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We propose an algorithm named best-scored random forest for binary classification problems. The terminology best-scored means to select the one with the best empirical performance out of a certain number of purely random tree candidates as each single tree in the forest. In this way, the resulting forest can be more accurate than the original purely random forest. From the theoretical perspective, within the framework of regularized empirical risk minimization penalized on the number of splits, we establish almost optimal convergence rates for the proposed best-scored random trees under certain conditions which can be extended to the best-scored random forest. In addition, we present a counterexample to illustrate that in order to ensure the consistency of the forest, every dimension must have the chance to be split. In the numerical experiments, for the sake of efficiency, we employ an adaptive random splitting criterion. Comparative experiments with other state-of-art classification methods demonstrate the accuracy of our best-scored random forest.
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This paper presents a brand new nonparametric density estimation strategy named the best-scored random forest density estimation whose effectiveness is supported by both solid theoretical analysis and significant experimental performance. The terminology best-scored stands for selecting one density tree with the best estimation performance out of a certain number of purely random density tree candidates and we then name the selected one the best-scored random density tree. In this manner, the ensemble of these selected trees that is the best-scored random density forest can achieve even better estimation results than simply integrating trees without selection. From the theoretical perspective, by decomposing the error term into two, we are able to carry out the following analysis: First of all, we establish the consistency of the best-scored random density trees under $L_1$-norm. Secondly, we provide the convergence rates of them under $L_1$-norm concerning with three different tail assumptions, respectively. Thirdly, the convergence rates under $L_{infty}$-norm is presented. Last but not least, we also achieve the above convergence rates analysis for the best-scored random density forest. When conducting comparative experiments with other state-of-the-art density estimation approaches on both synthetic and real data sets, it turns out that our algorithm has not only significant advantages in terms of estimation accuracy over other methods, but also stronger resistance to the curse of dimensionality.
Single-level density-based approach has long been widely acknowledged to be a conceptually and mathematically convincing clustering method. In this paper, we propose an algorithm called best-scored clustering forest that can obtain the optimal level and determine corresponding clusters. The terminology best-scored means to select one random tree with the best empirical performance out of a certain number of purely random tree candidates. From the theoretical perspective, we first show that consistency of our proposed algorithm can be guaranteed. Moreover, under certain mild restrictions on the underlying density functions and target clusters, even fast convergence rates can be achieved. Last but not least, comparisons with other state-of-the-art clustering methods in the numerical experiments demonstrate accuracy of our algorithm on both synthetic data and several benchmark real data sets.
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 TBRF leads to the almost optimal learning rates, and also makes ensemble learning possible, which resolves the boundary discontinuities long plaguing the existing algorithms. Secondly, the two-stage procedure paves the way for parallel computing, leading to computational efficiency. Last but not least, TBRF can serve as an inclusive framework where different mainstream regression strategies such as linear predictor and least squares support vector machines (LS-SVMs) can also be incorporated as value assignment approaches on leaves of the child trees, depending on the characteristics of the underlying data sets. Numerical assessments on comparisons with other state-of-the-art methods on several large-scale real data sets validate the promising prediction accuracy and high computational efficiency of our algorithm.
The random forest algorithm (RF) has several hyperparameters that have to be set by the user, e.g., the number of observations drawn randomly for each tree and whether they are drawn with or without replacement, the number of variables drawn randomly for each split, the splitting rule, the minimum number of samples that a node must contain and the number of trees. In this paper, we first provide a literature review on the parameters influence on the prediction performance and on variable importance measures. It is well known that in most cases RF works reasonably well with the default values of the hyperparameters specified in software packages. Nevertheless, tuning the hyperparameters can improve the performance of RF. In the second part of this paper, after a brief overview of tuning strategies we demonstrate the application of one of the most established tuning strategies, model-based optimization (MBO). To make it easier to use, we provide the tuneRanger R package that tunes RF with MBO automatically. In a benchmark study on several datasets, we compare the prediction performance and runtime of tuneRanger with other tuning implementations in R and RF with default hyperparameters.
Random forest (RF) missing data algorithms are an attractive approach for dealing with missing data. They have the desirable properties of being able to handle mixed types of missing data, they are adaptive to interactions and nonlinearity, and they have the potential to scale to big data settings. Currently there are many different RF imputation algorithms but relatively little guidance about their efficacy, which motivated us to study their performance. Using a large, diverse collection of data sets, performance of various RF algorithms was assessed under different missing data mechanisms. Algorithms included proximity imputation, on the fly imputation, and imputation utilizing multivariate unsupervised and supervised splitting---the latter class representing a generalization of a new promising imputation algorithm called missForest. Performance of algorithms was assessed by ability to impute data accurately. Our findings reveal RF imputation to be generally robust with performance improving with increasing correlation. Performance was good under moderate to high missingness, and even (in certain cases) when data was missing not at random.
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