Contents

Latent Dirichlet Allocation (LDA) is a generative probabilistic model for natural texts. It is used in problems such as automated topic discovery, collaborative filtering, and document classification.

In addition to an implementation of LDA, this MADlib module also provides a number of additional helper functions to interpret results of the LDA output.

Note: Topic modeling is often used as part of a larger text processing pipeline, which may include operations such as term frequency, stemming and stop word removal. You can use the function Term Frequency to generate the required vocabulary format from raw documents for the LDA training function. See the examples later on this page for more details.

Background

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 \( \alpha \) on each document's topic mixture. In addition, there is another (symmetric) Dirichlet prior with parameter \( \beta \) on the distribution of words for each topic.

The following generative process then defines a distribution over a corpus of documents:

Sample for each topic \( i \), a per-topic word distribution \( \phi_i \) from the Dirichlet ( \(\beta\)) prior.
For each document:
- Sample a document length N from a suitable distribution, say, Poisson.
- Sample a topic mixture \( \theta \) for the document from the Dirichlet ( \(\alpha\)) distribution.
- For each of the N words:
  - Sample a topic \( z_n \) from the multinomial topic distribution \( \theta \).
  - Sample a word \( w_n \) from the multinomial word distribution \( \phi_{z_n} \) associated with topic \( z_n \).

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 referred to as 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.

This implementation provides a parallel and scalable in-database solution for LDA based on Gibbs sampling. It takes advantage of the shared-nothing MPP architecture and is a different implementation than one would find for MPI or map/reduce.

Training Function

The LDA training function has the following syntax:

lda_train( data_table,
           model_table,
           output_data_table,
           voc_size,
           topic_num,
           iter_num,
           alpha,
           beta,
           evaluate_every,
           perplexity_tol
         )

Arguments

data_table

TEXT. Name of the table storing the training dataset. Each row is in the form <docid, wordid, count> where docid, wordid, and count are non-negative integers. The docid column refers to the document ID, the wordid column is the word ID (the index of a word in the vocabulary), and count is the number of occurrences of the word in the document. Please note:

wordid must be contiguous integers going from from 0 to voc_size − 1.
column names for docid, wordid, and count are currently fixed, so you must use these exact names in the data_table.

The function Term Frequency can be used to generate vocabulary in the required format from raw documents.

model_table

TEXT. This is an output table generated by LDA which contains the learned model. It has one row with the following columns:

voc_size	INTEGER. Size of the vocabulary. As mentioned above for the input table, `wordid` consists of contiguous integers going from 0 to `voc_size` − `1`.
topic_num	INTEGER. Number of topics.
alpha	DOUBLE PRECISION. Dirichlet prior for the per-document topic multinomial.
beta	DOUBLE PRECISION. Dirichlet prior for the per-topic word multinomial.
model	BIGINT[]. The encoded model description (not human readable).
num_iterations	INTEGER. Number of iterations that training ran for, which may be less than the maximum value specified in the parameter 'iter_num' if the perplexity tolerance was reached.
perplexity	DOUBLE PRECISION[]. Array of perplexity values as per the 'evaluate_every' parameter. For example, if 'evaluate_every=5' this would be an array of perplexity values for every 5th iteration, plus the last iteration.
perplexity_iters	INTEGER[]. Array indicating the iterations for which perplexity is calculated, as derived from the parameters 'iter_num' and 'evaluate_every'. For example, if 'iter_num=5' and 'evaluate_every=2', then 'perplexity_iters' value would be {2,4,5} indicating that perplexity is computed at iterations 2, 4 and 5 (at the end), unless of course it terminated earlier due to 'perplexity_tol'. If 'iter_num=5' and 'evaluate_every=1', then 'perplexity_iters' value would be {1,2,3,4,5} indicating that perplexity is computed at every iteration, again assuming it ran the full number of iterations.

output_data_table

TEXT. The name of the table generated by LDA that stores the output data. It has the following columns:

docid	INTEGER. Document id from input 'data_table'.
wordcount	INTEGER. Count of number of words in the document, including repeats. For example, if a word appears 3 times in the document, it is counted 3 times.
words	INTEGER[]. Array of `wordid` in the document, not including repeats. For example, if a word appears 3 times in the document, it appears only once in the `words` array.
counts	INTEGER[]. Frequency of occurance of a word in the document, indexed the same as the `words` array above. For example, if the 2nd element of the `counts` array is 4, it means that the word in the 2nd element of the `words` array occurs 4 times in the document.
topic_count	INTEGER[]. Array of the count of words in the document that correspond to each topic. This array is of length `topic_num`. Topic ids are continuous integers going from 0 to `topic_num` − `1`.
topic_assignment	INTEGER[]. Array indicating which topic each word in the document corresponds to. This array is of length `wordcount`. Words that are repeated `n` times in the document will show up consecutively `n` times in this array.

