We study the convergence of the predictive surface of regression trees and forests. To support our analysis we introduce a notion of adaptive concentration for regression trees. This approach breaks tree training into a model selection phase in which we pick the tree splits, followed by a model fitting phase where we find the best regression model consistent with these splits. We then show that the fitted regression tree concentrates around the optimal predictor with the same splits: as d and n get large, the discrepancy is with high probability bounded on the order of sqrt(log(d) log(n)/k) uniformly over the whole regression surface, where d is the dimension of the feature space, n is the number of training examples, and k is the minimum leaf size for each tree. We also provide rate-matching lower bounds for this adaptive concentration statement. From a practical perspective, our result enables us to prove consistency results for adaptively grown forests in high dimensions, and to carry out valid post-selection inference in the sense of Berk et al. [2013] for subgroups defined by tree leaves.
In this paper we present the practical benefits of a new random forest algorithm to deal withmissing values in the sample. The purpose of this work is to compare the different solutionsto deal with missing values with random forests and describe our new algorithm performanceas well as its algorithmic complexity. A variety of missing value mechanisms (such as MCAR,MAR, MNAR) are considered and simulated. We study the quadratic errors and the bias ofour algorithm and compare it to the most popular missing values random forests algorithms inthe literature. In particular, we compare those techniqu…
By seeking the narrowest prediction intervals (PIs) that satisfy the specified coverage probability requirements, the recently proposed quality-based PI learning principle can extract high-quality PIs that better summarize the predictive certainty in regression tasks, and has been widely applied to solve many practical problems. Currently, the state-of-the-art quality-based PI estimation methods are based on deep neural networks or linear models. In this paper, we propose Highest Density Interval Regression Forest (HDI-Forest), a novel quality-based PI estimation method that is instead based o…
The aim of this study is to compare two supervised classification methods on a crucial meteorological problem. The data consist of satellite measurements of cloud systems which are to be classified either in convective or non convective systems. Convective cloud systems correspond to lightning and detecting such systems is of main importance for thunderstorm monitoring and warning. Because the problem is highly unbalanced, we consider specific performance criteria and different strategies. This case study can be used in an advanced course of data mining in order to illustrate the use of logist…
Due to their accuracies, methods based on ensembles of regression trees are a popular approach for making predictions. Some common examples include Bayesian additive regression trees, boosting and random forests. This paper focuses on honest random forests, which add honesty to the original form of random forests and are proved to have better statistical properties. The main contribution is a new method that quantifies the uncertainties of the estimates and predictions produced by honest random forests. The proposed method is based on the generalized fiducial methodology, and provides a fiduci…
We characterize and study variable importance (VIMP) and pairwise variable associations in binary regression trees. A key component involves the node mean squared error for a quantity we refer to as a maximal subtree. The theory naturally extends from single trees to ensembles of trees and applies to methods like random forests. This is useful because while importance values from random forests are used to screen variables, for example they are used to filter high throughput genomic data in Bioinformatics, very little theory exists about their properties.
