arXiv subjects

Brandon M. Greenwell

Publications and source records attributed to Brandon M. Greenwell.

3 recordsLinked to original sources

arXiv2026

Random Triangles on Concentric Circles

Lewis Carroll's Pillow Problem asks for the probability that three points chosen at random in the plane form an obtuse triangle. The question has no answer until the sampling scheme is pinned down, and the cleanest way to pin it down is to put the points on a circle, which gives 3/4. This paper solves the version in which the three vertices lie on three concentric circles of radii $r_1$, $r_2$, and $r_3$. The answer is a sum of three terms, each the probability that a weighted sum of two independent arcsine variables exceeds a threshold, and it collapses to 3/4 when the radii are equal. Reading the formula gives closed forms in two subfamilies, one an arcsine and one the Legendre chi function; a Pythagorean condition deciding which vertex can carry the obtuse angle; and the bound $1/2 \leq P < 1$, with the minimum attained only when one vertex sits at the common center and the other two radii are equal. Conditioning on the radii shows that the same formula is the kernel for every independent rotationally symmetric sampling scheme, so the bound applies to all of them at once and averaging over Rayleigh radii recovers the known Gaussian value of 3/4 in closed form, supplying a link between the circular and Gaussian conventions that the literature records as missing. Simulation confirms the results throughout.

math.PR↗

arXiv2023

Explainable Boosting Machines with Sparsity -- Maintaining Explainability in High-Dimensional Settings

Compared to "black-box" models, like random forests and deep neural networks, explainable boosting machines (EBMs) are considered "glass-box" models that can be competitively accurate while also maintaining a higher degree of transparency and explainability. However, EBMs become readily less transparent and harder to interpret in high-dimensional settings with many predictor variables; they also become more difficult to use in production due to increases in scoring time. We propose a simple solution based on the least absolute shrinkage and selection operator (LASSO) that can help introduce sparsity by reweighting the individual model terms and removing the less relevant ones, thereby allowing these models to maintain their transparency and relatively fast scoring times in higher-dimensional settings. In short, post-processing a fitted EBM with many (i.e., possibly hundreds or thousands) of terms using the LASSO can help reduce the model's complexity and drastically improve scoring time. We illustrate the basic idea using two real-world examples with code.

stat.ML↗

arXiv2018

A Simple and Effective Model-Based Variable Importance Measure

In the era of "big data", it is becoming more of a challenge to not only build state-of-the-art predictive models, but also gain an understanding of what's really going on in the data. For example, it is often of interest to know which, if any, of the predictors in a fitted model are relatively influential on the predicted outcome. Some modern algorithms---like random forests and gradient boosted decision trees---have a natural way of quantifying the importance or relative influence of each feature. Other algorithms---like naive Bayes classifiers and support vector machines---are not capable of doing so and model-free approaches are generally used to measure each predictor's importance. In this paper, we propose a standardized, model-based approach to measuring predictor importance across the growing spectrum of supervised learning algorithms. Our proposed method is illustrated through both simulated and real data examples. The R code to reproduce all of the figures in this paper is available in the supplementary materials.

stat.ML↗