Random Forests

applications. There are two theoretical properties that need to be further exploration, one is the consistency that decides whether the algorithm could converge to an optimal solution as the sample size tends to infinity; the other one is to search an upper bound on the generalization error of the algorithms. In recent years, there are some works dedicated to prove the consistency of random forests. For example, an online random forests classification algorithm was proposed by Denil et al. [37], which not only had the consistency theory, but also had a good performance in practice; the random survival forests was proposed by Ishwaran et al. [38], which focused on the survival setting of the random forests; the reinforcement learning trees was proposed by Zhu et al. [39], which was a regression algorithm and had been proved to be consistent; a pure random forests regression algorithm was proposed by Genuer et al. [40], which had been proved to be consistent and had good performance. Besides, two simplified versions of the random forests were proposed by Biau et al. [20, 21]. But both of the algorithms were difficult to be applied in practice. It was obviously that the majority of existing works focused on the online or regression situation. Among these works, Biau et al. [21] presented an in-depth theoretical analysis for the off-line random forests algorithm. They proved the consistency of a simplified random forests algorithm by employing a second independent datasets to evaluate the importance of features in advance. Moreover, at each node, a fixed number of features were selected randomly. The midpoint of the most important feature was used as a split threshold to split on. In the work of Biau et al. [21], both the selection of the split threshold and the usage of the second sample set contributed to justification the consistency of random forests. For the original random forests algorithm, it used the bagging method and CART-splitting scheme on the actual samples, which all led to more difficulties to analyze the consistency of the algorithm. Therefore, the majority of the existing consistency analysis were based on a simplified version of the original random forests algorithm.

State-of-the-art learning algorithms, such as random forests or neural networks, are often qualied as black-boxes because of the high number and complexity of operations involved in their prediction mechanism. This lack of interpretability is a strong limitation for applications involving critical decisions, typically the analysis of production processes in the manufacturing industry. In such critical contexts, models have to be interpretable, i.e., simple, stable, and predictive. To address this issue, we design SIRUS (Stable and Interpretable RUle Set), a new classication algorithm based on random forests, which takes the form of a short list of rules. While simple models are usually unstable with respect to data perturbation, SIRUS achieves a remarkable stability improvement over cutting-edge methods. Furthermore, SIRUS inherits a predictive accuracy close to random forests, combined with the simplicity of decision trees. These properties are assessed both from a theoretical and empirical point of view, through extensive numerical experiments based on our R/C++ software implementation sirus available from CRAN.

The classification trees, from which the random forests are built, are built recursively in that the next splitting varia- ble is selected by means of locally optimizing a criterion (such as the Gini gain in the traditional CART algorithm [28]) within the current node. This current node is defined by a configuration of predictor values, that is determined by all previous splits in the same branch of the tree (see, e.g., [29] for illustrations). In this respect the evaluation of the next splitting variable can be considered conditional on the previously selected predictor variables, but regardless of any other predictor variable. In particu- lar, the selection of the first splitting variable involves only the marginal, univariate association between that predic- tor variable and the response, regardless of all other pre- dictor variables. However, this search strategy leads to a variable selection pattern where a predictor variable that is per se only weakly or not at all associated with the response, but is highly correlated with another influential predictor variable, may appear equally well suited for splitting as the truly influential predictor variable. We will illustrate this point in more detail in the following simu- lation study.

Owing to its notable predictive accuracy, many extensions have been proposed to further improve Random Forests. In [29], the use of five attribute good- ness measures was proposed, such that diversity in the ensemble is boosted. In addition to Gini index used in CART and random trees that make up Random Forests, Gain ratio, MDL (Minimum Description Length), Myopic ReliefF and ReliefF were used. Also unlike Random Forest in its traditional form, weighted voting was proposed. Both extensions have empirically shown potential in enhancing the predictive accuracy of Random Forests. In [21], McNemar non-parametric test of significance was used to limit the number of trees contributing to the majority voting. In a related work to this one, authors in [30] used more complex dynamic integration methods to replace majority voting. Stimulated by the low performance reported in high dimensional data sets, weighted sampling of features was proposed in [3]. In [6], each tree in a Random Forest is represented as a gene with the trees in that Random For- est represent an individual. Having a number of trained Random Forests, the problem has turned to be a Genetic Algorithm optimisation one. Extensive experimental study has shown the potential of this approach. For more infor- mation about these techniques, the reader is referred to the survey paper in [13]. In a more recent work, diversification using weighted random subspacing was proposed in [12].

