| /* ----------------------------------------------------------------------- *//** |
| * |
| * @file crf.sql_in |
| * |
| * @brief SQL functions for conditional random field |
| * @date July 2012 |
| * |
| * @sa For a brief introduction to conditional random field, see the |
| * module description \ref grp_crf. |
| * |
| *//* ----------------------------------------------------------------------- */ |
| |
| m4_include(`SQLCommon.m4') |
| |
| /** |
| @addtogroup grp_crf |
| |
| <div class="toc"><b>Contents</b> |
| <ul> |
| <li><a href="#train_feature">Training Feature Generation</a></li> |
| <li><a href="#train">CRF Training Function</a></li> |
| <li><a href="#test_feature">Testing Feature Generation</a></li> |
| <li><a href="#inference">Inference using Viterbi</a></li> |
| <li><a href="#usage">Using CRF</a></li> |
| <li><a href="#examples">Examples</a></li> |
| <li><a href="#background">Technical Background</a></li> |
| <li><a href="#literature">Literature</a></li> |
| <li><a href="#related">Related Topics</a></li> |
| </ul> |
| </div> |
| |
| @brief Constructs a Conditional Random Fields (CRF) model for labeling sequential data. |
| |
| A conditional random field (CRF) is a type of discriminative, undirected |
| probabilistic graphical model. A linear-chain CRF is a special type of CRF |
| that assumes the current state depends only on the previous state. |
| |
| Feature extraction modules are provided for text-analysis tasks such as part-of-speech |
| (POS) tagging and named-entity resolution (NER). Currently, six |
| feature types are implemented: |
| |
| - Edge Feature: transition feature that encodes the transition feature |
| weight from current label to next label. |
| - Start Feature: fired when the current token is the first token in a sequence. |
| - End Feature: fired when the current token is the last token in a sequence. |
| - Word Feature: fired when the current token is observed in the trained |
| dictionary. |
| - Unknown Feature: fired when the current token is not observed in the trained |
| dictionary for at least a certain number of times (default 1). |
| - Regex Feature: fired when the current token can be matched by a regular |
| expression. |
| |
| A Viterbi implementation is also provided |
| to get the best label sequence and the conditional probability |
| \f$ \Pr( \text{best label sequence} \mid \text{sequence}) \f$. |
| |
| Following steps are required for CRF Learning and Inference: |
| -# <li><a href="#train_feature">Training Feature Generation</a></li> |
| -# <li><a href="#train">CRF Training</a></li> |
| -# <li><a href="#test_feature">Testing Feature Generation</a></li> |
| -# <li><a href="#inference">Inference using Viterbi</a></li> |
| |
| @anchor train_feature |
| @par Training Feature Generation |
| The function takes \c train_segment_tbl and \c regex_tbl as input and does feature generation generating three tables |
| \c dictionary_tbl, \c train_feature_tbl and \c train_featureset_tbl, that are required as an input for CRF training. |
| <pre class="syntax"> |
| crf_train_fgen(train_segment_tbl, |
| regex_tbl, |
| label_tbl, |
| dictionary_tbl, |
| train_feature_tbl, |
| train_featureset_tbl) |
| </pre> |
| \b Arguments |
| <dl class="arglist"> |
| <dt>train_segment_tbl</dt> |
| <dd>TEXT. Name of the training segment table. The table is expected to have the following columns: |
| <table class="output"> |
| <tr> |
| <th>doc_id</th> |
| <td>INTEGER. Document id column</td> |
| </tr> |
| <tr> |
| <th>start_pos</th> |
| <td>INTEGER. Index of a particular term in the respective document</td> |
| </tr> |
| <tr> |
| <th>seg_text</th> |
| <td>TEXT. Term at the respective \c start_pos in the document</td> |
| </tr> |
| <tr> |
| <th>label</th> |
| <td>INTEGER. Label id for the term corresponding to the actual label from \c label_tbl</td> |
