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# Machine Learning, Volume 45, Number 1 - SpringerLink

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## Abstract

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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... ML has a plethora of classification algorithms, including RF, LR, Support Vector Machine (SVM), naive Bayes classifier, decision trees, and many more. In this study, we used the RF method (Breiman 2001), a supervised, ensemble learning, decision-tree-based algorithm for classification and regression. RF is one of the most popular classifier ML algorithms. ...
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... Finally, to understand what elements are most informative in the decision process, we calculate the permutation feature importance (Breiman 2001) of the classification pipeline. The feature importance for element X is defined as the decrease in cross-validation accuracy if we randomly shuffle the values of all abundances that include element X in the cross-validation data. ...
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Bagging predictors is a method for generating multiple versions of a predictor and using these to get an aggregated predictor. The aggregation averages over the versions when predicting a numerical outcome and does a plurality vote when predicting a class. The multiple versions are formed by making bootstrap replicates of the learning set and using these as new learning sets. Tests on real and simulated data sets using classification and regression trees and subset selection in linear regression show that bagging can give substantial gains in accuracy. The vital element is the instability of the prediction method. If perturbing the learning set can cause significant changes in the predictor constructed, then bagging can improve accuracy.
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We study the notions of bias and variance for classification rules. Following Efron (1978) we develop a decomposition of prediction error into its natural components. Then we derive bootstrap estimates of these components and illustrate how they can be used to describe the error behaviour of a classifier in practice. In the process we also obtain a bootstrap estimate of the error of a "bagged" classifier. Keywords: classification, prediction error, bias, variance, bootstrap 1 Introduction This article concerns classification rules that have been constructed from a set of training data. The training set X = (x 1 ; x 2 ; Delta Delta Delta ; x n ) consists of n observations x i = (t i ; g i ), with t i being the predictor or feature vector and g i being the response, taking values in f1; 2; : : : Kg. On the basis of X the Addresses: tibs@utstat.toronto.edu; http://www.utstat.toronto.edu/tibs 1 statistician constructs a classification rule C(t; X ). Our objective here is to un...
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The "minimum margin" of an ensemble classifier on a given training set is, roughly speaking, the smallest vote it gives to any correct training label. Recent work has shown that the Adaboost algorithm is particularly effective at producing ensembles with large minimum margins, and theory suggests that this may account for its success at reducing generalization error. We note, however, that the problem of finding good margins is closely related to linear programming, and we use this connection to derive and test new "LPboosting" algorithms that achieve better minimum margins than Adaboost.
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. One of the surprising recurring phenomena observed in experiments with boosting is that the test error of the generated classifier usually does not increase as its size becomes very large, and often is observed to decrease even after the training error reaches zero. In this paper, we show that this phenomenon is related to the distribution of margins of the training examples with respect to the generated voting classification rule, where the margin of an example is simply the difference between the number of correct votes and the maximumnumber of votes received by any incorrect label. We show that techniques used in the analysis of Vapnik's support vector classifiers and of neural networks with small weights can be applied to voting methods to relate the margin distribution to the test error. We also show theoretically and experimentally that boosting is especially effective at increasing the margins of the training examples. Finally, we compare our explanation to those based on the ...
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. Bagging and boosting are methods that generate a diverse ensemble of classifiers by manipulating the training data given to a "base" learning algorithm. Breiman has pointed out that they rely for their effectiveness on the instability of the base learning algorithm. An alternative approach to generating an ensemble is to randomize the internal decisions made by the base algorithm. This general approach has been studied previously by Ali and Pazzani and by Dietterich and Kong. This paper compares the effectiveness of randomization, bagging, and boosting for improving the performance of the decision-tree algorithm C4.5. The experiments show that in situations with little or no classification noise, randomization is competitive with (and perhaps slightly superior to) bagging but not as accurate as boosting. In situations with substantial classification noise, bagging is much better than boosting, and sometimes better than randomization. Keywords: Decision trees, ensemble learning, bagg...
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Introduction In recent research in combining predictors, it has been recognized that the critical thing to success in combining low-bias predictors such as trees and neural nets has been through methods that reduce the variability in the predictor due to training set variability. Assume that the training set consists of N independent draws from the same underlying distribution. Conceptually, training sets of size N can be drawn repeatedly and the same algorithm used to construct a predictor on each training set. These predictors will vary, and the extent of the variability is a dominant factor in the generalization prediction error. 2 Given a training set {(y n ,x n ),n=1,...N} where the y's are either class labels or numerical values, the most common way of reducing variability is by perturbing the training set to produce alternative training sets, growing a predictor on
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In bagging, predictors are constructed using bootstrap samples from the training set and then aggregated to form a bagged predictor. Each bootstrap sample leaves out about 37% of the examples. These left-out examples can be used to form accurate estimates of important quantities. For instance, they can be used to give much improved estimates of node probabilities and node error rates in decision trees. Using estimated outputs instead of the observed outputs improves accuracy in regression trees. They can also be used to give nearly optimal estimates of generalization errors for bagged predictors. * Partially supported by NSF Grant 1-444063-21445 Introduction: We assume that there is a training set T= {(y n ,x n ), n=1, ... ,N} and a method for constructing a predictor Q(x,T) using the given training set. The output variable y can either be a class label (classification) or numerical (regression). In bagging (Breiman[1996a]) a sequence of training sets T B,1 , ... , T B,K are generated ...
Multiple randomized classifiers: MRCL Technical Report, Depart-ment of Statistics An empirical comparison of voting classification algorithms
• Y Amit
• G Blanchard
• K Wilder
• E Bauer
• R Kohavi
Amit, Y., Blanchard, G., & Wilder, K. (1999). Multiple randomized classifiers: MRCL Technical Report, Depart-ment of Statistics, University of Chicago. Bauer, E. & Kohavi, R. (1999). An empirical comparison of voting classification algorithms. Machine Learning, 36(1/2), 105–139.
Out-of-bag estimation, ftp.stat.berkeley Arcing classifiers (discussion paper)
• L Breiman
Breiman, L. (1996b). Out-of-bag estimation, ftp.stat.berkeley.edu/pub/users/breiman/OOBestimation.ps Breiman, L. (1998a). Arcing classifiers (discussion paper). Annals of Statistics, 26, 801–824.
Using adaptive bagging to debias regressions Some infinity theory for predictor ensembles An experimental comparison of three methods for constructing ensembles of decision trees: Bagging, boosting and randomization
• L Breiman
• L Breiman
Breiman, L. 1999. Using adaptive bagging to debias regressions. Technical Report 547, Statistics Dept. UCB. Breiman, L. 2000. Some infinity theory for predictor ensembles. Technical Report 579, Statistics Dept. UCB. Dietterich, T. (1998). An experimental comparison of three methods for constructing ensembles of decision trees: Bagging, boosting and randomization, Machine Learning, 1–22.
An experimental comparison of three methods for constructing ensembles of decision trees: Bagging, boosting and randomization
• L Breiman
Breiman, L. 2000. Some infinity theory for predictor ensembles. Technical Report 579, Statistics Dept. UCB. Dietterich, T. (1998). An experimental comparison of three methods for constructing ensembles of decision trees: Bagging, boosting and randomization, Machine Learning, 1-22.
Experiments with a new boosting algorithm
• Y Freund
• R Schapire
Freund, Y. & Schapire, R. (1996). Experiments with a new boosting algorithm, Machine Learning: Proceedings of the Thirteenth International Conference, 148-156.
Some infinity theory for predictor ensembles
• L Breiman