This paper is primarily interested in random forests for vari- able selection. Mainly methodological the main contribution is twofold: to provide some insights about the behavior of the variable importance index based on random forests and to use it to propose a two-steps algorithm for two classical problems of variable selection starting from variable importance ranking. The first problem is to find important variables for interpretation and the second one is more restrictive and try to design a good prediction model. The general strategy involves a ranking of explanatory variables using the random forests score of impor- tance and a stepwise ascending variable introduction strategy. Let us mention that we propose an heuristic strategy which does not depend on specific model hypotheses but based on data- driven thresholds to take decisions.

It is not so clear what happens in this example if the successive cuts are made by minimizing the empirical error. Whether the middle square is ever cut will depend on the precise form of the stopping rule and the exact parameters of the distribution. The example is here to illustrate that consistency of greedily grown random forests is a delicate issue. Note however that if Breiman’s original algorithm is used in this example (i.e., when all cells with more than one data point in it are split) then one obtains a consistent classification rule. If, on the other hand, horizontal or vertical cuts are selected to minimize the probability of error, and k → ∞ in such a way that k = O(n 1/2 − ε )

Ensemble classification methods are learning algorithms that construct a set of classifiers instead of one classifier, and then classify new data points by taking a vote of their predictions is developed. Ensemble learning provides a more reliable mapping that can be obtained by combining the output of multiple classifiers. [16] Fig.2 illustrates ensemble learning. Random forest (or random forests) is an ensemble classifier that consists of many decision trees and outputs the class that is the mode of the class's output by individual trees. The algorithm was developed by Leo Breiman and Adele Cutler in the middle of 1990th.The method combines Breiman's "bagging" idea and the random selection of features.

The conclusion that “The random forest is clearly the best family of classifiers” is flawed. The paper gives three arguments for why random forests are the best family: “The eight random forest classifiers are included among the 25 best classifiers having all of them low ranks”, “The family RF has the lowest minimum rank (32.9) and mean (46.7), and also a narrow interval (up to 60.5), which means that all the RF classifiers work very well”, and “3 out of [the] 5” best classifiers are random forests.

First, the Prism family of algorithms has been introduced and compared with decision trees and next the well known Random Forests approach has been reviewed. Random Prism is inspired by the Prism family of algorithms, the Random Decision Forests and the Random Forests approaches. Random Prism uses the PrismTCS classifier as base classifier with some modifications called R-PrismTCS. The modifications were made in order to use the Random Decision Forests’ feature subset selection approach. Random Prism also incor- porates J-pruning for R-PrismTCS and Random Forests’ bagging approach. Contrary to Random Forests and Random Decision Forests, Random Prism uses a weighted majority voting system instead of a plain majority voting system, in order to take the individual classifier’s classification accuracy into account. Also, Random Prism does not take all classifiers into account, and the user can define the percentage of classifiers to be used for classification. Random Prism will select only the classifiers with the highest classification accuracy for the classification task.

LFS may have potential application on time-series data. Here time-series refers to data with each feature a record at a particular time point, not the response variable being time-series. The features of this type of data has a well structure given: they are organized in 1-D and features within a short time frame are highly dependent. We implemented LFS- randomTF that randomly sample features within a time frame to train each tree. However, we did not observe improvement in performance when applying LFS-randomTF on time- series datasets available on UCI. This is mainly because random forests or tree classifiers do not perform well on time-series data since the dimension is usually much higher than the sample size. In this case, the strength is the dominate term in determining the generalization error and one should aim to improve strength rather than reduce correlation in order to achieve a better prediction performance. However, we would expect to benefit from using LFS on time-series data when a sufficient large training set is given.