| </tr> |
| </table> |
| </dd> |
| <dt>regex_tbl</dt> |
| <dd>TEXT. Name of the regular expression table. The table is expected to have the following columns: |
| <table class="output"> |
| <tr> |
| <th>pattern</th> |
| <td>TEXT. Regular Expression</td> |
| </tr> |
| <tr> |
| <th>name</th> |
| <td>TEXT. Regular Expression name</td> |
| </tr> |
| </table> |
| </dd> |
| <dt>label_tbl</dt> |
| <dd>TEXT. Name of the table containing unique labels and their id's. The table is expected to have the following columns: |
| <table class="output"> |
| <tr> |
| <th>id</th> |
| <td>INTEGER. Unique label id. NOTE: Must range from 0 to total number of labels in the table - 1. |
| </td> |
| </tr> |
| <tr> |
| <th>label</th> |
| <td>TEXT. Label name</td> |
| </tr> |
| </table> |
| </dd> |
| <dt>dictionary_tbl</dt> |
| <dd>TEXT. Name of the dictionary table to be created containing unique terms along with their counts. The table will have the following columns: |
| <table class="output"> |
| <tr> |
| <th>token</th> |
| <td>TEXT. Contains all the unique terms found in \c train_segment_tbl</td> |
| </tr> |
| <tr> |
| <th>total</th> |
| <td>INTEGER. Respective counts for the terms</td> |
| </tr> |
| </table> |
| <dd> |
| <dt>train_feature_tbl<dt> |
| <dd>TEXT. Name of the training feature table to be created. The table will have the following columns: |
| <table class="output"> |
| <tr> |
| <th>doc_id</th> |
| <td>INTEGER. Document id</td> |
| </tr> |
| <tr> |
| <th>f_size</th> |
| <td>INTEGER. Feature set size. This value will be same for all the tuples in the table</td> |
| </tr> |
| <tr> |
| <th>sparse_r</th> |
| <td>DOUBLE PRECISION[]. Array union of individual single state features (previous label, label, feature index, start position, training existance indicator), ordered by their start position.</td> |
| </tr> |
| <tr> |
| <th>dense_m</th> |
| <td>DOUBLE PRECISION[]. Array union of (previous label, label, feature index, start position, training existance indicator) of edge features ordered by start position.</td> |
| </tr> |
| <tr> |
| <th>sparse_m</th> |
| <td>DOUBLE PRECISION[]. Array union of (feature index, previous label, label) of edge features ordered by feature index.</td> |
| </tr> |
| </table> |
| </dd> |
| |
| <dt>train_featureset_tbl</dt> |
| <dd>TEXT. Name of the table to be created containing distinct featuresets generated from training feature extraction. The table will have the following columns: |
| <table class="output"> |
| <tr> |
| <th>f_index</th> |
| <td>INTEGER. Column containing distinct featureset ids</td> |
| </tr> |
| <tr> |
| <th>f_name</th> |
| <td>TEXT. Feature name </td> |
| </tr> |
| <tr> |
| <th>feature</th> |
| <td>ARRAY. Feature value. The value is of the form [L1, L2] |
| \n - If L1 = -1: represents single state feature with L2 being the current label id. |
| \n - If L1 != -1: represents transition feature with L1 be the previous label and L2 be the current label. |
| </td> |
| </tr> |
| </dd> |
| </dl> |
| |
| @anchor train |
| @par Linear Chain CRF Training Function |
| The function takes \c train_feature_tbl and \c train_featureset_tbl tables generated in the training feature generation steps as input |
| along with other required parameters and produces two output tables \c crf_stats_tbl and \c crf_weights_tbl. |
| |
| <pre class="syntax"> |
| lincrf_train(train_feature_tbl, |
| train_featureset_tbl, |
| label_tbl, |
| crf_stats_tbl, |
| crf_weights_tbl |
| max_iterations |
| ) |
| </pre> |
| \b Arguments |
| <dl class="arglist"> |
| <dt>train_feature_tbl</dt> |
| <dd>TEXT. Name of the feature table generated during training feature generation</dd> |
| |
| <dt>train_featureset_tbl</dt> |
| <dd>TEXT. Name of the featureset table generated during training feature generation</dd> |
| |
| <dt>label_tbl</dt> |
