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class="vector-toc-numb">2.2</span> <span>Creating the bootstrap dataset</span> </div> </a> <ul id="toc-Creating_the_bootstrap_dataset-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Creating_the_out-of-bag_dataset" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Creating_the_out-of-bag_dataset"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Creating the out-of-bag dataset</span> </div> </a> <ul id="toc-Creating_the_out-of-bag_dataset-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Importance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Importance"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Importance</span> </div> </a> <ul id="toc-Importance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Creation_of_Decision_Trees" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Creation_of_Decision_Trees"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Creation of Decision Trees</span> </div> </a> <ul id="toc-Creation_of_Decision_Trees-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Random_Forests" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Random_Forests"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Random Forests</span> </div> </a> <ul id="toc-Random_Forests-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Improving_Random_Forests_and_Bagging" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Improving_Random_Forests_and_Bagging"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Improving Random Forests and Bagging</span> </div> </a> <ul id="toc-Improving_Random_Forests_and_Bagging-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algorithm_(classification)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algorithm_(classification)"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Algorithm (classification)</span> </div> </a> <ul id="toc-Algorithm_(classification)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example:_ozone_data" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Example:_ozone_data"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Example: ozone data</span> </div> </a> <ul id="toc-Example:_ozone_data-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Advantages_and_disadvantages" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Advantages_and_disadvantages"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Advantages and disadvantages</span> </div> </a> <ul id="toc-Advantages_and_disadvantages-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> 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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 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 class="mw-selflink selflink">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 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/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 href="/wiki/Local_outlier_factor" title="Local outlier factor">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 machine learning (ML), <b>bootstrap aggregating</b>, also called <b>bagging</b> (from <b>b</b>ootstrap <b>agg</b>regat<b>ing</b>) or <b>bootstrapping</b>, is an <a href="/wiki/Ensemble_learning" title="Ensemble learning">ensemble</a> <a href="/wiki/Metaheuristic" title="Metaheuristic">metaheuristic</a> for primarily reducing <a href="/wiki/Bias%E2%80%93variance_tradeoff" title="Bias–variance tradeoff">variance (as opposed to bias)</a>. It can also improve the <a href="/wiki/Stability_(learning_theory)" title="Stability (learning theory)">stability</a> and accuracy of ML <a href="/wiki/Statistical_classification" title="Statistical classification">classification</a> and <a href="/wiki/Regression_analysis" title="Regression analysis">regression</a> algorithms, and can reduce <a href="/wiki/Overfitting" title="Overfitting">overfitting</a>. Although it is usually applied to <a href="/wiki/Decision_tree_learning" title="Decision tree learning">decision tree</a> methods, it can be used with any type of method. Bagging is a special case of the <a href="/wiki/Ensemble_averaging_(machine_learning)" title="Ensemble averaging (machine learning)">ensemble averaging</a> approach. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Description_of_the_technique">Description of the technique</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrap_aggregating&action=edit&section=1" title="Edit section: Description of the technique"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a standard <a href="/wiki/Training,_validation,_and_test_data_sets" title="Training, validation, and test data sets">training set</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> of size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, bagging generates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> new training sets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f07b53d3212e08ca316a536c8aac0bbefa79ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle D_{i}}"></span>, each of size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d215ec5b3d3b48ac8ec46e7131e7b3c091c9114e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.079ex; height:2.509ex;" alt="{\displaystyle n'}"></span>, by <a href="/wiki/Sampling_(statistics)" title="Sampling (statistics)">sampling</a> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> <a href="/wiki/Probability_distribution#With_finite_support" title="Probability distribution">uniformly</a> and <a href="/wiki/Sampling_(statistics)#Replacement_of_selected_units" title="Sampling (statistics)">with replacement</a>. By sampling with replacement, some observations may