splines/0000775000175000017500000000000014425217637010712 5ustar nirnirsplines/NEWS0000644000175000017500000001016614425217567011415 0ustar nirnir Summary of important user-visible changes for splines 1.3.5: ------------------------------------------------------------------- ** bug fix in csape ** syntax made compatible with Octave 8 Summary of important user-visible changes for splines 1.3.4: ------------------------------------------------------------------- ** New functions regularization, regularization2D Summary of important user-visible changes for splines 1.3.3: ------------------------------------------------------------------- ** Demo and efficiency improvement (and accuracy fix for new fminbnd default settings) in csaps_sel ** Bug fix in csape Summary of important user-visible changes for splines 1.3.2: ------------------------------------------------------------------- ** bug fix in csaps Summary of important user-visible changes for splines 1.3.1: ------------------------------------------------------------------- ** bug fix in fnplt Summary of important user-visible changes for splines 1.3.0: ------------------------------------------------------------------- ** csape default is now Lagrange boundary conditions (Matlab compatible) ** csaps can return the fit degrees of freedom Summary of important user-visible changes for splines 1.2.9: ------------------------------------------------------------------- ** new function tps_val_der ** vectorization option for speedup of tps_val Summary of important user-visible changes for splines 1.2.8: ------------------------------------------------------------------- ** csaps now returns spline uncertainty at the fitting points xi, not at the data points x Summary of important user-visible changes for splines 1.2.7: ------------------------------------------------------------------- ** New regularization option (df_bound) in csaps_sel ** Bug fix in csape Summary of important user-visible changes for splines 1.2.6: ------------------------------------------------------------------- ** New preprocessing function bin_values Summary of important user-visible changes for splines 1.2.5: ------------------------------------------------------------------- ** Efficiency improvement in csaps_sel Summary of important user-visible changes for splines 1.2.4: ------------------------------------------------------------------- ** Bug fix in csape Summary of important user-visible changes for splines 1.2.3: ------------------------------------------------------------------- ** New regularization option (vm) in csaps_sel ** Checking for NaNs added in csaps and csaps_sel; dedup now can remove NaNs ** Bug fix in csape Summary of important user-visible changes for splines 1.2.2: ------------------------------------------------------------------- ** Bug fix in csaps and csaps_sel Summary of important user-visible changes for splines 1.2.1: ------------------------------------------------------------------- ** Points passed to csaps and csaps_sel now expected to be in strictly ascending order; new function dedup added to sort points and average values given at the same point Summary of important user-visible changes for splines 1.2.0: ------------------------------------------------------------------- ** The following functions are new: tpaps tps_val Summary of important user-visible changes for splines 1.1.2: ------------------------------------------------------------------- ** automatic smoothing parameter selection in csaps fixed ** calculation of spline coefficients for unequally spaced points in csaps fixed Summary of important user-visible changes for splines 1.1.1: ------------------------------------------------------------------- ** csape fixed to work correctly for matrix inputs ** csaps and csaps_sel fixed to extrapolate linearly Summary of important user-visible changes for splines 1.1.0: ------------------------------------------------------------------- ** The following functions are new: csaps csaps_sel ** fnder() and csape() have been fixed for compatibility with the latest Octave versions. ** splines package is now dependent on GNU Octave version 3.6.0 or later. ** Package is no longer automatically loaded. splines/CITATION0000644000175000017500000000031614425217567012047 0ustar nirnir@software{splines, author = {Krakauer, Nir Y. and others}, title = {Splines Package for GNU Octave}, url = {http://octave.sourceforge.net/splines}, version = {1.3.4}, date = {2021-02-23}, } splines/COPYING0000644000175000017500000010451314425217567011751 0ustar nirnir GNU GENERAL PUBLIC LICENSE Version 3, 29 June 2007 Copyright (C) 2007 Free Software Foundation, Inc. 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But first, please read . splines/INDEX0000644000175000017500000000030114425217567011476 0ustar nirniranalysis >> Data analysis Spline functions bin_values catmullrom csapi csape csaps csaps_sel dedup fnder fnplt fnval regularization regularization2D tpaps tps_val tps_val_der splines/inst/0000775000175000017500000000000014425217637011667 5ustar nirnirsplines/inst/csaps_sel.m0000644000175000017500000003077014425217567014030 0ustar nirnir## Copyright (C) 2012-2018 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 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 . ## -*- texinfo -*- ## @deftypefn{Function File}{[@var{yi} @var{p} @var{sigma2},@var{unc_y}] =} csaps_sel(@var{x}, @var{y}, @var{xi}, @var{w}=[], @var{crit}=[]) ## @deftypefnx{Function File}{[@var{pp} @var{p} @var{sigma2},@var{unc_y}] =} csaps_sel(@var{x}, @var{y}, [], @var{w}=[], @var{crit}=[]) ## ## Cubic spline approximation with smoothing parameter estimation @* ## Approximately interpolates [@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}. ## ## A selection criterion @var{crit} is used to find a suitable value for @var{p} (between 0 and 1); possible values for @var{crit} are `vm' (Vapnik's measure [Cherkassky and Mulier 2007] from statistical learning theory); `aicc' (corrected Akaike information criterion, the default); `aic' (original Akaike information criterion); `gcv' (generalized cross validation). ## ## If @var{crit} is a nonnegative scalar instead of a string, then @var{p} is chosen to so that the mean square scaled residual Mean_i (@var{w}_i*(@var{y}_i - @var{pp}(@var{x}_i))^2) is approximately equal to @var{crit}. If @var{crit} is a negative scalar, then @var{p} is chosen so that the effective number of degrees of freedom in the spline fit (which ranges from 2 when @var{p} = 0 to @var{n} when @var{p} = 1) is approximately equal to -@var{crit}. ## ## @var{x} and @var{w} should be @var{n} by 1 in size; @var{y} should be @var{n} by @var{m}; @var{xi} should be @var{k} by 1; the values in @var{x} should be distinct and in ascending order; the values in @var{w} should be nonzero. ## ## Returns the smoothing spline @var{pp} or its values @var{yi} at the desired @var{xi}; the selected @var{p}; the estimated data scatter (variance from the smooth trend) @var{sigma2}; the estimated uncertainty (SD) of the smoothing spline fit at each @var{x} value, @var{unc_y}; and the estimated number of degrees of freedom @var{df} (out of @var{n}) used in the fit. ## ## For small @var{n}, the optimization uses singular value decomposition of an @var{n} by @var{n} matrix in order to quickly compute the residual size and model degrees of freedom for many @var{p} values for the optimization (Craven and Wahba 1979). For large @var{n} (currently >300), an asymptotically more computation and storage efficient method that takes advantage of the sparsity of the problem's coefficient matrices is used (Hutchinson and de Hoog 1985). ## ## References: ## ## Vladimir Cherkassky and Filip Mulier (2007), Learning from Data: Concepts, Theory, and Methods. Wiley, Chapter 4 ## ## Carl de Boor (1978), A Practical Guide to Splines, Springer, Chapter XIV ## ## Clifford M. Hurvich, Jeffrey S. Simonoff, Chih-Ling Tsai (1998), Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion, J. Royal Statistical Society, 60B:271-293 ## ## M. F. Hutchinson and F. R. de Hoog (1985), Smoothing noisy data with spline functions, Numerische Mathematik, 47:99-106 ## ## M. F. Hutchinson (1986), Algorithm 642: A fast procedure for calculating minimum cross-validation cubic smoothing splines, ACM Transactions on Mathematical Software, 12:150-153 ## ## Grace Wahba (1983), Bayesian ``confidence intervals'' for the cross-validated smoothing