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<span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon 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Available in 11 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-11" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">11 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Factor_at%C3%ADpic_local" title="Factor atípic local – Catalan" lang="ca" hreflang="ca" data-title="Factor atípic local" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Local_Outlier_Factor" title="Local Outlier Factor – German" lang="de" hreflang="de" data-title="Local Outlier Factor" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Valor_at%C3%ADpico_local" title="Valor atípico local – Spanish" lang="es" hreflang="es" data-title="Valor atípico local" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%A7%DA%A9%D8%AA%D9%88%D8%B1_%D9%BE%D8%B1%D8%AA%DB%8C_%D9%85%D8%AD%D9%84%DB%8C" title="فاکتور پرتی محلی – Persian" lang="fa" hreflang="fa" data-title="فاکتور پرتی محلی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Faktor_pencilan_setempat" title="Faktor pencilan setempat – Indonesian" lang="id" hreflang="id" data-title="Faktor pencilan setempat" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B1%80%E6%89%80%E5%A4%96%E3%82%8C%E5%80%A4%E5%9B%A0%E5%AD%90%E6%B3%95" title="局所外れ値因子法 – Japanese" lang="ja" hreflang="ja" data-title="局所外れ値因子法" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9B%D0%BE%D0%BA%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D1%83%D1%80%D0%BE%D0%B2%D0%B5%D0%BD%D1%8C_%D0%B2%D1%8B%D0%B1%D1%80%D0%BE%D1%81%D0%B0" title="Локальный уровень выброса – Russian" lang="ru" hreflang="ru" data-title="Локальный уровень выброса" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Faktor_lokalnih_neobi%C4%8Dnih_vrednosti" title="Faktor lokalnih neobičnih vrednosti – Serbian" lang="sr" hreflang="sr" data-title="Faktor lokalnih neobičnih vrednosti" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Faktor_lokalnih_neobi%C4%8Dnih_vrednosti" title="Faktor lokalnih neobičnih vrednosti – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Faktor lokalnih neobičnih vrednosti" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%B5%D1%84%D1%96%D1%86%D1%96%D1%94%D0%BD%D1%82_%D0%BB%D0%BE%D0%BA%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B3%D0%BE_%D0%B2%D1%96%D0%B4%D1%85%D0%B8%D0%BB%D0%B5%D0%BD%D0%BD%D1%8F" title="Коефіцієнт локального відхилення – Ukrainian" lang="uk" hreflang="uk" data-title="Коефіцієнт локального відхилення" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Nh%C3%A2n_t%E1%BB%91_ngo%E1%BA%A1i_lai_c%E1%BB%A5c_b%E1%BB%99" title="Nhân tố 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Local_Outlier_Factor&redirect=no" class="mw-redirect" title="Local Outlier Factor">Local Outlier Factor</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Algorithm for anomaly detection</div> <style data-mw-deduplicate="TemplateStyles:r1244144826">.mw-parser-output .machine-learning-list-title{background-color:#ddddff}html.skin-theme-clientpref-night .mw-parser-output .machine-learning-list-title{background-color:#222}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .machine-learning-list-title{background-color:#222}}</style> <style 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class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Machine_learning" title="Machine learning">Machine learning</a><br />and <a href="/wiki/Data_mining" title="Data mining">data mining</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Paradigms</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Supervised_learning" title="Supervised learning">Supervised learning</a></li> <li><a href="/wiki/Unsupervised_learning" title="Unsupervised learning">Unsupervised learning</a></li> <li><a href="/wiki/Semi-supervised_learning" class="mw-redirect" title="Semi-supervised learning">Semi-supervised learning</a></li> <li><a href="/wiki/Self-supervised_learning" title="Self-supervised learning">Self-supervised learning</a></li> <li><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></li> <li><a href="/wiki/Meta-learning_(computer_science)" title="Meta-learning (computer science)">Meta-learning</a></li> <li><a href="/wiki/Online_machine_learning" title="Online machine learning">Online learning</a></li> <li><a href="/wiki/Batch_learning" class="mw-redirect" title="Batch learning">Batch learning</a></li> <li><a href="/wiki/Curriculum_learning" title="Curriculum learning">Curriculum learning</a></li> <li><a href="/wiki/Rule-based_machine_learning" title="Rule-based machine learning">Rule-based learning</a></li> <li><a href="/wiki/Neuro-symbolic_AI" title="Neuro-symbolic AI">Neuro-symbolic AI</a></li> <li><a