be repeated in each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f07b53d3212e08ca316a536c8aac0bbefa79ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle D_{i}}"></span>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n'=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n'=n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ceccf2717d935dbec3cb1cff872f1b88d2568e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.573ex; height:2.509ex;" alt="{\displaystyle n'=n}"></span>, then for large <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f07b53d3212e08ca316a536c8aac0bbefa79ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle D_{i}}"></span> is expected to have the fraction (1 - 1/<i><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a></i>) (~63.2%) of the unique samples of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>, the rest being duplicates.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> This kind of sample is known as a <a href="/wiki/Bootstrap_(statistics)" class="mw-redirect" title="Bootstrap (statistics)">bootstrap</a> sample. Sampling with replacement ensures each bootstrap is independent from its peers, as it does not depend on previous chosen samples when sampling. Then, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> models are fitted using the above bootstrap samples and combined by averaging the output (for regression) or voting (for classification). </p> <figure class="mw-default-size mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Ensemble_Bagging.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Ensemble_Bagging.svg/440px-Ensemble_Bagging.svg.png" decoding="async" width="440" height="248" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Ensemble_Bagging.svg/660px-Ensemble_Bagging.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Ensemble_Bagging.svg/880px-Ensemble_Bagging.svg.png 2x" data-file-width="512" data-file-height="289" /></a><figcaption>An illustration for the concept of bootstrap aggregating</figcaption></figure> <p>Bagging leads to "improvements for unstable procedures",<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> which include, for example, <a href="/wiki/Artificial_neural_networks" class="mw-redirect" title="Artificial neural networks">artificial neural networks</a>, <a href="/wiki/Classification_and_regression_tree" class="mw-redirect" title="Classification and regression tree">classification and regression trees</a>, and subset selection in <a href="/wiki/Linear_regression" title="Linear regression">linear regression</a>.<sup id="cite_ref-:1_3-0" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Bagging was shown to improve preimage learning.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><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> On the other hand, it can mildly degrade the performance of stable methods such as <a href="/wiki/K-nearest_neighbors_algorithm" title="K-nearest neighbors algorithm"><i>k</i>-nearest neighbors</a>.<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Process_of_the_algorithm">Process of the algorithm</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrap_aggregating&action=edit&section=2" title="Edit section: Process of the algorithm"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Key_Terms">Key Terms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrap_aggregating&action=edit&section=3" title="Edit section: Key Terms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are three types of datasets in bootstrap aggregating. These are the <b>original, bootstrap, and out-of-bag datasets.</b> Each section below will explain how each dataset is made except for the original dataset. The original dataset is whatever information is given. </p> <div class="mw-heading mw-heading3"><h3 id="Creating_the_bootstrap_dataset">Creating the bootstrap dataset</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrap_aggregating&action=edit&section=4" title="Edit section: Creating the bootstrap dataset"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The bootstrap dataset is made by randomly picking objects from the original dataset. Also, <b>it must be the same size as the original dataset.</b> However, the difference is that the bootstrap dataset can have duplicate objects. Here is a simple example to demonstrate how it works along with the illustration below: </p><p><span typeof="mw:File"><a href="/wiki/File:Bootstrap_Example_2.png" class="mw-file-description" title="Bootstrap Example"><img alt="Bootstrap Example" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Bootstrap_Example_2.png/672px-Bootstrap_Example_2.png" decoding="async" width="672" height="330" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/f/fe/Bootstrap_Example_2.png 1.5x" data-file-width="975" data-file-height="479" /></a></span> </p><p>Suppose the <b>original dataset</b> is a <b>group of 12 people.</b> Their names are <b>Emily, Jessie, George, Constantine, Lexi, Theodore, John, James, Rachel, Anthony, Ellie, and Jamal.</b> </p><p>By randomly picking a group of names, let us say <b>our bootstrap dataset</b> had <b>James, Ellie, Constantine, Lexi, John, Constantine, Theodore, Constantine, Anthony, Lexi, Constantine, and Theodore.</b> In this case, the bootstrap sample contained four duplicates for Constantine, and two duplicates for Lexi, and Theodore. </p> <div class="mw-heading mw-heading3"><h3 id="Creating_the_out-of-bag_dataset">Creating the out-of-bag dataset</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrap_aggregating&action=edit&section=5" title="Edit section: Creating the out-of-bag dataset"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The out-of-bag dataset <b>represents the remaining people who were not in the bootstrap dataset.