voc_size

INTEGER. Size of the vocabulary. As mentioned above for the input 'data_table', wordid consists of continuous integers going from 0 to voc_size − 1.

topic_num

INTEGER. Desired number of topics.

iter_num

INTEGER. Maximum number of iterations. If a 'perplexity_tol' is set, LDA may train for less than the maximum number of iterations if the tolerance is reached.

alpha

DOUBLE PRECISION. Dirichlet prior for the per-document topic multinomial (e.g., 50/topic_num is a reasonable value to start with as per Griffiths and Steyvers [2]).

beta

DOUBLE PRECISION. Dirichlet prior for the per-topic word multinomial (e.g., 0.01 is a reasonable value to start with).

evaluate_every (optional)

INTEGER, default: 0. How often to evaluate perplexity. Set it to 0 or a negative number to not evaluate perplexity in training at all. Evaluating perplexity can help you check convergence during the training process, but it will also increase total training time. For example, evaluating perplexity in every iteration might increase training time up to two-fold.

perplexity_tol (optional)

DOUBLE PRECISION, default: 0.1. Perplexity tolerance to stop iteration. Only used when the parameter 'evaluate_every' is greater than 0.

Prediction Function

Prediction involves labelling test documents using a learned LDA model:

lda_predict( data_table,
             model_table,
             output_predict_table
           );

Arguments

data_table: TEXT. Name of the table storing the test dataset (new document to be labeled).
model_table: TEXT. The model table generated by the training process.
output_predict_table: TEXT. The prediction output table. Each row in the table stores the topic distribution and the topic assignments for a document in the dataset. This table has the exact same columns and interpretation as the 'output_data_table' from the training function above.

Perplexity

Perplexity describes how well the model fits the data by computing word likelihoods averaged over the documents. This function returns a single perplexity value.

lda_get_perplexity( model_table,
                    output_data_table
                  );

Arguments

model_table: TEXT. The model table generated by the training process.
output_data_table: TEXT. Output table generated by the training or predict functions, containing the topic assignments by word.

Helper Functions

The helper functions can help to interpret the output from LDA training and LDA prediction.

Topic description by top-k words with highest probability

Applies to LDA training only.

lda_get_topic_desc( model_table,
                    vocab_table,
                    output_table,
                    top_k
                  )

Arguments

model_table

TEXT. The model table generated by the training process.

vocab_table

TEXT. The vocabulary table in the form <wordid, word>. Reminder that this table can be created using the term_frequency function (Term Frequency) with the parameter compute_vocab set to TRUE.

output_table

TEXT. The output table with per-topic description generated by this helper function. It has the following columns:

topicid	INTEGER. Topic id.
wordid	INTEGER. Word id.
prob	DOUBLE PRECISION. Probability that this topic will generate the word.
word	TEXT. Word in text form.

top_k

TEXT. The desired number of top words to show for each topic.

Per-word topic counts

Applies to LDA training only.

lda_get_word_topic_count( model_table,
                          output_table
                        )

Arguments

model_table

TEXT. The model table generated by the training process.

output_table

TEXT. The output table with per-word topic counts generated by this helper function. It has the following columns:

wordid	INTEGER. Word id.
topic_count	INTEGER[]. Count of word association with each topic, i.e., shows how many times a given word is assigned to a topic. Array is of length number of topics.

Per-topic word counts

Applies to LDA training only.

lda_get_topic_word_count( model_table,
                          output_table
                        )

Arguments

model_table

TEXT. The model table generated by the training process.

output_table

TEXT. The output table with per-topic word counts generated by this helper function. It has the following columns:

topicid	INTEGER. Topic id.
word_count	INTEGER[]. Array showing which words are associated with the topic by frequency. Array is of length number of words.