href="/wiki/Neuromorphic_engineering" class="mw-redirect" title="Neuromorphic engineering">Neuromorphic engineering</a></li> <li><a href="/wiki/Quantum_machine_learning" title="Quantum machine learning">Quantum machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Problems</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Statistical_classification" title="Statistical classification">Classification</a></li> <li><a href="/wiki/Generative_model" title="Generative model">Generative modeling</a></li> <li><a href="/wiki/Regression_analysis" title="Regression analysis">Regression</a></li> <li><a href="/wiki/Cluster_analysis" title="Cluster analysis">Clustering</a></li> <li><a href="/wiki/Dimensionality_reduction" title="Dimensionality reduction">Dimensionality reduction</a></li> <li><a href="/wiki/Density_estimation" title="Density estimation">Density estimation</a></li> <li><a href="/wiki/Anomaly_detection" title="Anomaly detection">Anomaly detection</a></li> <li><a href="/wiki/Data_cleaning" class="mw-redirect" title="Data cleaning">Data cleaning</a></li> <li><a href="/wiki/Automated_machine_learning" title="Automated machine learning">AutoML</a></li> <li><a href="/wiki/Association_rule_learning" title="Association rule learning">Association rules</a></li> <li><a href="/wiki/Semantic_analysis_(machine_learning)" title="Semantic analysis (machine learning)">Semantic analysis</a></li> <li><a href="/wiki/Structured_prediction" title="Structured prediction">Structured prediction</a></li> <li><a href="/wiki/Feature_engineering" title="Feature engineering">Feature engineering</a></li> <li><a href="/wiki/Feature_learning" title="Feature learning">Feature learning</a></li> <li><a href="/wiki/Learning_to_rank" title="Learning to rank">Learning to rank</a></li> <li><a href="/wiki/Grammar_induction" title="Grammar induction">Grammar induction</a></li> <li><a href="/wiki/Ontology_learning" title="Ontology learning">Ontology learning</a></li> <li><a href="/wiki/Multimodal_learning" title="Multimodal learning">Multimodal learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Supervised_learning" title="Supervised learning">Supervised learning</a><br /><span class="nobold"><span style="font-size:85%;">(<b><a href="/wiki/Statistical_classification" title="Statistical classification">classification</a></b> • <b><a href="/wiki/Regression_analysis" title="Regression analysis">regression</a></b>)</span></span> </div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Apprenticeship_learning" title="Apprenticeship learning">Apprenticeship learning</a></li> <li><a href="/wiki/Decision_tree_learning" title="Decision tree learning">Decision trees</a></li> <li><a href="/wiki/Ensemble_learning" title="Ensemble learning">Ensembles</a> <ul><li><a href="/wiki/Bootstrap_aggregating" title="Bootstrap aggregating">Bagging</a></li> <li><a href="/wiki/Boosting_(machine_learning)" title="Boosting (machine learning)">Boosting</a></li> <li><a href="/wiki/Random_forest" title="Random forest">Random forest</a></li></ul></li> <li><a href="/wiki/K-nearest_neighbors_algorithm" title="K-nearest neighbors algorithm"><i>k</i>-NN</a></li> <li><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></li> <li><a href="/wiki/Naive_Bayes_classifier" title="Naive Bayes classifier">Naive Bayes</a></li> <li><a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural networks</a></li> <li><a href="/wiki/Logistic_regression" title="Logistic regression">Logistic regression</a></li> <li><a href="/wiki/Perceptron" title="Perceptron">Perceptron</a></li> <li><a href="/wiki/Relevance_vector_machine" title="Relevance vector machine">Relevance vector machine (RVM)</a></li> <li><a href="/wiki/Support_vector_machine" title="Support vector machine">Support vector machine (SVM)</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Cluster_analysis" title="Cluster analysis">Clustering</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/BIRCH" title="BIRCH">BIRCH</a></li> <li><a href="/wiki/CURE_algorithm" title="CURE algorithm">CURE</a></li> <li><a href="/wiki/Hierarchical_clustering" title="Hierarchical clustering">Hierarchical</a></li> <li><a href="/wiki/K-means_clustering" title="K-means clustering"><i>k</i>-means</a></li> <li><a href="/wiki/Fuzzy_clustering" title="Fuzzy clustering">Fuzzy</a></li> <li><a href="/wiki/Expectation%E2%80%93maximization_algorithm" title="Expectation–maximization algorithm">Expectation–maximization (EM)</a></li> <li><br /><a href="/wiki/DBSCAN" title="DBSCAN">DBSCAN</a></li> <li><a href="/wiki/OPTICS_algorithm" title="OPTICS algorithm">OPTICS</a></li> <li><a href="/wiki/Mean_shift" title="Mean shift">Mean shift</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Dimensionality_reduction" title="Dimensionality reduction">Dimensionality