</b> It can be calculated by taking the difference between the original and the bootstrap datasets. In this case, the remaining samples who were not selected are <b>Emily, Jessie, George, Rachel, and Jamal.</b> Keep in mind that since both datasets are sets, when taking the difference the duplicate names are ignored in the bootstrap dataset. The illustration below shows how the math is done: </p><p><span typeof="mw:File"><a href="/wiki/File:Complete_Example_2.png" class="mw-file-description" title="Complete Example"><img alt="Complete Example" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Complete_Example_2.png/840px-Complete_Example_2.png" decoding="async" width="840" height="335" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/5/57/Complete_Example_2.png 1.5x" data-file-width="1083" data-file-height="432" /></a></span> </p> <div class="mw-heading mw-heading3"><h3 id="Importance">Importance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrap_aggregating&action=edit&section=6" title="Edit section: Importance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Creating the bootstrap and out-of-bag datasets is crucial since it is used to test the accuracy of a random forest algorithm. For example, a model that produces 50 trees using the bootstrap/out-of-bag datasets will have a better accuracy than if it produced 10 trees. Since the algorithm generates multiple trees and therefore multiple datasets the chance that an object is left out of the bootstrap dataset is low. The next few sections talk about how the random forest algorithm works in more detail. </p> <div class="mw-heading mw-heading3"><h3 id="Creation_of_Decision_Trees">Creation of Decision Trees</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrap_aggregating&action=edit&section=7" title="Edit section: Creation of Decision Trees"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The next step of the algorithm involves the generation of <a href="/wiki/Decision_tree" title="Decision tree">decision trees</a> from the bootstrapped dataset. To achieve this, the process examines each gene/feature and determines for how many samples the feature's presence or absence yields a positive or negative result. This information is then used to compute a <a href="/wiki/Confusion_matrix" title="Confusion matrix">confusion matrix</a>, which lists the true positives, false positives, true negatives, and false negatives of the feature when used as a classifier. These features are then ranked according to various <a href="/wiki/Decision_tree_learning" title="Decision tree learning">classification metrics</a> based on their confusion matrices. Some common metrics include estimate of positive correctness (calculated by subtracting false positives from true positives), measure of "goodness", and <a href="/wiki/Information_gain_in_decision_trees" class="mw-redirect" title="Information gain in decision trees">information gain</a>. These features are then used to partition the samples into two sets: those who possess the top feature, and those who do not. </p><p>The diagram below shows a decision tree of depth two being used to classify data. For example, a data point that exhibits Feature 1, but not Feature 2, will be given a "No". Another point that does not exhibit Feature 1, but does exhibit Feature 3, will be given a "Yes". </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Decision_Tree_Depth_2.png" class="mw-file-description" title="Decision Tree Depth 2"><img alt="Decision Tree Depth 2" src="//upload.wikimedia.org/wikipedia/commons/a/a8/Decision_Tree_Depth_2.png" decoding="async" width="881" height="401" class="mw-file-element" data-file-width="881" data-file-height="401" /></a></span> </p><p>This process is repeated recursively for successive levels of the tree until the desired depth is reached. At the very bottom of the tree, samples that test positive for the final feature are generally classified as positive, while those that lack the feature are classified as negative. These trees are then used as predictors to classify new data. </p> <div class="mw-heading mw-heading3"><h3 id="Random_Forests">Random Forests</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrap_aggregating&action=edit&section=8" title="Edit section: Random Forests"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The next part of the algorithm involves introducing yet another element of variability amongst the bootstrapped trees. In addition to each tree only examining a bootstrapped set of samples, only a small but consistent number of unique features are considered when ranking them as classifiers. This means that each tree only knows about the data pertaining to a small constant number of features, and a variable number of samples that is less than or equal to that of the original dataset. Consequently, the trees are more likely to return a wider array of answers, derived from more diverse knowledge. This results in a <a href="/wiki/Random_forest" title="Random forest">random forest</a>, which possesses numerous benefits over a single decision tree generated without randomness. In a random forest, each tree "votes" on whether or not to classify a sample as positive based on its features. The sample is then classified based on majority vote. An example of this is given in the diagram below, where the four trees in a random forest vote on whether or not a patient with mutations A, B, F, and G has cancer. Since three out of four trees vote yes, the patient is then classified as cancer positive. </p> <figure class="mw-halign-center" typeof="mw:File/Frameless"><a href="/wiki/File:Random_Forest_Diagram_Extra_Wide.