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