Random Forests
Open accessAbstract
Random forests are a combination of tree predictors such that each tree depends on the values of a random vector sampled independently and with the same distribution for all trees in the forest. The generalization error for forests converges a.s. to a limit as the number of trees in the forest becomes large. The generalization error of a forest of tree classifiers depends on the strength of the individual trees in the forest and the correlation between them. Using a random selection of features to split each node yields error rates that compare favorably to Adaboost (Y. Freund & R. Schapire, Machine Learning: Proceedings of the Thirteenth International conference, ***, 148–156), but are more robust with respect to noise. Internal estimates monitor error, strength, and correlation and these are used to show the response to increasing the number of features used in the splitting. Internal estimates are also used to measure variable importance. These ideas are also applicable to regression.
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Random forest uses majority voting, so the most common prediction across decision trees becomes the final classification.
“RF uses the principle of majority voting; the most common prediction from all decision trees is the final classification [17]”
From Application of predictive modeling tools for the identification of Ocimum spp. herbal products (2025) · cites this paper as [17]
Random forest is an ensemble method that combines predictions from multiple decision trees.
“RF is an ensemble approach that combines the predictive classifications of multiple decision trees.”
From Application of predictive modeling tools for the identification of Ocimum spp. herbal products (2025) · cites this paper as [17]
Random forests modify Breiman's (2001) traditional bagging approach.
“Random forests (rf) modify Breiman's ( 2001) traditional bagging approach.”
From Empirical asset pricing via machine learning: evidence from the European stock market (2021) · cites this paper as Breiman's (2001)
Breiman (2001), Friedman and Popescu (2008), and Hastie et al. (2009) introduced two important tools to interpret the GBM model: variable importance measures and partial dependence plots.
“To overcome the black-box nature of GBM, (Breiman 2001; Friedman and Popescu 2008; Hastie et al. 2009) introduced two important tools to interpret the GBM model: (i) variable importance measures and (ii) partial dependence plots.”
From Modelling Motor Insurance Claim Frequency and Severity Using Gradient Boosting (2023) · cites this paper as Breiman 2001
Random Forest was prioritised as the primary baseline due to its high interpretability and resistance to overfitting in high-dimensional feature spaces.
“We prioritised Random Forest (RF) [15] as our primary baseline due to its high interpretability and resistance to overfitting in high-dimensional feature spaces.”
From Safe Fall: Use of Predictive Modeling and Machine Vision Techniques for Fall Analysis and Fall Quality (2026) · cites this paper as [15]
Combining the predictions of many different models is a very successful way to reduce test errors.
“Combining the predictions of many different models is a very successful way to reduce test errors, 1, 3”
From ImageNet classification with deep convolutional neural networks (2017) · cites this paper as [3]
Random forest is investigated by integrating decisions from multiple tree-based classifiers to improve generalization performance of decision tree.
“In order to improve the generalization performance of decision tree, the random forest [306] is further investigated by integrating decisions from multiply tree-based classifiers.”
From Applications of machine learning to machine fault diagnosis: A review and roadmap (2020) · cites this paper as [306]
RF is an ensemble method based on decision trees and uses bagging to reduce overfitting.
“RF is an ensemble method based on decision trees and uses bagging to reduce overfitting (Breiman, 2001).”
From Spatial Optimization Methods for Malaria Risk Mapping in Sub‐Saharan African Cities Using Demographic and Health Surveys (2023) · cites this paper as Breiman, 2001
RF is a robust algorithm for dealing with multicollinearity and allows for non-linear relationships between covariates and the dependent variable.
“RF is a robust algorithm for dealing with multicollinearity and allows for non‐linear relationships between the covariates and the dependent variable (Breiman, 2001).”
From Spatial Optimization Methods for Malaria Risk Mapping in Sub‐Saharan African Cities Using Demographic and Health Surveys (2023) · cites this paper as Breiman, 2001
Non-parametric ML techniques like Random Forest have been successfully used to predict species richness and have shown significant error reduction with respect to traditional counterparts used in ecology, for example in fishes, spiders, and small mammals.
“Non-parametric ML techniques like Random Forest126 have been successfully used to predict species richness and have shown significant error reduction with respect to the traditional counterparts used in ecology, for example in the estimation of richness distributions of fishes127,128, spiders 129, and small mammals130.”
From Perspectives in machine learning for wildlife conservation (2022) · cites this paper as [126]
Breiman (2001) offers an upper bound on the generalization error of forests in terms of correlation and strength of the individual trees.
“The most celebrated theoretical result is that of Breiman (2001), which offers an upper bound on the generalization error of forests in terms of correlation and strength of the individual trees.”
From A random forest guided tour (2016) · cites this paper as Breiman (2001)
In Breiman's forests, the CART criterion is evaluated over a subset M_try of randomly selected coordinates, not over the whole range {1,...,p}.
“First of all, in Breiman's forests, the criterion (2) is evaluated over a subset M try of randomly selected coordinates, and not over the whole range { 1 , . . . , p } .”
