function [D] = DEnergyDist_DMMD_estimation(Y1,Y2,co) %Estimates the energy distance (D) according to the relation: D(f_1,f_2;rho) = 2 [MMD(f_1,f_2;k)]^2, where MMD denotes maximum mean discrepancy and k is a kernel that generates rho, a semimetric of negative type. % %Note: % 1)We use the naming convention 'D_estimation' to ease embedding new divergence estimation methods. % 2)This is a meta method: the MMD estimator can be arbitrary. % %INPUT: % Y1: Y1(:,t) is the t^th sample from the first distribution. % Y2: Y2(:,t) is the t^th sample from the second distribution. % co: divergence estimator object. % %REFERENCE: % Dino Sejdinovic, Arthur Gretton, Bharath Sriperumbudur, and Kenji Fukumizu. Hypothesis testing using pairwise distances and associated kernels. International Conference on Machine Learning (ICML), pages 1111-1118, 2012. (semimetric space; energy distance <=> MMD, with a suitable kernel) % Russell Lyons. Distance covariance in metric spaces. Annals of Probability, 2012. (To appear. http://php.indiana.edu/~rdlyons/pdf/dcov.pdf; http://arxiv.org/abs/1106.5758; energy distance, metric space of negative type; pre-equivalence to MMD). % Gabor J. Szekely and Maria L. Rizzo. A new test for multivariate normality. Journal of Multivariate Analysis, 93:58-80, 2005. (energy distance; metric space of negative type) % Gabor J. Szekely and Maria L. Rizzo. Testing for equal distributions in high dimension. InterStat, 5, 2004. (energy distance; R^d) % %Copyright (C) 2012 Zoltan Szabo ("http://nipg.inf.elte.hu/szzoli", "szzoli (at) cs (dot) elte (dot) hu") % %This file is part of the ITE (Information Theoretical Estimators) Matlab/Octave toolbox. % %ITE is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by %the Free Software Foundation, either version 3 of the License, or (at your option) any later version. % %This software is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of %MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. % %You should have received a copy of the GNU General Public License along with ITE. If not, see . %co.mult:OK. %verification: if size(Y1,1)~=size(Y2,1) error('The dimension of the samples in Y1 and Y2 must be equal.'); end D = 2 * ( D_estimation(Y1,Y2,co.member_co) )^2;