spline, J Royal Statistical Society, 45B:133-150 ## ## Herman J. Woltring (1986), A Fortran package for generalized, cross-validatory spline smoothing and differentiation, Advances in Engineering Software, 8(2):104–113 ## ## @end deftypefn ## @seealso{csaps, spline, csapi, ppval, dedup, bin_values, gcvspl} ## Author: Nir Krakauer function [ret,p,sigma2,unc_y,df]=csaps_sel(x,y,xi,w,crit) if (nargin < 5) crit = []; if(nargin < 4) w = []; if(nargin < 3) xi = []; endif endif endif if(columns(x) > 1) x = x'; y = y'; w = w'; endif if any (isnan ([x y w](:)) ) error('NaN values in inputs; pre-process to remove them') endif h = diff(x); if any(h <= 0) error('x must be strictly increasing; pre-process to achieve this') endif n = numel(x); if isempty(w) w = ones(n, 1); end if isscalar(crit) if crit == 0 || crit <= -n #return an exact cubic spline interpolation [ret,p]=csaps(x,y,1,xi,w); sigma2 = 0; unc_y = zeros(size(x)); return elseif crit > 0 w = w ./ crit; #adjust the sample weights so that the target mean square scaled residual is 1 crit = 'msr_bound'; else #negative value -- target degrees of freedom of fit if crit >= -2 #return linear regression p = 0; else global df_target df_target = -crit; crit = 'df_bound'; endif endif end if isempty(crit) crit = 'aicc'; end R = spdiags([h(2:end) 2*(h(1:end-1) + h(2:end)) h(1:end-1)], [-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); chol_method = (n > 300); #use a sparse Cholesky decomposition followed by solving for only the central bands of the inverse to solve for large n (faster), and singular value decomposition for small n (less prone to numerical error if data values are spaced close together) if chol_method penalty_function = @(p) penalty_compute_chol(p, QT, R, y, w, n, crit); else ##determine influence matrix for different p without repeated inversion [U D V] = svd(diag(1 ./ sqrt(w))*QT'*sqrtm(inv(R)), 0); D = diag(D).^2; penalty_function = @(p) penalty_compute(p, U, D, y, w, n, crit); end if ~exist("p", "var") ##choose p by minimizing the penalty function ##minimize over pt = log(-log(p)+1) to find p with good fractional accuracy if the optimum is very small, without making TolX very small tol_pt = 1E-6; pt_max = log(-log(realmin)+1) + 2*tol_pt; pt = fminbnd(@(pt) penalty_function(exp(1-exp(pt))), 0, pt_max, optimset ("TolX", tol_pt)); p = exp(1-exp(pt)); endif ## estimate the trend uncertainty if isargout (3) || isargout (4) || isargout (5) if chol_method [MSR, df] = penalty_terms_chol(p, QT, R, y, w, n); else H = influence_matrix(p, U, D, n, w); [MSR, df] = penalty_terms(H, y, w); end sigma2 = mean(MSR(:)) * (n / (n-df)); #estimated data error variance (wahba83) if isargout (4) if chol_method Hd = influence_matrix_diag_chol(p, QT, R, y, w, n); else Hd = diag(H); endif unc_y = sqrt(sigma2 * Hd ./ w); #uncertainty (SD) of fitted curve at each input x-value (hutchinson86) endif endif ## construct the fitted smoothing spline if isargout (1) ret = csaps (x,y,p,xi,w); endif endfunction function H = influence_matrix(p, U, D, n, w) #returns influence matrix for given p H = speye(n) - U * diag(D ./ (D + (p / (6*(1-p))))) * U'; H = diag(1 ./ sqrt(w)) * H * diag(sqrt(w)); #rescale to original units endfunction function [MSR, Ht] = penalty_terms(H, y, w) MSR = mean(w .* (y - (H*y)) .^ 2); #mean square residual Ht = trace(H); #effective number of fitted parameters endfunction function Hd = influence_matrix_diag_chol(p, QT, R, y, w, n) #LDL factorization of 6*(1-p)*QT*diag(1 ./ w)*QT' + p*R U = chol(6*(1-p)*QT*diag(1 ./ w)*QT' + p*R, 'upper'); d = 1 ./ diag(U); U = diag(d)*U; d = d .^ 2; #5 central bands in the inverse of 6*(1-p)*QT*diag(1 ./ w)*QT' + p*R Binv = banded_matrix_inverse(d, U, 2); Hd = full(diag(speye(n) - (6*(1-p))*diag(1 ./ w)*QT'*Binv*QT)); endfunction function [MSR, Ht] = penalty_terms_chol(p, QT, R, y, w, n) #LDL factorization of 6*(1-p)*QT*diag(1 ./ w)*QT' + p*R U = chol(6*(1-p)*QT*diag(1 ./ w)*QT' + p*R, 'upper'); d = 1 ./ diag(U); U = diag(d)*U; d = d .^ 2; Binv = banded_matrix_inverse(d, U, 2); #5 central bands in the inverse of 6*(1-p)*QT*diag(1 ./ w)*QT' + p*R Ht = 2 + trace(speye(n-2) - (6*(1-p))*QT*diag(1 ./ w)*QT'*Binv); MSR = mean(w .* ((6*(1-p)*diag(1 ./ w)*QT'*((6*(1-p)*QT*diag(1 ./ w)*QT' + p*R) \ (QT*y)))) .^ 2); endfunction function J = vm(MSR, Ht, n) #Vapnik-Chervonenkis penalization factor or Vapnik's measure in cherkassky07, p. 129 p = Ht/n; if p == 0 J = mean(log(MSR)(:)) - log(1 - sqrt(log(n)/(2*n))); elseif n == 0 || (p*(1 - log(p)) + log(n)/(2*n)) >= 1 J = Inf; else J = mean(log(MSR)(:)) - log(1 - sqrt(p*(1 - log(p)) + log(n)/(2*n))); endif endfunction function J = aicc(MSR, Ht, n) J = mean(log(MSR)(:)) + 2 * (Ht + 1) / max(n - Ht - 2, 0); #hurvich98, taking the average if there are multiple data sets as in woltring86 endfunction function J = aic(MSR, Ht, n) J = mean(log(MSR)(:)) + 2 * Ht / n; endfunction function J = gcv(MSR, Ht, n) J = mean(log(MSR)(:)) - 2 * log(1 - Ht / n); endfunction function J = msr_bound(MSR, Ht, n) J = mean(MSR(:) - 1) .^ 2; endfunction function J = df_bound(MSR, Ht, n) global df_target J = (Ht - df_target) .^ 2; endfunction function J = penalty_compute(p, U, D, y, w, n, crit) #evaluates a user-supplied penalty function crit at given p H = influence_matrix(p, U, D, n, w); [MSR, Ht] = penalty_terms(H, y, w); J = feval(crit, MSR, Ht, n); if ~isfinite(J) J = Inf; endif endfunction function J = penalty_compute_chol(p, QT, R, y, w, n, crit) #evaluates a user-supplied penalty function crit at given p [MSR, Ht] = penalty_terms_chol(p, QT, R, y, w, n); J = feval(crit, MSR, Ht, n); if ~isfinite(J) J = Inf; endif endfunction function Binv = banded_matrix_inverse(d, U, m) #given a (2m+1)-banded, symmetric n x n matrix B = U'*inv(diag(d))*U, where U is unit upper triangular with bandwidth (m+1), returns Binv, a sparse symmetric matrix containing the central 2m+1 bands of the inverse of B #Reference: Hutchinson and de Hoog 1985 Binv = sparse(diag(d)); n = rows(U); for i = n:(-1):1 p = min(m, n - i); for l = 1:p for k = 1:p Binv(i, i+l) -= U(i, i+k)*Binv(i + k, i + l); end Binv(i, i) -= U(i, i+l)*Binv(i, i+l); end Binv(i+(1:p), i) = Binv(i, i+(1:p))'; #add the lower triangular elements end endfunction %!shared x,y,ret,p,sigma2,unc_y %! x = [0:0.01:1]'; y = sin(x); %! [ret,p,sigma2,unc_y] = csaps_sel(x,y,x); %!assert (1 - p, 0, 1E-6); %!assert (sigma2, 0, 1E-10); %!assert (ret - y, zeros(size(y)), 1E-4); %!assert (ret, (csaps_sel(x,[y 2*y],x))'(:, 1), 1E-4); %!assert (unc_y, zeros(size(unc_y)), 1E-5); %!demo %! ni = 400; #number of evaluation points %! n = 40; #number of given sample points %! f = @(x) sin (2*pi*x); #function generating the synthetic data %! x = sort (rand (n, 1)); #sampled points %! y = f(x) + 0.1*randn (n, 1); %! xi = linspace (0, 1, ni); #evaluation points %! yi = csaps_sel (x,y,xi,[],'aicc'); %! plot (x, y, 's', xi, yi) %! grid on %! legend ('Given data', 'Spline fit') %! title ('Spline smoothing with synthetic data sampled from sine wave with noise') %{ #experiments comparing different selection criteria for recovering a function sampled with standard normal noise -- aicc was consistently better than aic, but otherwise which method does best is problem-specific tic m = 500; #number of replicates available ni = 400; #number of evaluation points ns = [5 10 20 40]; #number of given sample points nk = 100; #number of trials to average over f = @(x) sin(2*pi*x); #function generating the synthetic data mse = nan(4, numel(ns), nk); for i = 1:numel(ns) for k = 1:nk n = ns(i); x = linspace(0, 1, n)(:); y = f(x) + randn(n, m); xi = rand(ni, 1); yt = f(xi); yi = csaps_sel(x,y,xi,[],'vm'); mse(1, i, k) = meansq((yi - yt')(:)); yi = csaps_sel(x,y,xi,[],'aicc'); mse(2, i, k) = meansq((yi - yt')(:)); yi = csaps_sel(x,y,xi,[],'aic'); mse(3, i, k) = meansq((yi - yt')(:)); yi = csaps_sel(x,y,xi,[],'gcv'); mse(4, i, k) = meansq((yi - yt')(:)); endfor endfor msem = mean(mse, 3); toc %} splines/inst/fnplt.m0000644000175000017500000000341214425217567013170 0ustar nirnir## Copyright (C) 2000 Kai Habel ## ## 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 . ## -*- texinfo -*- ## @deftypefn {Function File} { } fnplt (@var{pp}, '@var{plt}') ## plots spline ## ## @seealso{ppval, spline, csape} ## @end deftypefn ## Author: Kai Habel ## Date: 3. dec 2000 ## 2001-02-19 Paul Kienzle ## * use pp.x rather than just x in linspace; add plt parameter ## * return points instead of plotting them if desired ## * also plot control