Random Forests
01 Oct 2001-Vol. 45, Iss: 1, pp 5-32
TL;DR: Internal estimates monitor error, strength, and correlation and these are used to show the response to increasing the number of features used in the forest, and are also applicable to regression.
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, aaa, 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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Book•
01 Jan 2015TL;DR: The aim of this textbook is to introduce machine learning, and the algorithmic paradigms it offers, in a principled way in an advanced undergraduate or beginning graduate course.
Abstract: Machine learning is one of the fastest growing areas of computer science, with far-reaching applications. The aim of this textbook is to introduce machine learning, and the algorithmic paradigms it offers, in a principled way. The book provides an extensive theoretical account of the fundamental ideas underlying machine learning and the mathematical derivations that transform these principles into practical algorithms. Following a presentation of the basics of the field, the book covers a wide array of central topics that have not been addressed by previous textbooks. These include a discussion of the computational complexity of learning and the concepts of convexity and stability; important algorithmic paradigms including stochastic gradient descent, neural networks, and structured output learning; and emerging theoretical concepts such as the PAC-Bayes approach and compression-based bounds. Designed for an advanced undergraduate or beginning graduate course, the text makes the fundamentals and algorithms of machine learning accessible to students and non-expert readers in statistics, computer science, mathematics, and engineering.
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TL;DR: This review paper introduces Bayesian optimization, highlights some of its methodological aspects, and showcases a wide range of applications.
Abstract: Big Data applications are typically associated with systems involving large numbers of users, massive complex software systems, and large-scale heterogeneous computing and storage architectures. The construction of such systems involves many distributed design choices. The end products (e.g., recommendation systems, medical analysis tools, real-time game engines, speech recognizers) thus involve many tunable configuration parameters. These parameters are often specified and hard-coded into the software by various developers or teams. If optimized jointly, these parameters can result in significant improvements. Bayesian optimization is a powerful tool for the joint optimization of design choices that is gaining great popularity in recent years. It promises greater automation so as to increase both product quality and human productivity. This review paper introduces Bayesian optimization, highlights some of its methodological aspects, and showcases a wide range of applications.
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Cites background or methods from "Random Forests"
...More precisely, the random forest is an ensemble method where the weak learners are decision trees trained on random subsamples of the data [24]....
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...Introduced in 2001 [24], random forests are a class of scalable and highly parallelizable regression models that have been very successful in practice [42]....
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Technische Universität München1, ETH Zurich2, University of Bern3, Harvard University4, National Institutes of Health5, University of Debrecen6, University Hospital Heidelberg7, McGill University8, University of Pennsylvania9, French Institute for Research in Computer Science and Automation10, University at Buffalo11, Microsoft12, University of Cambridge13, Stanford University14, University of Virginia15, Imperial College London16, Massachusetts Institute of Technology17, Columbia University18, Sabancı University19, Old Dominion University20, RMIT University21, Purdue University22, General Electric23
TL;DR: The Multimodal Brain Tumor Image Segmentation Benchmark (BRATS) as mentioned in this paper was organized in conjunction with the MICCAI 2012 and 2013 conferences, and twenty state-of-the-art tumor segmentation algorithms were applied to a set of 65 multi-contrast MR scans of low and high grade glioma patients.
Abstract: In this paper we report the set-up and results of the Multimodal Brain Tumor Image Segmentation Benchmark (BRATS) organized in conjunction with the MICCAI 2012 and 2013 conferences Twenty state-of-the-art tumor segmentation algorithms were applied to a set of 65 multi-contrast MR scans of low- and high-grade glioma patients—manually annotated by up to four raters—and to 65 comparable scans generated using tumor image simulation software Quantitative evaluations revealed considerable disagreement between the human raters in segmenting various tumor sub-regions (Dice scores in the range 74%–85%), illustrating the difficulty of this task We found that different algorithms worked best for different sub-regions (reaching performance comparable to human inter-rater variability), but that no single algorithm ranked in the top for all sub-regions simultaneously Fusing several good algorithms using a hierarchical majority vote yielded segmentations that consistently ranked above all individual algorithms, indicating remaining opportunities for further methodological improvements The BRATS image data and manual annotations continue to be publicly available through an online evaluation system as an ongoing benchmarking resource
3,699 citations
20 Jun 2011
TL;DR: This work takes an object recognition approach, designing an intermediate body parts representation that maps the difficult pose estimation problem into a simpler per-pixel classification problem, and generates confidence-scored 3D proposals of several body joints by reprojecting the classification result and finding local modes.