reduction</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></li> <li><a href="/wiki/Canonical_correlation" title="Canonical correlation">CCA</a></li> <li><a href="/wiki/Independent_component_analysis" title="Independent component analysis">ICA</a></li> <li><a href="/wiki/Linear_discriminant_analysis" title="Linear discriminant analysis">LDA</a></li> <li><a href="/wiki/Non-negative_matrix_factorization" title="Non-negative matrix factorization">NMF</a></li> <li><a href="/wiki/Principal_component_analysis" title="Principal component analysis">PCA</a></li> <li><a href="/wiki/Proper_generalized_decomposition" title="Proper generalized decomposition">PGD</a></li> <li><a href="/wiki/T-distributed_stochastic_neighbor_embedding" title="T-distributed stochastic neighbor embedding">t-SNE</a></li> <li><a href="/wiki/Sparse_dictionary_learning" title="Sparse dictionary learning">SDL</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Structured_prediction" title="Structured prediction">Structured prediction</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Graphical_model" title="Graphical model">Graphical models</a> <ul><li><a href="/wiki/Bayesian_network" title="Bayesian network">Bayes net</a></li> <li><a href="/wiki/Conditional_random_field" title="Conditional random field">Conditional random field</a></li> <li><a href="/wiki/Hidden_Markov_model" title="Hidden Markov model">Hidden Markov</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Anomaly_detection" title="Anomaly detection">Anomaly detection</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Random_sample_consensus" title="Random sample consensus">RANSAC</a></li> <li><a href="/wiki/K-nearest_neighbors_algorithm" title="K-nearest neighbors algorithm"><i>k</i>-NN</a></li> <li><a class="mw-selflink selflink">Local outlier factor</a></li> <li><a href="/wiki/Isolation_forest" title="Isolation forest">Isolation forest</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">Artificial neural network</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Autoencoder" title="Autoencoder">Autoencoder</a></li> <li><a href="/wiki/Deep_learning" title="Deep learning">Deep learning</a></li> <li><a href="/wiki/Feedforward_neural_network" title="Feedforward neural network">Feedforward neural network</a></li> <li><a href="/wiki/Recurrent_neural_network" title="Recurrent neural network">Recurrent neural network</a> <ul><li><a href="/wiki/Long_short-term_memory" title="Long short-term memory">LSTM</a></li> <li><a href="/wiki/Gated_recurrent_unit" title="Gated recurrent unit">GRU</a></li> <li><a href="/wiki/Echo_state_network" title="Echo state network">ESN</a></li> <li><a href="/wiki/Reservoir_computing" title="Reservoir computing">reservoir computing</a></li></ul></li> <li><a href="/wiki/Boltzmann_machine" title="Boltzmann machine">Boltzmann machine</a> <ul><li><a href="/wiki/Restricted_Boltzmann_machine" title="Restricted Boltzmann machine">Restricted</a></li></ul></li> <li><a href="/wiki/Generative_adversarial_network" title="Generative adversarial network">GAN</a></li> <li><a href="/wiki/Diffusion_model" title="Diffusion model">Diffusion model</a></li> <li><a href="/wiki/Self-organizing_map" title="Self-organizing map">SOM</a></li> <li><a href="/wiki/Convolutional_neural_network" title="Convolutional neural network">Convolutional neural network</a> <ul><li><a href="/wiki/U-Net" title="U-Net">U-Net</a></li> <li><a href="/wiki/LeNet" title="LeNet">LeNet</a></li> <li><a href="/wiki/AlexNet" title="AlexNet">AlexNet</a></li> <li><a href="/wiki/DeepDream" title="DeepDream">DeepDream</a></li></ul></li> <li><a href="/wiki/Neural_radiance_field" title="Neural radiance field">Neural radiance field</a></li> <li><a href="/wiki/Transformer_(machine_learning_model)" class="mw-redirect" title="Transformer (machine learning model)">Transformer</a> <ul><li><a href="/wiki/Vision_transformer" title="Vision transformer">Vision</a></li></ul></li> <li><a href="/wiki/Mamba_(deep_learning_architecture)" title="Mamba (deep learning architecture)">Mamba</a></li> <li><a href="/wiki/Spiking_neural_network" title="Spiking neural network">Spiking neural network</a></li> <li><a href="/wiki/Memtransistor" title="Memtransistor">Memtransistor</a></li> <li><a href="/wiki/Electrochemical_RAM" title="Electrochemical RAM">Electrochemical RAM</a> (ECRAM)</li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)"><a href="/wiki/Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Q-learning" title="Q-learning">Q-learning</a></li> <li><a href="/wiki/State%E2%80%93action%E2%80%93reward%E2%80%93state%E2%80%93action" title="State–action–reward–state–action">SARSA</a></li> <li><a href="/wiki/Temporal_difference_learning" title="Temporal difference learning">Temporal difference (TD)</a></li> <li><a