In this paper, we focus on methods based on the jackknife and the infinitesimal jackknife for bagging (Efron, 1992, 2013) that let us estimate standard errors based on the pre-existing bootstrap replicates. Other approaches that rely on forming second-order bootstrap repli- cates have been studied by Duan (2011) and Sexton and Laake (2009). Directly bootstrap- ping a random forest is usually not a good idea, as it requires forming a large number of base learners. Sexton and Laake (2009), however, propose a clever work-around to this problem. Their approach, which could have been called a bootstrap of little bags, involves bootstrap- ping small random forests with around B = 10 trees and then applying a bias correction to remove the extra Monte Carlo noise.

In the previous sections, we have demonstrated simple examples where random forests and AdaBoost yield the strongest performance with respect to the Bayes rule. We have argued that these algorithms are successful classifiers due to the fact that they fit initially complex models by interpolating the training data but also exhibit smoothing properties via self- averaging that stabilizes the fit in regions with signal, while continuing to keep localized the effect of noise points on the overall fit. While this smoothing mechanism is obvious for random forests via the averaging over decision trees, it is less obvious for AdaBoost. In this section we explain why the additional iterations in boosting way beyond the point at which perfect classification of the training data (i.e interpolation) has occurred actually has the effect of smoothing out the effects of noise rather than leading to more and more overfitting. To the best of our knowledge, this is a novel perspective on the algorithm. To explain our key idea, we will recall the pure noise example from before with p = .8, d = 20 and n = 5000.

Abstract: This paper proposes the use of Stacked Random Forests (SRF) for the classification of Polarimetric Synthetic Aperture Radar images. SRF apply several Random Forest instances in a sequence where each individual uses the class estimate of its predecessor as an additional feature. To this aim, the internal node tests are designed to work not only directly on the complex-valued image data, but also on spatially varying probability distributions and thus allow a seamless integration of RFs within the stacking framework. Experimental results show that the classification performance is consistently improved by the proposed approach, i.e., the achieved accuracy is increased by 4% and 7% for one fully- and one dual-polarimetric dataset. This increase only comes at the cost of a linear increased training and prediction time, which is rather limited as the method converges quickly.

Random Forests (RFs) are strong machine learning tools Breiman (2001), comparing well with state-of-the-art methods such as SVM or boosting algorithms Freund et al. (1996), and used in a wide range of domains Svetnik et al. (2003); D´ıaz-Uriarte and De Andres (2006); Genuer et al. (2010). These estimators fit a number of decision tree classifiers on different random sub-samples of the dataset. Each tree is built recursively, according to a splitting criterion based on some impurity measure of a node. The prediction is done by an average over each tree prediction. In classification the averaging is based on a majority vote. Practical and theoretical insights on RFs are given in Genuer et al. (2008); Biau et al. (2008); Louppe (2014); Biau and Scornet (2016).

While each of these studies mentions directing the students identified as at-risk to tutoring (Zhang et al., 2010), generic interventions (Delen, 2010; Macfayden and Dawson, 2010), or ad- ditional help or attention (Kotsiantis et al., 2004; Dekker et al., 2009), no analyses are performed to whether the intervention will help these students succeed or persist. Furthermore, misclassifi- cation of at-risk students, or merely limited definition of such a classification, may result in the intervention being applied to students who will not benefit or not being applied to students who will benefit from it. Superby et al. (2006) offers a possible correction for this phenomenon by instead categorizing students as “low-risk,” “medium-risk” and “high-risk”. In their study on stu- dent dropout using multiple data mining methods, including random forests and decision trees, the medium-risk group was defined as “students, who may succeed thanks to the measures taken by the university”. The outcome risk groups in Superby et al. (2006) were created by using grades obtained in the first month of class, simply categorizing students scoring less than 45% as high-risk and students scoring higher than 70% as low-risk. However, no discussion is made about the claim that the medium-risk students will benefit from measures taken by the university or what those measures might be. We propose the ITE approach as a method that directly identifies students who may benefit the most from a particular intervention, thus allowing for an effective allocation of resources.