Abstract: We propose a new method to quickly and accurately predict 3D positions of body joints from a single depth image, using no temporal information. We take an object recognition approach, designing an intermediate body parts representation that maps the difficult pose estimation problem into a simpler per-pixel classification problem. Our large and highly varied training dataset allows the classifier to estimate body parts invariant to pose, body shape, clothing, etc. Finally we generate confidence-scored 3D proposals of several body joints by reprojecting the classification result and finding local modes. The system runs at 200 frames per second on consumer hardware. Our evaluation shows high accuracy on both synthetic and real test sets, and investigates the effect of several training parameters. We achieve state of the art accuracy in our comparison with related work and demonstrate improved generalization over exact whole-skeleton nearest neighbor matching.
3,579 citations
TL;DR: High classification accuracy in all applications as measured by cross-validation and, in the case of the lichen data, by independent test data, when comparing RF to other common classification methods are observed.
Abstract: Classification procedures are some of the most widely used statistical methods in ecology. Random forests (RF) is a new and powerful statistical classifier that is well established in other disciplines but is relatively unknown in ecology. Advantages of RF compared to other statistical classifiers include (1) very high classification accuracy; (2) a novel method of determining variable importance; (3) ability to model complex interactions among predictor variables; (4) flexibility to perform several types of statistical data analysis, including regression, classification, survival analysis, and unsupervised learning; and (5) an algorithm for imputing missing values. We compared the accuracies of RF and four other commonly used statistical classifiers using data on invasive plant species presence in Lava Beds National Monument, California, USA, rare lichen species presence in the Pacific Northwest, USA, and nest sites for cavity nesting birds in the Uinta Mountains, Utah, USA. We observed high classification accuracy in all applications as measured by cross-validation and, in the case of the lichen data, by independent test data, when comparing RF to other common classification methods. We also observed that the variables that RF identified as most important for classifying invasive plant species coincided with expectations based on the literature.
3,368 citations
Cites background or methods from "Random Forests"
...Random forests (hereafter RF) is one such method (Breiman 2001)....
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...For the classification situation, Breiman (2001) showed that classification accuracy can be significantly improved by aggregating the results of many classifiers that have little bias by averaging or voting, if the classifiers have low pairwise correlations....
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References
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01 Aug 1996
TL;DR: 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.
Abstract: 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.
16,118 citations
Proceedings Article•
AT&T1
TL;DR: This paper describes experiments carried out to assess how well AdaBoost with and without pseudo-loss, performs on real learning problems and compared boosting to Breiman's "bagging" method when used to aggregate various classifiers.
Abstract: In an earlier paper, we introduced a new "boosting" algorithm called AdaBoost which, theoretically, can be used to significantly reduce the error of any learning algorithm that con- sistently generates classifiers whose performance is a little better than random guessing. We also introduced the related notion of a "pseudo-loss" which is a method for forcing a learning algorithm of multi-label concepts to concentrate on the labels that are hardest to discriminate. In this paper, we describe experiments we carried out to assess how well AdaBoost with and without pseudo-loss, performs on real learning problems. We performed two sets of experiments. The first set compared boosting to Breiman's "bagging" method when used to aggregate various classifiers (including decision trees and single attribute- value tests). We compared the performance of the two methods on a collection of machine-learning benchmarks. In the second set of experiments, we studied in more detail the performance of boosting using a nearest-neighbor classifier on an OCR problem.