| <dd>TEXT. Name of the label table used</dd> |
| |
| <dt>crf_stats_table</dt> |
| <dd>TEXT. Name of the table to be created containing statistics for CRF training. The table has the following columns: |
| <table class="output"> |
| <tr> |
| <th>coef</th> |
| <td>DOUBLE PRECISION[]. Array of coefficients</td> |
| </tr> |
| <tr> |
| <th>log_likelihood</th> |
| <td>DOUBLE. Log-likelihood</td> |
| </tr> |
| <tr> |
| <th>num_iterations</th> |
| <td>INTEGER. The number of iterations at which the algorithm terminated</td> |
| </tr> |
| </table> |
| </dd> |
| <dt>crf_weights_table</dt> |
| <dd>TEXT. Name of the table to be created creating learned feature weights. The table has the following columns: |
| <table class="output"> |
| <tr> |
| <th>id</th> |
| <td>INTEGER. Feature set id</td> |
| </tr> |
| <tr> |
| <th>name</th> |
| <td>TEXT. Feature name</td> |
| </tr> |
| <tr> |
| <th>prev_label_id</th> |
| <td>INTEGER. Label for the previous token encountered</td> |
| </tr> |
| <tr> |
| <th>label_id</th> |
| <td>INTEGER. Label of the token with the respective feature</td> |
| </tr> |
| <tr> |
| <th>weight</th> |
| <td>DOUBLE PRECISION. Weight for the respective feature set</td> |
| </tr> |
| </table> |
| </dd> |
| |
| <dt>max_iterations</dt> |
| <dd>INTEGER. The maximum number of iterations</dd> |
| </dl> |
| |
| @anchor test_feature |
| @par Testing Feature Generation |
| |
| <pre class="syntax"> |
| crf_test_fgen(test_segment_tbl, |
| dictionary_tbl, |
| label_tbl, |
| regex_tbl, |
| crf_weights_tbl, |
| viterbi_mtbl, |
| viterbi_rtbl |
| ) |
| </pre> |
| \b Arguments |
| <dl class="arglist"> |
| <dt>test_segment_tbl</dt> |
| <dd>TEXT. Name of the testing segment table. The table is expected to have the following columns: |
| <table class="output"> |
| <tr> |
| <th>doc_id</th> |
| <td>INTEGER. Document id column</td> |
| </tr> |
| <tr> |
| <th>start_pos</th> |
| <td>INTEGER. Index of a particular term in the respective document</td> |
| </tr> |
| <tr> |
| <th>seg_text</th> |
| <td>TEXT. Term at the respective \c start_pos in the document</td> |
| </tr> |
| </table> |
| </dd> |
| |
| <dt>dictionary_tbl</dt> |
| <dd>TEXT. Name of the dictionary table created during training feature generation (\c crf_train_fgen)</dd> |
| |
| <dt>label_tbl</dt> |
| <dd>TEXT. Name of the label table</dd> |
| |
| <dt>regex_tbl</dt> |
| <dd>TEXT. Name of the regular expression table</dd> |
| |
| <dt>crf_weights_tbl</dt> |
| <dd>TEXT. Name of the weights table generated during CRF training (\c lincrf_train) |
| |
| <dt>viterbi_mtbl</dt> |
| <dd>TEXT. Name of the Viterbi M table to be created</dd> |
| |
| <dt>viterbi_rtbl</dt> |
| <dd>TEXT. Name of the Viterbi R table to be created</dd> |
| </dl> |
| |
| @anchor inference |
| @par Inference using Viterbi |
| <pre class="syntax"> |
| vcrf_label(test_segment_tbl, |
| viterbi_mtbl, |
| viterbi_rtbl, |
| label_tbl, |
| result_tbl) |
| </pre> |
| \b Arguments |
| <dl class="arglist"> |
| <dt>test_segment_tbl</dt> |
| <dd>TEXT. Name of the testing segment table. For required table schema, please refer to arguments in previous section</dd> |
| <dt>viterbi_mtbl</dt> |
| <dd>TEXT. Name of the table \c viterbi_mtbl generated from testing feature generation \c crf_test_fgen. |
| <dt>viterbi_rtbl</dt> |
| <dd>TEXT. Name of the table \c viterbi_rtbl generated from testing feature generation \c crf_test_fgen. |
| <dt>label_tbl</dt> |
| <dd>TEXT. Name of the label table.</dd> |
| <dt>result_tbl</dt> |
| <dd>TEXT. Name of the result table to be created containing extracted best label sequences.</dd> |
| </dl> |
| |
| @anchor usage |
| @par Using CRF |
| |
| Generate text features, calculate their weights, and output the best label sequence for test data:\n |
| |
| -# Perform feature generation on training data i.e. \c train_segment_tbl generating \c train_feature_tbl and \c train_featureset_tbl. |
