 function [D] = DL2_kNN_k_estimation(Y1,Y2,co)
%Estimates the L2 divergence (D) of Y1 and Y2 (Y1(:,t), Y2(:,t) is the t^th sample)
%using the kNN method (S={k}). The number of samples in Y1 [=size(Y1,2)] and Y2 [=size(Y2,2)] can be different. Cost parameters are provided in the cost object co.
%
%We make use of the naming convention 'D<name>_estimation', to ease embedding new divergence estimation methods.
%
%REFERENCE:
% Barnabas Poczos, Zoltan Szabo, Jeff Schneider: Nonparametric divergence estimators for Independent Subspace Analysis. EUSIPCO2011, pages 18491853.
% Barnabas Poczos, Liang Xiong, Jeff Schneider. Nonparametric Divergence: Estimation with Applications to Machine Learning on Distributions. UAI2011.
%
%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 <http://www.gnu.org/licenses/>.
%co.mult:OK.
[dY2,num_of_samplesY2] = size(Y2);
[dY1,num_of_samplesY1] = size(Y1);
%verification:
if dY1~=dY2
error('The dimension of Y1 and Y2 must be equal.');
end
d = dY1;
c = volume_of_the_unit_ball(d);%volume of the ddimensional unit ball
squared_distancesY1Y1 = kNN_squared_distances(Y1,Y1,co,1);
squared_distancesY2Y1 = kNN_squared_distances(Y2,Y1,co,0);
dist_k_Y1Y1 = sqrt(squared_distancesY1Y1(end,:));
dist_k_Y2Y1 = sqrt(squared_distancesY2Y1(end,:));
term1 = mean(dist_k_Y1Y1.^(d)) * (co.k1) / ((num_of_samplesY11)*c);
term2 = mean(dist_k_Y2Y1.^(d)) * 2 * (co.k1) / (num_of_samplesY2*c);
term3 = mean((dist_k_Y1Y1.^d) ./ (dist_k_Y2Y1.^(2*d))) * (num_of_samplesY1  1) * (co.k2) * (co.k1) / (num_of_samplesY2^2*c*co.k);
L2 = term1term2+term3;
%D = sqrt(L2);%theoretically
D = sqrt(abs(L2));%due to the finite number of samples L2 can be negative. In this case: sqrt(L2) is complex; to avoid such values we take abs().