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Random_Forest_Diagram_Extra_Wide.png/1035px-Random_Forest_Diagram_Extra_Wide.png" decoding="async" width="1035" height="461" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Random_Forest_Diagram_Extra_Wide.png/1553px-Random_Forest_Diagram_Extra_Wide.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/76/Random_Forest_Diagram_Extra_Wide.png 2x" data-file-width="2021" data-file-height="901" /></a><figcaption></figcaption></figure> <p>Because of their properties, random forests are considered one of the most accurate data mining algorithms, are less likely to <a href="/wiki/Overfitting" title="Overfitting">overfit</a> their data, and run quickly and efficiently even for large datasets.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> They are primarily useful for classification as opposed to <a href="/wiki/Regression_analysis" title="Regression analysis">regression</a>, which attempts to draw observed connections between statistical variables in a dataset. This makes random forests particularly useful in such fields as banking, healthcare, the stock market, and <a href="/wiki/E-commerce" title="E-commerce">e-commerce</a> where it is important to be able to predict future results based on past data.<sup id="cite_ref-:4_7-0" class="reference"><a href="#cite_note-:4-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> One of their applications would be as a useful tool for predicting cancer based on genetic factors, as seen in the above example. </p><p>There are several important factors to consider when designing a random forest. If the trees in the random forests are too deep, overfitting can still occur due to over-specificity. If the forest is too large, the algorithm may become less efficient due to an increased runtime. Random forests also do not generally perform well when given sparse data with little variability.<sup id="cite_ref-:4_7-1" class="reference"><a href="#cite_note-:4-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> However, they still have numerous advantages over similar data classification algorithms such as <a href="/wiki/Neural_network" title="Neural network">neural networks</a>, as they are much easier to interpret and generally require less data for training.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (June 2024)">citation needed</span></a></i>]</sup> As an integral component of random forests, bootstrap aggregating is very important to classification algorithms, and provides a critical element of variability that allows for increased accuracy when analyzing new data, as discussed below. </p> <div class="mw-heading mw-heading2"><h2 id="Improving_Random_Forests_and_Bagging">Improving Random Forests and Bagging</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrap_aggregating&action=edit&section=9" title="Edit section: Improving Random Forests and Bagging"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While the techniques described above utilize <a href="/wiki/Random_forest" title="Random forest">random forests</a> and <a href="/wiki/Bootstrapping" title="Bootstrapping">bagging</a> (otherwise known as bootstrapping), there are certain techniques that can be used in order to improve their execution and voting time, their prediction accuracy, and their overall performance. The following are key steps in creating an efficient random forest: </p> <ol><li>Specify the maximum depth of trees: Instead of allowing your random forest to continue until all nodes are pure, it is better to cut it off at a certain point in order to further decrease chances of overfitting.</li> <li>Prune the dataset: Using an extremely large dataset may prove to create results that is less indicative of the data provided than a smaller set that more accurately represents what is being focused on. <ul><li>Continue pruning the data at each node split rather than just in the original bagging process.</li></ul></li> <li>Decide on accuracy or speed: Depending on the desired results, increasing or decreasing the number of trees within the forest can help. Increasing the number of trees generally provides more accurate results while decreasing the number of trees will provide quicker results.</li></ol> <table class="wikitable"> <caption>Pros and Cons of Random Forests and Bagging </caption> <tbody><tr> <th>Pros </th> <th>Cons </th></tr> <tr> <td>There are overall less requirements involved for normalization and scaling, making the use of random forests more convenient.<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> </td> <td>The algorithm may change significantly if there is a slight change to the data being bootstrapped and used within the forests.