assignments in each document in <code><em>output_data_table</em></code>. The table is in the following form: </p><pre>{TABLE} <em>output_data_table</em> (
+ <em>docid</em> INTEGER,
+ <em>wordcount</em> INTEGER,
+ <em>words</em> INTEGER[],
+ <em>counts</em> INTEGER[],
+ <em>topic_count</em> INTEGER[],
+ <em>topic_assignment</em> INTEGER[])
+ </pre></li>
+<li><p class="startli">The prediction (i.e. labelling of test documents using a learned LDA model) can be done with the following function: </p><pre>
+ SELECT <a class="el" href="lda_8sql__in.html#aaa89e30c8fd0ba41b6feee01ee195330">lda_predict</a>(
+ <em>'data_table'</em>,
+ <em>'model_table'</em>,
+ <em>'output_table'</em>);
+ </pre><p class="startli">This function stores the prediction results in <em>output_table</em>. Each row in the table stores the topic distribution and the topic assignments for a docuemnt in the dataset. The table is in the following form: </p><pre>{TABLE} <em>output_table</em> (
+ <em>docid</em> INTEGER,
+ <em>wordcount</em> INTEGER,
+ <em>words</em> INTEGER,
+ <em>counts</em> INTEGER,
+ <em>topic_count</em> INTEGER[],
+ <em>topic_assignment</em> INTEGER[])
+ </pre></li>
+<li>This module also provides a function for computing the perplexity: <pre>
+ SELECT <a class="el" href="lda_8sql__in.html#a25c3ef12d9808d8a38c5fd2630f3b5a9">lda_get_perplexity</a>(
+ <em>'model_table'</em>,
+ <em>'output_data_table'</em>);
+ </pre></li>
+</ul>
+</dd></dl>
+<dl class="section user"><dt>Implementation Notes</dt><dd>The input format requires the user to tokenize each document into an array of words. This process involves tokenizing and filtering documents - a process out-of-scope for this module. Internally, the input data will be validated and then converted to the following format for efficiency: <pre>{TABLE} <em>__internal_data_table__</em> (
+ <em>docid</em> INTEGER,
+ <em>wordcount</em> INTEGER,
+ <em>words</em> INTEGER[],
+ <em>counts</em> INTEGER[])
+</pre> where <code>docid</code> is the document ID, <code>wordcount</code> is the number of words in the document, <code>words</code> is the list of unique words in the document, and <code>counts</code> is a list of the number of occurrences of each unique word in the document.</dd></dl>
+<p><a class="anchor" id="train"></a></p><dl class="section user"><dt>Training Function</dt><dd>The LDA training function has the following syntax. <pre class="syntax">
+lda_train( data_table,
+ model_table,
+ output_data_table,
+ voc_size,
+ topic_num,
+ iter_num,
+ alpha,
+ beta
+ )
+</pre> <b>Arguments</b> <dl class="arglist">
+<dt>data_table </dt>
+<dd><p class="startdd">TEXT. The name of the table storing the training dataset. Each row is in the form <code><docid, wordid, count></code> where <code>docid</code>, <code>wordid</code>, and <code>count</code> are non-negative integers.</p>
+<p class="enddd">The <code>docid</code> column refers to the document ID, the <code>wordid</code> column is the word ID (the index of a word in the vocabulary), and <code>count</code> is the number of occurrences of the word in the document. </p>
+</dd>
+<dt>model_table </dt>
+<dd>TEXT. The name of the table storing the learned models. This table has one row and the following columns. <table class="output">
+<tr>
+<th>voc_size </th><td>INTEGER. Size of the vocabulary. Note that the <code>wordid</code> should be continous integers starting from 0 to <code>voc_size</code> − <code>1</code>. A data validation routine is called to validate the dataset. </td></tr>
+<tr>
+<th>topic_num </th><td>INTEGER. Number of topics. </td></tr>