Per-document word to topic mapping

Applies to both LDA training and LDA prediction.

lda_get_word_topic_mapping( output_data_table,  -- From training or prediction
                            output_table
                          )

Arguments

output_data_table

TEXT. The output data table generated by either LDA training or LDA prediction.

output_table

TEXT. The output table with word to topic mappings generated by this helper function. It has the following columns:

docid	INTEGER. Document id.
wordid	INTEGER. Word id.
topicid	INTEGER. Topic id.

Examples

Prepare a training dataset for LDA. The examples below are small strings extracted from various Wikipedia documents:

DROP TABLE IF EXISTS documents;
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.');

You can apply stemming, stop word removal and tokenization at this point in order to prepare the documents for text processing. Depending upon your database version, various tools are available. Databases based on more recent versions of PostgreSQL may do something like:

SELECT tsvector_to_array(to_tsvector('english',contents)) from documents;

                        tsvector_to_array
+-----------------------------------------------------------------------
 {analyz,bayesian,class,content,corpora,develop,document,larg,...}
 {1960s,1968,301,american,averag,balanc,bat,carl,favor,histori,...}
 {also,applic,close,comput,deliv,disciplin,domain,field,learn,...}
 {agricultur,area,california,center,central,coast,desert,divers,...}
(4 rows)

In this example, we assume a database based on an older version of PostgreSQL and just perform basic punctuation removal and tokenization. The array of words is added as a new column to the documents table:

ALTER TABLE documents ADD COLUMN words TEXT[];
UPDATE documents SET words =
    regexp_split_to_array(lower(
    regexp_replace(contents, E'[,.;\']','', 'g')
    ), E'[\\s+]');
\x on
SELECT * FROM documents ORDER BY docid;

-[ RECORD 1 ]---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
docid    | 0
contents | Statistical topic models are a class of Bayesian latent variable models, originally developed for analyzing the semantic content of large document corpora.
words    | {statistical,topic,models,are,a,class,of,bayesian,latent,variable,models,originally,developed,for,analyzing,the,semantic,content,of,large,document,corpora}
-[ RECORD 2 ]---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
docid    | 1
contents | 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.
words    | {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}
-[ RECORD 3 ]---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
docid    | 2
contents | 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.
words    | {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}
-[ RECORD 4 ]---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
docid    | 3
contents | 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.
words    | {californias,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}

Build a word count table by extracting the words and building a histogram for each document using the term_frequency function (Term Frequency).

DROP TABLE IF EXISTS documents_tf, documents_tf_vocabulary;
SELECT madlib.term_frequency('documents',    -- input table
                             'docid',        -- document id column
                             'words',        -- vector of words in document
                             'documents_tf', -- output documents table with term frequency
                             TRUE);          -- TRUE to created vocabulary table
\x off
SELECT * FROM documents_tf ORDER BY docid LIMIT 20;

 docid | wordid | count
-------+--------+-------
     0 |     71 |     1
     0 |     90 |     1
     0 |     56 |     1
     0 |     68 |     2
     0 |     85 |     1
     0 |     28 |     1
     0 |     35 |     1
     0 |     54 |     1
     0 |     64 |     2
     0 |      8 |     1
     0 |     29 |     1
     0 |     80 |     1
     0 |     24 |     1
     0 |     11 |     1
     0 |     17 |     1
     0 |     32 |     1
     0 |      3 |     1
     0 |     42 |     1
     0 |     97 |     1
     0 |     95 |     1
(20 rows)

Here is the associated vocabulary table. Note that wordid starts at 0:

SELECT * FROM documents_tf_vocabulary ORDER BY wordid LIMIT 20;

 wordid |     word
--------+--------------
      0 | 1960s
      1 | 1968
      2 | 301
      3 | a
      4 | agricultural
      5 | also
      6 | american
      7 | an
      8 | analyzing
      9 | and
     10 | application
     11 | are
     12 | area
     13 | areas
     14 | average
     15 | balance
     16 | batting
     17 | bayesian
     18 | between
     19 | by
(20 rows)

The total number of words in the vocabulary across all documents is:

SELECT COUNT(*) FROM documents_tf_vocabulary;

 count
+------
   103
(1 row)

Train LDA model. For Dirichlet priors we use initial rule-of-thumb values of 50/(number of topics) for alpha and 0.01 for beta. Reminder that column names for docid, wordid, and count are currently fixed, so you must use these exact names in the input table. After a successful run of the LDA training function two tables are generated, one for storing the learned model and the other for storing the output data table.

