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</li> + + + + + <li class="divider"></li> + + + + + <li class="header">TABLE OF CONTENTS</li> + + + + <li class="chapter " data-level="1.1" data-path="../"> + + <a href="../"> + + + <b>1.1.</b> + + Introduction + + </a> + + + + </li> + + <li class="chapter " data-level="1.2" data-path="../getting_started/"> + + <a href="../getting_started/"> + + + <b>1.2.</b> + + Getting Started + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="1.2.1" data-path="../getting_started/installation.html"> + + <a href="../getting_started/installation.html"> + + + <b>1.2.1.</b> + + Installation + + </a> + + + + </li> + + <li class="chapter " data-level="1.2.2" data-path="../getting_started/permanent-functions.html"> + + <a href="../getting_started/permanent-functions.html"> + + + <b>1.2.2.</b> + + Install as permanent functions + + </a> + + + + </li> + + <li class="chapter " data-level="1.2.3" data-path="../getting_started/input-format.html"> + + <a href="../getting_started/input-format.html"> + + + <b>1.2.3.</b> + + Input Format + + </a> + + + + </li> + + + </ul> + + </li> + + <li class="chapter " data-level="1.3" data-path="../misc/funcs.html"> + + <a href="../misc/funcs.html"> + + + <b>1.3.</b> + + List of Functions + + </a> + + + + </li> + + <li class="chapter " data-level="1.4" data-path="../tips/"> + + <a href="../tips/"> + + + <b>1.4.</b> + + Tips for Effective Hivemall + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="1.4.1" data-path="../tips/addbias.html"> + + <a href="../tips/addbias.html"> + + + <b>1.4.1.</b> + + Explicit add_bias() for better prediction + + </a> + + + + </li> + + <li class="chapter " data-level="1.4.2" data-path="../tips/rand_amplify.html"> + + <a href="../tips/rand_amplify.html"> + + + <b>1.4.2.</b> + + Use rand_amplify() to better prediction results + + </a> + + + + </li> + + <li class="chapter " data-level="1.4.3" data-path="../tips/rt_prediction.html"> + + <a href="../tips/rt_prediction.html"> + + + <b>1.4.3.</b> + + Real-time prediction on RDBMS + + </a> + + + + </li> + + <li class="chapter " data-level="1.4.4" data-path="../tips/ensemble_learning.html"> + + <a href="../tips/ensemble_learning.html"> + + + <b>1.4.4.</b> + + Ensemble learning for stable prediction + + </a> + + + + </li> + + <li class="chapter " data-level="1.4.5" data-path="../tips/mixserver.html"> + + <a href="../tips/mixserver.html"> + + + <b>1.4.5.</b> + + Mixing models for a better prediction convergence (MIX server) + + </a> + + + + </li> + + <li class="chapter " data-level="1.4.6" data-path="../tips/emr.html"> + + <a href="../tips/emr.html"> + + + <b>1.4.6.</b> + + Run Hivemall on Amazon Elastic MapReduce + + </a> + + + + </li> + + + </ul> + + </li> + + <li class="chapter " data-level="1.5" data-path="../tips/general_tips.html"> + + <a href="../tips/general_tips.html"> + + + <b>1.5.</b> + + General Hive/Hadoop Tips + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="1.5.1" data-path="../tips/rowid.html"> + + <a href="../tips/rowid.html"> + + + <b>1.5.1.</b> + + Adding rowid for each row + + </a> + + + + </li> + + <li class="chapter " data-level="1.5.2" data-path="../tips/hadoop_tuning.html"> + + <a href="../tips/hadoop_tuning.html"> + + + <b>1.5.2.</b> + + Hadoop tuning for Hivemall + + </a> + + + + </li> + + + </ul> + + </li> + + <li class="chapter " data-level="1.6" data-path="../troubleshooting/"> + + <a href="../troubleshooting/"> + + + <b>1.6.</b> + + Troubleshooting + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="1.6.1" data-path="../troubleshooting/oom.html"> + + <a href="../troubleshooting/oom.html"> + + + <b>1.6.1.</b> + + OutOfMemoryError in training + + </a> + + + + </li> + + <li class="chapter " data-level="1.6.2" data-path="../troubleshooting/mapjoin_task_error.html"> + + <a href="../troubleshooting/mapjoin_task_error.html"> + + + <b>1.6.2.</b> + + SemanticException generate map join task error: Cannot serialize object + + </a> + + + + </li> + + <li class="chapter " data-level="1.6.3" data-path="../troubleshooting/asterisk.html"> + + <a href="../troubleshooting/asterisk.html"> + + + <b>1.6.3.</b> + + Asterisk argument for UDTF does not work + + </a> + + + + </li> + + <li class="chapter " data-level="1.6.4" data-path="../troubleshooting/num_mappers.html"> + + <a href="../troubleshooting/num_mappers.html"> + + + <b>1.6.4.</b> + + The number of mappers is less than input splits in Hadoop 2.x + + </a> + + + + </li> + + <li class="chapter " data-level="1.6.5" data-path="../troubleshooting/mapjoin_classcastex.html"> + + <a href="../troubleshooting/mapjoin_classcastex.html"> + + + <b>1.6.5.</b> + + Map-side join causes ClassCastException on Tez + + </a> + + + + </li> + + + </ul> + + </li> + + + + + <li class="header">Part II - Generic Features</li> + + + + <li class="chapter " data-level="2.1" data-path="../misc/generic_funcs.html"> + + <a href="../misc/generic_funcs.html"> + + + <b>2.1.</b> + + List of Generic Hivemall Functions + + </a> + + + + </li> + + <li class="chapter " data-level="2.2" data-path="../misc/topk.html"> + + <a href="../misc/topk.html"> + + + <b>2.2.</b> + + Efficient Top-K Query Processing + + </a> + + + + </li> + + <li class="chapter " data-level="2.3" data-path="../misc/tokenizer.html"> + + <a href="../misc/tokenizer.html"> + + + <b>2.3.</b> + + Text Tokenizer + + </a> + + + + </li> + + <li class="chapter " data-level="2.4" data-path="../misc/approx.html"> + + <a href="../misc/approx.html"> + + + <b>2.4.