+<tr>
+<th>alpha </th><td>DOUBLE PRECISION. Dirichlet parameter for the per-doc topic multinomial (e.g. 50/topic_num). </td></tr>
+<tr>
+<th>beta </th><td>DOUBLE PRECISION. Dirichlet parameter for the per-topic word multinomial (e.g. 0.01). </td></tr>
+<tr>
+<th>model </th><td>BIGINT[]. </td></tr>
+</table>
+</dd>
+<dt>output_data_table </dt>
+<dd>TEXT. The name of the table to store the output data. It has the following columns: <table class="output">
+<tr>
+<th>docid </th><td>INTEGER. </td></tr>
+<tr>
+<th>wordcount </th><td>INTEGER. </td></tr>
+<tr>
+<th>words </th><td>INTEGER[]. </td></tr>
+<tr>
+<th>counts </th><td>INTEGER[]. </td></tr>
+<tr>
+<th>topic_count </th><td>INTEGER[]. </td></tr>
+<tr>
+<th>topic_assignment </th><td>INTEGER[]. </td></tr>
+</table>
+</dd>
+<dt>voc_size </dt>
+<dd>INTEGER. Size of the vocabulary. Note that the <code>wordid</code> should be continous integers starting from 0 to <code>voc_size</code> − <code>1</code>. A data validation routine is called to validate the dataset. </dd>
+<dt>topic_num </dt>
+<dd>INTEGER. Number of topics. </dd>
+<dt>iter_num </dt>
+<dd>INTEGER. Number of iterations (e.g. 60). </dd>
+<dt>alpha </dt>
+<dd>DOUBLE PRECISION. Dirichlet parameter for the per-doc topic multinomial (e.g. 50/topic_num). </dd>
+<dt>beta </dt>
+<dd>DOUBLE PRECISION. Dirichlet parameter for the per-topic word multinomial (e.g. 0.01). </dd>
+</dl>
+</dd></dl>
+<p><a class="anchor" id="predict"></a></p><dl class="section user"><dt>Prediction Function</dt><dd></dd></dl>
+<p>Prediction—labelling test documents using a learned LDA model—is accomplished with the following function: </p><pre class="syntax">
+lda_predict( data_table,
+ model_table,
+ output_table
+ );
+</pre><p>This function stores the prediction results in <code><em>output_table</em></code>. Each row in the table stores the topic distribution and the topic assignments for a document in the dataset. The table has the following columns: </p><table class="output">
+<tr>
+<th>docid </th><td>INTEGER. </td></tr>
+<tr>
+<th>wordcount </th><td>INTEGER. </td></tr>
+<tr>
+<th>words </th><td>INTEGER[]. List of word IDs in this document. </td></tr>
+<tr>
+<th>counts </th><td>INTEGER[]. List of word counts in this document. </td></tr>
+<tr>
+<th>topic_count </th><td>INTEGER[]. Of length topic_num, list of topic counts in this document. </td></tr>
+<tr>
+<th>topic_assignment </th><td>INTEGER[]. Of length wordcount, list of topic index for each word. </td></tr>
+</table>
+<p><a class="anchor" id="perplexity"></a></p><dl class="section user"><dt>Perplexity Function</dt><dd>This module provides a function for computing the perplexity. <pre class="syntax">
+lda_get_perplexity( model_table,
+ output_data_table
+ );
+</pre></dd></dl>
+<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
+<ol type="1">
+<li>Prepare a training dataset for LDA. The examples below are small strings extracted from various Wikipedia documents . <pre class="example">
+CREATE TABLE documents(docid INT4, contents TEXT);
+INSERT INTO documents VALUES
+(0, 'Statistical topic models are a class of Bayesian latent variable models, originally developed for analyzing the semantic content of large document corpora.'),
+(1, 'By the late 1960s, the balance between pitching and hitting had swung in favor of the pitchers. In 1968 Carl Yastrzemski won the American League batting title with an average of just .301, the lowest in history.'),
