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| <div class="title">Conditional Random Field<div class="ingroups"><a class="el" href="group__grp__super.html">Supervised Learning</a></div></div> </div> |
| </div><!--header--> |
| <div class="contents"> |
| <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><p>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.</p> |
| <p>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:</p> |
| <ul> |
| <li>Edge Feature: transition feature that encodes the transition feature weight from current label to next label.</li> |
| <li>Start Feature: fired when the current token is the first token in a sequence.</li> |
| <li>End Feature: fired when the current token is the last token in a sequence.</li> |
| <li>Word Feature: fired when the current token is observed in the trained dictionary.</li> |
| <li>Unknown Feature: fired when the current token is not observed in the trained dictionary for at least a certain number of times (default 1).</li> |
| <li>Regex Feature: fired when the current token can be matched by a regular expression.</li> |
| </ul> |
| <p>A Viterbi implementation is also provided to get the best label sequence and the conditional probability \( \Pr( \text{best label sequence} \mid \text{sequence}) \).</p> |
| <p>Following steps are required for CRF Learning and Inference:</p><ol type="1"> |
| <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> |
| </ol> |
| <p><a class="anchor" id="train_feature"></a></p><dl class="section user"><dt>Training Feature Generation</dt><dd>The function takes <code>train_segment_tbl</code> and <code>regex_tbl</code> as input and does feature generation generating three tables <code>dictionary_tbl</code>, <code>train_feature_tbl</code> and <code>train_featureset_tbl</code>, 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</b> <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 <code>start_pos</code> in the document </td></tr> |
| <tr> |
| <th>label </th><td>INTEGER. Label id for the term corresponding to the actual label from <code>label_tbl</code> </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 <code>train_segment_tbl</code> </td></tr> |
| <tr> |
| <th>total </th><td>INTEGER. Respective counts for the terms </td></tr> |
| </table> |
| </dd> |
| <dt>train_feature_tbl</dt> |
| <dd></dd> |
| <dt></dt> |
| <dd><p class="startdd">TEXT. Name of the training feature table to be created. The table will have the following columns: </p><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> |
| <p class="enddd"></p> |
| </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] <br /> |
| - If L1 = -1: represents single state feature with L2 being the current label id. <br /> |
| - If L1 != -1: represents transition feature with L1 be the previous label and L2 be the current label. </td></tr> |
| </table> |
| </dd> |
| </dl> |
| </dd></dl> |
| <p><a class="anchor" id="train"></a></p><dl class="section user"><dt>Linear Chain CRF Training Function</dt><dd>The function takes <code>train_feature_tbl</code> and <code>train_featureset_tbl</code> tables generated in the training feature generation steps as input along with other required parameters and produces two output tables <code>crf_stats_tbl</code> and <code>crf_weights_tbl</code>.</dd></dl> |
| <pre class="syntax"> |
| lincrf_train(train_feature_tbl, |
| train_featureset_tbl, |
| label_tbl, |
| crf_stats_tbl, |