</b> + + Approximate Aggregate Functions + + </a> + + + + </li> + + + + + <li class="header">Part III - Feature Engineering</li> + + + + <li class="chapter " data-level="3.1" data-path="scaling.html"> + + <a href="scaling.html"> + + + <b>3.1.</b> + + Feature Scaling + + </a> + + + + </li> + + <li class="chapter " data-level="3.2" data-path="hashing.html"> + + <a href="hashing.html"> + + + <b>3.2.</b> + + Feature Hashing + + </a> + + + + </li> + + <li class="chapter " data-level="3.3" data-path="selection.html"> + + <a href="selection.html"> + + + <b>3.3.</b> + + Feature Selection + + </a> + + + + </li> + + <li class="chapter " data-level="3.4" data-path="binning.html"> + + <a href="binning.html"> + + + <b>3.4.</b> + + Feature Binning + + </a> + + + + </li> + + <li class="chapter " data-level="3.5" data-path="pairing.html"> + + <a href="pairing.html"> + + + <b>3.5.</b> + + Feature Paring + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="3.5.1" data-path="polynomial.html"> + + <a href="polynomial.html"> + + + <b>3.5.1.</b> + + Polynomial features + + </a> + + + + </li> + + + </ul> + + </li> + + <li class="chapter " data-level="3.6" data-path="ft_trans.html"> + + <a href="ft_trans.html"> + + + <b>3.6.</b> + + Feature Transformation + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="3.6.1" data-path="vectorization.html"> + + <a href="vectorization.html"> + + + <b>3.6.1.</b> + + Feature vectorization + + </a> + + + + </li> + + <li class="chapter " data-level="3.6.2" data-path="quantify.html"> + + <a href="quantify.html"> + + + <b>3.6.2.</b> + + Quantify non-number features + + </a> + + + + </li> + + + </ul> + + </li> + + <li class="chapter " data-level="3.7" data-path="term_vector.html"> + + <a href="term_vector.html"> + + + <b>3.7.</b> + + Term Vector Model + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="3.7.1" data-path="tfidf.html"> + + <a href="tfidf.html"> + + + <b>3.7.1.</b> + + TF-IDF Term Weighting + + </a> + + + + </li> + + <li class="chapter active" data-level="3.7.2" data-path="bm25.html"> + + <a href="bm25.html"> + + + <b>3.7.2.</b> + + Okapi BM25 Term Weighting + + </a> + + + + </li> + + + </ul> + + </li> + + + + + <li class="header">Part IV - Evaluation</li> + + + + <li class="chapter " data-level="4.1" data-path="../eval/binary_classification_measures.html"> + + <a href="../eval/binary_classification_measures.html"> + + + <b>4.1.</b> + + Binary Classification Metrics + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="4.1.1" data-path="../eval/auc.html"> + + <a href="../eval/auc.html"> + + + <b>4.1.1.</b> + + Area under the ROC curve + + </a> + + + + </li> + + + </ul> + + </li> + + <li class="chapter " data-level="4.2" data-path="../eval/multilabel_classification_measures.html"> + + <a href="../eval/multilabel_classification_measures.html"> + + + <b>4.2.</b> + + Multi-label Classification Metrics + + </a> + + + + </li> + + <li class="chapter " data-level="4.3" data-path="../eval/regression.html"> + + <a href="../eval/regression.html"> + + + <b>4.3.</b> + + Regression Metrics + + </a> + + + + </li> + + <li class="chapter " data-level="4.4" data-path="../eval/rank.html"> + + <a href="../eval/rank.html"> + + + <b>4.4.</b> + + Ranking Measures + + </a> + + + + </li> + + <li class="chapter " data-level="4.5" data-path="../eval/datagen.html"> + + <a href="../eval/datagen.html"> + + + <b>4.5.</b> + + Data Generation + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="4.5.1" data-path="../eval/lr_datagen.html"> + + <a href="../eval/lr_datagen.html"> + + + <b>4.5.1.</b> + + Logistic Regression data generation + + </a> + + + + </li> + + + </ul> + + </li> + + + + + <li class="header">Part V - Supervised Learning</li> + + + + <li class="chapter " data-level="5.1" data-path="../supervised_learning/prediction.html"> + + <a href="../supervised_learning/prediction.html"> + + + <b>5.1.</b> + + How Prediction Works + + </a> + + + + </li> + + <li class="chapter " data-level="5.2" data-path="../supervised_learning/tutorial.html"> + + <a href="../supervised_learning/tutorial.html"> + + + <b>5.2.</b> + + Step-by-Step Tutorial on Supervised Learning + + </a> + + + + </li> + + + + + <li class="header">Part VI - Binary Classification</li> + + + + <li class="chapter " data-level="6.1" data-path="../binaryclass/general.html"> + + <a href="../binaryclass/general.html"> + + + <b>6.1.</b> + + Binary Classification + + </a> + + + + </li> + + <li class="chapter " data-level="6.2" data-path="../binaryclass/a9a.html"> + + <a href="../binaryclass/a9a.html"> + + + <b>6.2.</b> + + a9a Tutorial + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="6.2.1" data-path="../binaryclass/a9a_dataset.html"> + + <a href="../binaryclass/a9a_dataset.html"> + + + <b>6.2.1.</b> + + Data preparation + + </a> + + + + </li> + + <li class="chapter " data-level="6.2.2" data-path="../binaryclass/a9a_lr.html"> + + <a href="../binaryclass/a9a_lr.html"> + + + <b>6.2.2.</b> + + Logistic Regression + + </a> + + + + </li> + + <li class="chapter " data-level="6.2.3" data-path="../binaryclass/a9a_minibatch.html"> + + <a href="../binaryclass/a9a_minibatch.html"> + + + <b>6.2.3.</b> + + Mini-batch gradient descent + + </a> + + + + </li> + + + </ul> + + </li> + + <li class="chapter " data-level="6.3" data-path="../binaryclass/news20.html"> + + <a href="../binaryclass/news20.html"> + + + <b>6.3.