Tree ensemble methods such as random forests [Breiman, 2001] are very popular to handle high-dimensional tabular data sets, notably because of their good predictive accuracy. However, when machine learning is used for decision-making problems, settling for the best predictive procedures may not be reasonable since enlightened decisions require an in-depth comprehension of the algorithm prediction process. Unfortunately, random forests are not intrinsically interpretable since their prediction results from averaging several hundreds of decision trees. A classic approach to gain knowledge on this so-called black-box algorithm is to compute variable importances, that are employed to assess the predictive impact of each input variable. Variable importances are then used to rank or select variables and thus play a great role in data analysis. Nevertheless, there is no justification to use random forest variable importances in such way: we do not even know what these quantities estimate. In this paper, we analyze one of the two well-known random forest variable importances, the Mean Decrease Impurity (MDI). We prove that if input variables are independent and in absence of interactions, MDI provides a variance decomposition of the output, where the contribution of each variable is clearly identified. We also study models exhibiting dependence between input variables or interaction, for which the variable importance is intrinsically ill-defined. Our analysis shows that there may exist some benefits to use a forest compared to a single tree.
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…
The random forest algorithm, proposed by L. Breiman in 2001, has been extremely successful as a general-purpose classification and regression method. The approach, which combines several randomized decision trees and aggregates their predictions by averaging, has shown excellent performance in settings where the number of variables is much larger than the number of observations. Moreover, it is versatile enough to be applied to large-scale problems, is easily adapted to various ad-hoc learning tasks, and returns measures of variable importance. The present article reviews the most recent theor…
The default variable-importance measure in random Forests, Gini importance, has been shown to suffer from the bias of the underlying Gini-gain splitting criterion. While the alternative permutation importance is generally accepted as a reliable measure of variable importance, it is also computationally demanding and suffers from other shortcomings. We propose a simple solution to the misleading/untrustworthy Gini importance which can be viewed as an overfitting problem: we compute the loss reduction on the out-of-bag instead of the in-bag training samples.
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.
Augmented Functional Random Forests: Classifier Construction and Unbiased Functional Principal Components Importance through Ad-Hoc Conditional Permutations
Faculty of Technological and Innovation Sciences（技术与创新科学学院）；Faculty of Economics and Law（经济与法学院）
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…
We seek decision rules for prediction-time cost reduction, where complete data is available for training, but during prediction-time, each feature can only be acquired for an additional cost. We propose a novel random forest algorithm to minimize prediction error for a user-specified {\it average} feature acquisition budget. While random forests yield strong generalization performance, they do not explicitly account for feature costs and furthermore require low correlation among trees, which amplifies costs. Our random forest grows trees with low acquisition cost and high strength based on gre…
Random forests have been widely used for their ability to provide so-called importance measures, which give insight at a global (per dataset) level on the relevance of input variables to predict a certain output. On the other hand, methods based on Shapley values have been introduced to refine the analysis of feature relevance in tree-based models to a local (per instance) level. In this context, we first show that the global Mean Decrease of Impurity (MDI) variable importance scores correspond to Shapley values under some conditions. Then, we derive a local MDI importance measure of variable …
Tree ensembles such as Random Forests have achieved impressive empirical success across a wide variety of applications. To understand how these models make predictions, people routinely turn to feature importance measures calculated from tree ensembles. It has long been known that Mean Decrease Impurity (MDI), one of the most widely used measures of feature importance, incorrectly assigns high importance to noisy features, leading to systematic bias in feature selection. In this paper, we address the feature selection bias of MDI from both theoretical and methodological perspectives. Based on …
This paper introduces and develops a novel variable importance score function in the context of ensemble learning and demonstrates its appeal both theoretically and empirically. Our proposed score function is simple and more straightforward than its counterpart proposed in the context of random forest, and by avoiding permutations, it is by design computationally more efficient than the random forest variable importance function. Just like the random forest variable importance function, our score handles both regression and classification seamlessly. One of the distinct advantage of our propos…
Anomaly Detection is an unsupervised learning task aimed at detecting anomalous behaviours with respect to historical data. In particular, multivariate Anomaly Detection has an important role in many applications thanks to the capability of summarizing the status of a complex system or observed phenomenon with a single indicator (typically called `Anomaly Score') and thanks to the unsupervised nature of the task that does not require human tagging. The Isolation Forest is one of the most commonly adopted algorithms in the field of Anomaly Detection, due to its proven effectiveness and low comp…
This paper is about variable selection with the random forests algorithm in presence of correlated predictors. In high-dimensional regression or classification frameworks, variable selection is a difficult task, that becomes even more challenging in the presence of highly correlated predictors. Firstly we provide a theoretical study of the permutation importance measure for an additive regression model. This allows us to describe how the correlation between predictors impacts the permutation importance. Our results motivate the use of the Recursive Feature Elimination (RFE) algorithm for varia…