+(2, 'Machine learning is closely related to and often overlaps with computational statistics; a discipline that also specializes in prediction-making. It has strong ties to mathematical optimization, which deliver methods, theory and application domains to the field.'),
+(3, 'California''s diverse geography ranges from the Sierra Nevada in the east to the Pacific Coast in the west, from the Redwood\u2013Douglas fir forests of the northwest, to the Mojave Desert areas in the southeast. The center of the state is dominated by the Central Valley, a major agricultural area. ')
+</pre></li>
+<li>Build a word count table by extracting the words and building a histogram for each document using the <code>term_frequency</code> function (<a class="el" href="group__grp__text__utilities.html">Term Frequency</a>). <pre class="example">
+-- Convert a string to a list of words
+ALTER TABLE documents ADD COLUMN words TEXT[];
+UPDATE documents SET words = regexp_split_to_array(lower(contents), E'[\\s+\\.\\,]');
+
+-- Create the term frequency table
+DROP TABLE IF EXISTS my_training;
+SELECT madlib.term_frequency('documents', 'docid', 'words', 'my_training', TRUE);
+SELECT * FROM my_training order by docid limit 20;
+</pre> <pre class="result">
+ docid | wordid | count
+-------+--------+-------
+ 0 | 57 | 1
+ 0 | 86 | 1
+ 0 | 4 | 1
+ 0 | 55 | 1
+ 0 | 69 | 2
+ 0 | 81 | 1
+ 0 | 30 | 1
+ 0 | 33 | 1
+ 0 | 36 | 1
+ 0 | 43 | 1
+ 0 | 25 | 1
+ 0 | 65 | 2
+ 0 | 72 | 1
+ 0 | 9 | 1
+ 0 | 0 | 2
+ 0 | 29 | 1
+ 0 | 18 | 1
+ 0 | 12 | 1
+ 0 | 96 | 1
+ 0 | 91 | 1
+(20 rows)
+</pre> <pre class="example">
+SELECT * FROM my_training_vocabulary order by wordid limit 20;
+</pre> <pre class="result">
+ wordid | word
+--------+--------------
+ 0 |
+ 1 | 1960s
+ 2 | 1968
+ 3 | 301
+ 4 | a
+ 5 | agricultural
+ 6 | also
+ 7 | american
+ 8 | an
+ 9 | analyzing
+ 10 | and
+ 11 | application
+ 12 | are
+ 13 | area
+ 14 | areas
+ 15 | average
+ 16 | balance
+ 17 | batting
+ 18 | bayesian
+ 19 | between
+(20 rows)
+</pre></li>
+<li>Create an LDA model using the <code><a class="el" href="lda_8sql__in.html#aeb7593251a4dedb695494f65dc2d1f80" title="This UDF provides an entry for the lda training process. ">lda_train()</a></code> function. <pre class="example">
+SELECT madlib.lda_train( 'my_training',
+ 'my_model',
+ 'my_outdata',
+ 104,
+ 5,
+ 10,
+ 5,
+ 0.01
+ );
+</pre> After a successful run of the <a class="el" href="lda_8sql__in.html#aeb7593251a4dedb695494f65dc2d1f80" title="This UDF provides an entry for the lda training process. ">lda_train()</a> function two tables are generated, one for storing the learned model and the other for storing the output data table.</li>
+<li>To get the detailed information about the learned model, run these commands: <pre class="example">
+-- The topic description by top-k words
+SELECT madlib.lda_get_topic_desc( 'my_model',
+ 'my_training_vocabulary',
+ 'my_topic_desc',
+ 15);
+select * from my_topic_desc order by topicid, prob DESC;
+</pre> <pre class="result">
+ topicid | wordid | prob | word
+---------+--------+--------------------+-------------------
+ 1 | 69 | 0.181900726392252 | of
+ 1 | 52 | 0.0608353510895884 | is
+ 1 | 65 | 0.0608353510895884 | models
+ 1 | 30 | 0.0305690072639225 | corpora
+ 1 | 1 | 0.0305690072639225 | 1960s
+ 1 | 57 | 0.0305690072639225 | latent
+ 1 | 35 | 0.0305690072639225 | diverse
+ 1 | 81 | 0.0305690072639225 | semantic
+ 1 | 19 | 0.0305690072639225 | between
+ 1 | 75 | 0.0305690072639225 | pitchers
+ 1 | 43 | 0.0305690072639225 | for
+ 1 | 6 | 0.0305690072639225 | also
+ 1 | 40 | 0.0305690072639225 | favor
+ 1 | 47 | 0.0305690072639225 | had
+ 1 | 28 | 0.0305690072639225 | computational
+ ....