href="/wiki/Multi-agent_reinforcement_learning" title="Multi-agent reinforcement learning">Multi-agent</a> <ul><li><a href="/wiki/Self-play_(reinforcement_learning_technique)" class="mw-redirect" title="Self-play (reinforcement learning technique)">Self-play</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Learning with humans</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Active_learning_(machine_learning)" title="Active learning (machine learning)">Active learning</a></li> <li><a href="/wiki/Crowdsourcing" title="Crowdsourcing">Crowdsourcing</a></li> <li><a href="/wiki/Human-in-the-loop" title="Human-in-the-loop">Human-in-the-loop</a></li> <li><a href="/wiki/Reinforcement_learning_from_human_feedback" title="Reinforcement learning from human feedback">RLHF</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Model diagnostics</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Coefficient_of_determination" title="Coefficient of determination">Coefficient of determination</a></li> <li><a href="/wiki/Confusion_matrix" title="Confusion matrix">Confusion matrix</a></li> <li><a href="/wiki/Learning_curve_(machine_learning)" title="Learning curve (machine learning)">Learning curve</a></li> <li><a href="/wiki/Receiver_operating_characteristic" title="Receiver operating characteristic">ROC curve</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Mathematical foundations</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Kernel_machines" class="mw-redirect" title="Kernel machines">Kernel machines</a></li> <li><a href="/wiki/Bias%E2%80%93variance_tradeoff" title="Bias–variance tradeoff">Bias–variance tradeoff</a></li> <li><a href="/wiki/Computational_learning_theory" title="Computational learning theory">Computational learning theory</a></li> <li><a href="/wiki/Empirical_risk_minimization" title="Empirical risk minimization">Empirical risk minimization</a></li> <li><a href="/wiki/Occam_learning" title="Occam learning">Occam learning</a></li> <li><a href="/wiki/Probably_approximately_correct_learning" title="Probably approximately correct learning">PAC learning</a></li> <li><a href="/wiki/Statistical_learning_theory" title="Statistical learning theory">Statistical learning</a></li> <li><a href="/wiki/Vapnik%E2%80%93Chervonenkis_theory" title="Vapnik–Chervonenkis theory">VC theory</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Journals and conferences</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/ECML_PKDD" title="ECML PKDD">ECML PKDD</a></li> <li><a href="/wiki/Conference_on_Neural_Information_Processing_Systems" title="Conference on Neural Information Processing Systems">NeurIPS</a></li> <li><a href="/wiki/International_Conference_on_Machine_Learning" title="International Conference on Machine Learning">ICML</a></li> <li><a href="/wiki/International_Conference_on_Learning_Representations" title="International Conference on Learning Representations">ICLR</a></li> <li><a href="/wiki/International_Joint_Conference_on_Artificial_Intelligence" title="International Joint Conference on Artificial Intelligence">IJCAI</a></li> <li><a href="/wiki/Machine_Learning_(journal)" title="Machine Learning (journal)">ML</a></li> <li><a href="/wiki/Journal_of_Machine_Learning_Research" title="Journal of Machine Learning Research">JMLR</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed machine-learning-list-title"><div class="sidebar-list-title" style="border-top:1px solid #aaa; text-align:center;;color: var(--color-base)">Related articles</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Glossary_of_artificial_intelligence" title="Glossary of artificial intelligence">Glossary of artificial intelligence</a></li> <li><a href="/wiki/List_of_datasets_for_machine-learning_research" title="List of datasets for machine-learning research">List of datasets for machine-learning research</a> <ul><li><a href="/wiki/List_of_datasets_in_computer_vision_and_image_processing" title="List of datasets in computer vision and image processing">List of datasets in computer vision and image processing</a></li></ul></li> <li><a href="/wiki/Outline_of_machine_learning" title="Outline of machine learning">Outline of machine learning</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Machine_learning" title="Template:Machine learning"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Machine_learning" title="Template talk:Machine learning"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Machine_learning" title="Special:EditPage/Template:Machine learning"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Anomaly_detection" title="Anomaly detection">anomaly detection</a>, the <b>local outlier factor</b> (<b>LOF</b>) is an algorithm proposed by Markus M. Breunig, <a href="/wiki/Hans-Peter_Kriegel" title="Hans-Peter Kriegel">Hans-Peter Kriegel</a>, Raymond T. Ng and Jörg Sander in 2000 for finding anomalous data points by measuring the local deviation of a given data point with respect to its neighbours.