The selection of grouped variables using the random forest algorithm is considered. First a new importance measure adapted for groups of variables is proposed. Theoretical insights into this criterion are given for additive regression models. Second, an original method for selecting functional variables based on the grouped variable importance measure is developed. Using a wavelet basis, it is proposed to regroup all of the wavelet coefficients for a given functional variable and use a wrapper selection algorithm with these groups. Various other groupings which take advantage of the frequency …
Conditional Feature Importance with Generative Modeling Using Adversarial Random Forests
Leibniz Institute for Prevention Research & Epidemiology – BIPS（莱布尼茨预防研究与流行病学研究所 – BIPS）；Faculty of Mathematics and Computer Science, University of Bremen（不来梅大学数学与计算机科学学院）；Berlin School of Economics and Law（柏林经济与法律学院）；University of Copenhagen（哥本哈根大学）
This paper examines from an experimental perspective random forests, the increasingly used statistical method for classification and regression problems introduced by Leo Breiman in 2001. It first aims at confirming, known but sparse, advice for using random forests and at proposing some complementary remarks for both standard problems as well as high dimensional ones for which the number of variables hugely exceeds the sample size. But the main contribution of this paper is twofold: to provide some insights about the behavior of the variable importance index based on random forests and in add…
National Institute for Public Health and the Environment (RIVM)（荷兰国家公共卫生与环境研究所）
We give examples of data-generating models under which Breiman's random forest may be extremely slow to converge to the optimal predictor or even fail to be consistent. The evidence provided for these properties is based on mostly intuitive arguments, similar to those used earlier with simpler examples, and on numerical experiments. Although one can always choose models under which random forests perform very badly, we show that simple methods based on statistics of `variable use' and `variable importance' can often be used to construct a much better predictor based on a `many-armed' random fo…
Shapelet is a discriminative subsequence of time series. An advanced shapelet-based method is to embed shapelet into accurate and fast random forest. However, it shows several limitations. First, random shapelet forest requires a large training cost for split threshold searching. Second, a single shapelet provides limited information for only one branch of the decision tree, resulting in insufficient accuracy and interpretability. Third, randomized ensemble causes interpretability declining. For that, this paper presents Random Pairwise Shapelets Forest (RPSF). RPSF combines a pair of shapelet…
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…
Tsinghua Shenzhen International Graduate School, Tsinghua University（清华大学深圳国际研究生院）；Peng Cheng Laboratory（鹏城实验室）；Case Western Reserve University（凯斯西储大学）
Despite the impressive performance of random forests (RF), its theoretical properties have not been thoroughly understood. In this paper, we propose a novel RF framework, dubbed multinomial random forest (MRF), to analyze the \emph{consistency} and \emph{privacy-preservation}. Instead of deterministic greedy split rule or with simple randomness, the MRF adopts two impurity-based multinomial distributions to randomly select a split feature and a split value respectively. Theoretically, we prove the consistency of the proposed MRF and analyze its privacy-preservation within the framework of diff…
Variable importance measures are the main tools to analyze the black-box mechanisms of random forests. Although the mean decrease accuracy (MDA) is widely accepted as the most efficient variable importance measure for random forests, little is known about its statistical properties. In fact, the definition of MDA varies across the main random forest software. In this article, our objective is to rigorously analyze the behavior of the main MDA implementations. Consequently, we mathematically formalize the various implemented MDA algorithms, and then establish their limits when the sample size i…
Correcting Variable Importance Scored by Random Forests
Computer and Information Science（计算机与信息科学）；Mathematics and Data Science（数学与数据科学）；University of Massachusetts, Dartmouth, MA（马萨诸塞大学达特茅斯分校）；The Rivers School, Weston, MA（韦斯特on学校的河流学校）
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…
As the size, complexity, and availability of data continues to grow, scientists are increasingly relying upon black-box learning algorithms that can often provide accurate predictions with minimal a priori model specifications. Tools like random forests have an established track record of off-the-shelf success and even offer various strategies for analyzing the underlying relationships among variables. Here, motivated by recent insights into random forest behavior, we introduce the simple idea of augmented bagging (AugBagg), a procedure that operates in an identical fashion to classical baggin…
Columbia University（哥伦比亚大学）；Carnegie Mellon University（卡内基梅隆大学）
We study rare-event simulation for a class of problems where the target hitting sets of interest are defined via modern machine learning tools such as neural networks and random forests. This problem is motivated from fast emerging studies on the safety evaluation of intelligent systems, robustness quantification of learning models, and other potential applications to large-scale simulation in which machine learning tools can be used to approximate complex rare-event set boundaries. We investigate an importance sampling scheme that integrates the dominating point machinery in large deviations …
We use the output of a random forest to define a family of local smoothers with spatially adaptive bandwidth matrices. The smoother inherits the flexibility of the original forest but, since it is a simple, linear smoother, it is very interpretable and it can be used for tasks that would be intractable for the original forest. This includes bias correction, confidence intervals, assessing variable importance and methods for exploring the structure of the forest. We illustrate the method on some synthetic examples and on data related to Covid-19.