| <pre>SELECT madlib.crf_train_fgen( |
| '<em>train_segment_tbl</em>', |
| '<em>regex_tbl</em>', |
| '<em>label_tbl</em>', |
| '<em>dictionary_tbl</em>', |
| '<em>train_feature_tbl</em>', |
| '<em>train_featureset_tbl</em>');</pre> |
| -# Use linear-chain CRF for training providing \c train_feature_tbl and \c train_featureset_tbl generated from previous step as an input. |
| <pre>SELECT madlib.lincrf_train( |
| '<em>train_feature_tbl</em>', |
| '<em>train_featureset_tbl</em>', |
| '<em>label_tbl</em>', |
| '<em>crf_stats_tbl</em>', |
| '<em>crf_weights_tbl</em>', |
| <em>max_iterations</em>);</pre> |
| -# Perform feature generation on testing data \c test_segment_tbl generating \c viterbi_mtbl and \c viterbi_rtbl required for inferencing. |
| <pre>SELECT madlib.crf_test_fgen( |
| '<em>test_segment_tbl</em>', |
| '<em>dictionary_tbl</em>', |
| '<em>label_tbl</em>', |
| '<em>regex_tbl</em>', |
| '<em>crf_weights_tbl</em>', |
| '<em>viterbi_mtbl</em>', |
| '<em>viterbi_rtbl</em>');</pre> |
| -# Run the Viterbi function to get the best label sequence and the conditional |
| probability \f$ \Pr( \text{best label sequence} \mid \text{sequence}) \f$. |
| <pre>SELECT madlib.vcrf_label( |
| '<em>test_segment_tbl</em>', |
| '<em>viterbi_mtbl</em>', |
| '<em>viterbi_rtbl</em>', |
| '<em>label_tbl</em>', |
| '<em>result_tbl</em>');</pre> |
| |
| @anchor examples |
| @examp |
| This example uses a trivial training and test data set. |
| |
| -# Load the label table, the regular expressions table, and the training segment table: |
| <pre class="example"> |
| CREATE TABLE crf_label (id integer,label character varying); |
| INSERT INTO crf_label VALUES |
| (0,'CC'), (1,'CD'), (2,'DT'), (3,'EX'), (4,'FW'), (5,'IN'), (6,'JJ'), (7,'JJR'), (8,'JJS'), |
| (9,'LS'), (10,'MD'), (11,'NN'), (12,'NNS'), (13,'NNP'),(14,'NNPS'),(15,'PDT'),(16,'POS'),(17,'PRP'), |
| (18,'PRP$'),(19,'RB'), (20,'RBR'), (21,'RBS'), (22,'RP'), (23,'SYM'), (24,'TO'), (25,'UH'), (26,'VB'), |
| (27,'VBD'), (28,'VBG'),(29,'VBN'), (30,'VBP'), (31,'VBZ'),(32,'WDT'), (33,'WP'), (34,'WP$'),(35,'WRB'), |
| (36,'$'), (37,'#'), (38,''''''), (39,'``'), (40,'('), (41,')'), (42,','), (43,'.'), (44,':'); |
| CREATE TABLE crf_regex (pattern text,name text); |
| INSERT INTO crf_regex VALUES |
| ('^[A-Z][a-z]+$','InitCapital'), ('^[A-Z]+$','isAllCapital'), ('^.*[0-9]+.*$','containsDigit'), |
| ('^.+[.]$','endsWithDot'), ('^.+[,]$','endsWithComma'), ('^.+er$','endsWithER'), |
| ('^.+est$','endsWithEst'), ('^.+ed$','endsWithED'), ('^.+s$','endsWithS'), |
| ('^.+ing$','endsWithIng'), ('^.+ly$','endsWithly'), ('^.+-.+$','isDashSeparatedWords'), |
| ('^.*@.*$','isEmailId'); |
| CREATE TABLE train_segmenttbl(start_pos integer,doc_id integer,seg_text text,label integer,max_pos integer); |
| INSERT INTO train_segmenttbl VALUES |
| (0,1,'confidence',11,36), (1,1,'in',5,36), (2,1,'the',2,36), (3,1,'pound',11,36), |
| (4,1,'is',31,36), (5,1,'widely',19,36), (6,1,'expected',29,36), (7,1,'to',24,36), |
| (8,1,'take',26,36), (9,1,'another',2,36), (10,1,'sharp',6,36), (11,1,'dive',11,36), |
| (12,1,'if',5,36), (13,1,'trade',11,36), (14,1,'figures',12,36), (15,1,'for',5,36), |
| (16,1,'september',13,36), (17,1,',',42,36), (18,1,'due',6,36), (19,1,'for',5,36), |
| (20,1,'release',11,36), (21,1,'tomorrow',11,36), (22,1,',',42,36), (23,1,'fail',26,36), |
| (24,1,'to',24,36), (25,1,'show',26,36), (26,1,'a',2,36), (27,1,'substantial',6,36), |
| (28,1,'improvement',11,36),(29,1,'from',5,36), (30,1,'july',13,36), (31,1,'and',0,36), |
| (32,1,'august',13,36), (33,1,'''s',16,36), (34,1,'near-record',6,36),(35,1,'deficits',12,36), |
| (36,1,'.',43,36), (0,2,'chancellor',13,26),(1,2,'of',5,26), (2,2,'the',2,26), |
| (3,2,'exchequer',13,26), (4,2,'nigel',13,26), (5,2,'lawson',13,26), (6,2,'''s',16,26), |
