Topic

# Tree (graph theory)

About: Tree (graph theory) is a research topic. Over the lifetime, 5244 publications have been published within this topic receiving 190809 citations.

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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.

79,257 citations

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01 Jan 1983

TL;DR: The methodology used to construct tree structured rules is the focus of a monograph as mentioned in this paper, covering the use of trees as a data analysis method, and in a more mathematical framework, proving some of their fundamental properties.

Abstract: The methodology used to construct tree structured rules is the focus of this monograph. Unlike many other statistical procedures, which moved from pencil and paper to calculators, this text's use of trees was unthinkable before computers. Both the practical and theoretical sides have been developed in the authors' study of tree methods. Classification and Regression Trees reflects these two sides, covering the use of trees as a data analysis method, and in a more mathematical framework, proving some of their fundamental properties.

14,825 citations

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Bell Labs

^{1}TL;DR: In this article, the authors proposed a method to construct tree-based classifiers whose capacity can be arbitrarily expanded for increases in accuracy for both training and unseen data, which can be monotonically improved by building multiple trees in different subspaces of the feature space.

Abstract: Decision trees are attractive classifiers due to their high execution speed. But trees derived with traditional methods often cannot be grown to arbitrary complexity for possible loss of generalization accuracy on unseen data. The limitation on complexity usually means suboptimal accuracy on training data. Following the principles of stochastic modeling, we propose a method to construct tree-based classifiers whose capacity can be arbitrarily expanded for increases in accuracy for both training and unseen data. The essence of the method is to build multiple trees in randomly selected subspaces of the feature space. Trees in, different subspaces generalize their classification in complementary ways, and their combined classification can be monotonically improved. The validity of the method is demonstrated through experiments on the recognition of handwritten digits.

2,957 citations

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TL;DR: A hierarchical arrangement of stocks traded in a financial market is found by investigating the daily time series of the logarithm of stock price and the hierarchical tree of the subdominant ultrametric space associated with the graph provides a meaningful economic taxonomy.

Abstract: I find a hierarchical arrangement of stocks traded in a financial market by investigating the daily time series of the logarithm of stock price. The topological space is a subdominant ultrametric space associated with a graph connecting the stocks of the portfolio analyzed. The graph is obtained starting from the matrix of correlation coefficient computed between all pairs of stocks of the portfolio by considering the synchronous time evolution of the difference of the logarithm of daily stock price. The hierarchical tree of the subdominant ultrametric space associated with the graph provides a meaningful economic taxonomy.

1,808 citations

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TL;DR: This paper presents the latest release of the program RAxML-III for rapid maximum likelihood-based inference of large evolutionary trees which allows for computation of 1.000-taxon trees in less than 24 hours on a single PC processor.

Abstract: Motivation: The computation of large phylogenetic trees with statistical models such as maximum likelihood or bayesian inference is computationally extremely intensive. It has repeatedly been demonstrated that these models are able to recover the true tree or a tree which is topologically closer to the true tree more frequently than less elaborate methods such as parsimony or neighbor joining. Due to the combinatorial and computational complexity the size of trees which can be computed on a Biologist's PC workstation within reasonable time is limited to trees containing approximately 100 taxa.
Results: In this paper we present the latest release of our program RAxML-III for rapid maximum likelihood-based inference of large evolutionary trees which allows for computation of 1.000-taxon trees in less than 24 hours on a single PC processor. We compare RAxML-III to the currently fastest implementations for maximum likelihood and bayesian inference: PHYML and MrBayes. Whereas RAxML-III performs worse than PHYML and MrBayes on synthetic data it clearly outperforms both programs on all real data alignments used in terms of speed and final likelihood values.
Availability Supplementary information: RAxML-III including all alignments and final trees mentioned in this paper is freely available as open source code at http://wwwbode.cs.tum/~stamatak
Contact: stamatak@cs.tum.edu

1,423 citations