points ## * added demo function [x, y] = fnplt (pp, plt) if (nargin < 1 || nargin > 2) print_usage; endif if (nargin < 2) plt = "r;;"; endif xi = linspace(min(pp.breaks),max(pp.breaks),256)'; pts = ppval(pp,xi); if nargout == 2 x = xi; y = pts; elseif nargout == 1 x = [xi, pts]; else plot(xi,pts,plt,pp.breaks,ppval(pp,pp.breaks),"bx;;"); endif endfunction %!demo %! x = [ 0; sort(rand(25,1)); 1 ]; %! pp = csape (x, sin (2*pi*3*x), 'periodic'); %! axis([0,1,-2,2]); %! title('Periodic spline reconstruction of randomly sampled sine'); %! fnplt (pp,'r;reconstruction;'); %! t=linspace(0,1,100); y=sin(2*pi*3*t); %! hold on; plot(t,y,'g;ideal;'); hold off; %! axis; title(""); splines/inst/fnder.m0000644000175000017500000000240114425217567013140 0ustar nirnir## Copyright (C) 2001 Kai Habel ## ## 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 . ## -*- texinfo -*- ## @deftypefn {Function File} { } fnder (@var{pp}, @var{order}) ## differentiate the spline in pp-form ## ## @seealso{ppval} ## @end deftypefn ## Author: Kai Habel ## Date: 20. feb 2001 function dpp = fnder (pp, o) if (nargin < 1 || nargin > 2) print_usage; endif if (nargin < 2) o = 1; endif [X, P, N, K, D] = unmkpp (pp); c = columns (P); r = rows (P); for i = 1:o #pp.P = polyder (pp.P); matrix capable polyder is needed. P = P(:, 1:c - 1) .* kron ((c - 1):- 1:1, ones (r,1)); endfor dpp = mkpp (X, P); endfunction splines/inst/regularization.m0000644000175000017500000001505014425217567015105 0ustar nirnir## Copyright (C) 2019 Andreas Stahel ## ## 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 ## . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{grid},@var{u}] =} regularization (@var{data}, @var{interval}, @var{N}, @var{F1}) ## @deftypefnx {Function File} {[@var{grid},@var{u}] =} regularization (@var{data}, @var{interval}, @var{N}, @var{F1}, @var{F2}) ## ## Apply a Tikhonov regularization, the functional to be minimized is@* ## @var{F} = @var{FD} + @var{lambda1}*@var{F1} + @var{lambda2}*@var{F2} @* ## = sum_(i=1)^M (y_i-u(x_i))^2 ## + @var{lambda1}*int_a^b (u'(x) - @var{g1}(x))^2 dx ## + @var{lambda2}*int_a^b (u''(x) - @var{g2}(x))^2 dx ## ## With @var{lambda1} = 0 and @var{G2}(x) = 0 this leads to a smoothing spline. ## ## Parameters: ## @itemize ## @item @var{data} is a M*2 matrix with the x values in the first column and the y values in the second column. ## @item @var{interval} = [a,b] is the interval on which the regularization is applied. ## @item @var{N} is the number of subintervals of equal length. @var{grid} will consist of @var{N+1} grid points. ## @item @var{F1} is a structure containing the information on the first regularization term, integrating the square of the first derivative. ## @itemize ## @item @var{F1.lambda} is the value of the regularization parameter @var{lambda1}>=0. ## @item @var{F1.g} is the function handle for the function @var{g1(x)}. If not provided @var{G1=0} is used. ## @end itemize ## @item @var{F2} is a structure containing the information on the second regularization term, integrating the square of the second derivative. If @var{F2} is not provided @var{lambda2}=0 is assumed. ## @itemize ## @item @var{F2.lambda} is the value of the regularization parameter @var{lambda2}>=0. ## @item @var{F2.g} is the function handle for the function @var{g2(x)}. If not provided @var{G2=0} is used. ## @end itemize ## @end itemize ## ## Return values: ## @itemize ## @item @var{grid} is the grid on which @var{u} is evaluated. It consists of @var{N+1} equidistant points on the @var{interval}. ## @item @var{u} are the values of the regularized approximation to the @var{data} evaluated at @var{grid}. ## @end itemize ## @seealso{csaps, regularization2D, demo regularization} ## @end deftypefn ## Author: Andreas Stahel ## Created: 2019-03-26 function [grid,u] = regularization (data, interval, N, F1, F2) a = interval (1); b = interval (2); grid = linspace (a, b, N + 1)'; dx = grid (2) - grid (1); x = data (:, 1); ## select points in interval only ind = find ((x >= a) .* (x <= b)); x = x (ind); y = data (:, 2); y = y (ind); M = length (x); Interp = sparse (M, N + 1); ## interpolation matrix pos = floor ((x - a) / dx) + 1; theta = mod ((x - a) / dx, 1); for ii = 1:M if theta (ii) > 10*eps Interp (ii, pos (ii)) = 1 - theta (ii); Interp (ii, pos (ii) + 1) = theta (ii); else Interp (ii, pos (ii)) = 1; endif endfor mat = Interp' * Interp; rhs = (Interp' * y); if isfield (F1, 'lambda') A1 = spdiags ([-ones(N, 1), +ones(N, 1)], [0, 1], N, N + 1) / dx; mat = mat + F1.lambda * dx * A1' * A1; if isfield (F1, 'g') g1 = F1.g (grid (1:end - 1) + dx / 2); rhs = rhs + F1.lambda * dx * A1' * g1; endif endif if exist ('F2') A2 = spdiags (ones (N, 1) * [1, -2, 1], [0, 1, 2], N - 1, N + 1) / dx^2; mat = mat + F2.lambda * dx * A2' * A2; if isfield (F2, 'g') g2 = F2.g (grid (2:end - 1)); rhs = rhs + F2.lambda * dx * A2' * g2; endif endif u = mat \ rhs; endfunction %!demo %! N = 100; interval = [0,10]; %! x = [3.2,4,5,5.2,5.6]'; y = x; %! %! clear F1 F2 %! %% regularize towards slope 0.1, no smoothing %! F1.lambda = 1e-2; F1.g = @(x)0.1*ones(size(x)); %! [grid,u1] = regularization([x,y],interval,N,F1); %! %% regularize towards slope 0.1, with some smoothing %! F2.lambda = 1*1e-3; %! [grid,u2] = regularization([x,y],interval,N,F1,F2); %! %! figure(1) %! plot(grid,u1,'b',grid,u2,'g',x,y,'*r') %! xlabel('x'); ylabel('solution'); %! legend('regular1','regular2','data','location','northwest') %!demo %! N = 1000; interval = [0,pi]; %! x = linspace( pi/4,3*pi/4,15)'; y = sin(x)+ 0.03*randn(size(x)); %! %! clear F1 F2 %! %% regularize by smoothing only %! F1.lambda = 0; F2.lambda = 1e-3; %! [grid,u1] = regularization([x,y],interval,N,F1,F2); %! %% regularize by smoothing and aim for slope 0 %! F1.lambda = 1*1e-2; %! [grid,u2] = regularization([x,y],interval,N,F1,F2); %! %! figure(1) %! plot(grid,u1,'b',grid,u2,'g',x,y,'*r') %! xlabel('x'); ylabel('solution'); %! legend('regular1','regular2','data','location','northwest') %!demo %! interval = [0,1]; %! N = 400; %! x = rand(200,1); %! %% generate the data on four line segments, add some noise %! y = 2 - 2*x + (x>0.25) - 2*(x>0.5).*(x<0.65)+ 0.1*randn(length(x),1); %! clear F1 %! %% apply regularization with three different parameters %! F1.lambda = 1e-3; [grid,u1] = regularization([x,y],interval,N,F1); %! F1.lambda = 1e-1; [grid,u2] = regularization([x,y],interval,N,F1); %! F1.lambda = 3e+0; [grid,u3] = regularization([x,y],interval,N,F1); %! %! figure(1); plot(grid,u1,'b',grid,u2,'g',grid,u3,'m',x,y,'+r') %! xlabel('x'); ylabel('solution') %! legend('\lambda_1=0.001','\lambda_1=0.1','\lambda_1=3','data') %!demo %! %% generate a smoothing spline, see also csaps() in the package splines %! N = 1000; interval = [0,10.3]; %! x = [0 3 4 6 10]'; y = [0 1 0 1 0]'; %! clear F2 %! F2.lambda = 1e-2; %! %% apply regularization, the result is a smoothing spline %! [grid,u] = regularization([x,y],interval,N,0,F2); %! %! figure(1); %! plot(grid,u,'b',x,y,'*r') %! legend('spline','data') %! xlabel('x'); ylabel('solution') %!test %! data = [0,0;1,1;2,2]; %! F1.lambda = 0.0; F2.lambda = 0.1; %! [grid,u] = regularization(data,[0,1],10,F1,F2); %! assert(norm(grid-u),0,1e-12) %!test %! x = linspace(-1,1,11); %! F1.lambda = 0.01;F2.g = @(x) 2*ones(size(x)); F2.lambda = 0.01; %! [grid,u] = regularization([x;x.^2]',[-2,2],20,F1,F2); %! assert(u(11),7.330959483903200e-03,1e-8) splines/inst/tpaps.m0000644000175000017500000001257614425217567013207 0ustar nirnir## Copyright (C) 2013 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 . ## -*- texinfo -*- ## @deftypefn{Function File}{[@var{yi} @var{p}] =} tpaps(@var{x}, @var{y}, @var{p}, @var{xi}) ## @deftypefnx{Function File}{[@var{coefs} @var{p}] =} tpaps(@var{x}, @var{y}, @var{p}, []) ## ## Thin plate smoothing of scattered values in multi-D @* ## approximately interpolate [@var{x},@var{y}] at @var{xi} ## ## The chosen thin plate spline minimizes the sum of squared deviations from the given points plus a penalty term proportional to the curvature of the spline function ## ## @var{x} should be @var{n} by @var{d} in size, where @var{n} is the number of points and @var{d} the number of dimensions; @var{y} and @var{w} should be @var{n} by 1; @var{xi} should be @var{k} by @var{d}; the points in @var{x} should be distinct ## ## @table @asis ## @item @var{p}=0 ## maximum smoothing: flat surface ## @item @var{p}=1 ## no smoothing: interpolation ## @item @var{p}<0 or not given ## an intermediate amount of smoothing is chosen (such that the smoothing term and the interpolation term are of the same magnitude) ## @end table ## ## If @var{xi} is not specified, returns a vector @var{coefs} of the @var{n} + @var{d} + 1 fitted thin plate spline coefficients. ## Given @var{coefs}, the value of the thin-plate spline at any @var{xi} can be determined with @code{tps_val} ## ## Note: Computes the pseudoinverse of an @var{n} by @var{n} matrix, so not recommended for very large @var{n} ## ## Example usages: ## @example ## x = ([1:10 10.5 11.3])'; y = sin(x); xi = (0:0.1:12)'; ## yi = tpaps(x, y, 0.5, xi); ## plot(x, y, xi, yi) ## @end example ## ## @example ## x = rand(100, 2)*2 - 1; ## y = x(:, 1) .