A computationally fast variable importance test for random forests for high-dimensional data

Random forests are a commonly used tool for classification with high-dimensional data as well as for ranking candidate predictors based on the so-called variable importance measures. There are different importance measures for ranking predictor variables, the two most common measures are the Gini importance and the permutation importance. The latter has been found to be more reliable than the Gini importance. It is computed from the change in prediction accuracy when removing any association between the response and a predictor variable, with large changes indicating that the predictor variable is important. A drawback of those variable importance measures is that there is no natural cutoff that can be used to discriminate between important and non-important variables. Several approaches, for example approaches based on hypothesis testing, have been developed for addressing this problem. The existing testing approaches are permutation-based and require the repeated computation of forests. While for low-dimensional settings those permutation-based approaches might be computationally tractable, for high-dimensional settings typically including thousands of candidate predictors, computing time is enormous. A new computationally fast heuristic procedure of a variable importance test is proposed, that is appropriate for high-dimensional data where many variables do not carry any information. The testing approach is based on a modified version of the permutation variable importance measure, which is inspired by cross-validation procedures. The novel testing approach is tested and compared to the permutation-based testing approach of Altmann and colleagues using studies on complex high-dimensional binary classification settings. The new approach controlled the type I error and had at least comparable power at a substantially smaller computation time in our studies. The new variable importance test is implemented in the R package vita

Overview of Random Forest Methodology and Practical Guidance with Emphasis on Computational Biology and Bioinformatics

The Random Forest (RF) algorithm by Leo Breiman has become a standard data analysis tool in bioinformatics. It has shown excellent performance in settings where the number of variables is much larger than the number of observations, can cope with complex interaction structures as well as highly correlated variables and returns measures of variable importance. This paper synthesizes ten years of RF development with emphasis on applications to bioinformatics and computational biology. Special attention is given to practical aspects such as the selection of parameters, available RF implementations, and important pitfalls and biases of RF and its variable importance measures (VIMs). The paper surveys recent developments of the methodology relevant to bioinformatics as well as some representative examples of RF applications in this context and possible directions for future research

Kernel-Based Just-In-Time Learning for Passing Expectation Propagation Messages

We propose an efficient nonparametric strategy for learning a message operator in expectation propagation (EP), which takes as input the set of incoming messages to a factor node, and produces an outgoing message as output. This learned operator replaces the multivariate integral required in classical EP, which may not have an analytic expression. We use kernel-based regression, which is trained on a set of probability distributions representing the incoming messages, and the associated outgoing messages. The kernel approach has two main advantages: first, it is fast, as it is implemented using a novel two-layer random feature representation of the input message distributions; second, it has principled uncertainty estimates, and can be cheaply updated online, meaning it can request and incorporate new training data when it encounters inputs on which it is uncertain. In experiments, our approach is able to solve learning problems where a single message operator is required for multiple, substantially different data sets (logistic regression for a variety of classification problems), where it is essential to accurately assess uncertainty and to efficiently and robustly update the message operator.Comment: accepted to UAI 2015. Correct typos. Add more content to the appendix. Main results unchange

Forest Garrote

Variable selection for high-dimensional linear models has received a lot of attention lately, mostly in the context of l1-regularization. Part of the attraction is the variable selection effect: parsimonious models are obtained, which are very suitable for interpretation. In terms of predictive power, however, these regularized linear models are often slightly inferior to machine learning procedures like tree ensembles. Tree ensembles, on the other hand, lack usually a formal way of variable selection and are difficult to visualize. A Garrote-style convex penalty for trees ensembles, in particular Random Forests, is proposed. The penalty selects functional groups of nodes in the trees. These could be as simple as monotone functions of individual predictor variables. This yields a parsimonious function fit, which lends itself easily to visualization and interpretation. The predictive power is maintained at least at the same level as the original tree ensemble. A key feature of the method is that, once a tree ensemble is fitted, no further tuning parameter needs to be selected. The empirical performance is demonstrated on a wide array of datasets.Comment: 16 pages, 3 figure