</b> + + News20 Tutorial + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="6.3.1" data-path="../binaryclass/news20_dataset.html"> + + <a href="../binaryclass/news20_dataset.html"> + + + <b>6.3.1.</b> + + Data preparation + + </a> + + + + </li> + + <li class="chapter " data-level="6.3.2" data-path="../binaryclass/news20_pa.html"> + + <a href="../binaryclass/news20_pa.html"> + + + <b>6.3.2.</b> + + Perceptron, Passive Aggressive + + </a> + + + + </li> + + <li class="chapter " data-level="6.3.3" data-path="../binaryclass/news20_scw.html"> + + <a href="../binaryclass/news20_scw.html"> + + + <b>6.3.3.</b> + + CW, AROW, SCW + + </a> + + + + </li> + + <li class="chapter " data-level="6.3.4" data-path="../binaryclass/news20_adagrad.html"> + + <a href="../binaryclass/news20_adagrad.html"> + + + <b>6.3.4.</b> + + AdaGradRDA, AdaGrad, AdaDelta + + </a> + + + + </li> + + <li class="chapter " data-level="6.3.5" data-path="../binaryclass/news20_rf.html"> + + <a href="../binaryclass/news20_rf.html"> + + + <b>6.3.5.</b> + + Random Forest + + </a> + + + + </li> + + + </ul> + + </li> + + <li class="chapter " data-level="6.4" data-path="../binaryclass/kdd2010a.html"> + + <a href="../binaryclass/kdd2010a.html"> + + + <b>6.4.</b> + + KDD2010a Tutorial + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="6.4.1" data-path="../binaryclass/kdd2010a_dataset.html"> + + <a href="../binaryclass/kdd2010a_dataset.html"> + + + <b>6.4.1.</b> + + Data preparation + + </a> + + + + </li> + + <li class="chapter " data-level="6.4.2" data-path="../binaryclass/kdd2010a_scw.html"> + + <a href="../binaryclass/kdd2010a_scw.html"> + + + <b>6.4.2.</b> + + PA, CW, AROW, SCW + + </a> + + + + </li> + + + </ul> + + </li> + + <li class="chapter " data-level="6.5" data-path="../binaryclass/kdd2010b.html"> + + <a href="../binaryclass/kdd2010b.html"> + + + <b>6.5.</b> + + KDD2010b Tutorial + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="6.5.1" data-path="../binaryclass/kdd2010b_dataset.html"> + + <a href="../binaryclass/kdd2010b_dataset.html"> + + + <b>6.5.1.</b> + + Data preparation + + </a> + + + + </li> + + <li class="chapter " data-level="6.5.2" data-path="../binaryclass/kdd2010b_arow.html"> + + <a href="../binaryclass/kdd2010b_arow.html"> + + + <b>6.5.2.</b> + + AROW + + </a> + + + + </li> + + + </ul> + + </li> + + <li class="chapter " data-level="6.6" data-path="../binaryclass/webspam.html"> + + <a href="../binaryclass/webspam.html"> + + + <b>6.6.</b> + + Webspam Tutorial + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="6.6.1" data-path="../binaryclass/webspam_dataset.html"> + + <a href="../binaryclass/webspam_dataset.html"> + + + <b>6.6.1.</b> + + Data pareparation + + </a> + + + + </li> + + <li class="chapter " data-level="6.6.2" data-path="../binaryclass/webspam_scw.html"> + + <a href="../binaryclass/webspam_scw.html"> + + + <b>6.6.2.</b> + + PA1, AROW, SCW + + </a> + + + + </li> + + + </ul> + + </li> + + <li class="chapter " data-level="6.7" data-path="../binaryclass/titanic_rf.html"> + + <a href="../binaryclass/titanic_rf.html"> + + + <b>6.7.</b> + + Kaggle Titanic Tutorial + + </a> + + + + </li> + + <li class="chapter " data-level="6.8" data-path="../binaryclass/criteo.html"> + + <a href="../binaryclass/criteo.html"> + + + <b>6.8.</b> + + Criteo Tutorial + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="6.8.1" data-path="../binaryclass/criteo_dataset.html"> + + <a href="../binaryclass/criteo_dataset.html"> + + + <b>6.8.1.</b> + + Data preparation + + </a> + + + + </li> + + <li class="chapter " data-level="6.8.2" data-path="../binaryclass/criteo_ffm.html"> + + <a href="../binaryclass/criteo_ffm.html"> + + + <b>6.8.2.</b> + + Field-Aware Factorization Machines + + </a> + + + + </li> + + + </ul> + + </li> + + + + + <li class="header">Part VII - Multiclass Classification</li> + + + + <li class="chapter " data-level="7.1" data-path="../multiclass/news20.html"> + + <a href="../multiclass/news20.html"> + + + <b>7.1.</b> + + News20 Multiclass Tutorial + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="7.1.1" data-path="../multiclass/news20_dataset.html"> + + <a href="../multiclass/news20_dataset.html"> + + + <b>7.1.1.</b> + + Data preparation + + </a> + + + + </li> + + <li class="chapter " data-level="7.1.2" data-path="../multiclass/news20_one-vs-the-rest_dataset.html"> + + <a href="../multiclass/news20_one-vs-the-rest_dataset.html"> + + + <b>7.1.2.</b> + + Data preparation for one-vs-the-rest classifiers + + </a> + + + + </li> + + <li class="chapter " data-level="7.1.3" data-path="../multiclass/news20_pa.html"> + + <a href="../multiclass/news20_pa.html"> + + + <b>7.1.3.</b> + + PA + + </a> + + + + </li> + + <li class="chapter " data-level="7.1.4" data-path="../multiclass/news20_scw.html"> + + <a href="../multiclass/news20_scw.html"> + + + <b>7.1.4.</b> + + CW, AROW, SCW + + </a> + + + + </li> + + <li class="chapter " data-level="7.1.5" data-path="../multiclass/news20_ensemble.html"> + + <a href="../multiclass/news20_ensemble.html"> + + + <b>7.1.5.</b> + + Ensemble learning + + </a> + + + + </li> + + <li class="chapter " data-level="7.1.6" data-path="../multiclass/news20_one-vs-the-rest.html"> + + <a href="../multiclass/news20_one-vs-the-rest.html"> + + + <b>7.1.6.</b> + + one-vs-the-rest classifier + + </a> + + + + </li> + + + </ul> + + </li> + + <li class="chapter " data-level="7.2" data-path="../multiclass/iris.html"> + + <a href="../multiclass/iris.html"> + + + <b>7.2.</b> + + Iris Tutorial + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="7.2.1" data-path="../multiclass/iris_dataset.html"> + + <a href="../multiclass/iris_dataset.html"> + + + <b>7.2.1.