^ 2 + x(:, 2) .^ 2; ## scatter(x(:, 1), x(:, 2), 10, y, "filled") ## [x1 y1] = meshgrid((-1:0.2:1)', (-1:0.2:1)'); ## xi = [x1(:) y1(:)]; ## yi = tpaps(x, y, 1, xi); ## contourf(x1, y1, reshape(yi, 11, 11)) ## @end example ## ## Reference: ## David Eberly (2011), Thin-Plate Splines, www.geometrictools.com/Documentation/ThinPlateSplines.pdf ## Bouhamidi, A. (2005) Weighted thin plate splines, Analysis and Applications, 3: 297-324 ## ## @end deftypefn ## @seealso{csaps, tps_val, tps_val_der} ## Author: Nir Krakauer function [ret, p]=tpaps(x,y,p,xi) if(nargin < 4) xi = []; if(nargin < 3) || p < 0 p = []; endif endif [n d] = size(x); #d: number of dimensions; n: number of points [y should be n*1] dist = @(x1, x2) norm(x2 - x1, 2, "rows"); #Euclidian distance between points in d-dimensional space #form of the Green's function for solutions G = @(r) merge(r == 0, 0, r .^ 2 .* log(r)); N = [ones(n, 1) x]; if p == 0 #infinite regularization (no curvature allowed), solution is a regression plane b = N \ y; a = zeros(n, 1); else #coefficient matrices #need pairwise distances between points M = zeros(n); warn_state = warning ("query", "Octave:broadcast").state; warning ("off", "Octave:broadcast"); #turn off warning message for automatic broadcasting when dist is called unwind_protect for i = 1:(n-1) #M is symmetric, so only need to compute half M(i, (i+1):n) = G(dist(x(i, :), x((i+1):n, :))); endfor unwind_protect_cleanup warning (warn_state, "Octave:broadcast"); end_unwind_protect M = M + M'; if isempty(p) #choose an intermediate value for the regularization parameter lambda = mean(spdiags(M, 1:min(n, 3))(:)); p = 1 / (lambda + 1); else #use the given value lambda = (1 - p) / p; endif M = M + lambda*eye(n); #add regularization term M_inv = pinv(M); b = pinv(N' * M_inv * N) * N' * M_inv * y; a = M_inv * (y - N*b); endif if isempty(xi) #return the coefficients ret = [a' b']'; else #return the thin plate spline values at xi k = size(xi, 1); ret = [ones(k, 1) xi] * b; if p ~= 0 warn_state = warning ("query", "Octave:broadcast").state; warning ("off", "Octave:broadcast"); #turn off warning message for automatic broadcasting when dist is called unwind_protect if k > n ##choose from either of two ways of computing the values of the thin plate spline at xi for i = 1:n ret = ret + a(i)*G(dist(x(i, :), xi)); endfor else for i = 1:k ret(i) = ret(i) + dot(a, G(dist(x, xi(i, :)))); endfor endif unwind_protect_cleanup warning (warn_state, "Octave:broadcast"); end_unwind_protect endif endif endfunction %!shared x,y %! x = ([1:10 10.5 11.3])'; y = sin(x); %!assert (tpaps(x,y,1,x), y, 1E3*eps); %! x = ((1 ./ (1:100))' - 0.5) * ([0.2 0.6]); %! y = x(:, 1) .^ 2 + x(:, 2) .^ 2; %!assert (tpaps(x,y,1,x), y, 1E-10); splines/inst/fnval.m0000644000175000017500000000071514425217567013156 0ustar nirnir## Author: Paul Kienzle ## This program is granted to the public domain. ## r = fnval(pp,x) or r = fnval(x,pp) ## Compute the value of the piece-wise polynomial pp at points x. function r = fnval(a,b,left) if nargin == 2 || (nargin == 3 && left == 'l' && left == 'r') # XXX FIXME XXX ignoring left continuous vs. right continuous option if isstruct(a), r=ppval(a,b); else r=ppval(b,a); end else print_usage; end end splines/inst/csape.m0000644000175000017500000003420414425217567013143 0ustar nirnir## Copyright (C) 2000, 2001 Kai Habel ## ## 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 . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{pp} = } csape (@var{x}, @var{y}, @var{cond}, @var{valc}) ## cubic spline interpolation with various end conditions. ## creates the pp-form of the cubic spline. ## ## @var{x} should be @var{n} by 1, @var{y} should be @var{n} by @var{m}, ## @var{valc} should be 2 by @var{m} or 2 by 1 ## ## The following end conditions as given in @var{cond} are possible: ## @table @asis ## @item 'complete' ## match slopes at first and last point as given in @var{valc} ## (default; if @var{valc} is not given, the slopes matched are those ## of the cubic polynomials that interpolate the first and last four points) ## @item 'not-a-knot' ## third derivatives are continuous at the second and second last point ## @item 'periodic' ## match first and second derivative of first and last point ## @item 'second' ## match second derivative at first and last point as given in @var{valc} ## @item 'variational' ## set second derivative at first and last point to zero (natural cubic spline) ## @end table ## ## @seealso{ppval, spline} ## @end deftypefn ## Author: Kai Habel ## Date: 23 Nov 2000 ## Algorithms taken from G. Engeln-Muellges, F. Uhlig: ## "Numerical Algorithms with C", Springer, 1996 function pp = csape (x, y, cond, valc) x = x(:); n = length(x); if (n < 3) error ("csape requires at least 3 points"); endif ## Check the size and shape of y ndy = ndims (y); szy = size (y); if (ndy == 2 && (szy(1) == n || szy(2) == n)) if (szy(2) == n) a = y.'; else a = y; szy = fliplr (szy); endif else a = shiftdim (reshape (y, [prod(szy(1:end-1)), szy(end)]), 1); endif m = size (a, 2); if exist("valc", "var") && size(valc) == [1 2] valc = valc'; endif b = c = zeros (n, m); h = diff (x); idx = ones (columns(a),1); if (nargin < 3 || strcmp(cond,"complete")) # set first derivative at end points if (nargin < 4) valc = zeros(2, m); n_use = min(n, 4); for i = 1:m valc(1, i) = polyval(polyder(polyfit(x(1:n_use), a(1:n_use, i), n_use-1)), x(1)); valc(2, i) = polyval(polyder(polyfit(x(end-n_use+1:end), a(end-n_use+1:end, i), n_use-1)), x(end)); endfor endif if (n == 3) warning ("off", "Octave:broadcast", "local"); e = 2 * [h(1) h(1:(end-1))+h(2:end) h(end)]; A = spdiags([[h; 0], e(:), [0; h]], [-1,0,1], n, n); d = diff(a) ./ h; #uses broadcasting if columns(a) > 1 g = 3 * diff(d); A(1, 1) = 2 * h(1); A(1, 2) = h(1); A(n, n) = 2 * h(end); A(end, end-1) = h(end); g = [3*(d(1, :) - valc(1, :)); g; 3*(valc(2, :) - d(end, :))]; c = A \ g; else dg = 2 * (h(1:n - 2) + h(2:n - 1)); dg(1) = dg(1) - 0.5 * h(1); dg(n - 2) = dg(n-2) - 0.5 * h(n - 1); e = h(2:n - 2); g = 3 * diff (a(2:n,:)) ./ h(2:n - 1,idx)... - 3 * diff (a(1:n - 1,:)) ./ h(1:n - 2,idx); g(1,:) = 3 * (a(3,:) - a(2,:)) / h(2) ... - 3 / 2 * (3 * (a(2,:) - a(1,:)) / h(1) - valc(1,:)); g(n - 2,:) = 3 / 2 * (3 * (a(n,:) - a(n - 1,:)) / h(n - 1) - valc(2,:))... - 3 * (a(n - 1,:) - a(n - 2,:)) / h(n - 2); c(2:n - 1,:) = spdiags([[e(:);0],dg,[0;e(:)]],[-1,0,1],n-2,n-2) \ g; c(1,:) = (3 / h(1) * (a(2,:) - a(1,:)) - 3 * valc(1, :) - c(2,:) * h(1)) / (2 * h(1)); c(n,:) = - (3 / h(n - 1) * (a(n,:) - a(n - 1,:)) - 3 * valc(2, :) + c(n - 1,:) * h(n - 1)) / (2 * h(n - 1)); end b(1:n - 1,:) = diff (a) ./ h(1:n - 1, idx)... - h(1:n - 1,idx) / 3 .* (c(2:n,:) + 2 * c(1:n - 1,:)); d = diff (c) ./ (3 * h(1:n - 1, idx)); elseif (strcmp(cond,"variational") || strcmp(cond,"second")) if ((nargin < 4) || strcmp(cond,"variational")) ## set second derivatives at end points to zero valc = zeros (2, 1); endif c(1,:) = valc(1, :) / 2; c(n,:) = valc(2, :) / 2; g = 3 * diff (a(2:n,:)) ./ h(2:n - 1, idx)... - 3 * diff (a(1:n - 1,:)) ./ h(1:n - 2, idx); g(1,:) = g(1,:) - h(1) * c(1,:); g(n - 2,:) = g(n-2,:) - h(n - 1) * c(n,:); dg = 2 * (h(1:n - 2) + h(2:n - 1)); e = h(2:n - 2); c(2:n - 1,:) = spdiags([[e(:);0],dg,[0;e(:)]],[-1,0,1],n-2,n-2) \ g; b(1:n - 1,:) = diff (a) ./ h(1:n - 1,idx)... - h(1:n - 1,idx) / 3 .* (c(2:n,:) + 2 * c(1:n - 1,:)); d = diff (c) ./ (3 * h(1:n - 1, idx)); elseif (strcmp(cond,"periodic")) D = diff(a) ./ h; A = sparse (n-1); v = zeros (n-1, m); for i = 2:(n-2) A(i, i-1) = h(i-1); A(i, i) = 2 * (h(i-1) + h(i)); A(i, i+1) = h(i); v(i, :) = 3 * (D(i, :) - D(i-1, :)); endfor A(1, 1) = 2 * (h(1) + h(n-1)); A(1, 2) = h(1); A(1, n-1) += h(n-1); v(1, :) = 3 * (D(1, :) - D(n-1, :)); A(n-1, 1) = h(n-1); A(n-1, n-2) += h(n-2); A(n-1, n-1) = 2 * (h(n-2) + h(n-1)); v(n-1, :) = 3 * (D(n-1, :) - D(n-2, :)); c = A \ v; #this is a cyclic tridiagonal system -- the Sherwood-Morrison formula #could be used to solve it if \ is too slow in this case c(n,:) = c(1,:); b = diff (a) ./ h(1:n - 1,idx)... - h(1:n - 1,idx) / 3 .