<sup id="cite_ref-orig_paper_1-0" class="reference"><a href="#cite_note-orig_paper-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>LOF shares some concepts with <a href="/wiki/DBSCAN" title="DBSCAN">DBSCAN</a> and <a href="/wiki/OPTICS_algorithm" title="OPTICS algorithm">OPTICS</a> such as the concepts of "core distance" and "reachability distance", which are used for local density estimation.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Basic_idea">Basic idea</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Local_outlier_factor&action=edit&section=1" title="Edit section: Basic idea"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:LOF-idea.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/LOF-idea.svg/250px-LOF-idea.svg.png" decoding="async" width="250" height="262" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/LOF-idea.svg/375px-LOF-idea.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/LOF-idea.svg/500px-LOF-idea.svg.png 2x" data-file-width="124" data-file-height="130" /></a><figcaption>Basic idea of LOF: comparing the local density of a point with the densities of its neighbors. A has a much lower density than its neighbors.</figcaption></figure> <p>The local outlier factor is based on a concept of a local density, where locality is given by <i>k</i> nearest neighbors, whose distance is used to estimate the density. By comparing the local density of an object to the local densities of its neighbors, one can identify regions of similar density, and points that have a substantially lower density than their neighbors. These are considered to be <a href="/wiki/Outlier" title="Outlier">outliers</a>. </p><p>The local density is estimated by the typical distance at which a point can be "reached" from its neighbors. The definition of "reachability distance" used in LOF is an additional measure to produce more stable results within clusters. The "reachability distance" used by LOF has some subtle details that are often found incorrect in secondary sources, e.g., in the textbook of Ethem Alpaydin.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Formal">Formal</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Local_outlier_factor&action=edit&section=2" title="Edit section: Formal"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml"><i>k</i>-distance(<i>A</i>)</span> be the distance of the object <i>A</i> to the <i>k</i>-th nearest neighbor. Note that the set of the <i>k</i> nearest neighbors includes all objects at this distance, which can in the case of a "tie" be more than <i>k</i> objects. We denote the set of <i>k</i> nearest neighbors as <span class="texhtml">N<sub><i>k</i></sub>(A)</span>. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Reachability-distance.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Reachability-distance.svg/250px-Reachability-distance.svg.png" decoding="async" width="250" height="262" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Reachability-distance.svg/375px-Reachability-distance.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Reachability-distance.svg/500px-Reachability-distance.svg.png 2x" data-file-width="124" data-file-height="130" /></a><figcaption>Illustration of the reachability distance. Objects <i>B</i> and <i>C</i> have the same reachability distance (<span class="texhtml">k=3</span>), while <span class="texhtml"><i>D</i></span> is not a <i>k</i> nearest neighbor</figcaption></figure> <p>This distance is used to define what is called <i>reachability distance</i>: </p><p><span class="texhtml">reachability-distance<sub><i>k</i></sub>(<i>A</i>,<i>B</i>)=max{<i>k</i>-distance(<i>B</i>), d(<i>A</i>,<i>B</i>)} </span> </p><p>In words, the <i>reachability distance</i> of an object <i>A</i> <i>from</i> <i>B</i> is the true distance of the two objects, but at least the <span class="texhtml"><i>k</i>-distance</span> of <i>B</i>. Objects that belong to the <i>k</i> nearest neighbors of <i>B</i> (the "core" of <i>B</i>, see <a href="/wiki/DBSCAN" title="DBSCAN">DBSCAN cluster analysis</a>) are considered to be equally distant. The reason for this is to reduce the statistical fluctuations between all points <i>A</i> close to <i>B</i>, where increasing the value for <i>k</i> increases the smoothing effect.