DROP TABLE IF EXISTS lda_model, lda_output_data;
SELECT madlib.lda_train( 'documents_tf',     -- documents table in the form of term frequency
                         'lda_model',        -- model table created by LDA training (not human readable)
                         'lda_output_data',  -- readable output data table
                         103,                -- vocabulary size
                         5,                  -- number of topics
                         10,                 -- number of iterations
                         5,                  -- Dirichlet prior for the per-doc topic multinomial (alpha)
                         0.01                -- Dirichlet prior for the per-topic word multinomial (beta)
                       );
\x on
SELECT * FROM lda_output_data ORDER BY docid;

-[ RECORD 1 ]----+------------------------------------------------------------------------------------------------------
docid            | 0
wordcount        | 22
words            | {24,17,11,95,90,85,68,54,42,35,28,8,3,97,80,71,64,56,32,29}
counts           | {1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,2,1,1,1}
topic_count      | {4,2,4,3,9}
topic_assignment | {4,2,4,1,2,1,2,2,0,3,4,4,3,0,0,4,0,4,4,4,3,4}
-[ RECORD 2 ]----+------------------------------------------------------------------------------------------------------
docid            | 1
wordcount        | 37
words            | {1,50,49,46,19,16,14,9,7,0,90,68,57,102,101,100,93,88,75,74,59,55,53,48,39,21,18,15,6,2}
counts           | {1,3,1,1,1,1,1,1,1,1,5,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
topic_count      | {2,5,14,9,7}
topic_assignment | {0,3,3,3,1,4,2,2,2,1,3,1,2,2,2,2,2,2,2,1,4,3,2,0,4,2,4,2,3,4,3,1,3,4,3,2,4}
-[ RECORD 3 ]----+------------------------------------------------------------------------------------------------------
docid            | 2
wordcount        | 36
words            | {10,27,33,40,47,51,58,62,63,69,72,83,100,99,94,92,91,90,89,87,86,79,76,70,60,52,50,36,30,25,9,5,3}
counts           | {1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1}
topic_count      | {15,10,1,7,3}
topic_assignment | {0,3,1,3,0,0,3,3,1,0,1,0,0,0,0,1,1,0,4,2,0,4,1,0,1,0,0,4,3,3,3,0,1,1,1,0}
-[ RECORD 4 ]----+------------------------------------------------------------------------------------------------------
docid            | 3
wordcount        | 49
words            | {77,78,81,82,67,65,51,45,44,43,34,26,13,98,96,94,90,84,73,68,66,61,50,41,38,37,31,23,22,20,19,12,4,3}
counts           | {1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,2,11,1,1,2,1,1,3,1,1,1,1,1,1,1,1,1,1,1}
topic_count      | {5,5,26,5,8}
topic_assignment | {4,4,4,0,2,0,0,2,4,4,2,2,2,1,2,4,1,0,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,2,4,3,3,3,2,3,2,3,2,1,4,2,2,1,0}

Review summary table:

SELECT voc_size, topic_num, alpha, beta, num_iterations, perplexity, perplexity_iters from lda_model;

-[ RECORD 1 ]----+-----
voc_size         | 103
topic_num        | 5
alpha            | 5
beta             | 0.01
num_iterations   | 10
perplexity       |
perplexity_iters |

Review learned model using helper functions. First, we get topic description by top-k words. These are the k words with the highest probability for the topic. Note that if there are ties in probability, more than k words may actually be reported for each topic. Also note that topicid starts at 0:

DROP TABLE IF EXISTS helper_output_table;
SELECT madlib.lda_get_topic_desc( 'lda_model',                -- LDA model generated in training
                                  'documents_tf_vocabulary',  -- vocabulary table that maps wordid to word
                                  'helper_output_table',      -- output table for per-topic descriptions
                                  5);                         -- k: number of top words for each topic
\x off
SELECT * FROM helper_output_table ORDER BY topicid, prob DESC LIMIT 40;