* (c(2:n,:) + 2 * c(1:n - 1,:)); b(n,:) = b(1,:); d = diff (c) ./ (3 * h(1:n - 1, idx)); d(n,:) = d(1,:); elseif (strcmp(cond,"not-a-knot")) g = zeros(n - 2,columns(a)); g(1,:) = 3 / (h(1) + h(2)) * (a(3,:) - a(2,:)... - h(2) / h(1) * (a(2,:) - a(1,:))); g(n - 2,:) = 3 / (h(n - 1) + h(n - 2)) *... (h(n - 2) / h(n - 1) * (a(n,:) - a(n - 1,:)) -... (a(n - 1,:) - a(n - 2,:))); if (n > 4) g(2:n - 3,:) = 3 * diff (a(3:n - 1,:)) ./ h(3:n - 2,idx)... - 3 * diff (a(2:n - 2,:)) ./ h(2:n - 3,idx); dg = 2 * (h(1:n - 2) + h(2:n - 1)); dg(1) = dg(1) - h(1); dg(n - 2) = dg(n-2) - h(n - 1); ldg = udg = h(2:n - 2); udg(1) = udg(1) - h(1); ldg(n - 3) = ldg(n-3) - h(n - 1); c(2:n - 1,:) = spdiags([[ldg(:);0],dg,[0;udg(:)]],[-1,0,1],n-2,n-2) \ g; elseif (n == 4) dg = [h(1) + 2 * h(2); 2 * h(2) + h(3)]; ldg = h(2) - h(3); udg = h(2) - h(1); c(2:n - 1,:) = spdiags([[ldg(:);0],dg,[0;udg(:)]],[-1,0,1],n-2,n-2) \ g; else # n == 3 #with only 3 points, the not-a-knot cubic spline is not unique; #we choose the one where the cubic coefficients are zero, #which is the interpolating quadratic polynomial c = repmat ((a(1,:) - a(3,:)) / ((x(3) - x(1)) * (x(2) - x(3))) ... + (a(2,:) - a(1,:)) / ((x(2) - x(1)) * (x(2) - x(3))), [3 1]); endif if n == 3 d = zeros (n, m); else c(1,:) = c(2,:) + h(1) / h(2) * (c(2,:) - c(3,:)); c(n,:) = c(n - 1,:) + h(n - 1) / h(n - 2) * (c(n - 1,:) - c(n - 2,:)); d = diff (c) ./ (3 * h(1:n - 1, idx)); endif b = diff (a) ./ h(1:n - 1, idx)... - h(1:n - 1, idx) / 3 .* (c(2:n,:) + 2 * c(1:n - 1,:)); else msg = sprintf("unknown end condition: %s",cond); error (msg); endif d = d(1:n-1,:); c=c(1:n-1,:); b=b(1:n-1,:); a=a(1:n-1,:); pp = mkpp (x, cat (2, d'(:), c'(:), b'(:), a'(:)), szy(1:end-1)); endfunction %!shared x,x2,y,cond,pp,pp1,pp2,h,valc,xi,yi %! x = linspace(0,2*pi,5); y = sin(x); x2 = linspace(0,2*pi,16); %!assert (ppval(csape(x,y),x), y, 10*eps); %!assert (ppval(csape(x,y),x'), y', 10*eps); %!assert (ppval(csape(x',y'),x'), y', 10*eps); %!assert (ppval(csape(x',y'),x), y, 10*eps); %!assert (ppval(csape(x,[y;y]),x), ... %! [ppval(csape(x,y),x);ppval(csape(x,y),x)], 10*eps) %!assert (ppval(csape(x,[y;y]),x2), ... %! [ppval(csape(x,y),x2);ppval(csape(x,y),x2)], 10*eps) %!assert (ppval(csape([1 2 4],[2, 3, 6]), 3), 13/3, 10*eps); %!test cond='complete'; %!assert (ppval(csape(x,y,cond),x), y, 10*eps); %!assert (ppval(csape(x,y,cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x), y, 10*eps); %!assert (ppval(csape(x,[y;y],cond),x), ... %! [ppval(csape(x,y,cond),x);ppval(csape(x,y,cond),x)], 10*eps) %!assert (ppval(csape(x,[y;y],cond),x2), ... %! [ppval(csape(x,y,cond),x2);ppval(csape(x,y,cond),x2)], 10*eps) %!assert (ppval(csape([1 2 4],[2, 3, 6], cond, [2 1]), 3), 215/48, 10*eps); %!test cond='variational'; %!assert (ppval(csape(x,y,cond),x), y, 10*eps); %!assert (ppval(csape(x,y,cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x), y, 10*eps); %!assert (ppval(csape(x,[y;y],cond),x), ... %! [ppval(csape(x,y,cond),x);ppval(csape(x,y,cond),x)], 10*eps) %!assert (ppval(csape(x,[y;y],cond),x2), ... %! [ppval(csape(x,y,cond),x2);ppval(csape(x,y,cond),x2)], 10*eps) %!assert (ppval(csape([1 2 3],[2, 3, 5],cond), 1.5), 2.40625, 10*eps); %!assert (ppval(csape([1 2 4],[2, 3, 6], cond), 3), 4.375, 10*eps); %!test cond='second'; %!assert (ppval(csape(x,y,cond),x), y, 10*eps); %!assert (ppval(csape(x,y,cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x), y, 10*eps); %!assert (ppval(csape(x,[y;y],cond),x), ... %! [ppval(csape(x,y,cond),x);ppval(csape(x,y,cond),x)], 10*eps) %!assert (ppval(csape(x,[y;y],cond),x2), ... %! [ppval(csape(x,y,cond),x2);ppval(csape(x,y,cond),x2)], 10*eps) %!assert (ppval(csape([1 2 4],[2, 3, 6], cond, [1 2]), 3), 49/12, 10*eps); %!test cond='periodic'; %!assert (ppval(csape(x,y,cond),x), y, 10*eps); %!assert (ppval(csape(x,y,cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x), y, 10*eps); %!assert (ppval(csape(x,[y;y],cond),x), ... %! [ppval(csape(x,y,cond),x);ppval(csape(x,y,cond),x)], 10*eps) %!assert (ppval(csape(x,[y;y],cond),x2), ... %! [ppval(csape(x,y,cond),x2);ppval(csape(x,y,cond),x2)], 10*eps) %!test cond='not-a-knot'; %!assert (ppval(csape(x,y,cond),x), y, 10*eps); %!assert (ppval(csape(x,y,cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x'), y', 10*eps); %!assert (ppval(csape(x',y',cond),x), y, 10*eps); %!assert (ppval(csape(x,[y;y],cond),x), ... %! [ppval(csape(x,y,cond),x);ppval(csape(x,y,cond),x)], 10*eps) %!assert (ppval(csape(x,[y;y],cond),x2), ... %! [ppval(csape(x,y,cond),x2);ppval(csape(x,y,cond),x2)], 10*eps) %!assert (ppval(csape(x(1:4),y(1:4),cond),x(1:4)), y(1:4), 10*eps); %!assert (ppval(csape(x(1:4)',y(1:4)',cond),x(1:4)), y(1:4), 10*eps); %!test assert (ppval(csape([1 2 4],[2, 3, 6], 'not-a-knot'), 3), 13/3, 10*eps); %!test assert (ppval(csape([1 2 4 5],[2, 3, 6, 5], 'not-a-knot'), 3), 29/6, 10*eps); %!test assert (ppval(csape([1 2 4 5 6],[2, 3, 6, 5, 6], 'not-a-knot'), 3), 141/28, 10*eps); %!test cond='complete'; %! h = pi/6; n = 3; x = linspace(0,(n-1)*h,n)'; y = sin(x); valc = cos([x(1) x(end)]); pp = csape(x, y, cond, valc); %!assert (ppval(csape(x,[y y],cond, valc),x)', ... %! repmat(ppval(pp, x), [1 2]), 10*eps) %!assert (polyval(pp.coefs(1, :), [0 h])', y(1:2), 10*eps) %!assert (polyval(pp.coefs(2, :), [0 h])', y(2:3), 10*eps) %!assert (polyval([3*pp.coefs(1, 1) 2*pp.coefs(1, 2) pp.coefs(1, 3)], 0), valc(1), 10*eps) %!assert (polyval([3*pp.coefs(2, 1) 2*pp.coefs(2, 2) pp.coefs(2, 3)], h), valc(2), 10*eps) %!assert (polyval([3*pp.coefs(1, 1) 2*pp.coefs(1, 2) pp.coefs(1, 3)], h), polyval([3*pp.coefs(2, 1) 2*pp.coefs(2, 2) pp.coefs(2, 3)], 0), 10*eps) %!assert (polyval([6*pp.coefs(1, 1) 2*pp.coefs(1, 2)], h), polyval([6*pp.coefs(2, 1) 2*pp.coefs(2, 2)], 0), 10*eps) %! y = cos(x); valc = -sin([x(1) x(end)]); pp1 = csape(x, y, cond, valc); %!assert (ppval(csape(x,[y y],cond, valc),x)', ... %! repmat(ppval(pp1, x), [1 2]), 10*eps) %!assert (polyval(pp1.coefs(1, :), [0 h])', y(1:2), 10*eps) %!assert (polyval(pp1.coefs(2, :), [0 h])', y(2:3), 10*eps) %!assert (polyval([3*pp1.coefs(1, 1) 2*pp1.coefs(1, 2) pp1.coefs(1, 3)], 0), valc(1), 10*eps) %!assert (polyval([3*pp1.coefs(2, 1) 2*pp1.coefs(2, 2) pp1.coefs(2, 3)], h), valc(2), 10*eps) %!assert (polyval([3*pp1.coefs(1, 1) 2*pp1.coefs(1, 2) pp1.coefs(1, 3)], h), polyval([3*pp1.coefs(2, 1) 2*pp1.coefs(2, 2) pp1.coefs(2, 3)], 0), 10*eps) %!assert (polyval([6*pp1.coefs(1, 1) 2*pp1.coefs(1, 2)], h), polyval([6*pp1.coefs(2, 1) 2*pp1.coefs(2, 2)], 0), 10*eps) %! y = [sin(x) cos(x)]; valc = [cos([x(1); x(end)]) -sin([x(1); x(end)])]; pp2 = csape(x, y, cond, valc); %!assert (pp2.coefs([1 3], :), pp.coefs) %!assert (pp2.coefs([2 4], :), pp1.coefs) # more tests of correctness for periodic boundary conditions %!test cond='periodic'; %! x = [1 2 4 5 6]'; y = [2 3 6 5 6]'; xi = 3; yi = 129/26; pp = csape (x, y, cond); %!assert (ppval(pp, x), y, 10*eps); %!assert (ppval(pp, xi), yi, 10*eps); %!assert (ppval(ppder(pp), x(1)), ppval(ppder(pp), x(end)), 10*eps); %!assert (ppval(ppder(pp, 2), x(1)), ppval(ppder(pp, 2), x(end)), 10*eps); %! x = [1 2 4 6]'; y = [2 3 4 2]'; xi = 3; yi = 4 + 1/64; pp = csape (x, y, cond); %!assert (ppval(pp, x), y, 10*eps); %!assert (ppval(pp, xi), yi, 10*eps); %!assert (ppval(ppder(pp), x(1)), ppval(ppder(pp), x(end)), 10*eps); %!assert (ppval(ppder(pp, 2), x(1)), ppval(ppder(pp, 2), x(end)), 10*eps); %! x = [1 2 4 5]'; y = [2 3 6 5]'; xi = 3; yi = 5.1; pp = csape (x, y, cond); %!assert (ppval(pp, x), y, 10*eps); %!assert (ppval(pp, xi), yi, 10*eps); %!assert (ppval(ppder(pp), x(1)), ppval(ppder(pp), x(end)), 10*eps); %!assert (ppval(ppder(pp, 2), x(1)), ppval(ppder(pp, 2), x(end)), 10*eps); %! x = [1 2 4]'; y = [2 3 2]'; xi = 3; yi = 2.5; pp = csape (x, y, cond); %!assert (ppval(pp, x), y, 10*eps); %!assert (ppval(pp, xi), yi, 10*eps); %!assert (ppval(ppder(pp), x(1)), ppval(ppder(pp), x(end)), 10*eps); %!assert (ppval(ppder(pp, 2), x(1)), ppval(ppder(pp, 2), x(end)), 10*eps); splines/inst/csaps.m0000644000175000017500000002010514425217567013154 0ustar nirnir## Copyright (C) 2012-2021 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 . ## -*- texinfo -*- ## @deftypefn{Function