/usr/share/octave/packages/statistics-1.2.3/regress_gp.m is in octave-statistics 1.2.3-1.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 | ## 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,bsxfun(@plus, 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,bsxfun(@plus, yi, [-dy +dy]),'b-');
%! hold off
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