| (7,2,'restated',29,26), (8,2,'commitment',11,26),(9,2,'to',24,26), (10,2,'a',2,26), |
| (11,2,'firm',11,26), (12,2,'monetary',6,26), (13,2,'policy',11,26), (14,2,'has',31,26), |
| (15,2,'helped',29,26), (16,2,'to',24,26), (17,2,'prevent',26,26), (18,2,'a',2,26), |
| (19,2,'freefall',11,26), (20,2,'in',5,26), (21,2,'sterling',11,26), (22,2,'over',5,26), |
| (23,2,'the',2,26), (24,2,'past',6,26), (25,2,'week',11,26), (26,2,'.',43,26); |
| </pre> |
| -# Generate the training features: |
| <pre class="example"> |
| SELECT crf_train_fgen( 'train_segmenttbl', |
| 'crf_regex', |
| 'crf_label', |
| 'crf_dictionary', |
| 'train_featuretbl', |
| 'train_featureset' |
| ); |
| SELECT * from crf_dictionary; |
| </pre> |
| Result: |
| <pre class="result"> |
| token | total |
| ------------+------- |
| a | 3 |
| and | 1 |
| august | 1 |
| chancellor | 1 |
| dive | 1 |
| exchequer | 1 |
| ... |
| </pre> |
| <pre class="example"> |
| SELECT * from train_featuretbl; |
| </pre> |
| Result: |
| <pre class="result"> |
| doc_id | f_size | sparse_r | dense_m | sparse_m |
| -------+--------+------------------------------+----------------------------------+----------------------- |
| 1 | 115 | {-1,11,82,0,1,-1,2,32,0,...} | {11,5,11,1,1,5,2,8,2,1,2,11,...} | {5,2,13,11,11,5,13,...} |
| 2 | 115 | {-1,19,35,0,0,-1,26,38,0,..} | {13,5,66,1,1,5,2,8,2,1,2,13,...} | {5,2,13,11,11,5,13,...} |
| </pre> |
| <pre class="example"> |
| SELECT * from train_featureset; |
| </pre> |
| <pre class="result"> |
| f_index | f_name | feature |
| --------+---------------+--------- |
| 6 | W_the | {-1,2} |
| 9 | R_endsWithly | {-1,19} |
| 14 | W_figures | {-1,12} |
| 17 | W_helped | {-1,29} |
| 25 | W_show | {-1,26} |
| 28 | W_'s | {-1,16} |
| 33 | W_chancellor | {-1,13} |
| 43 | W_over | {-1,5} |
| 52 | W_trade | {-1,11} |
| 10 | W_july | {-1,13} |
| 21 | W_substantial | {-1,6} |
| 5 | E. | {2,13} |
| ... |
| </pre> |
| |
| -# Train using linear CRF: |
| <pre class="example"> |
| SELECT lincrf_train( 'train_featuretbl', |
| 'train_featureset', |
| 'crf_label', |
| 'crf_stats_tbl', |
| 'crf_weights_tbl', |
| 20 |
| ); |
| </pre> |
| <pre class="result"> |
| lincrf_train |
| ----------------------------------------------------------------------------------- |
| CRF Train successful. Results stored in the specified CRF stats and weights table |
| lincrf |
| </pre> |
| View the feature weight table. |
| <pre class="example"> |
| SELECT * from crf_weights_tbl; |
| </pre> |
| Result: |
| <pre class="result"> |
| id | name | prev_label_id | label_id | weight |
| ----+---------------+---------------+----------+------------------- |
| 4 | W_lawson | -1 | 13 | 1.73698153439171 |
| 3 | End. | -1 | 43 | 3.3198742329636 |
| 7 | W_has | -1 | 31 | 2.19831004450897 |
| 24 | W_tomorrow | -1 | 11 | 3.34106414300743 |
| 29 | W_. | -1 | 43 | 3.3198742329636 |
| 34 | W_from | -1 | 5 | 2.80284597986744 |
| 37 | W_august | -1 | 13 | 1.34455487966976 |
| 39 | W_due | -1 | 6 | 3.39258895715363 |
| 41 | W_exchequer | -1 | 13 | 1.82177698489335 |
| ... |
| </pre> |
| |
| -# To find the best labels for a test set using the trained linear CRF model, repeat steps #1-2 and generate the test features, except instead of creating a new dictionary, use the dictionary generated from the training set. |
| <pre class="example"> |
| CREATE TABLE test_segmenttbl (start_pos integer,doc_id integer,seg_text text,max_pos integer); |
| INSERT INTO test_segmenttbl VALUES |
| (0,1,'chancellor',26),(1,1,'of',26), (2,1,'the',26), (3,1,'exchequer',26), (4,1,'nigel',26), |
| (5,1,'lawson',26), (6,1,'''s',26), (7,1,'restated',26), (8,1,'commitment',26),(9,1,'to',26), |
| (10,1,'a',26), (11,1,'firm',26), (12,1,'monetary',26),(13,1,'policy',26), (14,1,'has',26), |
| (15,1,'helped',26), (16,1,'to',26), (17,1,'prevent',26), (18,1,'a',26), (19,1,'freefall',26), |
