Energy distance

Abstract

Energy distance is a metric that measures the distance between the distributions of random vectors. Energy distance is zero if and only if the distributions are identical, thus it characterizes equality of distributions and provides a theoretical foundation for statistical inference and analysis. Energy statistics are functions of distances between observations in metric spaces. As a statistic, energy distance can be applied to measure the difference between a sample and a hypothesized distribution or the difference between two or more samples in arbitrary, not necessarily equal dimensions. The name energy is inspired by the close analogy with Newton's gravitational potential energy. Applications include testing independence by distance covariance, goodness‐of‐fit, nonparametric tests for equality of distributions and extension of analysis of variance, generalizations of clustering algorithms, change point analysis, feature selection, and more. WIREs Comput Stat 2016, 8:27–38. doi: 10.1002/wics.1375 This article is categorized under: Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Statistical and Graphical Methods of Data Analysis > Nonparametric Methods

Multivariate quantile mapping bias correction: an N-dimensional probability density function transform for climate model simulations of multiple variables

“…The (squared) energy distance is a measure of statistical discrepancy between two multivariate distributions, defined between N-dimensional independent random vectors and with CDFs F and G, respectively, as where E denotes the expected value, ‖.‖ is the Euclidean norm, and ′ and ′ and independent and identically distributed copies of and . D(F, G) equals zero only when F equals G. Calculation details are given in Rizzo and Székely (2016).…”

Perceptual Decision-Making: Biases in Post-Error Reaction Times Explained by Attractor Network Dynamics

“…To assess if the distributions of two continuous variables are different, we make use of the Kolmogorov-Smirnov test (Hollander et al, 2014), and in the case of discrete variable distributions we use the Anderson-Darling test (Shorack and Wellner, 2009). For very large samples, we use the energy distance (Rizzo and Székely, 2016), which is a metric distance between the distributions of random vectors. We use the associated E-statistic (Szekely and Rizzo, 2013) for testing the null hypothesis that two random variables X and Y have the same cumulative distribution functions.…”

Temperature-dependent phase behaviour of tetrahydrofuran–water alters solubilization of xylan to improve co-production of furfurals from lignocellulosic biomass

“…The statistical energy distance is taken as the integral of the squared difference of the cumulative distribution functions of the distributions, i.e., D ¼ Ð 1 À1 ðFðxÞ À GðxÞÞ 2 dx, where F(x) and G(x) are the cumulative distribution functions of two different probability distributions. 38…”

“…Fig. 7 provides the statistical energy distance 38 between the distributions in Fig. 5, which quantifies the "distance" between these distributions and makes clear that indeed they approach one another in the miscibility gap.…”