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.
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…
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）
Department of Statistical Science, Cornell University（康奈尔大学统计科学系）
We propose a modification that corrects for split-improvement variable importance measures in Random Forests and other tree-based methods. These methods have been shown to be biased towards increasing the importance of features with more potential splits. We show that by appropriately incorporating split-improvement as measured on out of sample data, this bias can be corrected yielding better summaries and screening tools.
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Infinite-order U-statistics (IOUS) has been used extensively on subbagging ensemble learning algorithms such as random forests to quantify its uncertainty. While normality results of IOUS have been studied extensively, its variance estimation approaches and theoretical properties remain mostly unexplored. Existing approaches mainly utilize the leading term dominance property in the Hoeffding decomposition. However, such a view usually leads to biased estimation when the kernel size is large or the sample size is small. On the other hand, while several unbiased estimators exist in the literatur…
We attempt to give a unifying view of the various recent attempts to (i) improve the interpretability of tree-based models and (ii) debias the the default variable-importance measure in random Forests, Gini importance. In particular, we demonstrate a common thread among the out-of-bag based bias correction methods and their connection to local explanation for trees. In addition, we point out a bias caused by the inclusion of inbag data in the newly developed explainable AI for trees algorithms.
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 …
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…
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）
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…
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 thi…
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…
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…
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…
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…
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…
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）
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…
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.
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…
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）
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…
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…
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）
Survival random forest is a popular machine learning tool for modeling censored survival data. However, there is currently no statistically valid and computationally feasible approach for estimating its confidence band. This paper proposes an unbiased confidence band estimation by extending recent developments in infinite-order incomplete U-statistics. The idea is to estimate the variance-covariance matrix of the cumulative hazard function prediction on a grid of time points. We then generate the confidence band by viewing the cumulative hazard function estimation as a Gaussian process whose d…
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）
Department of Social Statistics and Demography, Univ. of Southampton（社会统计与人口学系，南安普顿大学）；University of Southampton & Statistisk sentralbyrå（南安普顿大学及瑞典统计局）；Australian Bureau of Statistics（澳大利亚统计局）；Instituto Nacional de Estadística（西班牙国家统计局）；Statistics Canada (Ret.)（加拿大统计局（退休））
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…
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）
Merck & Co., Inc.（默克公司）；Sanofi-Aventis US LLC（赛诺菲-安万特美国有限责任公司）
Precision medicine is becoming a focus in medical research recently, as its implementation brings values to all stakeholders in the healthcare system. Various statistical methodologies have been developed tackling problems in different aspects of this field, e.g., assessing treatment heterogeneity, identifying patient subgroups, or building treatment decision models. However, there is a lack of new tools devoted to selecting and prioritizing predictive biomarkers. We propose a novel tree-based ensemble method, random interaction forest (RIF), to generate predictive importance scores and priori…
The most popular approach for analyzing survival data is the Cox regression model. The Cox model may, however, be misspecified, and its proportionality assumption may not always be fulfilled. An alternative approach for survival prediction is random forests for survival outcomes. The standard split criterion for random survival forests is the log-rank test statistics, which favors splitting variables with many possible split points. Conditional inference forests avoid this split variable selection bias. However, linear rank statistics are utilized by default in conditional inference forests to…
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…
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）
University of Huelva（韦尔瓦大学）；Complutense University of Madrid（马德里康普顿斯大学）
Most representative decision tree ensemble methods have been used to examine the variable importance of Treasury term spreads to predict US economic recessions with a balance of generating rules for US economic recession detection. A strategy is proposed for training the classifiers with Treasury term spreads data and the results are compared in order to select the best model for interpretability. We also discuss the use of SHapley Additive exPlanations (SHAP) framework to understand US recession forecasts by analyzing feature importance. Consistently with the existing literature we find the m…
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…