| (20,1,'in',26), (21,1,'sterling',26),(22,1,'over',26), (23,1,'the',26), (24,1,'past',26), |
| (25,1,'week',26), (26,1,'.',26), (0,2,'but',28), (1,2,'analysts',28), (2,2,'reckon',28), |
| (3,2,'underlying',28),(4,2,'support',28), (5,2,'for',28), (6,2,'sterling',28), (7,2,'has',28), |
| (8,2,'been',28), (9,2,'eroded',28), (10,2,'by',28), (11,2,'the',28), (12,2,'chancellor',28), |
| (13,2,'''s',28), (14,2,'failure',28), (15,2,'to',28), (16,2,'announce',28), (17,2,'any',28), |
| (18,2,'new',28), (19,2,'policy',28), (20,2,'measures',28),(21,2,'in',28), (22,2,'his',28), |
| (23,2,'mansion',28), (24,2,'house',28), (25,2,'speech',28), (26,2,'last',28), (27,2,'thursday',28), |
| (28,2,'.',28), (0,3,'his',4), (1,3,'actions',4), (2,3,'prevent',4), (3,3,'disaster',4), |
| (4,3,'.',4); |
| </pre> |
| <pre class="example"> |
| SELECT crf_test_fgen( 'test_segmenttbl', |
| 'crf_dictionary', |
| 'crf_label', |
| 'crf_regex', |
| 'crf_weights_tbl', |
| 'viterbi_mtbl', |
| 'viterbi_rtbl' |
| ); |
| </pre> |
| -# Calculate the best label sequence and save in the table \c extracted_best_labels. |
| <pre class="example"> |
| SELECT vcrf_label( 'test_segmenttbl', |
| 'viterbi_mtbl', |
| 'viterbi_rtbl', |
| 'crf_label', |
| 'extracted_best_labels' |
| ); |
| </pre> |
| View the best labels. |
| <pre class="example"> |
| SELECT * FROM extracted_best_labels; |
| </pre> |
| Result: |
| <pre class="result"> |
| doc_id | start_pos | seg_text | label | id | max_pos | prob |
| -------+-----------+---------------+-------+----+---------+---------- |
| 1 | 4 | nigel | NNP | 13 | 26 | 0.387118 |
| 1 | 5 | lawson | NNP | 13 | 26 | 0.387118 |
| 1 | 7 | restated | VBN | 29 | 26 | 0.387118 |
| 1 | 8 | commitment | NN | 11 | 26 | 0.387118 |
| ... |
| 3 | 0 | his | NNP | 13 | 4 | 0.047757 |
| 3 | 2 | prevent | JJ | 6 | 4 | 0.047757 |
| 3 | 4 | . | . | 43 | 4 | 0.047757 |
| ... |
| </pre> |
| |
| @anchor background |
| @par Technical Background |
| |
| Specifically, a linear-chain CRF is a distribution defined by |
| \f[ |
| p_\lambda(\boldsymbol y | \boldsymbol x) = |
| \frac{\exp{\sum_{m=1}^M \lambda_m F_m(\boldsymbol x, \boldsymbol y)}}{Z_\lambda(\boldsymbol x)} |
| \,. |
| \f] |
| |
| where |
| - \f$ F_m(\boldsymbol x, \boldsymbol y) = \sum_{i=1}^n f_m(y_i,y_{i-1},x_i) \f$ is a global feature function that is a sum along a sequence |
| \f$ \boldsymbol x \f$ of length \f$ n \f$ |
| - \f$ f_m(y_i,y_{i-1},x_i) \f$ is a local feature function dependent on the current token label \f$ y_i \f$, the previous token label \f$ y_{i-1} \f$, |
| and the observation \f$ x_i \f$ |
| - \f$ \lambda_m \f$ is the corresponding feature weight |
| - \f$ Z_\lambda(\boldsymbol x) \f$ is an instance-specific normalizer |
| \f[ |
| Z_\lambda(\boldsymbol x) = \sum_{\boldsymbol y'} \exp{\sum_{m=1}^M \lambda_m F_m(\boldsymbol x, \boldsymbol y')} |
| \f] |
| |
| A linear-chain CRF estimates the weights \f$ \lambda_m \f$ by maximizing the log-likelihood |
| of a given training set \f$ T=\{(x_k,y_k)\}_{k=1}^N \f$. |
| |
| The log-likelihood is defined as |
| \f[ |
| \ell_{\lambda}=\sum_k \log p_\lambda(y_k|x_k) =\sum_k[\sum_{m=1}^M \lambda_m F_m(x_k,y_k) - \log Z_\lambda(x_k)] |
| \f] |
| |
| and the zero of its gradient |
| \f[ |
| \nabla \ell_{\lambda}=\sum_k[F(x_k,y_k)-E_{p_\lambda(Y|x_k)}[F(x_k,Y)]] |
| \f] |
| |
| is found since the maximum likelihood is reached when the empirical average of the global feature vector equals its model expectation. The MADlib implementation uses limited-memory BFGS (L-BFGS), a limited-memory variation of the Broyden–Fletcher–Goldfarb–Shanno (BFGS) update, a quasi-Newton method for unconstrained optimization. |
| |
| \f$E_{p_\lambda(Y|x)}[F(x,Y)]\f$ is found by using a variant of the forward-backward algorithm: |
| \f[ |
