function [Dtemp2] = estimate_Dtemp2(X,Y,co)
%Estimates Dtemp2 = \int p^a(u)q^b(u)p(u)du; the Hellinger distance and the Bhattacharyya distance are simple functions of this quantity.
%
%INPUT:
% X: X(:,t) is the t^th sample from the first distribution (X~p).
% Y: Y(:,t) is the t^th sample from the second distribution (Y~q).
% co: cost object (structure).
%
%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 .
%initialization:
[dY,num_of_samplesY] = size(Y);
[dX,num_of_samplesX] = size(X);
%verification:
if dX~=dY
error('The dimension of the samples in X and Y must be equal.');
end
%initialization - continued:
d = dX; %=dY
a = co.a;
b = co.b;
k = co.k;
squared_distancesXX = kNN_squared_distances(X,X,co,1);
squared_distancesYX = kNN_squared_distances(Y,X,co,0);
dist_k_XX = sqrt(squared_distancesXX(end,:));
dist_k_YX = sqrt(squared_distancesYX(end,:));
c = volume_of_the_unit_ball(d);
B = c^(-(a+b)) * gamma(k)^2 / (gamma(k-a)*gamma(k-b));
Dtemp2 = (num_of_samplesX-1)^(-a) * num_of_samplesY^(-b) * B * mean(dist_k_XX.^(-d*a).*dist_k_YX.^(-d*b));