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## Copyright (C) 2007 Andreas Stahel <Andreas.Stahel@bfh.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/>.
## general linear regression
##
## [p,y_var,r,p_var]=LinearRegression(F,y)
## [p,y_var,r,p_var]=LinearRegression(F,y,weight)
##
## determine the parameters p_j (j=1,2,...,m) such that the function
## f(x) = sum_(i=1,...,m) p_j*f_j(x) fits as good as possible to the
## given values y_i = f(x_i)
##
## parameters
## F n*m matrix with the values of the basis functions at the support points
## in column j give the values of f_j at the points x_i (i=1,2,...,n)
## y n column vector of given values
## weight n column vector of given weights
##
## return values
## p m vector with the estimated values of the parameters
## y_var estimated variance of the error
## r weighted norm of residual
## p_var estimated variance of the parameters p_j
function [p,y_var,r,p_var]=LinearRegression(F,y,weight)
if (nargin < 2 || nargin >= 4)
usage('wrong number of arguments in [p,y_var,r,p_var]=LinearRegression(F,y)');
end
[rF, cF] = size(F); [ry, cy] =size(y);
if (rF ~= ry || cy > 1)
error ('LinearRegression: incorrect matrix dimensions');
end
if (nargin==2) % set uniform weights if not provided
weight=ones(size(y));
end
%% Fw=diag(weight)*F;
wF=F;
for j=1:cF
wF(:,j)=weight.*F(:,j);
end
[Q,R]=qr(wF,0); % estimate the values of the parameters
p=R\(Q'*(weight.*y));
residual=F*p-y; % compute the residual vector
r=norm(weight.*residual); % and its weighted norm
% variance of the weighted y-errors
y_var= sum((residual.^2).*(weight.^4))/(rF-cF);
if nargout>3 % compute variance of parameters only if needed
%% M=inv(R)*Q'*diag(weight);
M=inv(R)*Q';
for j=1:cF
M(j,:)=M(j,:).*(weight');
end
M=M.*M; % square each entry in the matrix M
p_var=M*(y_var./(weight.^4)); % variance of the parameters
end
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