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function [beta,p,lli] = logist2(y,x,w)
% [beta,p,lli] = logist2(y,x)
%
% 2-class logistic regression.
%
% INPUT
% y Nx1 colum vector of 0|1 class assignments
% x NxK matrix of input vectors as rows
% [w] Nx1 vector of sample weights
%
% OUTPUT
% beta Kx1 column vector of model coefficients
% p Nx1 column vector of fitted class 1 posteriors
% lli log likelihood
%
% Class 1 posterior is 1 / (1 + exp(-x*beta))
%
% David Martin <dmartin@eecs.berkeley.edu>
% April 16, 2002
% Copyright (C) 2002 David R. Martin <dmartin@eecs.berkeley.edu>
%
% 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 2 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, write to the Free Software
% Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA
% 02111-1307, USA, or see http://www.gnu.org/copyleft/gpl.html.
error(nargchk(2,3,nargin));
% check inputs
if size(y,2) ~= 1,
error('Input y not a column vector.');
end
if size(y,1) ~= size(x,1),
error('Input x,y sizes mismatched.');
end
% get sizes
[N,k] = size(x);
% if sample weights weren't specified, set them to 1
if nargin < 3,
w = 1;
end
% normalize sample weights so max is 1
w = w / max(w);
% initial guess for beta: all zeros
beta = zeros(k,1);
% Newton-Raphson via IRLS,
% taken from Hastie/Tibshirani/Friedman Section 4.4.
iter = 0;
lli = 0;
while 1==1,
iter = iter + 1;
% fitted probabilities
p = 1 ./ (1 + exp(-x*beta));
% log likelihood
lli_prev = lli;
lli = sum( w .* (y.*log(p+eps) + (1-y).*log(1-p+eps)) );
% least-squares weights
wt = w .* p .* (1-p);
% derivatives of likelihood w.r.t. beta
deriv = x'*(w.*(y-p));
% Hessian of likelihood w.r.t. beta
% hessian = x'Wx, where W=diag(w)
% Do it this way to be memory efficient and fast.
hess = zeros(k,k);
for i = 1:k,
wxi = wt .* x(:,i);
for j = i:k,
hij = wxi' * x(:,j);
hess(i,j) = -hij;
hess(j,i) = -hij;
end
end
% make sure Hessian is well conditioned
if (rcond(hess) < eps),
error(['Stopped at iteration ' num2str(iter) ...
' because Hessian is poorly conditioned.']);
break;
end;
% Newton-Raphson update step
step = hess\deriv;
beta = beta - step;
% termination criterion based on derivatives
tol = 1e-6;
if abs(deriv'*step/k) < tol, break; end;
% termination criterion based on log likelihood
% tol = 1e-4;
% if abs((lli-lli_prev)/(lli+lli_prev)) < 0.5*tol, break; end;
end;
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