| crf_weights_tbl |
| max_iterations |
| ) |
| </pre><p> <b>Arguments</b> </p><dl class="arglist"> |
| <dt>train_feature_tbl </dt> |
| <dd><p class="startdd">TEXT. Name of the feature table generated during training feature generation</p> |
| <p class="enddd"></p> |
| </dd> |
| <dt>train_featureset_tbl </dt> |
| <dd><p class="startdd">TEXT. Name of the featureset table generated during training feature generation</p> |
| <p class="enddd"></p> |
| </dd> |
| <dt>label_tbl </dt> |
| <dd><p class="startdd">TEXT. Name of the label table used</p> |
| <p class="enddd"></p> |
| </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><p class="startdd">TEXT. Name of the table to be created creating learned feature weights. The table has the following columns: </p><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> |
| <p class="enddd"></p> |
| </dd> |
| <dt>max_iterations </dt> |
| <dd>INTEGER. The maximum number of iterations </dd> |
| </dl> |
| <p><a class="anchor" id="test_feature"></a></p><dl class="section user"><dt>Testing Feature Generation</dt><dd></dd></dl> |
| <pre class="syntax"> |
| crf_test_fgen(test_segment_tbl, |
| dictionary_tbl, |
| label_tbl, |
| regex_tbl, |
| crf_weights_tbl, |
| viterbi_mtbl, |
| viterbi_rtbl |
| ) |
| </pre><p> <b>Arguments</b> </p><dl class="arglist"> |
| <dt>test_segment_tbl </dt> |
| <dd><p class="startdd">TEXT. Name of the testing segment table. The table is expected to have the following columns: </p><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 <code>start_pos</code> in the document </td></tr> |
| </table> |
| <p class="enddd"></p> |
| </dd> |
| <dt>dictionary_tbl </dt> |
| <dd><p class="startdd">TEXT. Name of the dictionary table created during training feature generation (<code>crf_train_fgen</code>)</p> |
| <p class="enddd"></p> |
| </dd> |
| <dt>label_tbl </dt> |
| <dd><p class="startdd">TEXT. Name of the label table</p> |
| <p class="enddd"></p> |
| </dd> |
| <dt>regex_tbl </dt> |
| <dd><p class="startdd">TEXT. Name of the regular expression table</p> |
| <p class="enddd"></p> |
| </dd> |
| <dt>crf_weights_tbl </dt> |
| <dd><p class="startdd">TEXT. Name of the weights table generated during CRF training (<code>lincrf_train</code>)</p> |
| <p class="enddd"></p> |
| </dd> |
| <dt>viterbi_mtbl </dt> |
| <dd><p class="startdd">TEXT. Name of the Viterbi M table to be created</p> |
| <p class="enddd"></p> |
| </dd> |
| <dt>viterbi_rtbl </dt> |
| <dd>TEXT. Name of the Viterbi R table to be created </dd> |
| </dl> |
| <p><a class="anchor" id="inference"></a></p><dl class="section user"><dt>Inference using Viterbi</dt><dd><pre class="syntax"> |
| vcrf_label(test_segment_tbl, |
| viterbi_mtbl, |
| viterbi_rtbl, |
| label_tbl, |
| result_tbl) |
| </pre> <b>Arguments</b> <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 <code>viterbi_mtbl</code> generated from testing feature generation <code>crf_test_fgen</code>. </dd> |
| <dt>viterbi_rtbl </dt> |
| <dd>TEXT. Name of the table <code>viterbi_rtbl</code> generated from testing feature generation <code>crf_test_fgen</code>. </dd> |
| <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> |
| </dd></dl> |
| <p><a class="anchor" id="usage"></a></p><dl class="section user"><dt>Using CRF</dt><dd></dd></dl> |
| <p>Generate text features, calculate their weights, and output the best label sequence for test data:<br /> |
| </p><ol type="1"> |