In this article, a large data set containing every course taken by every undergraduate student in a major university in Canada over 10 years is analysed. Modern machine learning algorithms can use large data sets to build useful tools for the data provider, in this case, the university. In this article, two classifiers are constructed using random forests. To begin, the first two semesters of courses completed by a student are used to predict if they will obtain an undergraduate degree. Secondly, for the students that completed a program, their major is predicted using once again the first few…
Pan African University（泛非大学）；University of Nairobi（内罗毕大学）；Michigan State University（密歇根州立大学）
A reliable and accurate forecasting model for crop yields is of crucial importance for efficient decision-making process in the agricultural sector. However, due to weather extremes and uncertainties, most forecasting models for crop yield are not reliable and accurate. For measuring the uncertainty and obtaining further information of future crop yields, a probability density forecasting model based on quantile random forest and Epanechnikov kernel function (QRF-SJ) is proposed. The nonlinear structure of random forest is applied to change the quantile regression model for building the probab…
Data analysis and machine learning have become an integrative part of the modern scientific methodology, offering automated procedures for the prediction of a phenomenon based on past observations, unraveling underlying patterns in data and providing insights about the problem. Yet, caution should avoid using machine learning as a black-box tool, but rather consider it as a methodology, with a rational thought process that is entirely dependent on the problem under study. In particular, the use of algorithms should ideally require a reasonable understanding of their mechanisms, properties and …
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…
Interpretability of learning algorithms is crucial for applications involving critical decisions, and variable importance is one of the main interpretation tools. Shapley effects are now widely used to interpret both tree ensembles and neural networks, as they can efficiently handle dependence and interactions in the data, as opposed to most other variable importance measures. However, estimating Shapley effects is a challenging task, because of the computational complexity and the conditional expectation estimates. Accordingly, existing Shapley algorithms have flaws: a costly running time, or…
Multi forests: Variable importance for multi-class outcomes
Institute for Medical Information Processing, Biometry and Epidemiology, LMU Munich（慕尼黑大学医学信息处理、生物统计学与流行病学研究所）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；Institute of AI and Informatics in Medicine, TUM School of Medicine and Health, Technical University of Munich（慕尼黑工业大学医学与健康学院医学人工智能与信息学研究所）
University of British Columbia（不列颠哥伦比亚大学）；Oxford University（牛津大学）
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 o…
Principled Federated Random Forests for Heterogeneous Data
Inria（法国国家信息与自动化研究所）；PreMeDICaL（预医学实验室）；Inserm（法国国家医学研究院）；University of Montpellier（蒙彼利埃大学）；Sorbonne Université（索邦大学）；Université Paris Cité（巴黎城市大学）；CNRS（国家科学研究中心）；LPSM（巴黎高等师范学院）
INRIA（法国国家信息与自动化研究所）；INSERM（法国国家健康与医学研究院）；Univ. Bordeaux（波尔多大学）；Univ. Paris-Sud Orsay（巴黎第十一大学）；Univ. Paris Descartes（巴黎第五大学）；Université Côte d’Azur（蔚蓝海岸大学）；CNRS（法国国家科学研究中心）；Université de Toulouse（图卢兹大学）；INRA（法国国家农业研究院）
Big Data is one of the major challenges of statistical science and has numerous consequences from algorithmic and theoretical viewpoints. Big Data always involve massive data but they also often include online data and data heterogeneity. Recently some statistical methods have been adapted to process Big Data, like linear regression models, clustering methods and bootstrapping schemes. Based on decision trees combined with aggregation and bootstrap ideas, random forests were introduced by Breiman in 2001. They are a powerful nonparametric statistical method allowing to consider in a single and…
Nowadays new technologies, and especially artificial intelligence, are more and more established in our society. Big data analysis and machine learning, two sub-fields of artificial intelligence, are at the core of many recent breakthroughs in many application fields (e.g., medicine, communication, finance, ...), including some that are strongly related to our day-to-day life (e.g., social networks, computers, smartphones, ...). In machine learning, significant improvements are usually achieved at the price of an increasing computational complexity and thanks to bigger datasets. Currently, cut…
Tree ensembles, such as random forests and AdaBoost, are ubiquitous machine learning models known for achieving strong predictive performance across a wide variety of domains. However, this strong performance comes at the cost of interpretability (i.e. users are unable to understand the relationships a trained random forest has learned and why it is making its predictions). In particular, it is challenging to understand how the contribution of a particular feature, or group of features, varies as their value changes. To address this, we introduce Disentangled Attribution Curves (DAC), a method…
