function [I_alpha] = estimate_Ialpha(Y,co)
%Estimates I_alpha = \int p^{\alpha}(y)dy, the Renyi and the Tsallis entropies are simple functions of this quantity. Here, alpha:=co.alpha.
%
%INPUT:
% Y: Y(:,t) is the t^th sample from the distribution having density p.
% 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 <http://www.gnu.org/licenses/>.
[d,num_of_samples] = size(Y);
squared_distances = kNN_squared_distances(Y,Y,co,1);
V = pi^(d/2)/gamma(d/2+1);%volume of the d-dimensional unit ball
C = ( gamma(co.k)/gamma(co.k+1-co.alpha) )^(1/(1-co.alpha));
s = sum( squared_distances(co.k,:).^(d*(1-co.alpha)/2) ); %'/2' <= squared distances
I_alpha = (num_of_samples-1) / num_of_samples * V^(1-co.alpha) * C^(1-co.alpha) * s / (num_of_samples-1)^(co.alpha);