Random forests are a type of ensemble method which makes predictions by combining the results of several independent trees. However, the theory of random forests has long been outpaced by their application. In this paper, we propose a novel random forests algorithm based on cooperative game theory. Banzhaf power index is employed to evaluate the power of each feature by traversing possible feature coalitions. Unlike the previously used information gain rate of information theory, which simply chooses the most informative feature, the Banzhaf power index can be considered as a metric of the importance of each feature on the dependency among a group of features. More importantly, we have proved the consistency of the proposed algorithm, named Banzhaf random forests (BRF). This theoretical analysis takes a step towards narrowing the gap between the theory and practice of random forests for classification problems. Experiments on several UCI benchmark data sets show that BRF is competitive with state-of-the-art classifiers and dramatically outperforms previous consistent random forests. Particularly, it is much more efficient than previous consistent random forests.
Classic random forests introduced by Breiman @cite_27 combine several decision trees @cite_32 with bagging @cite_17 . The main idea of random forests is based on the early work of @cite_16 on the random subspace method, the feature selection work of @cite_3 , the way of random split selection of @cite_11 . Based on the seminal work of Breiman @cite_27 , @cite_25 suggests that it is best to average across sets of trees with different structures but not any of the constituent trees. @cite_31 present a unified, efficient model of random decision forests which can be applied to a number of machine learning, computer vision and medical image analysis tasks. With the development of random forests in recent years, they have been applied to a wide variety of real world problems @cite_26 , @cite_9 , @cite_5 , @cite_24 .
@cite_11 @cite_26 @cite_9 @cite_32 @cite_3 @cite_24 @cite_27 @cite_5 @cite_31 @cite_16 @cite_25 @cite_17
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Random forests are a scheme proposed by Leo Breiman in the 2000's for building a predictor ensemble with a set of decision trees that grow in randomly selected subspaces of data. Despite growing interest and practical use, there has been little exploration of the statistical properties of random forests, and little is known about the mathematical forces driving the algorithm. In this paper, we offer an in-depth analysis of a random forests model suggested by Breiman (2004), which is very close to the original algorithm. We show in particular that the procedure is consistent and adapts to sparsity, in the sense that its rate of convergence depends only on the number of strong features and not on how many noise variables are present. This review presents a unified, efficient model of random decision forests which can be applied to a number of machine learning, computer vision, and medical image analysis tasks. Our model extends existing forest-based techniques as it unifies classification, regression, density estimation, manifold learning, semi-supervised learning, and active learning under the same decision forest framework. This gives us the opportunity to write and optimize the core implementation only once, with application to many diverse tasks. The proposed model may be used both in a discriminative or generative way and may be applied to discrete or continuous, labeled or unlabeled data. The main contributions of this review are: (1) Proposing a unified, probabilistic and efficient model for a variety of learning tasks; (2) Demonstrating margin-maximizing properties of classification forests; (3) Discussing probabilistic regression forests in comparison with other nonlinear regression algorithms; (4) Introducing density forests for estimating probability density functions; (5) Proposing an efficient algorithm for sampling from a density forest; (6) Introducing manifold forests for nonlinear dimensionality reduction; (7) Proposing new algorithms for transductive learning and active learning. Finally, we discuss how alternatives such as random ferns and extremely randomized trees stem from our more general forest model. This document is directed at both students who wish to learn the basics of decision forests, as well as researchers interested in the new contributions. It presents both fundamental and novel concepts in a structured way, with many illustrative examples and real-world applications. Thorough comparisons with state-of-the-art algorithms such as support vector machines, boosting and Gaussian processes are presented and relative advantages and disadvantages discussed. The many synthetic examples and existing commercial applications demonstrate the validity of the proposed model and its flexibility. Random Forest is an ensemble supervised machine learning technique. Based on bagging and random feature selection, number of decision trees (base classifiers) is generated and majority voting is taken for classification. For effective learning and classification of Random Forest, there is need for reducing number of trees (Pruning) in Random Forest. We have presented here systematic survey of pruning efforts of Random Forest classifier along with the required theoretical background. Most of the work for pruning takes static approach while recently dynamic pruning is being targeted. We have also generated a Comparison Chart by taking relevant parameters. There is research scope for analyzing behavior of Random forest, generating accurate and diverse base decision trees, truly dynamic pruning algorithm for Random Forest classifier, and generating optimal subset of Random forest. Breiman (2001a,b) has recently developed an ensemble classification and regression approach that displayed outstanding performance with regard prediction error on a suite of benchmark datasets. As the base constituents of the ensemble are tree-structured predictors, and since each of these is constructed using an injection of randomness, the method is called ‘random forests’. That the exceptional performance is attained with seemingly only a single tuning parameter, to which sensitivity is minimal, makes the methodology all the more remarkable. The individual trees comprising the forest are all grown to maximal depth. While this helps with regard bias, there is the familiar tradeoff with variance. However, these variability concerns were potentially obscured because of an interesting feature of those benchmarking datasets extracted from the UCI machine learning repository for testing: all these datasets are hard to overfit using tree-structured methods. This raises issues about the scope of the repository. With this as motivation, and coupled with experience from boosting methods, we revisit the formulation of random forests and investigate prediction performance on real-world and simulated datasets for which maximally sized trees do overfit. These explorations reveal that gains can be realized by additional tuning to regulate tree size