 topicid | wordid |        prob        |       word
---------+--------+--------------------+-------------------
       0 |      3 |  0.111357750647429 | a
       0 |     51 |  0.074361820199778 | is
       0 |     94 |  0.074361820199778 | to
       0 |     70 | 0.0373658897521273 | optimization
       0 |     82 | 0.0373658897521273 | southeast
       0 |     60 | 0.0373658897521273 | machine
       0 |     71 | 0.0373658897521273 | originally
       0 |     69 | 0.0373658897521273 | often
       0 |     99 | 0.0373658897521273 | which
       0 |     83 | 0.0373658897521273 | specializes
       0 |      1 | 0.0373658897521273 | 1968
       0 |     97 | 0.0373658897521273 | variable
       0 |     25 | 0.0373658897521273 | closely
       0 |     93 | 0.0373658897521273 | title
       0 |     47 | 0.0373658897521273 | has
       0 |     65 | 0.0373658897521273 | mojave
       0 |     79 | 0.0373658897521273 | related
       0 |     89 | 0.0373658897521273 | that
       0 |     10 | 0.0373658897521273 | application
       0 |    100 | 0.0373658897521273 | with
       0 |     92 | 0.0373658897521273 | ties
       0 |     54 | 0.0373658897521273 | large
       1 |     94 |  0.130699088145897 | to
       1 |      9 |  0.130699088145897 | and
       1 |      5 | 0.0438558402084238 | also
       1 |     57 | 0.0438558402084238 | league
       1 |     49 | 0.0438558402084238 | hitting
       1 |     13 | 0.0438558402084238 | areas
       1 |     39 | 0.0438558402084238 | favor
       1 |     85 | 0.0438558402084238 | statistical
       1 |     95 | 0.0438558402084238 | topic
       1 |      0 | 0.0438558402084238 | 1960s
       1 |     76 | 0.0438558402084238 | prediction-making
       1 |     86 | 0.0438558402084238 | statistics
       1 |     84 | 0.0438558402084238 | state
       1 |     72 | 0.0438558402084238 | overlaps
       1 |     22 | 0.0438558402084238 | center
       1 |      4 | 0.0438558402084238 | agricultural
       1 |     63 | 0.0438558402084238 | methods
       1 |     33 | 0.0438558402084238 | discipline
(40 rows)

Get the per-word topic counts. This mapping shows how many times a given word is assigned to a topic. E.g., wordid 3 is assigned to topicid 0 three times:

DROP TABLE IF EXISTS helper_output_table;
SELECT madlib.lda_get_word_topic_count( 'lda_model',            -- LDA model generated in training
                                        'helper_output_table'); -- output table for per-word topic counts
SELECT * FROM helper_output_table ORDER BY wordid LIMIT 20;

 wordid | topic_count
--------+-------------
      0 | {0,1,0,0,0}
      1 | {1,0,0,0,0}
      2 | {1,0,0,0,0}
      3 | {3,0,0,0,0}
      4 | {0,0,0,0,1}
      5 | {0,1,0,0,0}
      6 | {1,0,0,0,0}
      7 | {0,0,0,1,0}
      8 | {0,1,0,0,0}
      9 | {0,0,0,3,0}
     10 | {1,0,0,0,0}
     11 | {1,0,0,0,0}
     12 | {0,0,1,0,0}
     13 | {0,0,0,0,1}
     14 | {0,1,0,0,0}
     15 | {0,0,0,0,1}
     16 | {0,1,0,0,0}
     17 | {0,0,1,0,0}
     18 | {1,0,0,0,0}
     19 | {2,0,0,0,0}
(20 rows)

Get the per-topic word counts. This mapping shows which words are associated with each topic by frequency:

DROP TABLE IF EXISTS topic_word_count;
SELECT madlib.lda_get_topic_word_count( 'lda_model',
                                        'topic_word_count');
\x on
SELECT * FROM topic_word_count ORDER BY topicid;

-[ RECORD 1 ]----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
topicid    | 1
word_count | {1,1,0,0,0,0,0,1,1,0,1,0,0,0,0,1,0,0,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,1,0,1,0,0,1,1,0,0,0,0,0,0,0,1,0}
-[ RECORD 2 ]----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
topicid    | 2
word_count | {0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,0,1,0,1,1,2,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,1,0,0,0,0,4,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,0,1,0,0,1,0,0,0}
-[ RECORD 3 ]----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
topicid    | 3
word_count | {0,0,0,0,0,0,0,0,0,3,0,1,0,1,1,0,0,0,0,2,0,0,0,0,1,0,0,1,0,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,2,1,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0}
-[ RECORD 4 ]----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
topicid    | 4
word_count | {0,0,1,0,0,1,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,1,0,0,1,0,0,1,0,0,0,1,0,0,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0,7,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,1,0,0,0,0,1,0,0,0,0,1,1,1,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1}
-[ RECORD 5 ]----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
topicid    | 5
word_count | {0,0,0,3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,1,0,2,0,0}