<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> In other words, random forests are incredibly dependent on their datasets, changing these can drastically change the individual trees' structures. </td></tr> <tr> <td>Easy data preparation. Data is prepared by creating a bootstrap set and a certain number of decision trees to build a random forest that also utilizes feature selection, as mentioned in the <u>Random Forests</u> section. </td> <td>Random Forests are more complex to implement than lone decision trees or other algorithms. This is because they take extra steps for bagging, as well as the need for recursion in order to produce an entire forest, which complicates implementation. Because of this, it requires much more computational power and computational resources. </td></tr> <tr> <td>Consisting of multiple <a href="/wiki/Decision_tree" title="Decision tree">decision trees</a>, forests are able to more accurately make predictions than single trees. </td> <td>Requires much more time to train the data compared to decision trees. Having a large forest can quickly begin to decrease the speed in which one's program operates because it has to traverse much more data even though each tree is using a smaller set of samples and features. </td></tr> <tr> <td>Works well with non-linear data. As most tree based algorithms use linear splits, using an ensemble of a set of trees works better than using a single tree on data that has nonlinear properties (i.e. most real world distributions). Working well with non-linear data is a huge advantage because other data mining techniques such as single decision trees do not handle this as well. </td> <td>Much easier to interpret than a random forest. A single tree can be walked by hand (by a human) leading to a somewhat "explainable" understanding for the analyst of what the tree is actually doing. As the number of trees and schemes grow for ensembling those trees into predictions, this reviewing becomes much more difficult if not impossible. </td></tr> <tr> <td>There is a lower risk of <a href="/wiki/Overfitting" title="Overfitting">overfitting</a> and runs efficiently on even large datasets.<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> This is the result of the random forest's use of bagging in conjunction with random feature selection. </td> <td>Does not predict beyond the range of the training data. This is a con because while bagging is often effective, all of the data is not being considered, therefore it cannot predict an entire dataset. </td></tr> <tr> <td>The random forest classifier operates with a high accuracy and speed.<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> Random forests are much faster than decision trees because of using a smaller dataset. </td> <td>To recreate specific results you need to keep track of the exact random seed used to generate the bootstrap sets. This may be important when collecting data for research or within a data mining class. Using random seeds is essential to the random forests, but can make it hard to support your statements based on forests if there is a failure to record the seeds. </td></tr> <tr> <td>Deals with <a href="/wiki/Missing_data" title="Missing data">missing data</a> and datasets with many outliers well. They deal with this by using <a href="/wiki/Binning_method" class="mw-redirect" title="Binning method">binning</a>, or by grouping values together to avoid values that are terribly far apart. </td> <td> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Algorithm_(classification)"><span id="Algorithm_.28classification.29"></span>Algorithm (classification)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrap_aggregating&action=edit&section=10" title="Edit section: Algorithm (classification)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Bagging_for_Classification_with_descripitons.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Bagging_for_Classification_with_descripitons.png/220px-Bagging_for_Classification_with_descripitons.png" decoding="async" width="220" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Bagging_for_Classification_with_descripitons.png/330px-Bagging_for_Classification_with_descripitons.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Bagging_for_Classification_with_descripitons.png/440px-Bagging_for_Classification_with_descripitons.png 2x" data-file-width="984" data-file-height="625" /></a><figcaption>Flow chart of the bagging algorithm when used for classification</figcaption></figure> <p>For classification, use a training set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>, Inducer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> and the number of bootstrap samples <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> as input. Generate a classifier <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fda87fa9eddc6a89e202bdebaa9a5e1a55dec9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.343ex;" alt="{\displaystyle C^{*}}"></span> as output<sup id="cite_ref-Bauer_12-0" class="reference"><a href="#cite_note-Bauer-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <ol><li>Create <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> new training sets <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f07b53d3212e08ca316a536c8aac0bbefa79ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle D_{i}}"></span>, from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> with replacement</li> <li>Classifier <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc49dc02c0ec8c86b67e7d10518ac791eda0bf22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.461ex; height:2.509ex;" alt="{\displaystyle C_{i}}"></span> is built from each set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f07b53d3212e08ca316a536c8aac0bbefa79ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle D_{i}}"></span> using <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> to determine the classification of set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f07b53d3212e08ca316a536c8aac0bbefa79ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle D_{i}}"></span></li> <li>Finally classifier <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fda87fa9eddc6a89e202bdebaa9a5e1a55dec9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.343ex;" alt="{\displaystyle C^{*}}"></span> is generated by using the previously created set of