</b> + + Data preparation + + </a> + + + + </li> + + <li class="chapter " data-level="7.2.2" data-path="../multiclass/iris_scw.html"> + + <a href="../multiclass/iris_scw.html"> + + + <b>7.2.2.</b> + + SCW + + </a> + + + + </li> + + <li class="chapter " data-level="7.2.3" data-path="../multiclass/iris_randomforest.html"> + + <a href="../multiclass/iris_randomforest.html"> + + + <b>7.2.3.</b> + + Random Forest + + </a> + + + + </li> + + + </ul> + + </li> + + + + + <li class="header">Part VIII - Regression</li> + + + + <li class="chapter " data-level="8.1" data-path="../regression/general.html"> + + <a href="../regression/general.html"> + + + <b>8.1.</b> + + Regression + + </a> + + + + </li> + + <li class="chapter " data-level="8.2" data-path="../regression/e2006.html"> + + <a href="../regression/e2006.html"> + + + <b>8.2.</b> + + E2006-tfidf Regression Tutorial + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="8.2.1" data-path="../regression/e2006_dataset.html"> + + <a href="../regression/e2006_dataset.html"> + + + <b>8.2.1.</b> + + Data preparation + + </a> + + + + </li> + + <li class="chapter " data-level="8.2.2" data-path="../regression/e2006_arow.html"> + + <a href="../regression/e2006_arow.html"> + + + <b>8.2.2.</b> + + Passive Aggressive, AROW + + </a> + + + + </li> + + + </ul> + + </li> + + <li class="chapter " data-level="8.3" data-path="../regression/kddcup12tr2.html"> + + <a href="../regression/kddcup12tr2.html"> + + + <b>8.3.</b> + + KDDCup 2012 Track 2 CTR Prediction Tutorial + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="8.3.1" data-path="../regression/kddcup12tr2_dataset.html"> + + <a href="../regression/kddcup12tr2_dataset.html"> + + + <b>8.3.1.</b> + + Data preparation + + </a> + + + + </li> + + <li class="chapter " data-level="8.3.2" data-path="../regression/kddcup12tr2_lr.html"> + + <a href="../regression/kddcup12tr2_lr.html"> + + + <b>8.3.2.</b> + + Logistic Regression, Passive Aggressive + + </a> + + + + </li> + + <li class="chapter " data-level="8.3.3" data-path="../regression/kddcup12tr2_lr_amplify.html"> + + <a href="../regression/kddcup12tr2_lr_amplify.html"> + + + <b>8.3.3.</b> + + Logistic Regression with amplifier + + </a> + + + + </li> + + <li class="chapter " data-level="8.3.4" data-path="../regression/kddcup12tr2_adagrad.html"> + + <a href="../regression/kddcup12tr2_adagrad.html"> + + + <b>8.3.4.</b> + + AdaGrad, AdaDelta + + </a> + + + + </li> + + + </ul> + + </li> + + + + + <li class="header">Part IX - Recommendation</li> + + + + <li class="chapter " data-level="9.1" data-path="../recommend/cf.html"> + + <a href="../recommend/cf.html"> + + + <b>9.1.</b> + + Collaborative Filtering + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="9.1.1" data-path="../recommend/item_based_cf.html"> + + <a href="../recommend/item_based_cf.html"> + + + <b>9.1.1.</b> + + Item-based collaborative filtering + + </a> + + + + </li> + + + </ul> + + </li> + + <li class="chapter " data-level="9.2" data-path="../recommend/news20.html"> + + <a href="../recommend/news20.html"> + + + <b>9.2.</b> + + News20 Related Article Recommendation Tutorial + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="9.2.1" data-path="../multiclass/news20_dataset.html"> + + <a href="../multiclass/news20_dataset.html"> + + + <b>9.2.1.</b> + + Data preparation + + </a> + + + + </li> + + <li class="chapter " data-level="9.2.2" data-path="../recommend/news20_jaccard.html"> + + <a href="../recommend/news20_jaccard.html"> + + + <b>9.2.2.</b> + + LSH/MinHash and Jaccard similarity + + </a> + + + + </li> + + <li class="chapter " data-level="9.2.3" data-path="../recommend/news20_knn.html"> + + <a href="../recommend/news20_knn.html"> + + + <b>9.2.3.</b> + + LSH/MinHash and brute-force search + + </a> + + + + </li> + + <li class="chapter " data-level="9.2.4" data-path="../recommend/news20_bbit_minhash.html"> + + <a href="../recommend/news20_bbit_minhash.html"> + + + <b>9.2.4.</b> + + kNN search using b-Bits MinHash + + </a> + + + + </li> + + + </ul> + + </li> + + <li class="chapter " data-level="9.3" data-path="../recommend/movielens.html"> + + <a href="../recommend/movielens.html"> + + + <b>9.3.</b> + + MovieLens Movie Recommendation Tutorial + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="9.3.1" data-path="../recommend/movielens_dataset.html"> + + <a href="../recommend/movielens_dataset.html"> + + + <b>9.3.1.</b> + + Data preparation + + </a> + + + + </li> + + <li class="chapter " data-level="9.3.2" data-path="../recommend/movielens_cf.html"> + + <a href="../recommend/movielens_cf.html"> + + + <b>9.3.2.</b> + + Item-based collaborative filtering + + </a> + + + + </li> + + <li class="chapter " data-level="9.3.3" data-path="../recommend/movielens_mf.html"> + + <a href="../recommend/movielens_mf.html"> + + + <b>9.3.3.</b> + + Matrix Factorization + + </a> + + + + </li> + + <li class="chapter " data-level="9.3.4" data-path="../recommend/movielens_fm.html"> + + <a href="../recommend/movielens_fm.html"> + + + <b>9.3.4.</b> + + Factorization Machine + + </a> + + + + </li> + + <li class="chapter " data-level="9.3.5" data-path="../recommend/movielens_slim.html"> + + <a href="../recommend/movielens_slim.html"> + + + <b>9.3.5.</b> + + SLIM for fast top-k recommendation + + </a> + + + + </li> + + <li class="chapter " data-level="9.3.6" data-path="../recommend/movielens_cv.html"> + + <a href="../recommend/movielens_cv.html"> + + + <b>9.3.6.