File}{[@var{yi} @var{p} @var{sigma2} @var{unc_yi} @var{df}] =} csaps(@var{x}, @var{y}, @var{p}=[], @var{xi}=[], @var{w}=[]) ## @deftypefnx{Function File}{[@var{pp} @var{p} @var{sigma2}] =} csaps(@var{x}, @var{y}, @var{p}=[], [], @var{w}=[]) ## ## Cubic spline approximation (smoothing)@* ## approximate [@var{x},@var{y}], weighted by @var{w} (inverse variance of the @var{y} values; 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} ## ## Outside the range of @var{x}, the cubic spline is a straight line ## ## @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 and in ascending order; the values in @var{w} should be nonzero ## ## @var{p} should be a scalar or empty:@* ## @table @asis ## @item @var{p}=0 ## maximum smoothing: straight line ## @item @var{p}=1 ## no smoothing: interpolation ## @item @var{p}<0 or empty ## an intermediate amount of smoothing is chosen @* ## and the corresponding @var{p} between 0 and 1 is returned @* ## (such that the smoothing term and the interpolation term are of the same magnitude) @* ## (csaps_sel provides other methods for automatically selecting the smoothing parameter @var{p}.) ## @end table ## ## @var{sigma2} is an estimate of the data error variance, assuming the smoothing parameter @var{p} is realistic ## ## @var{unc_yi} is an estimate of the standard error of the fitted curve(s) at the @var{xi}. ## Empty if @var{xi} is not provided. ## ## @var{df} is an estimate of the degrees of freedom used in the spline fit (2 for @var{p}=0, n for @var{p}=1) ## ## ## References: @* ## Carl de Boor (1978), A Practical Guide to Splines, Springer, Chapter XIV @* ## Grace Wahba (1983), Bayesian ``confidence intervals'' for the cross-validated smoothing spline, Journal of the Royal Statistical Society, 45B(1):133-150 ## ## @end deftypefn ## @seealso{spline, splinefit, csapi, ppval, dedup, bin_values, csaps_sel} ## Author: Nir Krakauer function [ret,p,sigma2,unc_yi,df]=csaps(x,y,p=[],xi=[],w=[]) if !(isscalar(p) || isempty(p)) error('p should be a scalar or empty') endif if(columns(x) > 1) x = x'; y = y'; w = w'; endif if any (isnan ([x y w](:)) ) error('NaN values in inputs; pre-process to remove them') endif h = diff(x); if !all(h > 0) && !all(h < 0) error('x must be strictly monotone; pre-process to achieve this') endif [n m] = size(y); #should also be that n = numel(x); if isempty(w) w = ones(n, 1); end R = spdiags([h(2:end) 2*(h(1:end-1) + h(2:end)) h(1:end-1)], [-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(p*R) if isempty(p) || (p < 0) r = full(6*trace(QT*diag(1 ./ w)*QT') / trace(R)); p = r ./ (1 + r); endif ## solve for the scaled second derivatives u and for the function values a at the knots ## (if p = 1, a = y; if p = 0, cc(:) = dd(:) = 0) ## QT*y can also be written as (y(3:n, :) - y(2:(n-1), :)) ./ h(2:end) - (y(2:(n-1), :) - y(1:(n-2), :)) ./ h(1:(end-1)) u = (6*(1-p)*QT*diag(1 ./ w)*QT' + p*R) \ (QT*y); a = y - 6*(1-p)*diag(1 ./ w)*QT'*u; ## derivatives for the piecewise cubic spline aa = bb = cc = dd = zeros (n+1, m); aa(2:end, :) = a; cc(3:n, :) = 6*p*u; #second derivative at endpoints is 0 [natural spline] dd(2:n, :) = diff(cc(2:(n+1), :)) ./ h; bb(2:n, :) = diff(a) ./ h - (h/3) .* (cc(2:n, :) + cc(3:(n+1), :)/2); ## add knots to either end of spline pp-form to ensure linear extrapolation dx_minus = eps(x(1)); dx_plus = eps(x(end)); xminus = x(1) - dx_minus; xplus = x(end) + dx_plus; x = [xminus; x; xplus]; slope_minus = bb(2, :); slope_plus = bb(n, :) + cc(n, :)*h(n-1) + (dd(n, :)/2)*h(n-1)^2; bb(1, :) = slope_minus; #linear extension of splines bb(n + 1, :) = slope_plus; aa(1, :) = a(1, :) - dx_minus*bb(1, :); ret = mkpp (x, cat (2, dd'(:)/6, cc'(:)/2, bb'(:), aa'(:)), m); clear a aa bb cc dd slope_minus slope_plus u #these values are no longer needed if ~isempty (xi) ret = ppval (ret, xi); endif if (isargout (4) && isempty (xi)) unc_yi = []; endif if isargout (3) || (isargout (4) && ~isempty (xi)) || isargout (5) if p == 1 #interpolation assumes no error in the given data sigma2 = 0; if isargout (4) && ~isempty (xi) unc_yi = zeros(numel(xi), 1); endif df = n; return endif [U D V] = svd (diag(1 ./ sqrt(w))*QT'*sqrtm(inv(R)), 0); D = diag(D).^2; #influence matrix for given p A = speye(n) - U * diag(D ./ (D + (p / (6*(1-p))))) * U'; A = diag (1 ./ sqrt(w)) * A * diag(sqrt(w)); #rescale to original units; a = A*y MSR = mean (w .* (y - (A*y)) .^ 2); #mean square residual df = trace (A); sigma2 = mean (MSR(:)) * (n / (n-df)); #estimated data error variance (wahba83) if isargout (4) && ~isempty (xi) ni = numel (xi); #dependence of spline values on each given point (to compute uncertainty) C = 6 * p * full ((6*(1-p)*QT*diag(1 ./ w)*QT' + p*R) \ QT); #cc(3:n, :) = C*y [sparsity is lost] D = diff ([zeros(n, 1) C' zeros(n, 1)]') ./ h; #dd(2:n, :) = D*y B = diff (A) ./ h - (h/3) .* ([zeros(n, 1) C']' + [C' zeros(n, 1)]' / 2); #bb(2:n, :) = B*y #add end-points C = [zeros(n, 2) C' zeros(n, 1)]'; D = [zeros(n, 1) D' zeros(n, 1)]'; B = [B(1, :)' B' B(end, :)' + C(n, :)'*h(n-1) + (D(n, :)'/2)*h(n-1)^2]'; A = [A(1, :)'-eps(x(1))*B(1, :)' A']'; #sum the squared dependence on each data value y at each requested point xi unc_yi = zeros (ni, 1); for i = 1:n unc_yi += (ppval (mkpp (x, cat (2, D(:, i)/6, C(:, i)/2, B(:, i), A(:, i))), xi(:))) .^ 2; endfor unc_yi = sqrt (sigma2 * unc_yi); #not exactly the same as unc_y as calculated in csaps_sel even if xi = x, but fairly close endif endif endfunction %!shared x,y,xi,yi,p,sigma2,unc_yi,df %! x = ([1:10 10.5 11.3])'; y = sin(x); xi = linspace(min(x), max(x), 30)'; %!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); %! [yi,p,sigma2,unc_yi,df] = csaps(x,y,1,xi); %!assert (yi, ppval(csape(x, y, "variational"), xi), eps); %!assert (p, 1); %!assert (unc_yi, zeros(size(xi))); %!assert (sigma2, 0); %!assert (df, numel(x)); %! [yi,p,~,~,df] = csaps(x,y,0,xi); %!assert (yi, polyval(polyfit(x, y, 1), xi), 10*eps); %!assert (p, 0); %!assert (df, 2, 100*eps); %{ test weighted smoothing: n = 500; a = 0; b = pi; f = @(x) sin(x); x = a + (b-a)*sort(rand(n, 1)); w = rand(n, 1); y = f(x) + randn(n, 1) ./ sqrt(w); xi = linspace(a, b, n)'; yi_target = f(xi); [~,p_sel] = csaps_sel(x, y, xi, w, 1); [yi,~,sigma2,unc_yi] = csaps(x,y,p_sel,xi,w); rmse = rms((yi - yi_target)); rmse_weighted = rms((yi - yi_target) ./ unc_yi); #worse results without the (correct) weighting: [~,p_sel] = csaps_sel(x, y, xi, []); [yi,~,sigma2,unc_yi] = csaps(x,y,p_sel,xi,[]); rmse_u = rms((yi - yi_target)); rmse_u_weighted = rms((yi - yi_target) ./ unc_yi); %} splines/inst/tps_val.m0000644000175000017500000000614714425217567013525 0ustar nirnir## Copyright (C) 2013 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 . ## -*- texinfo -*- ## @deftypefn{Function File}{[@var{yi}] =} tps_val(@var{x}, @var{coefs}, @var{xi}, @var{vectorize}=true) ## ## Evaluates a thin plate spline at given points @* ## @var{xi} ## ## @var{coefs} should be the vector of fitted coefficients returned from @code{tpaps(x, y, [p])} ## ## @var{x} should be @var{n} by @var{d} in size, where @var{n} is the number of points and @var{d} the number of dimensions; @var{coefs} should be @var{n} + @var{d} + 1 by 1; @var{xi} should be @var{k} by @var{d} ## ## The logical argument @var{vectorize} controls whether an @var{k} by @var{n} by @var{d} intermediate array is formed to speed up computation (the default) or whether looping is used to economize on memory ## ## The returned @var{yi} will be @var{k} by 1 ## ## See the documentation to @code{tpaps} for more information ## ## @end deftypefn ## @seealso{tpaps, tps_val_der} ## Author: Nir Krakauer function [yi]=tps_val(x,coefs,xi,vectorize=true) [n d] = size(x); #d: number of dimensions; n: number of points k = size(xi, 1); #number of points for which to find the spline function value #form of the Green's function for solutions G = @(r) merge(r == 0, 0, r .^ 2 .* log(r)); a = coefs(1:n); b = coefs((n+1):end); yi = [ones(k, 1) xi] * b; if vectorize if d == 1 yi = yi + G(abs(x' - xi)) * a; else yi = yi + G(sqrt(sumsq((reshape(x, 1, n, d) - reshape(xi, k, 1, d)), 3))) * a; endif else dist = @(x1, x2) norm(x2 - x1, 2, "rows"); #Euclidian distance between points in d-dimensional space warn_state = warning ("query", "Octave:broadcast").state; warning ("off", "Octave:broadcast"); #turn off warning message for automatic broadcasting when dist is called unwind_protect if k > n ##choose from either of two ways of computing the values of the thin plate spline at xi for i = 1:n yi = yi + a(i)*G(dist(x(i, :), xi)); endfor else for i = 1:k yi(i) = yi(i) + dot(a, G(dist(x, xi(i, :)))); endfor endif unwind_protect_cleanup warning (warn_state, "Octave:broadcast"); end_unwind_protect endif endfunction %!shared x,y,c,xi %! x = ([1:10 10.5 11.3])'; y = sin(x); %! c = tpaps(x,y,1); %!assert (tpaps(x,y,1,x), tps_val(x,c,x), 100*eps); %! x = ((1 ./ (1:100))' - 0.5) * ([0.2 0.6]); %! y = x(:, 1) .