Node harvest

When choosing a suitable technique for regression and classification with multivariate predictor variables, one is often faced with a tradeoff between interpretability and high predictive accuracy. To give a classical example, classification and regression trees are easy to understand and interpret. Tree ensembles like Random Forests provide usually more accurate predictions. Yet tree ensembles are also more difficult to analyze than single trees and are often criticized, perhaps unfairly, as `black box' predictors. Node harvest is trying to reconcile the two aims of interpretability and predictive accuracy by combining positive aspects of trees and tree ensembles. Results are very sparse and interpretable and predictive accuracy is extremely competitive, especially for low signal-to-noise data. The procedure is simple: an initial set of a few thousand nodes is generated randomly. If a new observation falls into just a single node, its prediction is the mean response of all training observation within this node, identical to a tree-like prediction. A new observation falls typically into several nodes and its prediction is then the weighted average of the mean responses across all these nodes. The only role of node harvest is to `pick' the right nodes from the initial large ensemble of nodes by choosing node weights, which amounts in the proposed algorithm to a quadratic programming problem with linear inequality constraints. The solution is sparse in the sense that only very few nodes are selected with a nonzero weight. This sparsity is not explicitly enforced. Maybe surprisingly, it is not necessary to select a tuning parameter for optimal predictive accuracy. Node harvest can handle mixed data and missing values and is shown to be simple to interpret and competitive in predictive accuracy on a variety of data sets.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS367 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

An Introduction to Recursive Partitioning: Rationale, Application and Characteristics of Classification and Regression Trees, Bagging and Random Forests

Recursive partitioning methods have become popular and widely used tools for nonparametric regression and classification in many scientific fields. Especially random forests, that can deal with large numbers of predictor variables even in the presence of complex interactions, have been applied successfully in genetics, clinical medicine and bioinformatics within the past few years. High dimensional problems are common not only in genetics, but also in some areas of psychological research, where only few subjects can be measured due to time or cost constraints, yet a large amount of data is generated for each subject. Random forests have been shown to achieve a high prediction accuracy in such applications, and provide descriptive variable importance measures reflecting the impact of each variable in both main effects and interactions. The aim of this work is to introduce the principles of the standard recursive partitioning methods as well as recent methodological improvements, to illustrate their usage for low and high dimensional data exploration, but also to point out limitations of the methods and potential pitfalls in their practical application. Application of the methods is illustrated using freely available implementations in the R system for statistical computing

Resolving transition metal chemical space: feature selection for machine learning and structure-property relationships

Machine learning (ML) of quantum mechanical properties shows promise for accelerating chemical discovery. For transition metal chemistry where accurate calculations are computationally costly and available training data sets are small, the molecular representation becomes a critical ingredient in ML model predictive accuracy. We introduce a series of revised autocorrelation functions (RACs) that encode relationships between the heuristic atomic properties (e.g., size, connectivity, and electronegativity) on a molecular graph. We alter the starting point, scope, and nature of the quantities evaluated in standard ACs to make these RACs amenable to inorganic chemistry. On an organic molecule set, we first demonstrate superior standard AC performance to other presently-available topological descriptors for ML model training, with mean unsigned errors (MUEs) for atomization energies on set-aside test molecules as low as 6 kcal/mol. For inorganic chemistry, our RACs yield 1 kcal/mol ML MUEs on set-aside test molecules in spin-state splitting in comparison to 15-20x higher errors from feature sets that encode whole-molecule structural information. Systematic feature selection methods including univariate filtering, recursive feature elimination, and direct optimization (e.g., random forest and LASSO) are compared. Random-forest- or LASSO-selected subsets 4-5x smaller than RAC-155 produce sub- to 1-kcal/mol spin-splitting MUEs, with good transferability to metal-ligand bond length prediction (0.004-5 {\AA} MUE) and redox potential on a smaller data set (0.2-0.3 eV MUE). Evaluation of feature selection results across property sets reveals the relative importance of local, electronic descriptors (e.g., electronegativity, atomic number) in spin-splitting and distal, steric effects in redox potential and bond lengths.Comment: 43 double spaced pages, 11 figures, 4 table