Search for: All records

Though introduced nearly 50 years ago, the infinitesimal jackknife (IJ) remains a popular modern tool for quantifying predictive uncertainty in complex estimation settings. In particular, when supervised learning ensembles are constructed via bootstrap samples, recent work demonstrated that the IJ estimate of variance is particularly convenient and useful. However, despite the algebraic simplicity of its final form, its derivation is rather complex. As a result, studies clarifying the intuition behind the estimator or rigorously investigating its properties have been severely lacking. This work aims to take a step forward on both fronts. We demonstrate that surprisingly, the exact form of the IJ estimator can be obtained via a straightforward linear regression of the individual bootstrap estimates on their respective weights or via the classical jackknife. The latter realization allows us to formally investigate the bias of the IJ variance estimator and better characterize the settings in which its use is appropriate. Finally, we extend these results to the case of U-statistics where base models are constructed via subsampling rather than bootstrapping and provide a consistent estimate of the resulting variance. 

Most scientific publications follow the familiar recipe of (i) obtain data, (ii) fit a model, and (iii) comment on the scientific relevance of the effects of particular covariates in that model. This approach, however, ignores the fact that there may exist a multitude of similarly-accurate models in which the implied effects of individual covariates may be vastly different. This problem of finding an entire collection of plausible models has also received relatively little attention in the statistics community, with nearly all of the proposed methodologies being narrowly tailored to a particular model class and/or requiring an exhaustive search over all possible models, making them largely infeasible in the current big data era. This work develops the idea of forward stability and proposes a novel, computationally-efficient approach to finding collections of accurate models we refer to as model path selection (MPS). MPS builds up a plausible model collection via a forward selection approach and is entirely agnostic to the model class and loss function employed. The resulting model collection can be displayed in a simple and intuitive graphical fashion, easily allowing practitioners to visualize whether some covariates can be swapped for others with minimal loss. 

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 bagging and random forests, but which operates on a larger, augmented space containing additional randomly generated noise features. Surprisingly, we demonstrate that this simple act of including extra noise variables in the model can lead to dramatic improvements in out-of-sample predictive accuracy, sometimes outperforming even an optimally tuned traditional random forest. As a result, intuitive notions of variable importance based on improved model accuracy may be deeply flawed, as even purely random noise can routinely register as statistically significant. Numerous demonstrations on both real and synthetic data are provided along with a proposed solution. 

Throughout the last decade, random forests have established themselves as among the most accurate and popular supervised learning methods. While their black-box nature has made their mathematical analysis difficult, recent work has established important statistical properties like consistency and asymptotic normality by considering subsampling in lieu of bootstrapping. Though such results open the door to traditional inference procedures, all formal methods suggested thus far place severe restrictions on the testing framework and their computational overhead often precludes their practical scientific use. Here we propose a hypothesis test to formally assess feature significance, which uses permutation tests to circumvent computationally infeasible estimates of nuisance parameters. This test is intended to be analogous to the F-test for linear regression. We establish asymptotic validity of the test via exchangeability arguments and show that the test maintains high power with orders of magnitude fewer computations. Importantly, the procedure scales easily to big data settings where large training and testing sets may be employed, conducting statistically valid inference without the need to construct additional models. Simulations and applications to ecological data, where random forests have recently shown promise, are provided. 

In 2001, Leo Breiman wrote of a divide between "data modeling" and "algorithmic modeling" cultures. Twenty years later this division feels far more ephemeral, both in terms of assigning individuals to camps, and in terms of intellectual boundaries. We argue that this is largely due to the "data modelers" incorporating algorithmic methods into their toolbox, particularly driven by recent developments in the statistical understanding of Breiman's own Random Forest methods. While this can be simplistically described as "Breiman won", these same developments also expose the limitations of the prediction-first philosophy that he espoused, making careful statistical analysis all the more important. This paper outlines these exciting recent developments in the random forest literature which, in our view, occurred as a result of a necessary blending of the two ways of thinking Breiman originally described. We also ask what areas statistics and statisticians might currently overlook. 

Abstract Due to their long‐standing reputation as excellent off‐the‐shelf predictors, random forests (RFs) continue to remain a go‐to model of choice for applied statisticians and data scientists. Despite their widespread use, however, until recently, little was known about their inner workings and about which aspects of the procedure were driving their success. Very recently, two competing hypotheses have emerged–one based on interpolation and the other based on regularization. This work argues in favor of the latter by utilizing the regularization framework to reexamine the decades‐old question of whether individual trees in an ensemble ought to be pruned. Despite the fact that default constructions of RFs use near full depth trees in most popular software packages, here we provide strong evidence that tree depth should be seen as a natural form of regularization across the entire procedure. In particular, our work suggests that RFs with shallow trees are advantageous when the signal‐to‐noise ratio in the data is low. In building up this argument, we also critique the newly popular notion of “double descent” in RFs by drawing parallels toU‐statistics and arguing that the noticeable jumps in random forest accuracy are the result of simple averaging rather than interpolation. 

Random forests remain among the most popular off-the-shelf supervised machine learning tools with a well-established track record of predictive accuracy in both regression and classification settings. Despite their empirical success as well as a bevy of recent work investigating their statistical properties, a full and satisfying explanation for their success has yet to be put forth. Here we aim to take a step forward in this direction by demonstrating that the additional randomness injected into individual trees serves as a form of implicit regularization, making random forests an ideal model in low signal-to-noise ratio (SNR) settings. Specifically, from a model-complexity perspective, we show that the mtry parameter in random forests serves much the same purpose as the shrinkage penalty in explicitly regularized regression procedures like lasso and ridge regression. To highlight this point, we design a randomized linear-model-based forward selection procedure intended as an analogue to tree-based random forests and demonstrate its surprisingly strong empirical performance. Numerous demonstrations on both real and synthetic data are provided. 

Summary Bird species’ migratory patterns have typically been studied through individual observations and historical records. In recent years, the eBird citizen science project, which solicits observations from thousands of bird watchers around the world, has opened the door for a data-driven approach to understanding the large-scale geographical movements. Here, we focus on the North American tree swallow (Tachycineta bicolor) occurrence patterns throughout the eastern USA. Migratory departure dates for this species are widely believed by both ornithologists and casual observers to vary substantially across years, but the reasons for this are largely unknown. In this work, we present evidence that maximum daily temperature is predictive of tree swallow occurrence. Because it is generally understood that species occurrence is a function of many complex, high order interactions between ecological covariates, we utilize the flexible modelling approach that is offered by random forests. Making use of recent asymptotic results, we provide formal hypothesis tests for predictive significance of various covariates and also develop and implement a permutation-based approach for formally assessing interannual variations by treating the prediction surfaces that are generated by random forests as functional data. Each of these tests suggest that maximum daily temperature is important in predicting migration patterns. 

Currently, a variety of popular quality metrics exist for NBA players such as Value Over Replacement Player (VORP), Player Efficiency Rating (PER), and Win Shares (WS). Despite their often relatively simple formulations, less is known regarding how traditional box score statistics may disproportionately affect player quality with respect to these different measures. Using traditional per-36 minute statistics along with more modern advanced metrics from the previous 15 seasons, we predict VORP, PER, and WS using a variety of approaches ranging from linear models to more advanced methods like random forests and tree-augmented regression. In addition to assessing fit, we pay particular attention to partial effects of individual statistics on each response, allowing us to investigate which types of players are favored by different measures. Finally, we test our models and conclusions by predicting VORP, PER, and WS for a sample of top NBA players from the 2016/2017 season.