| <li>Perform feature generation on training data i.e. <code>train_segment_tbl</code> generating <code>train_feature_tbl</code> and <code>train_featureset_tbl</code>. <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></li> |
| <li>Use linear-chain CRF for training providing <code>train_feature_tbl</code> and <code>train_featureset_tbl</code> 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></li> |
| <li>Perform feature generation on testing data <code>test_segment_tbl</code> generating <code>viterbi_mtbl</code> and <code>viterbi_rtbl</code> 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></li> |
| <li>Run the Viterbi function to get the best label sequence and the conditional probability \( \Pr( \text{best label sequence} \mid \text{sequence}) \). <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></li> |
| </ol> |
| <p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd>This example uses a trivial training and test data set.</dd></dl> |
| <ol type="1"> |
| <li>Load the label table, the regular expressions table, and the training segment table: <pre class="example"> |
| SELECT * FROM crf_label ORDER BY id; |
| </pre> Result: <pre class="result"> |
| id | label |
|  ---+------- |
| 0 | # |
| 1 | $ |
| 2 | '' |
| ... |
| 8 | CC |
| 9 | CD |
| 10 | DT |
| 11 | EX |
| 12 | FW |
| 13 | IN |
| 14 | JJ |
| ... |
| </pre> The regular expressions table: <pre class="example"> |
| SELECT * from crf_regex; |
| </pre> <pre class="result"> |
| pattern | name |
|  --------------+---------------------- |
| ^.+ing$ | endsWithIng |
| ^[A-Z][a-z]+$ | InitCapital |
| ^[A-Z]+$ | isAllCapital |
| ^.*[0-9]+.*$ | containsDigit |
| ... |
| </pre> The training segment table: <pre class="example"> |
| SELECT * from train_segmenttbl ORDER BY doc_id, start_pos; |
| </pre> <pre class="result"> |
| doc_id | start_pos | seg_text | label |
|  -------+-----------+------------+------- |
| 0 | 0 | Confidence | 18 |
| 0 | 1 | in | 13 |
| 0 | 2 | the | 10 |
| 0 | 3 | pound | 18 |
| 0 | 4 | is | 38 |
| 0 | 5 | widely | 26 |
| ... |
| 1 | 0 | Chancellor | 19 |
| 1 | 1 | of | 13 |
| 1 | 2 | the | 10 |
| 1 | 3 | Exchequer | 19 |
| 1 | 4 | Nigel | 19 |
| ... |
| </pre></li> |
| <li>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 |
|  ----------------+------- |
| Hawthorne | 1 |
| Mercedes-Benzes | 1 |
| Wolf | 3 |
| best-known | 1 |
| hairline | 1 |
| accepting | 2 |
| purchases | 14 |
| trash | 5 |
| co-venture | 1 |
| restaurants | 7 |
| ... |
| </pre> <pre class="example"> |
| SELECT * from train_featuretbl; |
| </pre> Result: <pre class="result"> |
| doc_id | f_size | sparse_r | dense_m | sparse_m |
|  -------+--------+-------------------------------+---------------------------------+----------------------- |
| 2 | 87 | {-1,13,12,0,1,-1,13,9,0,1,..} | {13,31,79,1,1,31,29,70,2,1,...} | {51,26,2,69,29,17,...} |
| 1 | 87 | {-1,13,0,0,1,-1,13,9,0,1,...} | {13,0,62,1,1,0,13,54,2,1,13,..} | {51,26,2,69,29,17,...} |
| </pre> <pre class="example"> |
| SELECT * from train_featureset; |
| </pre> <pre class="result"> |
| f_index | f_name | feature |
|  --------+---------------+--------- |
| 1 | R_endsWithED | {-1,29} |
| 13 | W_outweigh | {-1,26} |
| 29 | U | {-1,5} |
| 31 | U | {-1,29} |
| 33 | U | {-1,12} |
| 35 | W_a | {-1,2} |
| 37 | W_possible | {-1,6} |
| 15 | W_signaled | {-1,29} |
| 17 | End. | {-1,43} |
| 49 | W_'s | {-1,16} |
| 63 | W_acquire | {-1,26} |
| 51 | E. | {26,2} |
| 69 | E. | {29,17} |