^ 2 + x(:, 2) .^ 2; %! c = tpaps(x,y,1); %!assert (tpaps(x,y,1,x), tps_val(x,c,x), 100*eps); %!assert (tps_val(x,c,x,true), tps_val(x,c,x,false), 100*eps); splines/inst/regularization2D.m0000644000175000017500000001436114425217567015277 0ustar nirnir## Copyright (C) 2021 Andreas Stahel ## ## 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 ## . ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{grid},@var{u},@var{data_valid}] =} regularization2D (@var{data}, @var{box}, @var{N}, @var{lambda1},@var{lambda2}) ## ## Apply a Tikhonov regularization, the functional to be minimized is@* ## @var{F} = @var{FD} + @var{lambda1} * @var{F1} + @var{lambda2} * @var{F2} @* ## = sum_(i=1)^M (y_i-u(x_i))^2 +@* ## + @var{lambda1} * dintegral (du/dx)^2+(du/dy)^2 dA +@* ## + @var{lambda2} * dintegral (d^2u/dx^2)^2+(d^2u/dy^2)^2+2*(d^2u/dxdy) dA ## ## With @var{lambda1} = 0 and @var{lambda2}>0 this leads to a thin plate smoothing spline. ## ## Parameters: ## @itemize ## @item @var{data} is a M*3 matrix with the (x,y) values in the first two columns and the y values in the third column.@* ## Only data points strictly inside the @var{box} are used ## @item @var{box} = [x0,x1;y0,y1] is the rectangle x0= 0 is the value of the first regularization parameter ## @item @var{lambda2} > 0 is the value of the secondregularization parameter ## @end itemize ## ## Return values: ## @itemize ## @item @var{grid} is the grid on which @var{u} is evaluated. It consists of ## (@var{N1+1})x(@var{N2}+1) equidistant points on the the rectangle @var{box}. ## @item @var{u} are the values of the regularized approximation to the @var{data} evaluated on the @var{grid}. ## @item @var{data_valid} returns the values data points used and the values of the regularized function at these points ## @end itemize ## @seealso {tpaps, regularization, demo regularization2D} ## @end deftypefn ## Author: Andreas Stahel ## Created: 2021-01-13 function [grid,u,data_valid] = regularization2D (data, box, N, lambda1, lambda2) %% generate the grid N = N+1; %%% now N is the number of grid points in either direction x = linspace(box(1,1),box(1,2),N(1)); y = linspace(box(2,1),box(2,2),N(2)); [xx,yy] = meshgrid(x,y); dx = diff(box(1,:))/(N(1)-1); dy = diff(box(2,:))/(N(2)-1); grid.x = xx; grid.y = yy; x = data (:,1); y = data(:,2); z = data(:,3); ## select points in box only ind = find ((x > box(1,1)) .* (x < box(1,2)) .* (y > box(2,1)) .* (y < box(2,2))); x = x (ind); y = y (ind); z = z (ind); %% generate the sparse interpolation matrix M = length (x); x_ind = floor((x-box(1,1))/dx); %% xi = mod(x,dx)/dx; xi = (x-box(1,1)-x_ind*dx)/dx; y_ind = floor((y-box(2,1))/dy); %%nu = mod(y,dy)/dy; nu = (y-box(2,1)-y_ind*dy)/dy; row = ones(4,1)*[1:M]; index_base = N(2)*x_ind+y_ind+1; index = index_base + [0,N(2),1,N(2)+1]; index = index'; coeff = [(1-xi).*(1-nu),xi.*(1-nu),(1-xi).*nu,xi.*nu]; coeff = coeff'; Interp = sparse(row(:),index(:),coeff(:),M,N(1)*N(2)); mat = Interp' * Interp; rhs = (Interp' * z); %%% derivative with respect to x Dx = kron(spdiags(ones(N(1),1)*[-1 1],[0 1],N(1)-1,N(1))/dx,speye(N(2))); Wx = ones(N(2),1); Wx(1) = 1/2; Wx(N(2)) = 1/2; Wx = kron(speye(N(1)-1),diag(Wx))*dx*dy; %%% derivative with respect to y Wy = ones(N(1),1); Wy(1) = 1/2; Wy(N(1)) = 1/2; Wy = kron(diag(Wy),speye(N(2)-1))*dx*dy; if lambda1 > 0 Dy = kron(speye(N(1)),spdiags(ones(N(2),1)*[-1 1],[0 1],N(2)-1,N(2))/dy); mat += lambda1*(Dx'*Wx*Dx + Dy'*Wy*Dy); endif %%% second derivative with respect to x Dxx = spdiags(ones(N(1),1)*[1 -1 -1 1],[-1 0 1 2],N(1)-1,N(1)); Dxx(1,1:4) = [3 -7 5 -1]; Dxx(N(1)-1,N(1)-3:N(1)) = [-1 5 -7 3]; Dxx = Dxx/(2*dx^2); Dxx = kron(Dxx,speye(N(2))); %%% second derivative with respect to y Dyy = spdiags(ones(N(2),1)*[1 -1 -1 1],[-1 0 1 2],N(2)-1,N(2)); Dyy(1,1:4) = [3 -7 5 -1]; Dyy(N(2)-1,N(2)-3:N(2)) = [-1 5 -7 3]; Dyy = Dyy/(2*dy^2); Dyy = kron(speye(N(1)),Dyy); %%% mixed second derivative Dy2 = kron(speye(N(1)-1),spdiags(ones(N(2),1)*[-1 1],[0 1],N(2)-1,N(2))/dy); Dxy = Dy2*Dx*sqrt(dx*dy); mat += lambda2*(Dxx'*Wx*Dxx + Dyy'*Wy*Dyy + 2*Dxy'*Dxy); %%% solve u = reshape(mat\rhs,N(2),N(1)); if nargout>2 data_valid =[x,y,Interp*u(:)]; endif endfunction %!demo %! M = 100; %! lambda1 = 0; lambda2 = 0.05; %! x = 2*rand(M,1)-1; y = 2*rand(M,1)-1; %! z = x.*y + 0.1*randn(M,1); %! data = [x,y,z]; %! [grid,u] = regularization2D(data,[-1 1;-1 1],[50 50],lambda1,lambda2); %! figure() %! mesh(grid.x, grid.y,u) %! xlabel('x'); ylabel('y'); %! hold on %! plot3(data(:,1),data(:,2),data(:,3),'*b','Markersize',2) %! hold off %! view([30,30]); %!demo %! lambda1 = 0; lambda2 = 0.01; %! M = 4; angles = [1:M]/M*2*pi; %! data = zeros(M+1,3); data(M+1,3) = 1; %! data(1:M,1) = cos(angles); data(1:M,2) = sin(angles); %! [grid,u] = regularization2D(data,[-1.5 1.5;-1.5 1.5],[50 50],lambda1,lambda2); %! figure() %! mesh(grid.x, grid.y,u) %! xlabel('x'); ylabel('y'); %! hold on %! plot3(data(:,1),data(:,2),data(:,3),'*b','Markersize',2) %! hold off %!test %! data = [0,0,0;1,2,3;2,0,2;0,2,2]; % data on z = x+y %! lambda1 = 0.0 ; lambda2 = 0.1; %! [grid,u,u_valid] = regularization2D(data,[-0.2 2.2; -0.2,2.2],[11,12],lambda1,lambda2); %! assert(norm(data-u_valid),0,1e-12) %! assert(norm(grid.x(:) + grid.y(:) - u(:)),0,1e-10) %!test %! data = [0,0,0;1,1,3;2,1,2;0,2,-2;-1.5,-1,0;-2,2,0]; %! lambda1 = 0.001 ; lambda2 = 0.01; %! [grid,u,u_valid] = regularization2D(data,[-2.1 2.1;-2.1,2.1],[40,50],lambda1,lambda2); %! value_at_data = [ 1.233351091378741e-01 %! 2.694454049778803e+00 %! 2.102786670571836e+00 %! -1.870655633783656e+00 %! -1.635243922246946e-02 %! -3.356775648311422e-02]; %! assert(norm(value_at_data-u_valid(:,3)),0,1e-12) splines/inst/tps_val_der.m0000644000175000017500000001014514425217567014350 0ustar nirnir## Copyright (C) 2015 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 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 . ## -*- texinfo -*- ## @deftypefn{Function File}{[@var{dyi}] =} tps_val_der(@var{x}, @var{coefs}, @var{xi}, @var{vectorize}=true ) ## ## Evaluates the first derivative of a thin plate spline at given points @* ## @var{xi} ## ## @var{coefs} should be the vector of fitted coefficients returned from @code{tpaps(x, y, [p])} ## ## @var{x} should be @var{n} by @var{d} in size, where @var{n} is the number of points and @var{d} the number of dimensions; @var{coefs} should be (@var{n} + @var{d} + 1) by 1; @var{xi} should be @var{k} by @var{d} ## ## The logical argument @var{vectorize} controls whether @var{k} by @var{n} by @var{d} intermediate arrays are formed to speed up computation (the default) or whether looping is used to economize on memory ## ## The returned @var{dyi} will be @var{k} by @var{d}, containing the first partial derivatives of the thin plate spline at @var{xi} ## ## ## Example usages: ## @example ## x = ([1:10 10.5 11.3])'; y = sin(x); dy = cos(x); xi = (0:0.1:12)'; ## coefs = tpaps(x, y, 0.5); ## [dyi] = tps_val_der(x,coefs,xi); ## subplot(1, 1, 1) ## plot(x, dy, 's', xi, dyi) ## legend('original', 'tps') ## @end example ## ## @example ## x = rand(100, 2)*2 - 1; ## y = x(:, 1) .^ 2 + x(:, 2) .^ 2; ## [x1 y1] = meshgrid((-1:0.2:1)', (-1:0.2:1)'); ## xi = [x1(:) y1(:)]; ## coefs = tpaps(x, y, 1); ## dyio = [2*x1(:) 2*y1(:)]; ## [dyi] = tps_val_der(x,coefs,xi); ## subplot(2, 2, 1) ## contourf(x1, y1, reshape(dyio(:, 1), 11, 11)); colorbar ## title('original x1 partial derivative') ## subplot(2, 2, 2) ## contourf(x1, y1, reshape(dyi(:, 1), 11, 11)); colorbar ## title('tps x1 partial derivative') ## subplot(2, 2, 3) ## contourf(x1, y1, reshape(dyio(:, 2), 11, 11)); colorbar ## title('original x2 partial derivative') ## subplot(2, 2, 4) ## contourf(x1, y1, reshape(dyi(:, 2), 11, 11)); colorbar ## title('tps x2 partial derivative') ## @end example ## ## See the documentation to @code{tpaps} for more information ## ## @end deftypefn ## @seealso{tpaps, tpaps_val, tps_val_der} ## Author: Nir Krakauer function [dyi]=tps_val_der(x,coefs,xi,vectorize=true) [n d] = size(x); #d: number of dimensions; n: number of points k = size(xi, 1); #number of points for which to find the spline derivative value #derivative of the spline Green's function, divided by radial distance dG_scaled = @(r) merge(r == 0, 0, 1 + 2 .