</b> + + 10-fold cross validation (Matrix Factorization) + + </a> + + + + </li> + + + </ul> + + </li> + + + + + <li class="header">Part X - Anomaly Detection</li> + + + + <li class="chapter " data-level="10.1" data-path="../anomaly/lof.html"> + + <a href="../anomaly/lof.html"> + + + <b>10.1.</b> + + Outlier Detection using Local Outlier Factor (LOF) + + </a> + + + + </li> + + <li class="chapter " data-level="10.2" data-path="../anomaly/sst.html"> + + <a href="../anomaly/sst.html"> + + + <b>10.2.</b> + + Change-Point Detection using Singular Spectrum Transformation (SST) + + </a> + + + + </li> + + <li class="chapter " data-level="10.3" data-path="../anomaly/changefinder.html"> + + <a href="../anomaly/changefinder.html"> + + + <b>10.3.</b> + + ChangeFinder: Detecting Outlier and Change-Point Simultaneously + + </a> + + + + </li> + + + + + <li class="header">Part XI - Clustering</li> + + + + <li class="chapter " data-level="11.1" data-path="../clustering/lda.html"> + + <a href="../clustering/lda.html"> + + + <b>11.1.</b> + + Latent Dirichlet Allocation + + </a> + + + + </li> + + <li class="chapter " data-level="11.2" data-path="../clustering/plsa.html"> + + <a href="../clustering/plsa.html"> + + + <b>11.2.</b> + + Probabilistic Latent Semantic Analysis + + </a> + + + + </li> + + + + + <li class="header">Part XII - GeoSpatial Functions</li> + + + + <li class="chapter " data-level="12.1" data-path="../geospatial/latlon.html"> + + <a href="../geospatial/latlon.html"> + + + <b>12.1.</b> + + Lat/Lon functions + + </a> + + + + </li> + + + + + <li class="header">Part XIII - Hivemall on Spark</li> + + + + <li class="chapter " data-level="13.1" data-path="../spark/getting_started/"> + + <a href="../spark/getting_started/"> + + + <b>13.1.</b> + + Getting Started + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="13.1.1" data-path="../spark/getting_started/installation.html"> + + <a href="../spark/getting_started/installation.html"> + + + <b>13.1.1.</b> + + Installation + + </a> + + + + </li> + + + </ul> + + </li> + + <li class="chapter " data-level="13.2" data-path="../spark/binaryclass/"> + + <a href="../spark/binaryclass/"> + + + <b>13.2.</b> + + Binary Classification + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="13.2.1" data-path="../spark/binaryclass/a9a_df.html"> + + <a href="../spark/binaryclass/a9a_df.html"> + + + <b>13.2.1.</b> + + a9a tutorial for DataFrame + + </a> + + + + </li> + + <li class="chapter " data-level="13.2.2" data-path="../spark/binaryclass/a9a_sql.html"> + + <a href="../spark/binaryclass/a9a_sql.html"> + + + <b>13.2.2.</b> + + a9a tutorial for SQL + + </a> + + + + </li> + + + </ul> + + </li> + + <li class="chapter " data-level="13.3" data-path="../spark/binaryclass/"> + + <a href="../spark/binaryclass/"> + + + <b>13.3.</b> + + Regression + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="13.3.1" data-path="../spark/regression/e2006_df.html"> + + <a href="../spark/regression/e2006_df.html"> + + + <b>13.3.1.</b> + + E2006-tfidf regression tutorial for DataFrame + + </a> + + + + </li> + + <li class="chapter " data-level="13.3.2" data-path="../spark/regression/e2006_sql.html"> + + <a href="../spark/regression/e2006_sql.html"> + + + <b>13.3.2.</b> + + E2006-tfidf regression tutorial for SQL + + </a> + + + + </li> + + + </ul> + + </li> + + <li class="chapter " data-level="13.4" data-path="../spark/misc/misc.html"> + + <a href="../spark/misc/misc.html"> + + + <b>13.4.</b> + + Generic features + + </a> + + + + <ul class="articles"> + + + <li class="chapter " data-level="13.4.1" data-path="../spark/misc/topk_join.html"> + + <a href="../spark/misc/topk_join.html"> + + + <b>13.4.1.</b> + + Top-k join processing + + </a> + + + + </li> + + <li class="chapter " data-level="13.4.2" data-path="../spark/misc/functions.html"> + + <a href="../spark/misc/functions.html"> + + + <b>13.4.2.</b> + + Other utility functions + + </a> + + + + </li> + + + </ul> + + </li> + + + + + <li class="header">Part XIV - Hivemall on Docker</li> + + + + <li class="chapter " data-level="14.1" data-path="../docker/getting_started.html"> + + <a href="../docker/getting_started.html"> + + + <b>14.1.</b> + + Getting Started + + </a> + + + + </li> + + + + + <li class="header">Part XIV - External References</li> + + + + <li class="chapter " data-level="15.1" > + + <a target="_blank" href="https://github.com/daijyc/hivemall/wiki/PigHome";> + + + <b>15.1.</b> + + Hivemall on Apache Pig + + </a> + + + + </li> + + + + + <li class="divider"></li> + + <li> + <a href="https://www.gitbook.com"; target="blank" class="gitbook-link"> + Published with GitBook + </a> + </li> +</ul> + + + </nav> + + + </div> + + <div class="book-body"> + + <div class="body-inner"> + + + +<div class="book-header" role="navigation"> + + + <!-- Title --> + <h1> + <i class="fa fa-circle-o-notch fa-spin"></i> + <a href=".." >Okapi BM25 Term Weighting</a> + </h1> +</div> + + + + + <div class="page-wrapper" tabindex="-1" role="main"> + <div class="page-inner"> + +<div id="book-search-results"> + <div class="search-noresults"> + + <section class="normal markdown-section"> + + <!