+</pre> <pre class="example">
+-- The per-word topic counts (sorted by topic id)
+SELECT madlib.lda_get_word_topic_count( 'my_model',
+ 'my_word_topic_count');
+</pre> <pre class="result">
+ wordid | topic_count
+--------+--------------
+ 0 | {0,17,0,0,0}
+ 1 | {1,0,0,0,0}
+ 2 | {0,0,0,0,1}
+ 3 | {0,0,0,0,1}
+ 4 | {0,0,0,0,3}
+ 5 | {0,1,0,0,0}
+ 6 | {1,0,0,0,0}
+ 7 | {1,0,0,0,0}
+ 8 | {0,0,0,1,0}
+ 9 | {1,0,0,0,0}
+ 10 | {0,0,0,0,3}
+ 11 | {0,0,1,0,0}
+ ....
+</pre></li>
+<li>To get the topic counts and the topic assignments for each doucment, run the following commands: <pre class="example">
+-- The per-document topic assignments and counts:
+SELECT docid, topic_assignment, topic_count FROM my_outdata;
+</pre> <pre class="result">
+ docid | topic_assignment | topic_count
+-------+-----------------------------------------------------------------------------------------------------------------+----------------
+ 1 | {1,1,1,1,1,1,2,4,1,4,4,4,1,0,2,1,0,2,2,3,4,2,1,1,4,2,4,3,0,0,2,4,4,3,3,3,3,3,0,1,0,4} | {6,12,7,7,10}
+ 3 | {1,1,1,1,1,1,4,0,2,3,1,2,0,0,0,1,2,2,1,3,3,2,2,1,2,2,2,0,3,0,4,1,0,0,1,4,3,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,3} | {8,12,10,21,4}
+ 0 | {1,1,4,2,1,4,4,4,1,3,1,0,0,0,0,0,0,0,0,1,1,3,0,1} | {9,8,1,2,4}
+ 2 | {1,1,1,1,4,1,4,4,2,0,2,4,1,1,4,1,2,0,1,3,1,2,4,3,2,4,4,3,1,2,0,3,3,1,4,3,3,3,2,1} | {3,13,7,8,9}
+(4 rows)
+</pre></li>
+<li>To use a learned LDA model for prediction (that is, to label new documents), use the following command: <pre class="example">
+SELECT madlib.lda_predict( 'my_testing',
+ 'my_model',
+ 'my_pred'
+ );
+</pre> The test table (<em>my_testing</em>) is expected to be in the same form as the training table (<em>my_training</em>) and can be created with the same process. After a successful run of the <a class="el" href="lda_8sql__in.html#af1fde06c39dd12bb9e5544997f815323" title="This UDF provides an entry for the lda predicton process. ">lda_predict()</a> function, the prediction results are generated and stored in <em>my_pred</em>. This table has the same schema as the <em>my_outdata</em> table generated by the <a class="el" href="lda_8sql__in.html#aeb7593251a4dedb695494f65dc2d1f80" title="This UDF provides an entry for the lda training process. ">lda_train()</a> function.</li>
+<li>Use the following command to compute the perplexity of the result. <pre class="example">
+SELECT madlib.lda_get_perplexity( 'my_model',
+ 'my_pred'
+ );
+</pre></li>
+</ol>
+<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
+<p>[1] D.M. Blei, A.Y. Ng, M.I. Jordan, <em>Latent Dirichlet Allocation</em>, Journal of Machine Learning Research, vol. 3, pp. 993-1022, 2003.</p>
+<p>[2] T. Griffiths and M. Steyvers, <em>Finding scientific topics</em>, PNAS, vol. 101, pp. 5228-5235, 2004.</p>
+<p>[3] Y. Wang, H. Bai, M. Stanton, W-Y. Chen, and E.Y. Chang, <em>lda: Parallel Dirichlet Allocation for Large-scale Applications</em>, AAIM, 2009.</p>
+<p>[4] <a href="http://en.wikipedia.org/wiki/Latent_Dirichlet_allocation">http://en.wikipedia.org/wiki/Latent_Dirichlet_allocation</a></p>
+<p>[5] J. Chang, Collapsed Gibbs sampling methods for topic models, R manual, 2010.</p>
+<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd>File <a class="el" href="lda_8sql__in.html" title="SQL functions for Latent Dirichlet Allocation. ">lda.sql_in</a> documenting the SQL functions. </dd></dl>
+</div><!-- contents -->
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+<!-- start footer part -->
+<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
+ <ul>
+ <li class="footer">Generated on Tue Sep 20 2016 11:27:01 for MADlib by
+ <a href="http://www.doxygen.org/index.html">
+ <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.10 </li>
+ </ul>
+</div>
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+</html>
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+ <div id="projectname">
+ <span id="projectnumber">1.9.1</span>
+ </div>
+ <div id="projectbrief">User Documentation for MADlib</div>
