function p=mdKDE(X,xi,bw_coeff) %%%%% % This function implements multidimensional kernel density estimation with % a Gaussian kernel. % % Inputs: % X: data matrix, rows are observations and columns are dimensions. % xi: coordinates at which joint probabilities are to be estimated, rows are data points and columns are dimensions. % bw_coeff: multiplier that adjusts the rule of thumb Gaussian bandwidth, a value of 1 means no change in the rule of thumb bandwidth % % Output: % p: joint probability at xi % % % Copyright 2011 Joon Lee % % This program 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 program 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 this program. If not, see . %%%%% N=size(X,1); d=size(X,2); % pick bandwidth for each dimension h=zeros(1,d); for i=1:d h(i)=bw_coeff*1.06*N^(-1/5)*std(X(:,i)); % rule of thumb that assumes a Gaussian distribution end % estimate probability at xi p=zeros(size(xi,1),1); for k=1:length(p) for i=1:N prod=1; for j=1:d u=(xi(k,j)-X(i,j))/h(j); prod=prod*1/sqrt(2*pi)*exp(-0.5*u^2)/h(j); end p(k)=p(k)+prod; end end p=p./N;