| 71 | E. | {2,11} |
| 83 | W_the | {-1,2} |
| 85 | E. | {16,11} |
| 4 | W_return | {-1,11} |
| ... |
| </pre></li> |
| <li>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 |
|  ---+---------------+---------------+----------+------------------- |
| 1 | R_endsWithED | -1 | 29 | 1.54128249293937 |
| 13 | W_outweigh | -1 | 26 | 1.70691232223653 |
| 29 | U | -1 | 5 | 1.40708515869008 |
| 31 | U | -1 | 29 | 0.830356200936407 |
| 33 | U | -1 | 12 | 0.769587378281239 |
| 35 | W_a | -1 | 2 | 2.68470625883726 |
| 37 | W_possible | -1 | 6 | 3.41773107604468 |
| 15 | W_signaled | -1 | 29 | 1.68187039165771 |
| 17 | End. | -1 | 43 | 3.07687845517082 |
| 49 | W_'s | -1 | 16 | 2.61430312229883 |
| 63 | W_acquire | -1 | 26 | 1.67247047385797 |
| 51 | E. | 26 | 2 | 3.0114240119435 |
| 69 | E. | 29 | 17 | 2.82385531733866 |
| 71 | E. | 2 | 11 | 3.00970493772732 |
| 83 | W_the | -1 | 2 | 2.58742315259326 |
| ... |
| </pre></li> |
| <li>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"> |
| SELECT * from test_segmenttbl ORDER BY doc_id, start_pos; |
| </pre> Result: <pre class="result"> |
| doc_id | start_pos | seg_text |
|  -------+-----------+--------------- |
| 0 | 0 | Rockwell |
| 0 | 1 | International |
| 0 | 2 | Corp. |
| 0 | 3 | 's |
| 0 | 4 | Tulsa |
| 0 | 5 | unit |
| 0 | 6 | said |
| ... |
| 1 | 0 | Rockwell |
| 1 | 1 | said |
| 1 | 2 | the |
| 1 | 3 | agreement |
| 1 | 4 | calls |
| ... |
| </pre> <pre class="example"> |
| SELECT crf_test_fgen( 'test_segmenttbl', |
| 'crf_dictionary', |
| 'crf_label', |
| 'crf_regex', |
| 'crf_weights_tbl', |
| 'viterbi_mtbl', |
| 'viterbi_rtbl' |
| ); |
| </pre></li> |
| <li>Calculate the best label sequence and save in the table <code>extracted_best_labels</code>. <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 |
|  -------+-----------+---------------+-------+----+---------+---------- |
| 0 | 0 | Rockwell | NNP | 19 | 27 | 0.000269 |
| 0 | 1 | International | NNP | 19 | 27 | 0.000269 |
| 0 | 2 | Corp. | NNP | 19 | 27 | 0.000269 |
| 0 | 3 | 's | NNP | 19 | 27 | 0.000269 |
| ... |
| 1 | 0 | Rockwell | NNP | 19 | 16 | 0.000168 |
| 1 | 1 | said | NNP | 19 | 16 | 0.000168 |
| 1 | 2 | the | DT | 10 | 16 | 0.000168 |
| 1 | 3 | agreement | JJ | 14 | 16 | 0.000168 |
| 1 | 4 | calls | NNS | 21 | 16 | 0.000168 |
| ... |
| </pre></li> |
| </ol> |
| <p><a class="anchor" id="background"></a></p><dl class="section user"><dt>Technical Background</dt><dd></dd></dl> |
| <p>Specifically, a linear-chain CRF is a distribution defined by </p><p class="formulaDsp"> |
| \[ p_\lambda(\boldsymbol y | \boldsymbol x) = \frac{\exp{\sum_{m=1}^M \lambda_m F_m(\boldsymbol x, \boldsymbol y)}}{Z_\lambda(\boldsymbol x)} \,. \] |
| </p> |
| <p>where</p><ul> |
| <li>\( F_m(\boldsymbol x, \boldsymbol y) = \sum_{i=1}^n f_m(y_i,y_{i-1},x_i) \) is a global feature function that is a sum along a sequence \( \boldsymbol x \) of length \( n \)</li> |
| <li>\( f_m(y_i,y_{i-1},x_i) \) is a local feature function dependent on the current token label \( y_i \), the previous token label \( y_{i-1} \), and the observation \( x_i \)</li> |
| <li>\( \lambda_m \) is the corresponding feature weight</li> |
| <li>\( Z_\lambda(\boldsymbol x) \) is an instance-specific normalizer <p class="formulaDsp"> |
| \[ Z_\lambda(\boldsymbol x) = \sum_{\boldsymbol y'} \exp{\sum_{m=1}^M \lambda_m F_m(\boldsymbol x, \boldsymbol y')} \] |
| </p> |
| </li> |
| </ul> |