classifiers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc49dc02c0ec8c86b67e7d10518ac791eda0bf22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.461ex; height:2.509ex;" alt="{\displaystyle C_{i}}"></span> on the original dataset <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>, the classification predicted most often by the sub-classifiers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc49dc02c0ec8c86b67e7d10518ac791eda0bf22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.461ex; height:2.509ex;" alt="{\displaystyle C_{i}}"></span> is the final classification</li></ol> <pre>for i = 1 to m { D' = bootstrap sample from D (sample with replacement) Ci = I(D') } C*(x) = argmax #{i:Ci(x)=y} (most often predicted label y) y∈Y </pre> <div class="mw-heading mw-heading2"><h2 id="Example:_ozone_data">Example: ozone data</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrap_aggregating&action=edit&section=11" title="Edit section: Example: ozone data"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To illustrate the basic principles of bagging, below is an analysis on the relationship between <a href="/wiki/Ozone" title="Ozone">ozone</a> and temperature (data from <a href="/wiki/Peter_Rousseeuw" title="Peter Rousseeuw">Rousseeuw</a> and Leroy<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (May 2021)">clarification needed</span></a></i>]</sup> (1986), analysis done in <a href="/wiki/R_(programming_language)" title="R (programming language)">R</a>). </p><p>The relationship between temperature and ozone appears to be nonlinear in this dataset, based on the scatter plot. To mathematically describe this relationship, <a href="/wiki/Local_regression" title="Local regression">LOESS</a> smoothers (with bandwidth 0.5) are used. Rather than building a single smoother for the complete dataset, 100 <a href="/wiki/Bootstrap_(statistics)" class="mw-redirect" title="Bootstrap (statistics)">bootstrap</a> samples were drawn. Each sample is composed of a random subset of the original data and maintains a semblance of the master set's distribution and variability. For each bootstrap sample, a LOESS smoother was fit. Predictions from these 100 smoothers were then made across the range of the data. The black lines represent these initial predictions. The lines lack agreement in their predictions and tend to overfit their data points: evident by the wobbly flow of the lines. </p> <figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/File:Ozone.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/de/Ozone.png" decoding="async" width="431" height="325" class="mw-file-element" data-file-width="431" data-file-height="325" /></a><figcaption></figcaption></figure> <p>By taking the average of 100 smoothers, each corresponding to a subset of the original dataset, we arrive at one bagged predictor (red line). The red line's flow is stable and does not overly conform to any data point(s). </p> <div class="mw-heading mw-heading2"><h2 id="Advantages_and_disadvantages">Advantages and disadvantages</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrap_aggregating&action=edit&section=12" title="Edit section: Advantages and disadvantages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Advantages: </p> <ul><li>Many weak learners aggregated typically outperform a single learner over the entire set, and have less overfit</li> <li>Reduces variance in high-variance <a href="/wiki/Bias_(statistics)" title="Bias (statistics)">low-bias</a> weak learner,<sup id="cite_ref-:3_13-0" class="reference"><a href="#cite_note-:3-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> which can improve <a href="/wiki/Efficiency_(statistics)" title="Efficiency (statistics)">efficiency (statistics)</a></li> <li>Can be performed in <a href="/wiki/Parallel_Computing" class="mw-redirect" title="Parallel Computing">parallel</a>, as each separate bootstrap can be processed on its own before aggregation.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></li></ul> <p>Disadvantages: </p> <ul><li>For a weak learner with high bias, bagging will also carry high bias into its aggregate<sup id="cite_ref-:3_13-1" class="reference"><a href="#cite_note-:3-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li> <li>Loss of interpretability of a model.</li> <li>Can be computationally expensive depending on the dataset.</li></ul> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrap_aggregating&action=edit&section=13" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The concept of bootstrap aggregating is derived from the concept of bootstrapping which was developed by Bradley Efron.<sup id="cite_ref-:8_15-0" class="reference"><a href="#cite_note-:8-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Bootstrap aggregating was proposed by <a href="/wiki/Leo_Breiman" title="Leo Breiman">Leo Breiman</a> who also coined the abbreviated term "bagging" (<b>b</b>ootstrap <b>agg</b>regat<b>ing</b>). Breiman developed the concept of bagging in 1994 to improve classification by combining classifications of randomly generated training sets. He argued, "If perturbing the learning set can cause significant changes in the predictor constructed, then bagging can improve accuracy".