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## Copyright (C) 2008 Jonathan Stickel <jonathan.stickel@nrel.gov>
##
## 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{yhat}, @var{v}] =} rgdtsmcore (@var{x}, @var{y}, @var{d}, @var{lambda}, [@var{options}])
##
## Smooths @var{y} vs. @var{x} values by Tikhonov regularization.
## Although this function can be used directly, the more feature rich
## function "regdatasmooth" should be used instead. In addition to
## @var{x} and @var{y}, required input includes the smoothing derivative
## @var{d} and the regularization parameter @var{lambda}. The smooth
## y-values are returned as @var{yhat}. The generalized cross
## validation variance @var{v} may also be returned.
##
## Note: the options have changed!
## Currently supported input options are (multiple options are allowed):
##
## @table @code
## @item "xhat", @var{vector}
## A vector of x-values to use for the smooth curve; must be
## monotonically increasing and must at least span the data
## @item "weights", @var{vector}
## A vector of weighting values for fitting each point in the data.
## @item "relative"
## use relative differences for the goodnes of fit term. Conflicts
## with the "weights" option.
## @item "midpointrule"
## use the midpoint rule for the integration terms rather than a direct
## sum; this option conflicts with the option "xhat"
## @end table
##
## References: Anal. Chem. (2003) 75, 3631; AIChE J. (2006) 52, 325
## @seealso{regdatasmooth}
## @end deftypefn
function [yhat, v] = rgdtsmcore (x, y, d, lambda, varargin)
if (nargin < 4)
print_usage;
endif
## Defaults if not provided
xhatprov = 0;
xhat = x;
weights = 0;
relative = 0;
midpr = 0;
## parse the provided options
if ( length(varargin) )
for i = 1:length(varargin)
arg = varargin{i};
if ischar(arg)
switch arg
case "xhat"
xhatprov = 1;
xhat = varargin{i+1};
case "weights"
weights = 1;
weightv = varargin{i+1};
case "relative"
relative = 1;
case "midpointrule"
midpr = 1;
otherwise
printf("Option '%s' is not implemented;\n", arg)
endswitch
endif
endfor
endif
if (xhatprov && midpr)
warning("midpointrule is currently not used if xhat is provided (since x,y may be scattered)")
midpr = 0;
endif
if (weights && relative)
warning("relative differences is not used if a weighting vector is provided")
endif
N = length(x);
Nhat = length(xhat);
## test that xhat is increasing
if !all(diff(xhat)>0)
if xhatprov
error("xhat must be monotonically increasing")
else
error("x must be monotonically increasing if xhat is not provided")
endif
endif
## test that xhat spans x
if ( min(x) < min(xhat) || max(xhat) < max(x) )
error("xhat must at least span the data")
endif
## construct M, D
M = speye(Nhat);
idx = interp1(xhat,1:Nhat,x,"nearest"); # works for unequally spaced xhat
M = M(idx,:);
D = ddmat(xhat,d);
## construct "weighting" matrices W and U
if (weights)
## use arbitrary weighting as provided
W = diag(weightv);
elseif (relative)
## use relative differences
Yinv = sparse(diag(1./y));
W = Yinv^2;
else
W = speye(N);
endif
## use midpoint rule integration (rather than simple sums)
if (midpr)
Bhat = sparse(diag(-ones(N-1,1),-1)) + sparse(diag(ones(N-1,1),1));
Bhat(1,1) = -1;
Bhat(N,N) = 1;
B = 1/2*sparse(diag(Bhat*x));
if ( floor(d/2) == d/2 ) # test if d is even
dh = d/2;
Btilda = B(dh+1:N-dh,dh+1:N-dh);
else # d is odd
dh = ceil(d/2);
Btilda = B(dh:N-dh,dh:N-dh);
endif
W = W*B;
U = Btilda;
else
## W = W*speye(Nhat);
U = speye(Nhat-d);
endif
## Smoothing
delta = trace(D'*D)/Nhat^(2+d); # using "relative" or other weighting affects this!
yhat = (M'*W*M + lambda*delta^(-1)*D'*U*D) \ M'*W*y;
#[R,P,S] = splchol(M'*W*M + lambda*delta^(-1)*D'*U*D);
#yhat = S*(R'\(R\(S'*M'*W*y)));
## Computation of hat diagonal and cross-validation
if (nargout > 1)
## from AIChE J. (2006) 52, 325
## note: chol factorization does not help speed up the computation of H;
## should implement Eiler's partial H computation if many point smoothing by GCV is needed
##H = M*(S*(R'\(R\(S'*M'*W))));
H = M*((M'*W*M + lambda*delta^(-1)*D'*U*D)\M'*W);
## note: this is variance, squared of the standard error that Eilers uses
v = (M*yhat - y)'*(M*yhat - y)/N / (1 - trace(H)/N)^2;
endif
## test mapping
##figure(5)
##plot(x,y,'o',x,M*yhat,'x')
endfunction