7,601 citations
"Random Forests" refers background or methods in this paper
...But none of these these three forests do as well as Adaboost (Freund & Schapire, 1996) or other algorithms that work by adaptive reweighting (arcing) of the training set (see Breiman, 1998b; Dieterrich, 1998; Bauer & Kohavi, 1999)....
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...In its original version, Adaboost (Freund & Schapire, 1996) is a deterministic algorithm that selects the weights on the training set for input to the next classifier based on the misclassifications in the previous classifiers....
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TL;DR: A method to construct a decision tree based classifier is proposed that maintains highest accuracy on training data and improves on generalization accuracy as it grows in complexity.
Abstract: Much of previous attention on decision trees focuses on the splitting criteria and optimization of tree sizes. The dilemma between overfitting and achieving maximum accuracy is seldom resolved. A method to construct a decision tree based classifier is proposed that maintains highest accuracy on training data and improves on generalization accuracy as it grows in complexity. The classifier consists of multiple trees constructed systematically by pseudorandomly selecting subsets of components of the feature vector, that is, trees constructed in randomly chosen subspaces. The subspace method is compared to single-tree classifiers and other forest construction methods by experiments on publicly available datasets, where the method's superiority is demonstrated. We also discuss independence between trees in a forest and relate that to the combined classification accuracy.
5,984 citations
"Random Forests" refers background in this paper
...Ho (1998) has written a number of papers on “the random subspace” method which does a random selection of a subset of features to use to grow each tree....
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...Keywords: classification, regression, ensemble...
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TL;DR: In this article, the authors compared the effectiveness of randomization, bagging, and boosting for improving the performance of the decision-tree algorithm C4.5 and found that in situations with little or no classification noise, randomization is competitive with bagging but not as accurate as boosting.
Abstract: 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.
2,919 citations
TL;DR: It is found that Bagging improves when probabilistic estimates in conjunction with no-pruning are used, as well as when the data was backfit, and that Arc-x4 behaves differently than AdaBoost if reweighting is used instead of resampling, indicating a fundamental difference.
Abstract: Methods for voting classification algorithms, such as Bagging and AdaBoost, have been shown to be very successful in improving the accuracy of certain classifiers for artificial and real-world datasets. We review these algorithms and describe a large empirical study comparing several variants in conjunction with a decision tree inducer (three variants) and a Naive-Bayes inducer. The purpose of the study is to improve our understanding of why and when these algorithms, which use perturbation, reweighting, and combination techniques, affect classification error. We provide a bias and variance decomposition of the error to show how different methods and variants influence these two terms. This allowed us to determine that Bagging reduced variance of unstable methods, while boosting methods (AdaBoost and Arc-x4) reduced both the bias and variance of unstable methods but increased the variance for Naive-Bayes, which was very stable. We observed that Arc-x4 behaves differently than AdaBoost if reweighting is used instead of resampling, indicating a fundamental difference. Voting variants, some of which are introduced in this paper, include: pruning versus no pruning, use of probabilistic estimates, weight perturbations (Wagging), and backfitting of data. We found that Bagging improves when probabilistic estimates in conjunction with no-pruning are used, as well as when the data was backfit. We measure tree sizes and show an interesting positive correlation between the increase in the average tree size in AdaBoost trials and its success in reducing the error. We compare the mean-squared error of voting methods to non-voting methods and show that the voting methods lead to large and significant reductions in the mean-squared errors. Practical problems that arise in implementing boosting algorithms are explored, including numerical instabilities and underflows. We use scatterplots that graphically show how AdaBoost reweights instances, emphasizing not only “hard” areas but also outliers and noise.
2,686 citations
"Random Forests" refers background or methods in this paper
...But none of these these three forests do as well as Adaboost (Freund & Schapire, 1996) or other algorithms that work by adaptive reweighting (arcing) of the training set (see Breiman, 1998b; Dieterrich, 1998; Bauer & Kohavi, 1999)....
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...The second is that bagging can be used to give ongoing estimates of the generalization error (PE∗) of the combined ensemble of trees, as well as estimates for the strength and correlation....
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