-- + Licensed to the Apache Software Foundation (ASF) under one + or more contributor license agreements. See the NOTICE file + distributed with this work for additional information + regarding copyright ownership. The ASF licenses this file + to you under the Apache License, Version 2.0 (the + "License"); you may not use this file except in compliance + with the License. You may obtain a copy of the License at + + http://www.apache.org/licenses/LICENSE-2.0 + + Unless required by applicable law or agreed to in writing, + software distributed under the License is distributed on an + "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY + KIND, either express or implied. See the License for the + specific language governing permissions and limitations + under the License. +--> +<p><a href="https://en.wikipedia.org/wiki/Okapi_BM25"; target="_blank">Okapi BM25</a> is a ranking function for documents for a given query.</p> +<p>It can also be used for a better replacement of <a href="https://en.wikipedia.org/wiki/Tf%E2%80%93idf"; target="_blank">TF-IDF</a> and can be used for term-weight for each document.</p> +<!-- toc --><div id="toc" class="toc"> + +<ul> +<li><a href="#the-ranking-function">The ranking function</a></li> +<li><a href="#data-preparation">Data preparation</a></li> +<li><a href="#define-views-of-termdoc-frequency">Define views of term/doc frequency</a></li> +<li><a href="#compute-okapi-bm25-score">Compute Okapi BM25 score</a><ul> +<li><a href="#hyperparameters">Hyperparameters</a></li> +<li><a href="#show-important-terms-for-each-document">Show important terms for each document</a></li> +</ul> +</li> +<li><a href="#retrive-relevant-documents-for-a-given-search-terms">Retrive relevant documents for a given search terms</a></li> +</ul> + +</div><!-- tocstop --> +<h1 id="the-ranking-function">The ranking function</h1> +<p>Given a query <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit">Q</span></span></span></span>, containing keywords <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>q</mi><mn>1</mn><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>q</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">q1,....,q_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.03588em;">q</span><span class="mord mathrm">1</span><span class="mpunct">,</span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mord mathrm">.</span><span class="mpunct">,</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight">n</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></ span></span></span>, the BM25 score of a document <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.02778em;">D</span></span></span></span> is:</p> +<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi><mo>(</mo><mi>Q</mi><mo separator="true">,</mo><mi>D</mi><mo>)</mo><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mi>I</mi><mi>D</mi><mi>F</mi><mo>(</mo><msub><mi>q</mi><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>⋅</mo><mfrac><mrow><mi>t</mi><mi>f</mi><mo>(</mo><msub><mi>q</mi><mrow><mi>i</mi></mrow></msub><mo separator="true">,</mo><mi>D</mi><mo>)</mo><mo>⋅</mo><mo>(</mo><msub><mi>k</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>t</mi><mi>f</mi><mo>(</mo><msub><mi>q</mi><mrow><mi>i</mi></mrow></msub><mo separator="true">,</mo><mi>D</mi><mo>)</mo><mo>+</mo><msub><mi>k</mi><mrow><mn>1</mn></mrow></msub><mo>⋅</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>b</mi><mo>+</mo><mi>b</mi><mo>⋅</mo><mfrac><mrow><mi mathv ariant="normal">∣</mi><mi>D</mi><mi mathvariant="normal">∣</mi></mrow><mrow><mi>a</mi><mi>v</mi><mi>g</mi><mi>d</mi><mi>l</mi></mrow></mfrac><mo>)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex"> +score(Q, D) = \sum_{i=1}^{n}IDF(q_{i}) \cdot \frac{tf(q_{i},D) \cdot (k_{1}+1)}{tf(q_{i},D) + k_{1} \cdot (1 - b + b \cdot \frac{|D|}{avgdl})} +</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.6513970000000002em;"></span><span class="strut bottom" style="height:3.0325050000000005em;vertical-align:-1.381108em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord mathit">s</span><span class="mord mathit">c</span><span class="mord mathit">o</span><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="mord mathit">e</span><span class="mopen">(</span><span class="mord mathit">Q</span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.02778em;">D</span><span class="mclose">)</span><span class="mrel">=</span><span class="mop op-limits"><span class="vlist"><span style="top:1.1776689999999999em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord script style cramped mtight"><span class="mord mathit mtight">i</span><span class="mrel mtight">=</span><span class="mord mathrm mtight">1</span></span></span></span><span style="top:-0.000005000000000143778em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-1.2500050000000003em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped mtight"><span class="mord scriptstyle uncramped mtight"><span class="mord mathit mtight">n</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mathit" style="margin-right:0.07847em;">I</span><span class="mord mathit" style="margin-right:0.02778em;">D</span><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">i</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mclose">)</span><span class="mbin">⋅</span><span class="mord reset-textstyle displaystyle textstyle uncramped"><span class="mopen sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style= "top:0.9em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle cramped"><span class="mord textstyle cramped"><span class="mord mathit">t</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">i</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.02778em;">D</span><span class="mclose">)</span><span