Mercurial > forge
view main/splines/inst/csaps.m @ 9889:1f75709e4156 octave-forge
Added new self-contained csaps function in splines for producing cubic smoothing splines
author | nir-krakauer |
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date | Tue, 27 Mar 2012 17:50:14 +0000 |
parents | |
children | 28912aa15ea3 |
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## Copyright (C) 2012 Nir Krakauer ## ## 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, see <http://www.gnu.org/licenses/>. ## -*- texinfo -*- ## @deftypefn{Function File}{[@var{yi} @var{p}] =} csaps(@var{x}, @var{y}, @var{p}, @var{xi}, @var{w}=[]) ## @deftypefnx{Function File}{[@var{pp} @var{p}] =} csaps(@var{x}, @var{y}, @var{p}, [], @var{w}=[]) ## ## Cubic spline approximation (smoothing)@* ## approximate [@var{x},@var{y}], weighted by @var{w} (inverse variance; if not given, equal weighting is assumed), at @var{xi} ## ## The chosen cubic spline with natural boundary conditions @var{pp}(@var{x}) minimizes @var{p} Sum_i @var{w}_i*(@var{y}_i - @var{pp}(@var{x}_i))^2 + (1-@var{p}) Int @var{pp}''(@var{x}) d@var{x} ## ## @var{x} and @var{w} should be n by 1 in size; @var{y} should be n by m; @var{xi} should be k by 1; the values in @var{x} should be distinct; the values in @var{w} should be nonzero ## ## @table @asis ## @item @var{p}=0 ## maximum smoothing: straight line ## @item @var{p}=1 ## no smoothing: interpolation ## @item @var{p}<0 or not given ## an intermediate amount of smoothing is chosen (the smoothing term and the interpolation term are scaled to be of the same magnitude) ## @end table ## ## @end deftypefn ## @seealso{spline, csapi, ppval, gcvspl} ## Author: Nir Krakauer <nkrakauer@ccny.cuny.edu> function [ret,p]=csaps(x,y,p,xi,w) if(nargin < 5) w = []; if(nargin < 4) xi = []; if(nargin < 3) p = []; endif endif endif if(columns(x) > 1) x = x.'; y = y.'; w = w.'; endif [x,i] = sort(x); y = y(i, :); n = numel(x); if isempty(w) w = ones(n, 1); end h = diff(x); R = spdiags([h(1:end-1) 2*(h(1:end-1) + h(2:end)) h(2:end)], [-1 0 1], n-2, n-2); QT = spdiags([1 ./ h(1:end-1) -(1 ./ h(1:end-1) + 1 ./ h(2:end)) 1 ./ h(2:end)], [0 1 2], n-2, n); ## if not given, choose p so that trace(6*(1-p)*QT*diag(1 ./ w)*QT') = trace(pR) if isempty(p) || (p < 0) r = 6*trace(QT*diag(1 ./ w)*QT') / trace(R); p = 1 ./ (1 + r); endif ## solve for the scaled second derivatives u and for the function values a at the knots (if p = 1, a = y) u = (6*(1-p)*QT*diag(1 ./ w)*QT' + p*R) \ (QT*y); a = y - 6*(1-p)*diag(1 ./ w)*QT'*u; ## derivatives at all but the last knot for the piecewise cubic spline aa = a(1:(end-1), :); cc = zeros(size(y)); cc(2:(n-1), :) = 6*p*u; #cc([1 n], :) = 0 [natural spline] dd = diff(cc) ./ h; cc = cc(1:(end-1), :); bb = diff(a) ./ h - (cc/2).*h - (dd/6).*(h.^2); ret = mkpp (x, cat (2, dd'(:)/6, cc'(:)/2, bb'(:), aa'(:)), size(y, 2)); if ~isempty(xi) ret = ppval (ret, xi); endif endfunction %!shared x,y %! x = ([1:10 10.5 11.3])'; y = sin(x); %!assert (csaps(x,y,1,x), y, 10*eps); %!assert (csaps(x,y,1,x'), y', 10*eps); %!assert (csaps(x',y',1,x'), y', 10*eps); %!assert (csaps(x',y',1,x), y, 10*eps); %!assert (csaps(x,[y 2*y],1,x)', [y 2*y], 10*eps);