Random forest (RF) methodology is one of the most popular machine learning techniques for prediction problems. In this article, we discuss some cases where random forests may suffer and propose a novel generalized RF method, namely regression-enhanced random forests (RERFs), that can improve on RFs by borrowing the strength of penalized parametric regression. The algorithm for constructing RERFs and selecting its tuning parameters is described. Both simulation study and real data examples show that RERFs have better predictive performance than RFs in important situations often encountered in p…
Random Forests [Breiman:2001] (RF) are a fully non-parametric statistical method requiring no distributional assumptions on covariate relation to the response. RF are a robust, nonlinear technique that optimizes predictive accuracy by fitting an ensemble of trees to stabilize model estimates. The randomForestSRC package (http://cran.r-project.org/package=randomForestSRC) is a unified treatment of Breiman's random forests for survival, regression and classification problems. Predictive accuracy make RF an attractive alternative to parametric models, though complexity and interpretability of the…
Université d’Abomey-Calavi（阿波美卡拉维大学）；Université Paris 1 Panthéon Sorbonne（巴黎第一大学（先贤祠-索邦大学））
This paper deals with prediction of anopheles number, the main vector of malaria risk, using environmental and climate variables. The variables selection is based on an automatic machine learning method using regression trees, and random forests combined with stratified two levels cross validation. The minimum threshold of variables importance is accessed using the quadratic distance of variables importance while the optimal subset of selected variables is used to perform predictions. Finally the results revealed to be qualitatively better, at the selection, the prediction , and the CPU time p…
Sorbonne Université（索邦大学）；CNRS（法国国家科学研究中心）；Ecole Polytechnique（巴黎综合理工学院）；University College London（伦敦大学学院）
Given an ensemble of randomized regression trees, it is possible to restructure them as a collection of multilayered neural networks with particular connection weights. Following this principle, we reformulate the random forest method of Breiman (2001) into a neural network setting, and in turn propose two new hybrid procedures that we call neural random forests. Both predictors exploit prior knowledge of regression trees for their architecture, have less parameters to tune than standard networks, and less restrictions on the geometry of the decision boundaries than trees. Consistency results …
Variable selection in sparse regression models is an important task as applications ranging from biomedical research to econometrics have shown. Especially for higher dimensional regression problems, for which the link function between response and covariates cannot be directly detected, the selection of informative variables is challenging. Under these circumstances, the Random Forest method is a helpful tool to predict new outcomes while delivering measures for variable selection. One common approach is the usage of the permutation importance. Due to its intuitive idea and flexible usage, it…
Random Forest (Breiman, 2001) is a successful and widely used regression and classification algorithm. Part of its appeal and reason for its versatility is its (implicit) construction of a kernel-type weighting function on training data, which can also be used for targets other than the original mean estimation. We propose a novel forest construction for multivariate responses based on their joint conditional distribution, independent of the estimation target and the data model. It uses a new splitting criterion based on the MMD distributional metric, which is suitable for detecting heterogene…
It has been postulated and observed in practice that for prediction problems in which covariate data can be naturally partitioned into clusters, ensembling algorithms based on suitably aggregating models trained on individual clusters often perform substantially better than methods that ignore the clustering structure in the data. In this paper, we provide theoretical support to these empirical observations by asymptotically analyzing linear least squares and random forest regressions under a linear model. Our main results demonstrate that the benefit of ensembling compared to training a singl…
Environmental data may be "large" due to number of records, number of covariates, or both. Random forests has a reputation for good predictive performance when using many covariates with nonlinear relationships, whereas spatial regression, when using reduced rank methods, has a reputation for good predictive performance when using many records that are spatially autocorrelated. In this study, we compare these two techniques using a data set containing the macroinvertebrate multimetric index (MMI) at 1859 stream sites with over 200 landscape covariates. A primary application is mapping MMI pred…
Statistical wisdom suggests that very complex models, interpolating training data, will be poor at predicting unseen examples.Yet, this aphorism has been recently challenged by the identification of benign overfitting regimes, specially studied in the case of parametric models: generalization capabilities may be preserved despite model high complexity.While it is widely known that fully-grown decision trees interpolate and, in turn, have bad predictive performances, the same behavior is yet to be analyzed for Random Forests (RF).In this paper, we study the trade-off between interpolation and c…
As a flexible nonparametric learning tool, the random forests algorithm has been widely applied to various real applications with appealing empirical performance, even in the presence of high-dimensional feature space. Unveiling the underlying mechanisms has led to some important recent theoretical results on the consistency of the random forests algorithm and its variants. However, to our knowledge, almost all existing works concerning random forests consistency in high dimensional setting were established for various modified random forests models where the splitting rules are independent of…
An increasing array of biomedical and computer vision applications requires the predictive modeling of complex data, for example images and shapes. The main challenge when predicting such objects lies in the fact that they do not comply to the assumptions of Euclidean geometry. Rather, they occupy non-linear spaces, a.k.a. manifolds, where it is difficult to define concepts such as coordinates, vectors and expected values. In this work, we construct a non-parametric predictive methodology for manifold-valued objects, based on a distance modification of the Random Forest algorithm. Our method i…
In this paper we study asymptotic properties of random forests within the framework of nonlinear time series modeling. While random forests have been successfully applied in various fields, the theoretical justification has not been considered for their use in a time series setting. Under mild conditions, we prove a uniform concentration inequality for regression trees built on nonlinear autoregressive processes and, subsequently, we use this result to prove consistency for a large class of random forests. The results are supported by various simulations.