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+ onmouseover="return searchBox.OnSearchSelectShow()"
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+ name="MSearchResults" id="MSearchResults">
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+
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+ <div class="headertitle">
+<div class="title">Norms and Distance functions<div class="ingroups"><a class="el" href="group__grp__datatrans.html">Data Types and Transformations</a> » <a class="el" href="group__grp__arraysmatrix.html">Arrays and Matrices</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> </p><ul>
+<li class="level1">
+<a href="#functions">Linear Algebra Utility Functions</a> </li>
+<li class="level1">
+<a href="#examples">Examples</a> </li>
+<li class="level1">
+<a href="#literature">Literature</a> </li>
+<li class="level1">
+<a href="#related">Related Functions</a> </li>
+</ul>
+</div><p>The linalg module consists of useful utility functions for basic linear algebra operations. Several of these functions can be used while implementing new algorithms. These functions operate on vectors (1-D FLOAT8 array) and matrices (2-D FLOAT8 array). Note that other array types may need to be casted into FLOAT8[] before calling the functions.</p>
+<p>Refer to the <a class="el" href="linalg_8sql__in.html" title="SQL functions for linear algebra. ">linalg.sql_in</a> file for documentation on each of the utility functions.</p>
+<p><a class="anchor" id="functions"></a></p><dl class="section user"><dt>Linear Algebra Utility Functions</dt><dd><table class="output">
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a300300fe4b8576ba0b97b95d8dea3057" title="1-norm of a vector ">norm1()</a> </th><td><p class="starttd">1-norm of a vector, <img class="formulaInl" alt="$\|\vec{a}\|_1$" src="form_150.png"/>.</p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a50fdfe30cc0edc6888a909dbb4b4c239" title="2-norm of a vector ">norm2()</a> </th><td><p class="starttd">2-norm of a vector, <img class="formulaInl" alt="$\|\vec{a}\|_2$" src="form_151.png"/>. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#aad193850e79c4b9d811ca9bc53e13476" title="1-norm of the difference between two vectors ">dist_norm1()</a> </th><td><p class="starttd">1-norm of the difference between two vectors, <img class="formulaInl" alt="$\|\vec{a} - \vec{b}\|_1$" src="form_152.png"/>. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#aa58e51526edea6ea98db30b6f250adb4" title="2-norm of the difference between two vectors ">dist_norm2()</a> </th><td><p class="starttd">2-norm of the difference between two vectors, <img class="formulaInl" alt="$\|\vec{a} - \vec{b}\|_2$" src="form_153.png"/>. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#ad9cc156ae57bf7c0a2fe90798259105a" title="p-norm of the difference between two vectors ">dist_pnorm()</a> </th><td><p class="starttd">Generic p-norm of the difference between two vectors, <img class="formulaInl" alt="$\|\vec{a} - \vec{b}\|_p, p > 0$" src="form_154.png"/>. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a31fa9f2f5b45507c09f136464fdad1db" title="Infinity-norm of the difference between two vectors. ">dist_inf_norm()</a> </th><td><p class="starttd">Infinity-norm of the difference between two vectors, <img class="formulaInl" alt="$\|\vec{a} - \vec{b}\|_\infty$" src="form_155.png"/>. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a00a08e69f27524f2096032214e15b668" title="Squared 2-norm of the difference between two vectors. ">squared_dist_norm2()</a> </th><td><p class="starttd">Squared 2-norm of the difference between two vectors, <img class="formulaInl" alt="$\|\vec{a} - \vec{b}\|_2^2$" src="form_156.png"/>. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a1782f2ba00d9f9fab20894a576079f87" title="cosine similarity score between two vectors ">cosine_similarity()</a> </th><td><p class="starttd">Cosine score between two vectors, <img class="formulaInl" alt="$\frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|_2 \|\vec{b}\|_2}$" src="form_157.png"/>. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a8c7b9281a72ff22caf06161701b27e84" title="Angle between two vectors. ">dist_angle()</a> </th><td><p class="starttd">Angle between two vectors in an Euclidean space, <img class="formulaInl" alt="$\cos^{-1}(\frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|_2 \|\vec{b}\|_2})$" src="form_158.png"/>. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#afa13b4c6122b99422d666dedea136c18" title="Tanimoto distance between two vectors. ">dist_tanimoto()</a> </th><td><p class="starttd">Tanimoto distance between two vectors. [1] </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#ac1397ac9f4a35b3b67c3be05b5e1a828" title="Jaccard distance between two vectors (treated as sets) ">dist_jaccard()</a> </th><td><p class="starttd">Jaccard distance between two varchar vectors treated as sets. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#af6b905fcac7746ef0ed0c36df4a1e070" title="Get an indexed row of the given matrix (2-D array) ">get_row()</a> </th><td><p class="starttd">Return the indexed row of a matrix (2-D array). </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a20f34c9e661191e5225cca7bc23252c5" title="Get an indexed col of the given matrix (2-D array) ">get_col()</a> </th><td><p class="starttd">Return the indexed col of a matrix (2-D array). </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a1aa37f73fb1cd8d7d106aa518dd8c0b4" title="Compute the average of vectors. ">avg()</a> </th><td><p class="starttd">Compute the average of vectors. </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a0b04663ca206f03e66aed5ea2b4cc461" title="Compute the normalized average of vectors. ">normalized_avg()</a> </th><td><p class="starttd">Compute the normalized average of vectors (unit vector in an Euclidean space). </p>
+<p class="endtd"></p>
+</td></tr>
+<tr>
+<th><a class="el" href="linalg_8sql__in.html#a9c439706f35d6cac89f151d553a5f111" title="Combine vectors to a matrix. ">matrix_agg()</a> </th><td><p class="starttd">Combine vectors to a matrix. </p>
+<p class="endtd"></p>
+</td></tr>
+</table>
+</dd></dl>
+<p><a class="anchor" id="examples"></a></p>
+<p><b>Vector Norms and Distances</b></p>
+<ol type="1">
+<li>Create a database table with two vector columns and add some data. <pre class="example">
+CREATE TABLE two_vectors(
+ id integer,
+ a float8[],
+ b float8[]);
+</pre> <pre class="example">
+INSERT INTO two_vectors VALUES
+(1, '{3,4}', '{4,5}'),
+(2, '{1,1,0,-4,5,3,4,106,14}', '{1,1,0,6,-3,1,2,92,2}');
+</pre></li>
+<li>Invoke norm functions. <pre class="example">
+SELECT
+ id,
+ madlib.norm1(a),
+ madlib.norm2(a)
+FROM two_vectors;
+</pre> Result: <pre class="result">
+ id | norm1 | norm2
+----+-------+------------------
+ 1 | 7 | 5
+ 2 | 138 | 107.238052947636
+(2 rows)
+</pre></li>
+<li>Invoke distance functions. <pre class="example">
+SELECT
+ id,
+ madlib.dist_norm1(a, b),
+ madlib.dist_norm2(a, b),
+ madlib.dist_pnorm(a, b, 5) AS norm5,
+ madlib.dist_inf_norm(a, b),
+ madlib.squared_dist_norm2(a, b) AS sq_dist_norm2,
+ madlib.cosine_similarity(a, b),
+ madlib.dist_angle(a, b),
+ madlib.dist_tanimoto(a, b),
+ madlib.dist_jaccard(a::text[], b::text[])
+FROM two_vectors;
+</pre> Result: <pre class="result">
+ id | dist_norm1 | dist_norm2 | norm5 | dist_inf_norm | sq_dist_norm2 | cosine_similarity | dist_angle | dist_tanimoto | dist_jaccard
+----+------------+------------------+------------------+---------------+---------------+-------------------+--------------------+--------------------+-------------------
+ 1 | 2 | 1.4142135623731 | 1.14869835499704 | 1 | 2 | 0.999512076087079 | 0.0312398334302684 | 0.0588235294117647 | 0.666666666666667
+ 2 | 48 | 22.6274169979695 | 15.585086360695 | 14 | 512 | 0.985403348449008 | 0.17106899659286 | 0.0498733684005455 | 0.833333333333333
+(2 rows)
+</pre></li>
+</ol>
+<p><b>Matrix Functions</b></p>
+<ol type="1">
+<li>Create a database table with a matrix column. <pre class="example">