From A random forest guided tour (2016) · cites this paper as Breiman's (2001)
Picking a large M does not lead to overfitting.
“Interestingly, picking a large M does not lead to overfitting.”
From A random forest guided tour (2016) · cites this paper as Breiman (2001)
In Breiman's (2001) forests, each leaf of individual trees contains a small number of observations, typically between 1 and 5.
“As well as this, in Breiman's (2001) forests, each leaf (that is, a terminal node) of individual trees contains a small number of observations, typically between 1 and 5.”
From A random forest guided tour (2016) · cites this paper as Breiman's (2001)
In Breiman's forests, each tree is constructed on a subset of a_n examples picked within the initial sample, not on the whole sample D_n, and only these a_n observations are used to calculate the estimation.
“At last, each tree is constructed on a subset of a n examples picked within the initial sample, not on the whole sample D n ; and only these a n observations are used to calculate the estimation.”
From A random forest guided tour (2016) · cites this paper as Breiman's (2001)
Scornet et al. (2015) show that Breiman's (2001) forests are consistent in an additive regression framework.
“Scornet et al. (2015), who show that Breiman's (2001) forests are consistent in an additive regression framework.”
From A random forest guided tour (2016) · cites this paper as Breiman's (2001)
In Breiman's forests, individual trees are not pruned, and the final cells do not contain more than nodesize observations unless all data points in the cell have the same X_i.
“Besides, the individual trees are not pruned, and the final cells do not contain more than nodesize observations (unless all data points in the cell have the same X i ).”
From A random forest guided tour (2016) · cites this paper as Breiman's (2001)
The choice k approximately equal to log n corresponds to obtaining a small number of examples in the leaves, in accordance with Breiman's (2001) idea that the individual trees should not be pruned.
“In particular, the choice k ≈ log n corresponds to obtaining a small number of examples in the leaves, in accordance with Breiman's (2001) idea that the individual trees should not be pruned.”
From A random forest guided tour (2016) · cites this paper as Breiman's (2001)
Breiman (2001) shows that the error of the forest is small as soon as the predictive power of each tree is good and the correlation between the tree errors is low.
“A seminal result by Breiman (2001) shows that the error of the forest is small as soon as the predictive power of each tree is good and the correlation between the tree errors is low.”
From A random forest guided tour (2016) · cites this paper as Breiman (2001)
If Breiman's (2001) original algorithm is used in the Biau et al. (2008) example (each cell contains exactly one data point), then one obtains a consistent classification rule.
“We note however that if Breiman's (2001) original algorithm is used in this example (that is, each cell contains exactly one data point) then one obtains a consistent classification rule.”
From A random forest guided tour (2016) · cites this paper as Breiman (2001)
Independently of the type of forest, the forest error satisfies an inequality involving the correlation between tree errors and the individual tree error.
“More precisely, independently of the type of forest, one has”
From A random forest guided tour (2016) · cites this paper as Breiman (2001)
Random forests offer a powerful nonparametric discriminative method for multi-class classification that offers state-of-the-art performance.
“For activity classification we use random forests 50 which offer a powerful nonparametric discriminative method for multi-class classification that offers state-of-the-art performance 51 .”
From Statistical machine learning of sleep and physical activity phenotypes from sensor data in 96,220 UK Biobank participants (2018) · cites this paper as [50]
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“One of the noteworthy methods is the permutation feature importance (PFI) by Breiman [108] for random forests.”
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- Authors
- Breiman, L.
- Affiliations
- University of California, Berkeley (US)
- Corresponding author
- Breiman, Leo
- Venue
- Machine Learning
- Published
- 2001-10
- Volume
- 45
- Issue
- 1
- Pages
- 5-32
- ISSN
- 0885-6125 (ISSN-L)
- 1573-0565
- OpenAlex
- W2911964244
- Cited by
- 133,538
- References
- 15
- Citations by year
- 2026: 18528 · 2025: 16751 · 2024: 14088 · 2023: 12732 · 2022: 11712 · 2021: 11577 · 2020: 9483 · 2019: 7661 · 2018: 6275 · 2017: 4956 · 2016: 4152 · 2015: 3623 · 2014: 2944 · 2013: 2268 · 2012: 1761
- FWCI
- 74.54
- Type
- journal article
- Language
- en
- Open access
- green
- www.ine.es
- Licence
- Locations
- Machine Learning (publishedVersion)
- Repository of Digital Objects for Teaching Research and Culture (University of Valencia) (submittedVersion, cc-by-sa)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Physical Sciences › Computer Science › Artificial Intelligence › Machine Learning and Data Classification
- Physical Sciences › Computer Science › Artificial Intelligence › Imbalanced Data Classification Techniques
- Keywords
- random forest, ensemble learning, classification trees, generalization error, asymptotic convergence, feature selection, node splitting, AdaBoost comparison, noise robustness, feature importance, regression forests
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Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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