Renmin University of China（中国人民大学）；KU Leuven（鲁汶大学）
We propose a novel method designed for large-scale regression problems, namely the two-stage best-scored random forest (TBRF). "Best-scored" means to select one regression tree with the best empirical performance out of a certain number of purely random regression tree candidates, and "two-stage" means to divide the original random tree splitting procedure into two: In stage one, the feature space is partitioned into non-overlapping cells; in stage two, child trees grow separately on these cells. The strengths of this algorithm can be summarized as follows: First of all, the pure randomness in…
Random forests, introduced by Leo Breiman in 2001, are a very effective statistical method. The complex mechanism of the method makes theoretical analysis difficult. Therefore, a simplified version of random forests, called purely random forests, which can be theoretically handled more easily, has been considered. In this paper we introduce a variant of this kind of random forests, that we call purely uniformly random forests. In the context of regression problems with a one-dimensional predictor space, we show that both random trees and random forests reach minimax rate of convergence. In add…
École Polytechnique（巴黎综合理工学院）；CNRS（法国国家科学研究中心）；Université de Paris（巴黎大学）；Ecole normale supérieure（巴黎高等师范学院）
Random Forests (RF) is one of the algorithms of choice in many supervised learning applications, be it classification or regression. The appeal of such tree-ensemble methods comes from a combination of several characteristics: a remarkable accuracy in a variety of tasks, a small number of parameters to tune, robustness with respect to features scaling, a reasonable computational cost for training and prediction, and their suitability in high-dimensional settings. The most commonly used RF variants however are "offline" algorithms, which require the availability of the whole dataset at once. In…
Regression models for supervised learning problems with a continuous target are commonly understood as models for the conditional mean of the target given predictors. This notion is simple and therefore appealing for interpretation and visualisation. Information about the whole underlying conditional distribution is, however, not available from these models. A more general understanding of regression models as models for conditional distributions allows much broader inference from such models, for example the computation of prediction intervals. Several random forest-type algorithms aim at est…
Random forests are a very effective and commonly used statistical method, but their full theoretical analysis is still an open problem. As a first step, simplified models such as purely random forests have been introduced, in order to shed light on the good performance of random forests. In this paper, we study the approximation error (the bias) of some purely random forest models in a regression framework, focusing in particular on the influence of the number of trees in the forest. Under some regularity assumptions on the regression function, we show that the bias of an infinite forest decre…
Frankfurt Institute for Advanced Studies（法兰克福高等研究院）；Goethe University Frankfurt（法兰克福大学）；TU Dortmund University（多特蒙德工业大学）
In many studies, we want to determine the influence of certain features on a dependent variable. More specifically, we are interested in the strength of the influence -- i.e., is the feature relevant? -- and, if so, how the feature influences the dependent variable. Recently, data-driven approaches such as \emph{random forest regression} have found their way into applications (Boulesteix et al., 2012). These models allow to directly derive measures of feature importance, which are a natural indicator of the strength of the influence. For the relevant features, the correlation or rank correlati…
Random forests remain among the most popular off-the-shelf supervised machine learning tools with a well-established track record of predictive accuracy in both regression and classification settings. Despite their empirical success as well as a bevy of recent work investigating their statistical properties, a full and satisfying explanation for their success has yet to be put forth. Here we aim to take a step forward in this direction by demonstrating that the additional randomness injected into individual trees serves as a form of implicit regularization, making random forests an ideal model…
Department of Statistics & Data Science Carnegie Mellon University（卡内基梅隆大学统计与数据科学系）
Random forests is a common non-parametric regression technique which performs well for mixed-type data and irrelevant covariates, while being robust to monotonic variable transformations. Existing random forest implementations target regression or classification. We introduce the RFCDE package for fitting random forest models optimized for nonparametric conditional density estimation, including joint densities for multiple responses. This enables analysis of conditional probability distributions which is useful for propagating uncertainty and of joint distributions that describe relationships …