<sup id="cite_ref-:1_3-1" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrap_aggregating&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Boosting_(machine_learning)" title="Boosting (machine learning)">Boosting (machine learning)</a></li> <li><a href="/wiki/Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Bootstrapping (statistics)</a></li> <li><a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross-validation (statistics)</a></li> <li><a href="/wiki/Out-of-bag_error" title="Out-of-bag error">Out-of-bag error</a></li> <li><a href="/wiki/Random_forest" title="Random forest">Random forest</a></li> <li><a href="/wiki/Random_subspace_method" title="Random subspace method">Random subspace method</a> (attribute bagging)</li> <li><a href="/wiki/Resampled_efficient_frontier" title="Resampled efficient frontier">Resampled efficient frontier</a></li> <li><a href="/wiki/Predictive_analytics" title="Predictive analytics">Predictive analysis: Classification and regression trees</a></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=Bootstrap_aggregating&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Aslam, Javed A.; Popa, Raluca A.; and Rivest, Ronald L. (2007); <a rel="nofollow" class="external text" href="http://people.csail.mit.edu/rivest/pubs/APR07.pdf"><i>On Estimating the Size and Confidence of a Statistical Audit</i></a>, Proceedings of the Electronic Voting Technology Workshop (EVT '07), Boston, MA, August 6, 2007. More generally, when drawing with replacement <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d215ec5b3d3b48ac8ec46e7131e7b3c091c9114e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.079ex; height:2.509ex;" alt="{\displaystyle n'}"></span> values out of a set of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> (different and equally likely), the expected number of unique draws is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(1-e^{-n'/n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>n</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(1-e^{-n'/n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26da6de761b567e2f32908ee83e6be3cb6d97222" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.127ex; height:3.343ex;" alt="{\displaystyle n(1-e^{-n'/n})}"></span>.</span> </li> <li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><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="CITEREFBreiman1996" class="citation journal cs1"><a href="/wiki/Leo_Breiman" title="Leo Breiman">Breiman, Leo</a> (1996). 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Retrieved <span class="nowrap">2021-11-26</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Corporate+Finance+Institute&rft.atitle=Random+Forest&rft_id=https%3A%2F%2Fcorporatefinanceinstitute.com%2Fresources%2Fknowledge%2Fother%2Frandom-forest%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrap+aggregating" class="Z3988"></span></span> </li> <li id="cite_note-Bauer-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bauer_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBauerKohavi1999" class="citation journal cs1">Bauer, Eric; Kohavi, Ron (1999). <a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1007515423169">"An Empirical Comparison of Voting Classification Algorithms: Bagging, Boosting, and Variants"</a>. <i>Machine Learning</i>. <b>36</b>: 108–109. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1007515423169">10.1023/A:1007515423169</a></span>. <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:1088806">1088806</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Machine+Learning&rft.atitle=An+Empirical+Comparison+of+Voting+Classification+Algorithms%3A+Bagging%2C+Boosting%2C+and+Variants&rft.volume=36&rft.pages=108-109&rft.date=1999&rft_id=info%3Adoi%2F10.1023%2FA%3A1007515423169&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A1088806%23id-name%3DS2CID&rft.aulast=Bauer&rft.aufirst=Eric&rft.au=Kohavi%2C+Ron&rft_id=https%3A%2F%2Fdoi.org%2F10.1023%252FA%253A1007515423169&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrap+aggregating" class="Z3988"></span></span> </li> <li id="cite_note-:3-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-:3_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:3_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://corporatefinanceinstitute.com/resources/knowledge/other/bagging-bootstrap-aggregation/">"What is Bagging (Bootstrap Aggregation)?"</a>. <i>CFI</i>. Corporate Finance Institute<span class="reference-accessdate">. Retrieved <span class="nowrap">December 5,</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=CFI&rft.atitle=What+is+Bagging+%28Bootstrap+Aggregation%29%3F&rft_id=https%3A%2F%2Fcorporatefinanceinstitute.com%2Fresources%2Fknowledge%2Fother%2Fbagging-bootstrap-aggregation%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrap+aggregating" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZoghni2020" class="citation web cs1">Zoghni, Raouf (September 5, 2020). <a rel="nofollow" class="external text" href="https://medium.com/swlh/bagging-bootstrap-aggregating-overview-b73ca019e0e9">"Bagging (Bootstrap Aggregating), Overview"</a>. The Startup – via Medium.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Bagging+%28Bootstrap+Aggregating%29%2C+Overview&rft.pub=The+Startup&rft.date=2020-09-05&rft.aulast=Zoghni&rft.aufirst=Raouf&rft_id=https%3A%2F%2Fmedium.com%2Fswlh%2Fbagging-bootstrap-aggregating-overview-b73ca019e0e9&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrap+aggregating" class="Z3988"></span></span> </li> <li id="cite_note-:8-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-:8_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEfron1979" class="citation journal cs1"><a href="/wiki/Bradley_Efron" title="Bradley Efron">Efron, B.</a> (1979). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176344552">"Bootstrap methods: Another look at the jackknife"</a>. <i><a href="/wiki/The_Annals_of_Statistics" class="mw-redirect" title="The Annals of Statistics">The Annals of Statistics</a></i>. <b>7</b> (1): 1–26. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176344552">10.1214/aos/1176344552</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Annals+of+Statistics&rft.atitle=Bootstrap+methods%3A+Another+look+at+the+jackknife&rft.volume=7&rft.issue=1&rft.pages=1-26&rft.date=1979&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176344552&rft.aulast=Efron&rft.aufirst=B.&rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faos%252F1176344552&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrap+aggregating" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bootstrap_aggregating&action=edit&section=16" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBreiman1996" class="citation journal cs1"><a href="/wiki/Leo_Breiman" title="Leo Breiman">Breiman, Leo</a> (1996). "Bagging predictors". <i><a href="/wiki/Machine_Learning_(journal)" title="Machine Learning (journal)">Machine Learning</a></i>. <b>24</b> (2): 123–140. <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=10.1.1.32.9399">10.1.1.32.9399</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/10.1007%2FBF00058655">10.1007/BF00058655</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:47328136">47328136</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Machine+Learning&rft.atitle=Bagging+predictors&rft.volume=24&rft.issue=2&rft.pages=123-140&rft.date=1996&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.32.9399%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A47328136%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF00058655&rft.aulast=Breiman&rft.aufirst=Leo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrap+aggregating" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlfaro2012" class="citation journal cs1">Alfaro, E., Gámez, M. and García, N. (2012). <a rel="nofollow" class="external text" href="https://cran.r-project.org/package=adabag">"adabag: An R package for classification with AdaBoost.M1, AdaBoost-SAMME and Bagging"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=adabag%3A+An+R+package+for+classification+with+AdaBoost.M1%2C+AdaBoost-SAMME+and+Bagging&rft.date=2012&rft.aulast=Alfaro&rft.aufirst=E.%2C+G%C3%A1mez%2C+M.+and+Garc%C3%ADa%2C+N.&rft_id=https%3A%2F%2Fcran.r-project.org%2Fpackage%3Dadabag&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrap+aggregating" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Cite journal requires <code class="cs1-code">|journal=</code> (<a href="/wiki/Help:CS1_errors#missing_periodical" title="Help:CS1 errors">help</a>)</span><span class="cs1-maint citation-comment">CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKotsiantis2014" class="citation journal cs1">Kotsiantis, Sotiris (2014). "Bagging and boosting variants for handling classifications problems: a survey". <i>Knowledge Eng. Review</i>. <b>29</b> (1): 78–100. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0269888913000313">10.1017/S0269888913000313</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:27301684">27301684</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Knowledge+Eng.+Review&rft.atitle=Bagging+and+boosting+variants+for+handling+classifications+problems%3A+a+survey&rft.volume=29&rft.issue=1&rft.pages=78-100&rft.date=2014&rft_id=info%3Adoi%2F10.1017%2FS0269888913000313&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A27301684%23id-name%3DS2CID&rft.aulast=Kotsiantis&rft.aufirst=Sotiris&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrap+aggregating" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoehmkeGreenwell2019" class="citation book cs1">Boehmke, Bradley; Greenwell, Brandon (2019). "Bagging". <i>Hands-On Machine Learning with R</i>. Chapman & Hall. pp. 191–202. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-138-49568-5" title="Special:BookSources/978-1-138-49568-5"><bdi>978-1-138-49568-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Bagging&rft.btitle=Hands-On+Machine+Learning+with+R&rft.pages=191-202&rft.pub=Chapman+%26+Hall&rft.date=2019&rft.isbn=978-1-138-49568-5&rft.aulast=Boehmke&rft.aufirst=Bradley&rft.au=Greenwell%2C+Brandon&rfr_id=info%3Asid%2Fen.wikipedia.org%3ABootstrap+aggregating" class="Z3988"></span></li></ul>    </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=Bootstrap_aggregating&oldid=1255351719">https://en.wikipedia.org/w/index.php?title=Bootstrap_aggregating&oldid=1255351719</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:Ensemble_learning" title="Category:Ensemble learning">Ensemble learning</a></li><li><a href="/wiki/Category:Machine_learning_algorithms" title="Category:Machine learning algorithms">Machine learning algorithms</a></li><li><a href="/wiki/Category:Computational_statistics" title="Category:Computational statistics">Computational statistics</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: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><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_June_2024" title="Category:Articles with unsourced statements from June 2024">Articles with unsourced statements from June 2024</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_clarification_from_May_2021" title="Category:Wikipedia articles needing clarification from May 2021">Wikipedia articles needing clarification from May 2021</a></li><li><a href="/wiki/Category:CS1_errors:_missing_periodical" title="Category:CS1 errors: missing periodical">CS1 errors: missing periodical</a></li><li><a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">CS1 maint: multiple names: authors list</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 4 November 2024, at 14:05<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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