| E_{p_\lambda(Y|x)}[F(x,Y)] = \sum_y p_\lambda(y|x)F(x,y) |
| = \sum_i\frac{\alpha_{i-1}(f_i*M_i)\beta_i^T}{Z_\lambda(x)} |
| \f] |
| \f[ |
| Z_\lambda(x) = \alpha_n.1^T |
| \f] |
| where \f$\alpha_i\f$ and \f$ \beta_i\f$ are the forward and backward state cost vectors defined by |
| \f[ |
| \alpha_i = |
| \begin{cases} |
| \alpha_{i-1}M_i, & 0<i<=n\\ |
| 1, & i=0 |
| \end{cases}\\ |
| \f] |
| \f[ |
| \beta_i^T = |
| \begin{cases} |
| M_{i+1}\beta_{i+1}^T, & 1<=i<n\\ |
| 1, & i=n |
| \end{cases} |
| \f] |
| |
| To avoid overfitting, we penalize the likelihood with a spherical Gaussian weight prior: |
| \f[ |
| \ell_{\lambda}^\prime=\sum_k[\sum_{m=1}^M \lambda_m F_m(x_k,y_k) - \log Z_\lambda(x_k)] - \frac{\lVert \lambda \rVert^2}{2\sigma ^2} |
| \f] |
| |
| \f[ |
| \nabla \ell_{\lambda}^\prime=\sum_k[F(x_k,y_k) - E_{p_\lambda(Y|x_k)}[F(x_k,Y)]] - \frac{\lambda}{\sigma ^2} |
| \f] |
| |
| |
| |
| @literature |
| [1] F. Sha, F. Pereira. Shallow Parsing with Conditional Random Fields, http://www-bcf.usc.edu/~feisha/pubs/shallow03.pdf |
| |
| [2] Wikipedia, Conditional Random Field, http://en.wikipedia.org/wiki/Conditional_random_field |
| |
| [3] A. Jaiswal, S.Tawari, I. Mansuri, K. Mittal, C. Tiwari (2012), CRF, http://crf.sourceforge.net/ |
| |
| [4] D. Wang, ViterbiCRF, http://www.cs.berkeley.edu/~daisyw/ViterbiCRF.html |
| |
| [5] Wikipedia, Viterbi Algorithm, http://en.wikipedia.org/wiki/Viterbi_algorithm |
| |
| [6] J. Nocedal. Updating Quasi-Newton Matrices with Limited Storage (1980), Mathematics of Computation 35, pp. 773-782 |
| |
| [7] J. Nocedal, Software for Large-scale Unconstrained Optimization, http://users.eecs.northwestern.edu/~nocedal/lbfgs.html |
| |
| @anchor related |
| @par Related Topics |
| |
| File crf.sql_in crf_feature_gen.sql_in viterbi.sql_in (documenting the SQL functions) |
| |
| */ |
| |
| DROP TYPE IF EXISTS MADLIB_SCHEMA.lincrf_result CASCADE; |
| CREATE TYPE MADLIB_SCHEMA.lincrf_result AS ( |
| coef DOUBLE PRECISION[], |
| log_likelihood DOUBLE PRECISION, |
| num_iterations INTEGER |
| ); |
| |
| CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.lincrf_lbfgs_step_transition( |
| DOUBLE PRECISION[], |
| DOUBLE PRECISION[], |
| DOUBLE PRECISION[], |
| DOUBLE PRECISION[], |
| DOUBLE PRECISION, |
| DOUBLE PRECISION, |
| DOUBLE PRECISION[]) |
| RETURNS DOUBLE PRECISION[] |
| AS 'MODULE_PATHNAME' |
| LANGUAGE C IMMUTABLE; |
| |
| CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.lincrf_lbfgs_step_merge_states( |
| state1 DOUBLE PRECISION[], |
| state2 DOUBLE PRECISION[]) |
| RETURNS DOUBLE PRECISION[] |
| AS 'MODULE_PATHNAME' |
| LANGUAGE C IMMUTABLE STRICT; |
| |
| CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.lincrf_lbfgs_step_final( |
| state DOUBLE PRECISION[]) |
| RETURNS DOUBLE PRECISION[] |
| AS 'MODULE_PATHNAME' |
| LANGUAGE C IMMUTABLE STRICT; |
| |
| CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.internal_lincrf_lbfgs_converge( |
| /*+ state */ DOUBLE PRECISION[]) |
| RETURNS DOUBLE PRECISION AS |
| 'MODULE_PATHNAME' |
| LANGUAGE c IMMUTABLE STRICT; |
| |
| |
| CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.internal_lincrf_lbfgs_result( |
| /*+ state */ DOUBLE PRECISION[]) |
| RETURNS MADLIB_SCHEMA.lincrf_result AS |
| 'MODULE_PATHNAME' |
| LANGUAGE c IMMUTABLE STRICT; |
| |
| /** |
| * @internal |
| * @brief Perform one iteration of the L-BFGS method for computing |
| * conditional random field |
| */ |
| DROP AGGREGATE IF EXISTS MADLIB_SCHEMA.lincrf_lbfgs_step( |
| DOUBLE PRECISION[], |
| DOUBLE PRECISION[], |
| DOUBLE PRECISION[], |
| DOUBLE PRECISION, |
| DOUBLE PRECISION, |
| DOUBLE PRECISION[] |
| ) CASCADE; |
| CREATE AGGREGATE MADLIB_SCHEMA.lincrf_lbfgs_step( |
| /* sparse_r columns */ DOUBLE PRECISION[], |
| /* dense_m columns */ DOUBLE PRECISION[], |
| /* sparse_m columns */ DOUBLE PRECISION[], |