For the observed dataset used in this study, posterior expectations and quantiles of the parameters of interest ra and N2/Na are reported in Tables S6 and S7. Expectation and CI values substantially vary for both parameters, depending on the method used. The impact of the tolerance levels is noteworthy for both the rejection and local linear adjustment ABC methods. The posterior expectation of ra obtained using ABC- RF was equal to 0.221 with a relatively narrow associated 95% CI of [0.112,0.287]. The latter estimation lays well within previous estimates of the mean proportion of genes of European ancestry within African American individuals, which typically ranged from 0.070 to 0.270 − with most estimates around 0.200 −, depending on individual exclusions, the population samples and sets of genetic markers considered, as well as the evolutionary models assumed and inferential methods used (reviewed in Bryc et al., 2015). Interestingly, a recent genomic analysis using a conditional random field parametrized by random forests trained on reference panels (Maples et al., 2013) and 500 000 SNPs provided a similar expectation value of ra for the same African American population ASW (i.e. ra = 0.213), with a somewhat smaller 95% CI (i.e. [0.195,0.232]), probably due to the ten times larger number of SNPs in their dataset (Baharian et al., 2016).

Before we begin to discuss the intricacies of a random forest, we will first consider a single tree. Random forests are constructed from a combination of decision trees. A decision tree is a method of classifying a feature of interest, denoted T , through the use of other features available. In the case of detecting spam tweets, knowing whether a URL is present in a tweet, the users account age, and if the tweet has been reported as spam could all be assessed before deciding on whether a tweet is spam or not spam. Table 1 gives examples of these features.

Social lending, also known as peer-to-peer lending, provides customers with a platform to borrow and lend money online. It is now rapidly gaining its pop- ularity for its superior monetary advantage comparing to banks for both bor- rowers and lenders. Thus, choosing a reliable is very important, whereas the only method most of the platforms use now is a grading system. In order to better prevent the risks, we propose a method of combining Random Forests and Neural Network for predicting the borrowers’ status. Our data are from Lending Club, a popular social lending platform, and our results indicate that our method outperforms the lending Club good borrower grades.

In this work, we focus on one commonly used class of classifiers: decision trees and random forests [43, 19]. Decision trees are simple classifiers that consist of a collection of decision nodes arranged in a tree structure. As the name suggests, each decision node is associated with a pred- icate or test on the query (for example, a possible predicate could be “age > 55”). Decision tree evaluation simply corresponds to tree traversal. These models are often favored by users for their ease of interpretability. In fact, there are numerous web APIs [2, 1] that enable users to both train and query decision trees as part of a machine learning as a service platform. In spite of their simple structure, decision trees are widely used in machine learning, and have been successfully applied to many scenarios such as disease diagnosis [62, 5] and credit-risk assessment [49].

Background: Clustering plays a crucial role in several application domains, such as bioinformatics. In bioinformatics, clustering has been extensively used as an approach for detecting interesting patterns in genetic data. One application is population structure analysis, which aims to group individuals into subpopulations based on shared genetic variations, such as single nucleotide polymorphisms. Advances in DNA sequencing technology have facilitated the obtainment of genetic datasets with exceptional sizes. Genetic data usually contain hundreds of thousands of genetic markers genotyped for thousands of individuals, making an efficient means for handling such data desirable. Results: Random Forests (RFs) has emerged as an efficient algorithm capable of handling high-dimensional data. RFs provides a proximity measure that can capture different levels of co-occurring relationships between variables. RFs has been widely considered a supervised learning method, although it can be converted into an unsupervised learning method. Therefore, RF-derived proximity measure combined with a clustering technique may be well suited for determining the underlying structure of unlabeled data. This paper proposes, RFcluE, a cluster ensemble approach for determining the underlying structure of genetic data based on RFs. The approach comprises a cluster ensemble framework to combine multiple runs of RF clustering. Experiments were conducted on high-dimensional, real genetic dataset to evaluate the