| <p>A linear-chain CRF estimates the weights \( \lambda_m \) by maximizing the log-likelihood of a given training set \( T=\{(x_k,y_k)\}_{k=1}^N \).</p> |
| <p>The log-likelihood is defined as </p><p class="formulaDsp"> |
| \[ \ell_{\lambda}=\sum_k \log p_\lambda(y_k|x_k) =\sum_k[\sum_{m=1}^M \lambda_m F_m(x_k,y_k) - \log Z_\lambda(x_k)] \] |
| </p> |
| <p>and the zero of its gradient </p><p class="formulaDsp"> |
| \[ \nabla \ell_{\lambda}=\sum_k[F(x_k,y_k)-E_{p_\lambda(Y|x_k)}[F(x_k,Y)]] \] |
| </p> |
| <p>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.</p> |
| <p>\(E_{p_\lambda(Y|x)}[F(x,Y)]\) is found by using a variant of the forward-backward algorithm: </p><p class="formulaDsp"> |
| \[ E_{p_\lambda(Y|x)}[F(x,Y)] = \sum_y p_\lambda(y|x)F(x,y) = \sum_i\frac{\alpha_{i-1}(f_i*M_i)\beta_i^T}{Z_\lambda(x)} \] |
| </p> |
| <p class="formulaDsp"> |
| \[ Z_\lambda(x) = \alpha_n.1^T \] |
| </p> |
| <p> where \(\alpha_i\) and \( \beta_i\) are the forward and backward state cost vectors defined by </p><p class="formulaDsp"> |
| \[ \alpha_i = \begin{cases} \alpha_{i-1}M_i, & 0<i<=n\\ 1, & i=0 \end{cases}\\ \] |
| </p> |
| <p class="formulaDsp"> |
| \[ \beta_i^T = \begin{cases} M_{i+1}\beta_{i+1}^T, & 1<=i<n\\ 1, & i=n \end{cases} \] |
| </p> |
| <p>To avoid overfitting, we penalize the likelihood with a spherical Gaussian weight prior: </p><p class="formulaDsp"> |
| \[ \ell_{\lambda}^\prime=\sum_k[\sum_{m=1}^M \lambda_m F_m(x_k,y_k) - \log Z_\lambda(x_k)] - \frac{\lVert \lambda \rVert^2}{2\sigma ^2} \] |
| </p> |
| <p class="formulaDsp"> |
| \[ \nabla \ell_{\lambda}^\prime=\sum_k[F(x_k,y_k) - E_{p_\lambda(Y|x_k)}[F(x_k,Y)]] - \frac{\lambda}{\sigma ^2} \] |
| </p> |
| <dl class="section user"><dt>Literature</dt><dd>[1] F. Sha, F. Pereira. Shallow Parsing with Conditional Random Fields, <a href="http://www-bcf.usc.edu/~feisha/pubs/shallow03.pdf">http://www-bcf.usc.edu/~feisha/pubs/shallow03.pdf</a></dd></dl> |
| <p>[2] Wikipedia, Conditional Random Field, <a href="http://en.wikipedia.org/wiki/Conditional_random_field">http://en.wikipedia.org/wiki/Conditional_random_field</a></p> |
| <p>[3] A. Jaiswal, S.Tawari, I. Mansuri, K. Mittal, C. Tiwari (2012), CRF, <a href="http://crf.sourceforge.net/">http://crf.sourceforge.net/</a></p> |
| <p>[4] D. Wang, ViterbiCRF, <a href="http://www.cs.berkeley.edu/~daisyw/ViterbiCRF.html">http://www.cs.berkeley.edu/~daisyw/ViterbiCRF.html</a></p> |
| <p>[5] Wikipedia, Viterbi Algorithm, <a href="http://en.wikipedia.org/wiki/Viterbi_algorithm">http://en.wikipedia.org/wiki/Viterbi_algorithm</a></p> |
| <p>[6] J. Nocedal. Updating Quasi-Newton Matrices with Limited Storage (1980), Mathematics of Computation 35, pp. 773-782</p> |
| <p>[7] J. Nocedal, Software for Large-scale Unconstrained Optimization, <a href="http://users.eecs.northwestern.edu/~nocedal/lbfgs.html">http://users.eecs.northwestern.edu/~nocedal/lbfgs.html</a></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="crf_8sql__in.html" title="SQL functions for conditional random field. ">crf.sql_in</a> <a class="el" href="crf__feature__gen_8sql__in.html" title="SQL function for POS/NER feature extraction. ">crf_feature_gen.sql_in</a> <a class="el" href="viterbi_8sql__in.html" title="concatenate a set of input values into arrays to feed into viterbi c function and create a human read...">viterbi.sql_in</a> (documenting the SQL functions) </p> |
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