Get the per-document word to topic mapping:

DROP TABLE IF EXISTS helper_output_table;
SELECT madlib.lda_get_word_topic_mapping('lda_output_data',  -- Output table from training
                                         'helper_output_table');
\x off
SELECT * FROM helper_output_table ORDER BY docid LIMIT 40;

 docid | wordid | topicid
-------+--------+---------
     0 |     56 |       1
     0 |     54 |       1
     0 |     42 |       2
     0 |     35 |       1
     0 |     32 |       1
     0 |     29 |       3
     0 |     28 |       4
     0 |     24 |       3
     0 |     17 |       2
     0 |     11 |       0
     0 |      8 |       1
     0 |      3 |       0
     0 |     97 |       0
     0 |     95 |       3
     0 |     90 |       0
     0 |     85 |       0
     0 |     80 |       2
     0 |     71 |       2
     0 |     68 |       0
     0 |     64 |       1
     1 |      2 |       0
     1 |      1 |       0
     1 |      0 |       1
     1 |    102 |       4
     1 |    101 |       2
     1 |    100 |       1
     1 |     93 |       3
     1 |     90 |       2
     1 |     90 |       0
     1 |     88 |       1
     1 |     75 |       1
     1 |     74 |       3
     1 |     68 |       0
     1 |     59 |       2
     1 |     57 |       4
     1 |     55 |       3
     1 |     53 |       3
     1 |     50 |       0
     1 |     49 |       1
     1 |     48 |       0
(40 rows)

Use a learned LDA model for prediction (that is, to label new documents). In this example, we use the same input table as we used to train, just for demonstration purpose. Normally, the test document is a new one that we want to predict on.

DROP TABLE IF EXISTS outdata_predict;
SELECT madlib.lda_predict( 'documents_tf',          -- Document to predict
                           'lda_model',             -- LDA model from training
                           'outdata_predict'        -- Output table for predict results
                         );
\x on
SELECT * FROM outdata_predict;

-[ RECORD 1 ]----+------------------------------------------------------------------------------------------------------
docid            | 0
wordcount        | 22
words            | {17,11,28,29,95,3,32,97,85,35,54,80,64,90,8,24,42,71,56,68}
counts           | {1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,2}
topic_count      | {1,3,16,1,1}
topic_assignment | {2,2,1,0,2,2,2,3,2,2,2,2,2,2,4,2,2,2,2,2,1,1}
-[ RECORD 2 ]----+------------------------------------------------------------------------------------------------------
docid            | 1
wordcount        | 37
words            | {90,101,2,88,6,7,75,46,74,68,39,9,48,49,102,50,59,53,55,57,100,14,15,16,18,19,93,21,0,1}
counts           | {5,1,1,1,1,1,1,1,1,2,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
topic_count      | {0,1,11,6,19}
topic_assignment | {4,4,4,4,4,4,4,4,4,2,4,2,2,1,3,2,2,4,4,4,3,3,3,4,3,3,2,4,4,2,2,4,2,4,2,4,2}
-[ RECORD 3 ]----+------------------------------------------------------------------------------------------------------
docid            | 2
wordcount        | 36
words            | {90,3,5,9,10,25,27,30,33,36,40,47,50,51,52,58,60,62,63,69,70,72,76,79,83,86,87,89,91,92,94,99,100}
counts           | {1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1}
topic_count      | {26,3,5,1,1}
topic_assignment | {4,0,0,2,2,0,0,0,0,2,0,0,0,3,0,0,0,0,0,0,0,0,0,2,0,2,0,0,0,0,0,1,1,1,0,0}
-[ RECORD 4 ]----+------------------------------------------------------------------------------------------------------
docid            | 3
wordcount        | 49
words            | {41,38,3,77,78,94,37,81,82,19,84,34,96,13,31,98,90,51,26,61,23,22,50,65,66,67,45,44,68,4,12,43,20,73}
counts           | {1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,11,1,1,1,1,1,3,1,1,1,1,2,2,1,1,1,1,1}
topic_count      | {0,28,0,4,17}
topic_assignment | {1,1,4,1,1,1,1,1,1,4,1,1,1,3,1,1,1,4,4,4,4,4,4,4,4,4,4,4,4,1,1,1,4,3,3,3,1,1,4,4,1,1,1,1,1,1,1,1,1}

The test table is expected to be in the same form as the training table and can be created with the same process. The LDA prediction results have the same format as the output table generated by the LDA training function.