Peter the Great St.Petersburg Polytechnic University（圣彼得堡彼得大帝理工大学）
A new approach called ABRF (the attention-based random forest) and its modifications for applying the attention mechanism to the random forest (RF) for regression and classification are proposed. The main idea behind the proposed ABRF models is to assign attention weights with trainable parameters to decision trees in a specific way. The weights depend on the distance between an instance, which falls into a corresponding leaf of a tree, and instances, which fall in the same leaf. This idea stems from representation of the Nadaraya-Watson kernel regression in the form of a RF. Three modificatio…
The issue of estimating residual variance in regression models has experienced relatively little attention in the machine learning community. However, the estimate is of primary interest in many practical applications, e.g. as a primary step towards the construction of prediction intervals. Here, we consider this issue for the random forest. Therein, the functional relationship between covariates and response variable is modeled by a weighted sum of the latter. The dependence structure is, however, involved in the weights that are constructed during the tree construction process making the mod…
Breiman's random forest (RF) can be interpreted as an implicit kernel generator,where the ensuing proximity matrix represents the data-driven RF kernel. Kernel perspective on the RF has been used to develop a principled framework for theoretical investigation of its statistical properties. However, practical utility of the links between kernels and the RF has not been widely explored and systematically evaluated.Focus of our work is investigation of the interplay between kernel methods and the RF. We elucidate the performance and properties of the data driven RF kernels used by regularized lin…
Idaho State University（爱达荷州立大学）；Utah State University（犹他州立大学）
Random forests are considered one of the best out-of-the-box classification and regression algorithms due to their high level of predictive performance with relatively little tuning. Pairwise proximities can be computed from a trained random forest and measure the similarity between data points relative to the supervised task. Random forest proximities have been used in many applications including the identification of variable importance, data imputation, outlier detection, and data visualization. However, existing definitions of random forest proximities do not accurately reflect the data ge…
Statistical inference for high-dimensional regression heteroskedasticity is an important but under-explored problem. The current paper aims at filling this gap by proposing two tests, namely the variance difference test and the variance difference Breusch-Pagan test, for assessing high-dimensional regression heteroskedasticity. The former tests whether an explanatory feature of interest is associated with the conditional variance of a response variable, while the latter tests heteroskedasticity in the regression, which is known to be the Breusch-Pagan test problem. To formally establish the te…
University of Tübingen（图宾根大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Assume we are given a set of items from a general metric space, but we neither have access to the representation of the data nor to the distances between data points. Instead, suppose that we can actively choose a triplet of items (A,B,C) and ask an oracle whether item A is closer to item B or to item C. In this paper, we propose a novel random forest algorithm for regression and classification that relies only on such triplet comparisons. In the theory part of this paper, we establish sufficient conditions for the consistency of such a forest. In a set of comprehensive experiments, we then de…
Peter the Great St.Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
New models of random forests jointly using the attention and self-attention mechanisms are proposed for solving the regression problem. The models can be regarded as extensions of the attention-based random forest whose idea stems from applying a combination of the Nadaraya-Watson kernel regression and the Huber's contamination model to random forests. The self-attention aims to capture dependencies of the tree predictions and to remove noise or anomalous predictions in the random forest. The self-attention module is trained jointly with the attention module for computing weights. It is shown …
Random forests are one of the most popular machine learning methods due to their accuracy and variable importance assessment. However, random forests only provide variable importance in a global sense. There is an increasing need for such assessments at a local level, motivated by applications in personalized medicine, policy-making, and bioinformatics. We propose a new nonparametric estimator that pairs the flexible random forest kernel with local sufficient dimension reduction to adapt to a regression function's local structure. This allows us to estimate a meaningful directional local varia…
