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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> &raquo; <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>&lt;docid, wordid, count&gt;</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> &minus; <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> &minus; <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&mdash;labelling test documents using a learned LDA model&mdash;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–Douglas 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>
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