Sequential Permutation Testing of Random Forest Variable Importance Measures
Institute of General Practice and Health Services Research（全科医学与卫生服务研究所）；Institute of AI and Informatics in Medicine（医学人工智能与信息学研究所）；Technical University of Munich（慕尼黑工业大学）；Institute for Medical Information Processing, Biometry and Epidemiology（医学信息处理、生物统计学与流行病学研究所）；University of Munich（慕尼黑大学）
Jet Propulsion Laboratory, California Institute of Technology（加州理工学院喷气推进实验室）；Columbia University（哥伦比亚大学）
Have you ever wondered how your feature space is impacting the prediction of a specific sample in your dataset? In this paper, we introduce Single Sample Feature Importance (SSFI), which is an interpretable feature importance algorithm that allows for the identification of the most important features that contribute to the prediction of a single sample. When a dataset can be learned by a Random Forest classifier or regressor, SSFI shows how the Random Forest's prediction path can be utilized for low-level feature importance calculation. SSFI results in a relative ranking of features, highlight…
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…
Lassoed Forests: Random Forests with Adaptive Lasso Post-selection
Department of Statistics（统计学系）；Department of Biomedical Data Science（生物医学数据科学系）；Princess Margaret Cancer Centre（荣格癌症中心）；Department of Medical Biophysics（医学生物物理系）
Integrating Random Forests and Generalized Linear Models for Improved Accuracy and Interpretability
Department of Statistics University of California, Berkeley（加州大学伯克利分校统计学系）；Department of Statistics University of California, Irvine（加州大学伊尔市统计学系）；Department of Statistics and Data Science National University of Singapore（新加坡国立大学统计学与数据科学系）；Department of Applied and Computational Mathematics and Statistics University of Notre Dame（圣约翰大学应用与计算数学与统计学系）；Department of Statistics, EECS, CCB University of California, Berkeley（加州大学伯克利分校统计学系、电子工程与计算机科学系、中心计算中心）
Following the line of classification-based two-sample testing, tests based on the Random Forest classifier are proposed. The developed tests are easy to use, require almost no tuning, and are applicable for any distribution on $\mathbb{R}^d$. Furthermore, the built-in variable importance measure of the Random Forest gives potential insights into which variables make out the difference in distribution. An asymptotic power analysis for the proposed tests is developed. Finally, two real-world applications illustrate the usefulness of the introduced methodology. To simplify the use of the method, …
Erasmus University Rotterdam（鹿特丹伊拉斯姆斯大学）；University of Chicago（芝加哥大学）；Migros T.A.Ş（Migros公司）
We propose two algorithms for interpretation and boosting of tree-based ensemble methods. Both algorithms make use of mathematical programming models that are constructed with a set of rules extracted from an ensemble of decision trees. The objective is to obtain the minimum total impurity with the least number of rules that cover all the samples. The first algorithm uses the collection of decision trees obtained from a trained random forest model. Our numerical results show that the proposed rule covering approach selects only a few rules that could be used for interpreting the random forest …
Randomization Can Reduce Both Bias and Variance: A Case Study in Random Forests
Operations Research Center Massachusetts Institute of Technology（麻省理工学院运筹学研究中心）；Sloan School of Management Massachusetts Institute of Technology（麻省理工学院斯隆管理学院）
University of Liège（列日大学）；New York University（纽约大学）
In many cases, feature selection is often more complicated than identifying a single subset of input variables that would together explain the output. There may be interactions that depend on contextual information, i.e., variables that reveal to be relevant only in some specific circumstances. In this setting, the contribution of this paper is to extend the random forest variable importances framework in order (i) to identify variables whose relevance is context-dependent and (ii) to characterize as precisely as possible the effect of contextual information on these variables. The usage and t…
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 imp…
Decision trees with binary splits are popularly constructed using Classification and Regression Trees (CART) methodology. For binary classification and regression models, this approach recursively divides the data into two near-homogenous daughter nodes according to a split point that maximizes the reduction in sum of squares error (the impurity) along a particular variable. This paper aims to study the bias and adaptive properties of regression trees constructed with CART. In doing so, we derive an interesting connection between the bias and the mean decrease in impurity (MDI) measure of vari…