Review prediction using helper function. (This is the same per-document word to topic mapping that we used on the learned model.)

DROP TABLE IF EXISTS helper_output_table;
SELECT madlib.lda_get_word_topic_mapping('outdata_predict',  -- Output table from prediction
                                         'helper_output_table');
\x off
SELECT * FROM helper_output_table ORDER BY docid LIMIT 40;

 docid | wordid | topicid
-------+--------+---------
     0 |     54 |       4
     0 |     42 |       1
     0 |     35 |       4
     0 |     32 |       4
     0 |     29 |       4
     0 |     28 |       1
     0 |     24 |       4
     0 |     17 |       1
     0 |     11 |       4
     0 |      8 |       4
     0 |      3 |       0
     0 |     97 |       4
     0 |     95 |       1
     0 |     90 |       2
     0 |     85 |       4
     0 |     80 |       0
     0 |     71 |       0
     0 |     68 |       0
     0 |     64 |       4
     0 |     64 |       1
     0 |     56 |       4
     1 |      2 |       4
     1 |      1 |       4
     1 |      0 |       2
     1 |    102 |       4
     1 |    101 |       4
     1 |    100 |       4
     1 |     93 |       4
     1 |     90 |       2
     1 |     90 |       0
     1 |     88 |       2
     1 |     75 |       2
     1 |     74 |       0
     1 |     68 |       0
     1 |     59 |       4
     1 |     57 |       2
     1 |     55 |       2
     1 |     53 |       1
     1 |     50 |       0
     1 |     49 |       2
(40 rows)

Call the perplexity function to see how well the model fits the data. Perplexity computes word likelihoods averaged over the test documents.

SELECT madlib.lda_get_perplexity( 'lda_model',        -- LDA model from training
                                  'outdata_predict'   -- Prediction output
                                );

 lda_get_perplexity
+--------------------
    79.481894411824
(1 row)

Perplexity by iteration. Now let's look at how perplexity changes from one iteration to the next:

DROP TABLE IF EXISTS lda_model_perp, lda_output_data_perp;
SELECT madlib.lda_train( 'documents_tf',          -- documents table in the form of term frequency
                         'lda_model_perp',        -- model table created by LDA training (not human readable)
                         'lda_output_data_perp',  -- readable output data table
                         103,                     -- vocabulary size
                         5,                       -- number of topics
                         30,                      -- number of iterations
                         5,                       -- Dirichlet prior for the per-doc topic multinomial (alpha)
                         0.01,                    -- Dirichlet prior for the per-topic word multinomial (beta)
                         2,                       -- Evaluate perplexity every n iterations
                         0.3                      -- Tolerance to stop iteration
                       );
\x on
SELECT voc_size, topic_num, alpha, beta, num_iterations, perplexity, perplexity_iters from lda_model_perp;

-[ RECORD 1 ]----+----------------------------------------------------------------------------------------------------
voc_size         | 103
topic_num        | 5
alpha            | 5
beta             | 0.01
num_iterations   | 14
perplexity       | {70.0297335165,65.6497887327,70.2040806534,68.2594871716,70.3816093812,67.9193935299,67.6325562682}
perplexity_iters | {2,4,6,8,10,12,14}

Iterating stops at 14 since the tolerance is reached. There are 7 perplexity values because we computed it only every 2nd iteration to save time. As expected, the perplexity on the training data is that same as the final iteration value:

\x off
SELECT madlib.lda_get_perplexity( 'lda_model_perp',
                                  'lda_output_data_perp'
                                );

 lda_get_perplexity
--------------------+
    67.632556268157

Literature

[1] D.M. Blei, A.Y. Ng, M.I. Jordan, Latent Dirichlet Allocation, Journal of Machine Learning Research, vol. 3, pp. 993-1022, 2003.

[2] T. Griffiths and M. Steyvers, Finding scientific topics, PNAS, vol. 101, pp. 5228-5235, 2004.

[3] Y. Wang, H. Bai, M. Stanton, W-Y. Chen, and E.Y. Chang, lda: Parallel Dirichlet Allocation for Large-scale Applications, AAIM, 2009.

[4] http://en.wikipedia.org/wiki/Latent_Dirichlet_allocation

[5] J. Chang, Collapsed Gibbs sampling methods for topic models, R manual, 2010.

Related Topics: File lda.sql_in documenting the SQL functions.