38.7 Examples

The following program computes a linear least squares fit to data using cubic B-spline basis functions with uniform breakpoints. The data is generated from the curve y(x) = \cos(x) \exp(-x/10) on the interval [0, 15] with gaussian noise added.

     #include <stdio.h>
     #include <stdlib.h>
     #include <math.h>
     #include <gsl/gsl_bspline.h>
     #include <gsl/gsl_multifit.h>
     #include <gsl/gsl_rng.h>
     #include <gsl/gsl_randist.h>
     #include <gsl/gsl_statistics.h>
     
     /* number of data points to fit */
     #define N        200
     
     /* number of fit coefficients */
     #define NCOEFFS  12
     
     /* nbreak = ncoeffs + 2 - k = ncoeffs - 2 since k = 4 */
     #define NBREAK   (NCOEFFS - 2)
     
     int
     main (void)
     {
       const size_t n = N;
       const size_t ncoeffs = NCOEFFS;
       const size_t nbreak = NBREAK;
       size_t i, j;
       gsl_bspline_workspace *bw;
       gsl_vector *B;
       double dy;
       gsl_rng *r;
       gsl_vector *c, *w;
       gsl_vector *x, *y;
       gsl_matrix *X, *cov;
       gsl_multifit_linear_workspace *mw;
       double chisq, Rsq, dof, tss;
     
       gsl_rng_env_setup();
       r = gsl_rng_alloc(gsl_rng_default);
     
       /* allocate a cubic bspline workspace (k = 4) */
       bw = gsl_bspline_alloc(4, nbreak);
       B = gsl_vector_alloc(ncoeffs);
     
       x = gsl_vector_alloc(n);
       y = gsl_vector_alloc(n);
       X = gsl_matrix_alloc(n, ncoeffs);
       c = gsl_vector_alloc(ncoeffs);
       w = gsl_vector_alloc(n);
       cov = gsl_matrix_alloc(ncoeffs, ncoeffs);
       mw = gsl_multifit_linear_alloc(n, ncoeffs);
     
       printf("#m=0,S=0\n");
       /* this is the data to be fitted */
       for (i = 0; i < n; ++i)
         {
           double sigma;
           double xi = (15.0 / (N - 1)) * i;
           double yi = cos(xi) * exp(-0.1 * xi);
     
           sigma = 0.1 * yi;
           dy = gsl_ran_gaussian(r, sigma);
           yi += dy;
     
           gsl_vector_set(x, i, xi);
           gsl_vector_set(y, i, yi);
           gsl_vector_set(w, i, 1.0 / (sigma * sigma));
     
           printf("%f %f\n", xi, yi);
         }
     
       /* use uniform breakpoints on [0, 15] */
       gsl_bspline_knots_uniform(0.0, 15.0, bw);
     
       /* construct the fit matrix X */
       for (i = 0; i < n; ++i)
         {
           double xi = gsl_vector_get(x, i);
     
           /* compute B_j(xi) for all j */
           gsl_bspline_eval(xi, B, bw);
     
           /* fill in row i of X */
           for (j = 0; j < ncoeffs; ++j)
             {
               double Bj = gsl_vector_get(B, j);
               gsl_matrix_set(X, i, j, Bj);
             }
         }
     
       /* do the fit */
       gsl_multifit_wlinear(X, w, y, c, cov, &chisq, mw);
     
       dof = n - ncoeffs;
       tss = gsl_stats_wtss(w->data, 1, y->data, 1, y->size);
       Rsq = 1.0 - chisq / tss;
     
       fprintf(stderr, "chisq/dof = %e, Rsq = %f\n", 
                        chisq / dof, Rsq);
     
       /* output the smoothed curve */
       {
         double xi, yi, yerr;
     
         printf("#m=1,S=0\n");
         for (xi = 0.0; xi < 15.0; xi += 0.1)
           {
             gsl_bspline_eval(xi, B, bw);
             gsl_multifit_linear_est(B, c, cov, &yi, &yerr);
             printf("%f %f\n", xi, yi);
           }
       }
     
       gsl_rng_free(r);
       gsl_bspline_free(bw);
       gsl_vector_free(B);
       gsl_vector_free(x);
       gsl_vector_free(y);
       gsl_matrix_free(X);
       gsl_vector_free(c);
       gsl_vector_free(w);
       gsl_matrix_free(cov);
       gsl_multifit_linear_free(mw);
     
       return 0;
     } /* main() */

The output can be plotted with gnu graph.

     $ ./a.out > bspline.dat
     chisq/dof = 1.118217e+00, Rsq = 0.989771
     $ graph -T ps -X x -Y y -x 0 15 -y -1 1.3 < bspline.dat > bspline.ps