<sup id="cite_ref-orig_paper_1-1" class="reference"><a href="#cite_note-orig_paper-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Note that this is not a <a href="/wiki/Distance" title="Distance">distance</a> in the mathematical definition, since it is not symmetric. (While it is a common mistake<sup id="cite_ref-generalized_4-0" class="reference"><a href="#cite_note-generalized-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> to always use the <span class="texhtml"><i>k</i>-distance(A)</span>, this yields a slightly different method, referred to as Simplified-LOF<sup id="cite_ref-generalized_4-1" class="reference"><a href="#cite_note-generalized-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>) </p><p>The <i>local reachability density</i> of an object <i>A</i> is defined by </p><p><span class="texhtml">lrd<sub><i>k</i></sub>(A):=1<big><big>/(</big></big><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">Σ<sub>B ∈ N<sub><i>k</i></sub>(A)</sub>reachability-distance<sub><i>k</i></sub>(A, B)</span><span class="sr-only">/</span><span class="den">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">N<sub><i>k</i></sub>(A)</span>|</span></span>⁠</span><big><big>)</big></big></span> </p><p>which is the inverse of the average reachability distance of the object <i>A</i> <i>from</i> its neighbors. Note that it is not the average reachability of the neighbors from <i>A</i> (which by definition would be the <span class="texhtml"><i>k</i>-distance(A)</span>), but the distance at which <i>A</i> can be "reached" <i>from</i> its neighbors. With duplicate points, this value can become infinite. </p><p>The local reachability densities are then compared with those of the neighbors using </p><p><span class="texhtml"> LOF<sub><i>k</i></sub>(A):=<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">Σ<sub>B ∈ N<sub><i>k</i></sub>(A)</sub><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">lrd<sub><i>k</i></sub>(B)</span><span class="sr-only">/</span><span class="den">lrd<sub><i>k</i></sub>(A)</span></span>⁠</span></span><span class="sr-only">/</span><span class="den">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">N<sub><i>k</i></sub>(A)</span>|</span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">Σ<sub>B ∈ N<sub><i>k</i></sub>(A)</sub>lrd<sub><i>k</i></sub>(B)</span><span class="sr-only">/</span><span class="den">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">N<sub><i>k</i></sub>(A)</span>| · lrd<sub><i>k</i></sub>(A)</span></span>⁠</span> </span> </p><p>which is the <i>average local reachability density of the neighbors</i> divided by the object's own local reachability density. A value of approximately <span class="texhtml">1</span> indicates that the object is comparable to its neighbors (and thus not an outlier). A value below <span class="texhtml">1</span> indicates a denser region (which would be an inlier), while values significantly larger than <span class="texhtml">1</span> indicate outliers. </p><p><span class="texhtml">LOF(k) ~ 1</span> means <b>Similar density as neighbors,</b> </p><p><span class="texhtml">LOF(k) < 1</span> means <b>Higher density than neighbors (Inlier),</b> </p><p><span class="texhtml">LOF(k) > 1</span> means <b>Lower density than neighbors (Outlier)</b> </p> <div class="mw-heading mw-heading2"><h2 id="Advantages">Advantages</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Local_outlier_factor&action=edit&section=3" title="Edit section: Advantages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:LOF.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/LOF.svg/400px-LOF.svg.png" decoding="async" width="400" height="357" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/LOF.svg/600px-LOF.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/59/LOF.svg/800px-LOF.svg.png 2x" data-file-width="655" data-file-height="585" /></a><figcaption>LOF scores as visualized by <a href="/wiki/Environment_for_DeveLoping_KDD-Applications_Supported_by_Index-Structures" class="mw-redirect" title="Environment for DeveLoping KDD-Applications Supported by Index-Structures">ELKI</a>. While the upper right cluster has a comparable density to the outliers close to the bottom left cluster, they are detected correctly.</figcaption></figure> <p>Due to the local approach, LOF is able to identify outliers in a data set that would not be outliers in another area of the data set. For example, a point at a "small" distance to a very dense cluster is an outlier, while a point within a sparse cluster might exhibit similar distances to its neighbors. </p><p>While the geometric intuition of LOF is only applicable to low-dimensional vector spaces, the algorithm can be applied in any context a dissimilarity function can be defined. It has experimentally been shown to work very well in numerous setups, often outperforming the competitors, for example in <a href="/wiki/Network_intrusion_detection_system" class="mw-redirect" title="Network intrusion detection system">network intrusion detection</a><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and on processed classification benchmark data.