| /* feature size */ DOUBLE PRECISION, |
| /* tag size */ DOUBLE PRECISION, |
| /* previous_state */ DOUBLE PRECISION[]) ( |
| |
| STYPE=DOUBLE PRECISION[], |
| SFUNC=MADLIB_SCHEMA.lincrf_lbfgs_step_transition, |
| m4_ifdef(`__POSTGRESQL__', `', `prefunc=MADLIB_SCHEMA.lincrf_lbfgs_step_merge_states,') |
| FINALFUNC=MADLIB_SCHEMA.lincrf_lbfgs_step_final, |
| INITCOND='{0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}' |
| ); |
| |
| m4_ifdef(`USE_COMPATIBLE_ARRAY', |
| `DROP AGGREGATE IF EXISTS MADLIB_SCHEMA.array_union(anycompatiblearray) CASCADE;', |
| `DROP AGGREGATE IF EXISTS MADLIB_SCHEMA.array_union(anyarray) CASCADE;') |
| |
| CREATE m4_ifdef(`__POSTGRESQL__', `', |
| m4_ifdef(`__HAS_ORDERED_AGGREGATES__', `ORDERED')) AGGREGATE |
| m4_ifdef(`USE_COMPATIBLE_ARRAY', |
| `MADLIB_SCHEMA.array_union(anycompatiblearray) (SFUNC = array_cat, STYPE = anycompatiblearray);', |
| `MADLIB_SCHEMA.array_union(anyarray) (SFUNC = array_cat, STYPE = anyarray);' |
| ) |
| -- We only need to document the last one (unfortunately, in Greenplum we have to |
| -- use function overloading instead of default arguments). |
| CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.compute_lincrf( |
| "source" TEXT, |
| "sparse_R" TEXT, |
| "dense_M" TEXT, |
| "sparse_M" TEXT, |
| "featureSize" TEXT, |
| "tagSize" INTEGER, |
| "maxNumIterations" INTEGER) |
| RETURNS INTEGER |
| AS $$PythonFunction(crf, crf, compute_lincrf)$$ |
| LANGUAGE plpython3u VOLATILE |
| m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `'); |
| |
| /** |
| * @brief Compute linear-chain crf coefficients and diagnostic statistics |
| * |
| * @param source Name of the source relation containing the training data |
| * @param sparse_R Name of the sparse single state feature column (of type DOUBLE PRECISION[]) |
| * @param dense_M Name of the dense two state feature column (of type DOUBLE PRECISION[]) |
| * @param sparse_M Name of the sparse two state feature column (of type DOUBLE PRECISION[]) |
| * @param featureSize Name of feature size column (of type DOUBLE PRECISION) |
| * @param tagSize The number of tags in the tag set |
| * @param featureset The unique feature set |
| * @param crf_feature The Name of output feature table |
| * @param maxNumIterations The maximum number of iterations |
| * |
| * @return a composite value: |
| * - <tt>coef FLOAT8[]</tt> - Array of coefficients, \f$ \boldsymbol c \f$ |
| * - <tt>log_likelihood FLOAT8</tt> - Log-likelihood \f$ l(\boldsymbol c) \f$ |
| * - <tt>num_iterations INTEGER</tt> - The number of iterations before the |
| * algorithm terminated \n\n |
| * A 'crf_feature' table is used to store all the features and corresponding weights |
| * |
| * @note This function starts an iterative algorithm. It is not an aggregate |
| * function. Source and column names have to be passed as strings (due to |
| * limitations of the SQL syntax). |
| * |
| * @internal |
| * @sa This function is a wrapper for crf::compute_lincrf(), which |
| * sets the default values. |
| */ |
| |
| CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.lincrf_train( |
| train_feature_tbl TEXT, |
| train_featureset_tbl TEXT, |
| label_tbl TEXT, |
| crf_stats_tbl TEXT, |
| crf_weights_tbl TEXT, |
| max_iterations INTEGER /* DEFAULT 20 */ |
| ) RETURNS TEXT AS $$ |
| PythonFunction(crf, crf, lincrf_train) |
| $$ LANGUAGE plpython3u |
| m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `'); |
| |
| CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.lincrf_train( |
| train_feature_tbl TEXT, |
| train_featureset_tbl TEXT, |
| label_tbl TEXT, |
| crf_stats_tbl TEXT, |
| crf_weights_tbl TEXT |
| ) RETURNS TEXT AS |
| $$ |
| SELECT MADLIB_SCHEMA.lincrf_train($1, $2, $3, $4, $5, 20); |
| $$ LANGUAGE sql VOLATILE |
| m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `'); |