via limiting the number of splits and or the size of nodes for which splitting is allowed. Nonetheless, even in these settings, good performance for random forests can be attained by using larger (than default) primary tuning parameter values. In this article we study random forests through their connection with a new framework of adaptive nearest-neighbor methods. We introduce a concept of potential nearest neighbors (k-PNNs) and show that random forests can be viewed as adaptively weighted k-PNN methods. Various aspects of random forests can be studied from this perspective. We study the effect of terminal node sizes on the prediction accuracy of random forests. We further show that random forests with adaptive splitting schemes assign weights to k-PNNs in a desirable way: for the estimation at a given target point, these random forests assign voting weights to the k-PNNs of the target point according to the local importance of different input variables. We propose a new simple splitting scheme that achieves desirable adaptivity in a straightforward fashion. This simple scheme can be combined with existing algorithms. The resulting algorithm is computationally faster and gives comparable results. Other possible aspects of random forests, such... Random forests were introduced as a machine learning tool in Breiman (2001) and have since proven to be very popular and powerful for high-dimensional regression and classification. For regression, random forests give an accurate approximation of the conditional mean of a response variable. It is shown here that random forests provide information about the full conditional distribution of the response variable, not only about the conditional mean. Conditional quantiles can be inferred with quantile regression forests, a generalisation of random forests. Quantile regression forests give a non-parametric and accurate way of estimating conditional quantiles for high-dimensional predictor variables. The algorithm is shown to be consistent. Numerical examples suggest that the algorithm is competitive in terms of predictive power. A new classification and regression tool, Random Forest, is introduced and investigated for predicting a compound's quantitative or categorical biological activity based on a quantitative description of the compound's molecular structure. Random Forest is an ensemble of unpruned classification or regression trees created by using bootstrap samples of the training data and random feature selection in tree induction. Prediction is made by aggregating (majority vote or averaging) the predictions of the ensemble. We built predictive models for six cheminformatics data sets. Our analysis demonstrates that Random Forest is a powerful tool capable of delivering performance that is among the most accurate methods to date. We also present three additional features of Random Forest: built-in performance assessment, a measure of relative importance of descriptors, and a measure of compound similarity that is weighted by the relative importance of descriptors. It is the combination of relatively high prediction accu... Random forests are ensemble methods which grow trees as base learners and combine their predictions by averaging. Random forests are known for their good practical performance, particularly in high-dimensional settings. On the theoretical side, several studies highlight the potentially fruitful connection between the random forests and the kernel methods. In this paper, we work out this connection in detail. In particular, we show that by slightly modifying their definition, random forests can be rewritten as kernel methods (called KeRF for kernel based on random forests) which are more interpretable and easier to analyze. Explicit expressions of KeRF estimates for some specific random forest models are given, together with upper bounds on their rate of consistency. We also show empirically that the KeRF estimates compare favourably to the random forest estimates. Despite widespread interest and practical use, the theoretical properties of random forests are still not well understood. In this paper we contribute to this understanding in two ways. We present a new theoretically tractable variant of random regression forests and prove that our algorithm is consistent. We also provide an empirical evaluation, comparing our algorithm and other theoretically tractable random forest models to the random forest algorithm used in practice. Our experiments provide insight into the relative importance of different simplifications that theoreticians have made to obtain tractable models for analysis. In the last years of his life, Leo Breiman promoted random forests for use in classification. He suggested using averaging as a means of obtaining good discrimination rules. The base classifiers used for averaging are simple and randomized, often based on random samples from the data. He left a few questions unanswered regarding the consistency of such rules. In this paper, we give a number of theorems that establish the universal consistency of averaging rules. We also show that some popular classifiers, including one suggested by Breiman, are not universally consistent. This work investigates the use of Random Forests for class based pixel-wise segmentation of images. The contribution of this paper is three-fold. First, we show that apparently quite dissimilar classifiers (such as nearest neighbour matching to texton class histograms) can be mapped onto a Random Forest architecture. Second, based on this insight, we show that the performance of such classifiers can be improved by incorporating the spatial context and discriminative learning that arises naturally in the Random Forest framework. Finally, we show that the ability of Random Forests to combine multiple features leads to a further increase in performance when textons, colour, filterbanks, and HOG features are used simultaneously. The benefit of the multi-feature classifier is demonstrated with extensive experimentation on existing labelled image datasets. The method equals or exceeds the state of the art on these datasets. AbstractMany scientific and engineering challenges—ranging from personalized medicine to customized marketing recommendations—require an understanding of treatment effect heterogeneity. In this paper, we develop a non-parametric causal forest for estimating heterogeneous treatment effects that extends Breiman's widely used random forest algorithm. In the potential outcomes framework with unconfoundedness, we show that causal forests are pointwise consistent for the true treatment effect, and have an asymptotically Gaussian and centered sampling distribution. We also discuss a practical method for constructing asymptotic confidence intervals for the true treatment effect that are centered at the causal forest estimates. Our theoretical results rely on a generic Gaussian theory for a large family of random forest algorithms. To our knowledge, this is the first set of results that allows any type of random forest, including classification and regression forests, to be used for provably valid statistical infe...