<sup id="cite_ref-CamposZimek2016_6-0" class="reference"><a href="#cite_note-CamposZimek2016-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>The LOF family of methods can be easily generalized and then applied to various other problems, such as detecting outliers in geographic data, video streams or authorship networks.<sup id="cite_ref-generalized_4-2" class="reference"><a href="#cite_note-generalized-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Disadvantages_and_Extensions">Disadvantages and Extensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Local_outlier_factor&action=edit&section=4" title="Edit section: Disadvantages and Extensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The resulting values are <a href="/wiki/Quotient" title="Quotient">quotient</a>-values and hard to interpret. A value of 1 or even less indicates a clear inlier, but there is no clear rule for when a point is an outlier. In one data set, a value of 1.1 may already be an outlier, in another dataset and parameterization (with strong local fluctuations) a value of 2 could still be an inlier. These differences can also occur within a dataset due to the locality of the method. There exist extensions of LOF that try to improve over LOF in these aspects: </p> <ul><li><i>Feature Bagging for Outlier Detection</i><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> runs LOF on multiple projections and combines the results for improved detection qualities in high dimensions. This is the first <a href="/wiki/Ensemble_learning" title="Ensemble learning">ensemble learning</a> approach to outlier detection, for other variants see ref.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li> <li><i>Local Outlier Probability</i> (LoOP)<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> is a method derived from LOF but using inexpensive local statistics to become less sensitive to the choice of the parameter <i>k</i>. In addition, the resulting values are scaled to a value range of <span class="texhtml">[0:1]</span>.</li> <li><i>Interpreting and Unifying Outlier Scores</i><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> proposes a normalization of the LOF outlier scores to the interval <span class="texhtml">[0:1]</span> using statistical scaling to increase <a href="/wiki/Usability" title="Usability">usability</a> and can be seen an improved version of the LoOP ideas.</li> <li><i>On Evaluation of Outlier Rankings and Outlier Scores</i><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> proposes methods for measuring similarity and diversity of methods for building advanced outlier detection <a href="/wiki/Ensemble_learning" title="Ensemble learning">ensembles</a> using LOF variants and other algorithms and improving on the Feature Bagging approach discussed above.</li> <li><i>Local outlier detection reconsidered: a generalized view on locality with applications to spatial, video, and network outlier detection</i><sup id="cite_ref-generalized_4-3" class="reference"><a href="#cite_note-generalized-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> discusses the general pattern in various local outlier detection methods (including, e.g., LOF, a simplified version of LOF and LoOP) and abstracts from this into a general framework. This framework is then applied, e.g., to detecting outliers in geographic data, video streams and authorship networks.</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Local_outlier_factor&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-orig_paper-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-orig_paper_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-orig_paper_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBreunigKriegelNgSander2000" class="citation conference cs1">Breunig, M. M.; <a href="/wiki/Hans-Peter_Kriegel" title="Hans-Peter Kriegel">Kriegel, H.-P.</a>; Ng, R. T.; Sander, J. (2000). <a rel="nofollow" class="external text" href="http://www.dbs.ifi.lmu.de/Publikationen/Papers/LOF.pdf"><i>LOF: Identifying Density-based Local Outliers</i></a> <span class="cs1-format">(PDF)</span>. <i>Proceedings of the 2000 ACM SIGMOD International Conference on Management of Data</i>. <a href="/wiki/SIGMOD" title="SIGMOD">SIGMOD</a>. pp. 93–104. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F335191.335388">10.1145/335191.335388</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-58113-217-4" title="Special:BookSources/1-58113-217-4"><bdi>1-58113-217-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=conference&rft.jtitle=Proceedings+of+the+2000+ACM+SIGMOD+International+Conference+on+Management+of+Data&rft.atitle=LOF%3A+Identifying+Density-based+Local+Outliers&rft.pages=93-104&rft.date=2000&rft_id=info%3Adoi%2F10.1145%2F335191.335388&rft.isbn=1-58113-217-4&rft.aulast=Breunig&rft.aufirst=M.