+CREATE TABLE matrix(
+ id integer,
+ m float8[]);
+</pre> <pre class="example">
+INSERT INTO matrix VALUES
+(1, '{{4,5},{3,5},{9,0}}');
+</pre></li>
+<li>Invoke matrix functions. <pre class="example">
+SELECT
+ madlib.get_row(m, 1) AS row_1,
+ madlib.get_row(m, 2) AS row_2,
+ madlib.get_row(m, 3) AS row_3,
+ madlib.get_col(m, 1) AS col_1,
+ madlib.get_col(m, 2) AS col_2
+FROM matrix;
+</pre> Result: <pre class="result">
+ row_1 | row_2 | row_3 | col_1 | col_2
+-------+-------+-------+---------+---------
+ {4,5} | {3,5} | {9,0} | {4,3,9} | {5,5,0}
+(1 row)
+</pre></li>
+</ol>
+<p><b>Aggregate Functions</b></p>
+<ol type="1">
+<li>Create a database table with a vector column. <pre class="example">
+CREATE TABLE vector(
+ id integer,
+ v float8[]);
+</pre> <pre class="example">
+INSERT INTO vector VALUES
+(1, '{4,3}'),
+(2, '{8,6}'),
+(3, '{12,9}');
+</pre></li>
+<li>Invoke aggregate functions. <pre class="example">
+SELECT
+ madlib.avg(v),
+ madlib.normalized_avg(v),
+ madlib.matrix_agg(v)
+FROM vector;
+</pre> Result: <pre class="result">
+ avg | normalized_avg | matrix_agg
+-------+----------------+----------------------
+ {8,6} | {0.8,0.6} | {{4,3},{8,6},{12,9}}
+(1 row)
+</pre></li>
+</ol>
+<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
+<p>[1] <a href="http://en.wikipedia.org/wiki/Jaccard_index#Tanimoto_similarity_and_distance">http://en.wikipedia.org/wiki/Jaccard_index#Tanimoto_similarity_and_distance</a></p>
+<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd>File <a class="el" href="linalg_8sql__in.html" title="SQL functions for linear algebra. ">linalg.sql_in</a> documenting the SQL functions. </dd></dl>
+</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 Tue Sep 20 2016 11:27:01 for MADlib by
+ <a href="http://www.doxygen.org/index.html">
+ <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.10 </li>
+ </ul>
+</div>
+</body>
+</html>
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+<title>MADlib: Linear Solvers</title>
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+ <td style="padding-left: 0.5em;">
+ <div id="projectname">
+ <span id="projectnumber">1.9.1</span>
+ </div>
+ <div id="projectbrief">User Documentation for MADlib</div>
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+<a href="#groups">Modules</a> </div>
+ <div class="headertitle">
+<div class="title">Linear Solvers<div class="ingroups"><a class="el" href="group__grp__utility__functions.html">Utility Functions</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
+<p>A collection of methods that implement solutions for systems of consistent linear equations. </p>
+<table class="memberdecls">
+<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="groups"></a>
+Modules</h2></td></tr>
+<tr class="memitem:group__grp__dense__linear__solver"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__dense__linear__solver.html">Dense Linear Systems</a></td></tr>
+<tr class="memdesc:group__grp__dense__linear__solver"><td class="mdescLeft"> </td><td class="mdescRight">Implements solution methods for large dense linear systems. Currently, restricted to problems that fit in memory. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:group__grp__sparse__linear__solver"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__sparse__linear__solver.html">Sparse Linear Systems</a></td></tr>
+<tr class="memdesc:group__grp__sparse__linear__solver"><td class="mdescLeft"> </td><td class="mdescRight">Implements solution methods for linear systems with sparse matrix input. Currently, restricted to problems that fit in memory. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
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http://git-wip-us.apache.org/repos/asf/incubator-madlib-site/blob/bed9253d/docs/v1.9.1/group__grp__linear__solver.js
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+ [ "Sparse Linear Systems", "group__grp__sparse__linear__solver.html", null ]
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