* log(r)); a = coefs(1:n); b = coefs((n+1):end); dyi = ones(k, 1) * b(2:end)'; #derivatives of linear part if vectorize dists = reshape(xi, k, 1, d) - reshape(x, 1, n, d); dist = sqrt(sumsq(dists, 3)); #Euclidian distance between points in d-dimensional space dyi += squeeze(sum(reshape(a, 1, n) .* dG_scaled(dist) .* dists, 2)); else for i = 1:k for l = 1:n dists = xi(i, :) - x(l, :); dist = sqrt(sumsq(dists)); dyi(i, :) += a(l) * dG_scaled(dist) * dists; endfor endfor endif endfunction #check with linear functions (derivatives should be constant) %!shared a,b,x,y,x1,x2,y1,c,dy,dy0 %! a = 2; b = -3; x = ([1:2:10 10.5 11.3])'; y = a*x; %! c = tpaps(x,y,1); %!assert (a*ones(size(x)), tps_val_der(x,c,x), 1E3*eps); %! [x1 x2] = meshgrid(x, x); %! y1 = a*x1+b*x2; %! c = tpaps([x1(:) x2(:)],y1(:),0.5); %! dy = tps_val_der([x1(:) x2(:)],c,[x1(:) x2(:)]); %! dy0 = tps_val_der([x1(:) x2(:)],c,[x1(:) x2(:)],false); %!assert (a*ones(size(x1(:))), dy(:, 1), 1E3*eps); %!assert (b*ones(size(x2(:))), dy(:, 2), 1E3*eps); %!assert (dy0, dy, 1E3*eps); splines/inst/dedup.m0000644000175000017500000000531114425217567013146 0ustar nirnir## Copyright (C) 2013 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 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 . ## -*- texinfo -*- ## @deftypefn{Function File}{[@var{x_new} @var{y_new} @var{w_new}] =} dedup(@var{x}, @var{y}, @var{w}, @var{tol}, @var{nan_remove}=true) ## ## De-duplication and sorting to facilitate spline smoothing@* ## Points are sorted in ascending order of @var{x}, with each set of duplicates (values with the same @var{x}, within @var{tol}) replaced by a weighted average. ## Any NaN values are removed (if the flag @var{nan_remove} is set). ## ## Useful, for example, as a preprocessor to spline smoothing ## ## Inputs:@* ## @var{x}: @var{n}*1 real array@* ## @var{y}: @var{n}*@var{m} array of values at the coordinates @var{x}@* ## @var{w}: @var{n}*1 array of positive weights (inverse error variances); @code{ones(size(x))} by default@* ## @var{tol}: if the difference between two @var{x} values is no more than this scalar, merge them; 0 by default ## ## Outputs: ## De-duplicated and sorted @var{x}, @var{y}, @var{w} ## @end deftypefn ## @seealso{csaps, bin_values} ## Author: Nir Krakauer function [x, y, w] = dedup(x, y, w=ones(size(x)), tol=0, nan_remove=true) warning ("off", "Octave:broadcast", "local"); if isempty(w) w = ones(size(x)); endif if isempty(tol) tol = 0; endif if nan_remove #remove any rows with missing entries notnans = !any (isnan ([x y w]) , 2); x = x(notnans); y = y(notnans, :); w = w(notnans); endif [x,i] = sort(x); y = y(i, :); w = w(i); h = diff(x); if any(h <= tol) hh = ones(size(x)); hh(2:end) = cumsum(h > tol) + 1; #any elements tol or less apart are placed in the same equivalence class #replace original points with equivalence classes, using weighted averages wnew = accumarray(hh, w); x = accumarray(hh, x .* w) ./ wnew; y = accumdim(hh, y .* w, 1) ./ wnew; w = wnew; endif %!shared x, y, w %! x = [1; 2; 2; 3; 4]; %! y = [0 0; 1 1; 2 1; 3 4; 5 NaN]; %! w = [1; 0.1; 1; 0.5; 1]; %!assert (nthargout(1:3, @dedup, x, y, ones(size(x))), nthargout(1:3, @dedup, x, y)) %! [x y w] = dedup(x, y, w); %!assert (x, [1; 2; 3]); %!assert (y, [0 0; 21/11 1; 3 4], 10*eps); %!assert (w, [1; 1.1; 0.5]); splines/inst/catmullrom.m0000644000175000017500000000355014425217567014227 0ustar nirnir## Copyright (C) 2008 Carlo de Falco ## ## 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 . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{pp}} = catmullrom( @var{x},@ ## @var{f}, @var{v}) ## ## Returns the piecewise polynomial form of the Catmull-Rom cubic ## spline interpolating @var{f} at the points @var{x}. ## If the input @var{v} is supplied it will be interpreted as the ## values of the tangents at the extremals, if it is ## missing, the values will be computed from the data via one-sided ## finite difference formulas. See the wikipedia page for "Cubic ## Hermite spline" for a description of the algorithm. ## ## @seealso{ppval} ## @end deftypefn function pp = catmullrom(x,f,v) if ( nargin < 2 ) print_usage(); endif h00 = [2 -3 0 1]; h10 = [1 -2 1 0]; h01 = [-2 3 0 0]; h11 = [1 -1 0 0]; h = diff(x(:)'); p0 = f(:)'(1:end-1); p1 = f(:)'(2:end); if (nargin < 3) v(1) = (p1(1)-p0(1))./h(1); v(2) = (p1(end)-p0(end))./h(end); endif m = (p1(2:end)-p0(1:end-1))./(h(2:end)+h(1:end-1)); m0 = [v(1) m]; m1 = [m v(2)]; for ii = 1:4 coeff(:,ii) = ((h00(ii)*p0 + h10(ii)*h.*m0 +... h01(ii)*p1 + h11(ii)*h.*m1 )./h.^(4-ii))' ; end pp = mkpp (x, coeff); endfunction splines/inst/csapi.m0000644000175000017500000000215114425217567013143 0ustar nirnir## Copyright (C) 2000 Kai Habel ## ## 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 . ## -*- texinfo -*- ## @deftypefn {Function File} {@var{pp} = } csapi (@var{x}, @var{y}) ## @deftypefnx {Function File} {@var{yi} = } csapi (@var{x}, @var{y}, @var{xi}) ## cubic spline interpolation ## ## @seealso{ppval, spline, csape} ## @end deftypefn ## Author: Kai Habel ## Date: 3. dec 2000 function ret = csapi (x, y, xi) ret = csape(x,y,'not-a-knot'); if (nargin == 3) ret = ppval(ret,xi); endif endfunction splines/inst/bin_values.m0000644000175000017500000000627414425217567014205 0ustar nirnir## Copyright (C) 2011-2013 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 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 . # -*- texinfo -*- ## @deftypefn{Function File}{[@var{x_bin} @var{y_bin} @var{w_bin} @var{n_bin}] =} bin_values(@var{x}, @var{y}, @var{k}) ## ## Average values over ranges of one variable@* ## Given @var{x} (size @var{n}*1) and @var{y} (@var{n}*@var{m}), this function splits the range of @var{x} into up to @var{k} intervals (bins) containing approximately equal numbers of elements, and for each part of the range computes the mean of y. ## ## Any NaN values are removed. ## ## Useful for detecting possible nonlinear dependence of @var{y} on @var{x} and as a preprocessor for spline fitting. ## E.g., to make a plot of the average behavior of y versus x: @code{errorbar(x_bin, y_bin, 1 ./ sqrt(w_bin)); grid on} ## ## Inputs:@* ## @var{x}: @var{n}*1 real array@* ## @var{y}: @var{n}*@var{m} array of values at the coordinates @var{x}@* ## @var{k}: Desired number of bins, @code{floor(sqrt(n))} by default ## ## Outputs:@* ## @var{x_bin}, @var{y_bin}: Mean values by bin (ordered by increasing @var{x}) @* ## @var{w_bin}: Weights (inverse standard error of each element in @var{y_bin}; note: will be NaN or Inf where @var{n_bin} = 1)@* ## @var{n_bin}: Number of elements of @var{x} per bin ## @end deftypefn ## @seealso{csaps, dedup} ## Author: Nir Krakauer function [x_bin y_bin w_bin n_bin] = bin_values(x, y, k=[]) #remove any rows with missing entries notnans = !any (isnan ([x y]) , 2); x = x(notnans); y = y(notnans, :); [n, m] = size(y); #x should be n by 1 if isempty(k) k = floor(sqrt(n)); #reasonable default end if k <= 1 #only a single bin x_bin_mean = mean(x); y_bin_mean = mean(y); w = n / var(y_bin_mean); n_bin = n; return end #arrange values in increasing order of x [x, i] = sort (x); y = y(i, :); #decide where to separate bins bound_inds = 1 + (n-1)*(1:(k-1))/k; bound_x = unique(interp1((1:n)', x, bound_inds)); #assign each point an index corresponding to its bin idx = lookup(bound_x, x); #get number of elements in each bin [ids, ~, j] = unique(idx); #k = numel(ids); #calculate the desired outputs n_bin = accumarray(j, 1); x_bin = accumarray(j, x, [], @mean); y_bin = accumdim(j, y, 1, [], @mean); f = @(x, dim) var(x, [], dim); warning ("off", "Octave:broadcast", "local"); w_bin = n_bin ./ accumdim(j, y, 1, [], f); %!shared x, y, x_bin, y_bin, w_bin, n_bin %! x = [1; 2; 2; 3; 4]; %! y = [0 0; 1 1; 2 1; 3 4; 5 NaN]; %! [x_bin y_bin w_bin n_bin] = bin_values(x, y); %!assert (x_bin, [1; 7/3]); %!assert (y_bin, [0 0; 2 2]); %!assert (!any(isfinite(w_bin(1, :)))); %!assert (w_bin(2, :), [3 1]); %!assert (n_bin, [1; 3]); splines/DESCRIPTION0000644000175000017500000000045014425217567012417 0ustar nirnirName: splines Version: 1.3.5 Date: 2023-05-05 Author: various authors Maintainer: Nir Krakauer Title: Splines. Description: Additional spline functions. Categories: Splines Depends: octave (>= 3.6.0) Autoload: no License: GPLv3+, public domain Url: http://octave.sf.net