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39.7 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

Next: , Previous: Working with the Greville abscissae, Up: Basis Splines   [Index]