class="mbin">+</span><span class="mord"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03148em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mbin">⋅</span><span class="mopen">(</span><span class="mord mathrm">1</span><span class="mbin">−</span><span class="mord mathit">b</span><span class="mbin">+</span><span class="mord mathit">b</span><span class="mbin">⋅</span><span clas s="mord reset-textstyle textstyle cramped"><span class="mopen sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">a</span><span class="mord mathit mtight" style="margin-right:0.03588em;">v</span><span class="mord mathit mtight" style="margin-right:0.03588em;">g</span><span class="mord mathit mtight">d</span><span class="mord mathit mtight" style="margin-right:0.01968em;">l</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fon tsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">∣</span><span class="mord mathit mtight" style="margin-right:0.02778em;">D</span><span class="mord mathrm mtight">∣</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mclose">)</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.677em;"><span class="fontsize-ensurer reset-size5 size5"><span style ="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped"><span class="mord textstyle uncramped"><span class="mord mathit">t</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">i</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.02778em;">D</span><span class="mclose"> )</span><span class="mbin">⋅</span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03148em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mbin">+</span><span class="mord mathrm">1</span><span class="mclose">)</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose sizing reset-siz e5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span></span></span></span></span></p> +<p>where <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi><mi>f</mi><mo>(</mo><msub><mi>q</mi><mrow><mi>i</mi></mrow></msub><mo separator="true">,</mo><mi>D</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">tf(q_{i}, D)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit">t</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03588em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">i</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.02778em;">D</span><span class="mclose">)</span></span></span></span> is <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>q</mi><mrow><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">q_{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;mar gin-right:0.05em;margin-left:-0.03588em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">i</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span>'s term frequency in the document <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.02778em;">D</span></span></span></sp an>, <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">∣</mi><mi>D</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|D|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">∣</span><span class="mord mathit" style="margin-right:0.02778em;">D</span><span class="mord mathrm">∣</span></span></span></span> is the length of the document <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;" ></span><span class="base textstyle uncramped"><span class="mord mathit" >style="margin-right:0.02778em;">D</span></span></span></span> in words, and ><span class="katex"><span >class="katex-mathml"><math><semantics><mrow><mi>a</mi><mi>v</mi><mi>g</mi><mi>d</mi><mi>l</mi></mrow><annotation > >encoding="application/x-tex">avgdl</annotation></semantics></math></span><span > class="katex-html" aria-hidden="true"><span class="strut" >style="height:0.69444em;"></span><span class="strut bottom" >style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span >class="base textstyle uncramped"><span class="mord mathit">a</span><span >class="mord mathit" style="margin-right:0.03588em;">v</span><span class="mord >mathit" style="margin-right:0.03588em;">g</span><span class="mord >mathit">d</span><span class="mord mathit" >style="margin-right:0.01968em;">l</span></span></span></span> is the average >document length in the text collection from which documents are drawn. <span >class="katex"><span cla ss="katex-mathml"><math><semantics><mrow><msub><mi>k</mi><mrow><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">k_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03148em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</s pan></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">b</span></span></span></span> are free parameters, usually chosen, in absence of an advanced optimization, as <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>k</mi><mrow><mn>1</mn></mrow></msub><mo>∈</mo><mo>[</mo><mn>1</mn><mi mathvariant="normal">.</mi><mn>2</mn><mo separator="true">,</mo><mn>2</mn><mi mathvariant="normal">.</mi><mn>0</mn><mo>]</mo></mrow><annotation encoding="application/x-tex">k_{1} \in [1.2,2.0]</annotation></semantics></math></span><span class="katex-html" aria-hidden="t rue"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03148em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mrel">∈</span><span class="mopen">[</span><span class="mord mathrm">1</span><span class="mord mathrm">.</span><span class="mord mathrm">2</span><span class="mpunct">,</span><span cla ss="mord mathrm">2</span><span class="mord mathrm">.