Fox Chase Cancer Center（福克斯蔡斯癌症中心）；University of Kansas（堪萨斯大学）
Over the past decade, random forest models have become widely used as a robust method for high-dimensional data regression tasks. In part, the popularity of these models arises from the fact that they require little hyperparameter tuning and are not very susceptible to overfitting. Random forest regression models are comprised of an ensemble of decision trees that independently predict the value of a (continuous) dependent variable; predictions from each of the trees are ultimately averaged to yield an overall predicted value from the forest. Using a suite of representative real-world datasets…
Random forests have become an important tool for improving accuracy in regression and classification problems since their inception by Leo Breiman in 2001. In this paper, we revisit a historically important random forest model originally proposed by Breiman in 2004 and later studied by Gérard Biau in 2012, where a feature is selected at random and the splits occurs at the midpoint of the node along the chosen feature. If the regression function is Lipschitz and depends only on a small subset of $ S $ out of $ d $ features, we show that, given access to $ n $ observations and properly tuned spl…
A fully nonparametric approach for making probabilistic predictions in multi-response regression problems is introduced. Random forests are used as marginal models for each response variable and, as novel contribution of the present work, the dependence between the multiple response variables is modeled by a generative neural network. This combined modeling approach of random forests, corresponding empirical marginal residual distributions and a generative neural network is referred to as RafterNet. Multiple datasets serve as examples to demonstrate the flexibility of the approach and its impa…
We introduce canonical correlation forests (CCFs), a new decision tree ensemble method for classification and regression. Individual canonical correlation trees are binary decision trees with hyperplane splits based on local canonical correlation coefficients calculated during training. Unlike axis-aligned alternatives, the decision surfaces of CCFs are not restricted to the coordinate system of the inputs features and therefore more naturally represent data with correlated inputs. CCFs naturally accommodate multiple outputs, provide a similar computational complexity to random forests, and in…
Civil Aviation Flight University of China（中国民用航空飞行学院）；Central University of Finance and Economics（中央财经大学）
In this paper, we propose a model-free feature selection method for ultra-high dimensional data with mass features. This is a two phases procedure that we propose to use the fused Kolmogorov filter with the random forest based RFE to remove model limitations and reduce the computational complexity. The method is fully nonparametric and can work with various types of datasets. It has several appealing characteristics, i.e., accuracy, model-free, and computational efficiency, and can be widely used in practical problems, such as multiclass classification, nonparametric regression, and Poisson re…
Random Forests (RF) is a popular machine learning method for classification and regression problems. It involves a bagging application to decision tree models. One of the primary advantages of the Random Forests model is the reduction in the variance of the forecast. In large scale applications of the model with millions of data points and hundreds of features, the size of the fitted objects can get very large and reach the limits on the available space in production setups, depending on the number and depth of the trees. This could be especially challenging when trained models need to be down…
University of Pennsylvania（宾夕法尼亚大学）；Wharton School（沃顿商学院）
A random forest is a popular tool for estimating probabilities in machine learning classification tasks. However, the means by which this is accomplished is unprincipled: one simply counts the fraction of trees in a forest that vote for a certain class. In this paper, we forge a connection between random forests and kernel regression. This places random forest probability estimation on more sound statistical footing. As part of our investigation, we develop a model for the proximity kernel and relate it to the geometry and sparsity of the estimation problem. We also provide intuition and recom…
Random forests have become an established tool for classification and regression, in particular in high-dimensional settings and in the presence of complex predictor-response relationships. For bounded outcome variables restricted to the unit interval, however, classical random forest approaches may severely suffer as they do not account for the heteroscedasticity in the data. A random forest approach is proposed for relating beta distributed outcomes to explanatory variables. The approach explicitly makes use of the likelihood function of the beta distribution for the selection of splits duri…
