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## Copyright (c) 2012 Juan Pablo Carbajal <carbajal@ifi.uzh.ch>
##
## 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 <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
## @deftypefn {Function File} {[@var{m}, @var{K}] =} regress_gp (@var{x}, @var{y}, @var{Sp})
## @deftypefnx {Function File} {[@dots{} @var{yi} @var{dy}] =} sqp (@dots{}, @var{xi})
## Linear scalar regression using gaussian processes.
##
## It estimates the model @var{y} = @var{x}'*m for @var{x} R^D and @var{y} in R.
## The information about errors of the predictions (interpolation/extrapolation) is given
## by the covarianve matrix @var{K}. If D==1 the inputs must be column vectors,
## if D>1 then @var{x} is n-by-D, with n the number of data points. @var{Sp} defines
## the prior covariance of @var{m}, it should be a (D+1)-by-(D+1) positive definite matrix,
## if it is empty, the default is @code{Sp = 100*eye(size(x,2)+1)}.
##
## If @var{xi} inputs are provided, the model is evaluated and returned in @var{yi}.
## The estimation of the variation of @var{yi} are given in @var{dy}.
##
## Run @code{demo regress_gp} to see an examples.
##
## The function is a direc implementation of the formulae in pages 11-12 of
## Gaussian Processes for Machine Learning. Carl Edward Rasmussen and @
## Christopher K. I. Williams. The MIT Press, 2006. ISBN 0-262-18253-X.
## available online at @url{http://gaussianprocess.org/gpml/}.
##
## @seealso{regress}
## @end deftypefn
function [wm K yi dy] = regress_gp (x,y,Sp=[],xi=[])
if isempty(Sp)
Sp = 100*eye(size(x,2)+1);
end
x = [ones(1,size(x,1)); x'];
## Juan Pablo Carbajal <carbajal@ifi.uzh.ch>
## Note that in the book the equation (below 2.11) for the A reads
## A = (1/sy^2)*x*x' + inv (Vp);
## where sy is the scalar variance of the of the residuals (i.e y = x' * w + epsilon)
## and epsilon is drawn from N(0,sy^2). Vp is the variance of the parameters w.
## Note that
## (sy^2 * A)^{-1} = (1/sy^2)*A^{-1} = (x*x' + sy^2 * inv(Vp))^{-1};
## and that the formula for the w mean is
## (1/sy^2)*A^{-1}*x*y
## Then one obtains
## inv(x*x' + sy^2 * inv(Vp))*x*y
## Looking at the formula bloew we see that Sp = (1/sy^2)*Vp
## making the regression depend on only one parameter, Sp, and not two.
A = x*x' + inv (Sp);
K = inv (A);
wm = K*x*y;
yi =[];
dy =[];
if !isempty (xi);
xi = [ones(size(xi,1),1) xi];
yi = xi*wm;
dy = diag (xi*K*xi');
end
endfunction
%!demo
%! % 1D Data
%! x = 2*rand (5,1)-1;
%! y = 2*x -1 + 0.3*randn (5,1);
%!
%! % Points for interpolation/extrapolation
%! xi = linspace (-2,2,10)';
%!
%! [m K yi dy] = regress_gp (x,y,[],xi);
%!
%! plot (x,y,'xk',xi,yi,'r-',xi,yi+[-dy +dy],'b-');
%!demo
%! % 2D Data
%! x = 2*rand (4,2)-1;
%! y = 2*x(:,1)-3*x(:,2) -1 + 1*randn (4,1);
%!
%! % Mesh for interpolation/extrapolation
%! [xi yi] = meshgrid (linspace (-1,1,10));
%!
%! [m K zi dz] = regress_gp (x,y,[],[xi(:) yi(:)]);
%! zi = reshape (zi, 10,10);
%! dz = reshape (dz,10,10);
%!
%! plot3 (x(:,1),x(:,2),y,'.g','markersize',8);
%! hold on;
%! h = mesh (xi,yi,zi,zeros(10,10));
%! set(h,'facecolor','none');
%! h = mesh (xi,yi,zi+dz,ones(10,10));
%! set(h,'facecolor','none');
%! h = mesh (xi,yi,zi-dz,ones(10,10));
%! set(h,'facecolor','none');
%! hold off
%! axis tight
%! view(80,25)
%!demo
%! % Projection over basis function
%! pp = [2 2 0.3 1];
%! n = 10;
%! x = 2*rand (n,1)-1;
%! y = polyval(pp,x) + 0.3*randn (n,1);
%!
%! % Powers
%! px = [sqrt(abs(x)) x x.^2 x.^3];
%!
%! % Points for interpolation/extrapolation
%! xi = linspace (-1,1,100)';
%! pxi = [sqrt(abs(xi)) xi xi.^2 xi.^3];
%!
%! Sp = 100*eye(size(px,2)+1);
%! Sp(2,2) = 1; # We don't believe the sqrt is present
%! [m K yi dy] = regress_gp (px,y,Sp,pxi);
%! disp(m)
%!
%! plot (x,y,'xk;Data;',xi,yi,'r-;Estimation;',xi,polyval(pp,xi),'g-;True;');
%! axis tight
%! axis manual
%! hold on
%! plot (xi,yi+[-dy +dy],'b-');
%! hold off