</span><span class="mord mathrm">0</span><span class="mclose">]</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>b</mi><mo>=</mo><mn>0</mn><mi mathvariant="normal">.</mi><mn>7</mn><mn>5</mn></mrow><annotation encoding="application/x-tex">b = 0.75</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">b</span><span class="mrel">=</span><span class="mord mathrm">0</span><span class="mord mathrm">.</span><span class="mord mathrm">7</span><span class="mord mathrm">5</span></span></span></span>.</p> +<p>BM25 can also be applied for term weighing, showing how important a word is to a document in a collection or corpus, as follows:</p> +<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi><mi>c</mi><mi>o</mi><mi>r</mi><mi>e</mi><mo>(</mo><msub><mi>t</mi><mrow><mi>i</mi></mrow></msub><mo separator="true">,</mo><mi>D</mi><mo>)</mo><mo>=</mo><mi>I</mi><mi>D</mi><mi>F</mi><mo>(</mo><msub><mi>t</mi><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>⋅</mo><mfrac><mrow><mi>t</mi><mi>f</mi><mo>(</mo><msub><mi>t</mi><mrow><mi>i</mi></mrow></msub><mo separator="true">,</mo><mi>D</mi><mo>)</mo><mo>⋅</mo><mo>(</mo><msub><mi>k</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>t</mi><mi>f</mi><mo>(</mo><msub><mi>t</mi><mrow><mi>i</mi></mrow></msub><mo separator="true">,</mo><mi>D</mi><mo>)</mo><mo>+</mo><msub><mi>k</mi><mrow><mn>1</mn></mrow></msub><mo>⋅</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>b</mi><mo>+</mo><mi>b</mi><mo>⋅</mo><mfrac><mrow><mi mathvariant="normal">∣</mi><mi>D</mi><mi mathvariant="normal">&# x2223;</mi></mrow><mrow><mi>a</mi><mi>v</mi><mi>g</mi><mi>d</mi><mi>l</mi></mrow></mfrac><mo>)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex"> +score(t_{i}, D) = IDF(t_{i}) \cdot \frac{tf(t_{i},D) \cdot (k_{1}+1)}{tf(t_{i},D) + k_{1} \cdot (1 - b + b \cdot \frac{|D|}{avgdl})} +</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:1.427em;"></span><span class="strut bottom" style="height:2.808108em;vertical-align:-1.381108em;"></span><span class="base displaystyle textstyle uncramped"><span class="mord mathit">s</span><span class="mord mathit">c</span><span class="mord mathit">o</span><span class="mord mathit" style="margin-right:0.02778em;">r</span><span class="mord mathit">e</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">t</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">i</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 si ze5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.02778em;">D</span><span class="mclose">)</span><span class="mrel">=</span><span class="mord mathit" style="margin-right:0.07847em;">I</span><span class="mord mathit" style="margin-right:0.02778em;">D</span><span class="mord mathit" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">t</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">i</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">& #x200B;</span></span>​</span></span></span></span><span class="mclose">)</span><span class="mbin">⋅</span><span class="mord reset-textstyle displaystyle textstyle uncramped"><span class="mopen sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.9em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle cramped"><span class="mord textstyle cramped"><span class="mord mathit">t</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">t</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mo rd scriptstyle cramped mtight"><span class="mord mathit mtight">i</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mpunct">,</span><span class="mord mathit" style="margin-right:0.02778em;">D</span><span class="mclose">)</span><span class="mbin">+</span><span class="mord"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.03148em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>&# x200B;</span></span></span></span><span class="mbin">⋅</span><span class="mopen">(</span><span class="mord mathrm">1</span><span class="mbin">−</span><span class="mord mathit">b</span><span class="mbin">+</span><span class="mord mathit">b</span><span class="mbin">⋅</span><span class="mord reset-textstyle textstyle cramped"><span class="mopen sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span><span class="mfrac"><span class="vlist"><span style="top:0.345em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathit mtight">a</span><span class="mord mathit mtight" style="margin-right:0.03588em;">v</span><span class="mord mathit mtight" style="margin-right:0.03588em;">g</span><span class="mord mathit mtight">d</span><span class="mord mathit mtight" style="margin-righ t:0.01968em;">l</span></span></span></span><span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.485em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord scriptstyle cramped mtight"><span class="mord mathrm mtight">∣</span><span class="mord mathit mtight" style="margin-right:0.02778em;">D</span><span class="mord mathrm mtight">∣</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose sizing reset-size5 size5 reset-textstyle textstyle uncramped nulldelimiter"></span></span><span class="mclose">)</span></span></span></span>< span style="top:-0.22999999999999998em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped frac-line"></span></span><span style="top:-0.677em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle textstyle uncramped"><span class="mord textstyle uncramped"><span class="mord mathit">t</span><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathit">t</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-texts
<TRUNCATED>