+M.&rft.au=Kriegel%2C+H.-P.&rft.au=Ng%2C+R.+T.&rft.au=Sander%2C+J.&rft_id=http%3A%2F%2Fwww.dbs.ifi.lmu.de%2FPublikationen%2FPapers%2FLOF.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocal+outlier+factor" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBreunigKriegelNgSander1999" class="citation book cs1">Breunig, M. M.; <a href="/wiki/Hans-Peter_Kriegel" title="Hans-Peter Kriegel">Kriegel, H.-P.</a>; Ng, R. T.; Sander, J. R. (1999). <a rel="nofollow" class="external text" href="http://www.dbs.ifi.lmu.de/Publikationen/Papers/PKDD99-Outlier.pdf">"OPTICS-OF: Identifying Local Outliers"</a> <span class="cs1-format">(PDF)</span>. <i>Principles of Data Mining and Knowledge Discovery</i>. Lecture Notes in Computer Science. Vol. 1704. pp. 262–270. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-48247-5_28">10.1007/978-3-540-48247-5_28</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-66490-1" title="Special:BookSources/978-3-540-66490-1"><bdi>978-3-540-66490-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=OPTICS-OF%3A+Identifying+Local+Outliers&rft.btitle=Principles+of+Data+Mining+and+Knowledge+Discovery&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=262-270&rft.date=1999&rft_id=info%3Adoi%2F10.1007%2F978-3-540-48247-5_28&rft.isbn=978-3-540-66490-1&rft.aulast=Breunig&rft.aufirst=M.+M.&rft.au=Kriegel%2C+H.-P.&rft.au=Ng%2C+R.+T.&rft.au=Sander%2C+J.+R.&rft_id=http%3A%2F%2Fwww.dbs.ifi.lmu.de%2FPublikationen%2FPapers%2FPKDD99-Outlier.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocal+outlier+factor" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlpaydin2020" class="citation book cs1">Alpaydin, Ethem (2020). <i>Introduction to machine learning</i> (Fourth ed.). 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"Local outlier detection reconsidered: A generalized view on locality with applications to spatial, video, and network outlier detection". <i>Data Mining and Knowledge Discovery</i>. <b>28</b>: 190–237. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10618-012-0300-z">10.1007/s10618-012-0300-z</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:19036098">19036098</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Data+Mining+and+Knowledge+Discovery&rft.atitle=Local+outlier+detection+reconsidered%3A+A+generalized+view+on+locality+with+applications+to+spatial%2C+video%2C+and+network+outlier+detection&rft.volume=28&rft.pages=190-237&rft.date=2012&rft_id=info%3Adoi%2F10.1007%2Fs10618-012-0300-z&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A19036098%23id-name%3DS2CID&rft.aulast=Schubert&rft.aufirst=E.&rft.au=Zimek%2C+A.&rft.au=Kriegel%2C+H.+-P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocal+outlier+factor" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLazarevic,_A.Ozgur,_A.Ertoz,_L.Srivastava,_J.2003" class="citation journal cs1">Lazarevic, A.; Ozgur, A.; Ertoz, L.; Srivastava, J.; Kumar, V. 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Proceedings of the 2012 SIAM International Conference on Data Mining. pp. 1047–1058. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.300.7205">10.1.1.300.7205</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F1.9781611972825.90">10.1137/1.9781611972825.90</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-61197-232-0" title="Special:BookSources/978-1-61197-232-0"><bdi>978-1-61197-232-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=On+Evaluation+of+Outlier+Rankings+and+Outlier+Scores&rft.pages=1047-1058&rft.date=2012&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.300.7205%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1137%2F1.9781611972825.90&rft.isbn=978-1-61197-232-0&rft.aulast=Schubert&rft.aufirst=E.&rft.au=Wojdanowski%2C+R.&rft.au=Zimek%2C+A.&rft.au=Kriegel%2C+H.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALocal+outlier+factor" class="Z3988"></span></span> </li> </ol></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Local_outlier_factor&oldid=1224921028">https://en.wikipedia.org/w/index.php?title=Local_outlier_factor&oldid=1224921028</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Statistical_outliers" title="Category:Statistical outliers">Statistical outliers</a></li><li><a href="/wiki/Category:Data_mining" title="Category:Data mining">Data mining</a></li><li><a href="/wiki/Category:Machine_learning_algorithms" title="Category:Machine learning algorithms">Machine learning algorithms</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">CS1 maint: location missing publisher</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 21 May 2024, at 08:43<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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