#### 38.8.2 Multi-parameter Linear Regression Example

The following program performs a quadratic fit y = c_0 + c_1 x + c_2 x^2 to a weighted dataset using the generalised linear fitting function `gsl_multifit_wlinear`. The model matrix X for a quadratic fit is given by,

    X = [ 1   , x_0  , x_0^2 ;
          1   , x_1  , x_1^2 ;
          1   , x_2  , x_2^2 ;
          ... , ...  , ...   ]

where the column of ones corresponds to the constant term c_0. The two remaining columns corresponds to the terms c_1 x and c_2 x^2.

The program reads n lines of data in the format (x, y, err) where err is the error (standard deviation) in the value y.

    #include <stdio.h>
    #include <gsl/gsl_multifit.h>

    int
    main (int argc, char **argv)
    {
      int i, n;
      double xi, yi, ei, chisq;
      gsl_matrix *X, *cov;
      gsl_vector *y, *w, *c;

      if (argc != 2)
        {
          fprintf (stderr,"usage: fit n < data\n");
          exit (-1);
        }

      n = atoi (argv[1]);

      X = gsl_matrix_alloc (n, 3);
      y = gsl_vector_alloc (n);
      w = gsl_vector_alloc (n);

      c = gsl_vector_alloc (3);
      cov = gsl_matrix_alloc (3, 3);

      for (i = 0; i < n; i++)
        {
          int count = fscanf (stdin, "%lg %lg %lg",
                              &xi, &yi, &ei);

          if (count != 3)
            {
              fprintf (stderr, "error reading file\n");
              exit (-1);
            }

          printf ("%g %g +/- %g\n", xi, yi, ei);

          gsl_matrix_set (X, i, 0, 1.0);
          gsl_matrix_set (X, i, 1, xi);
          gsl_matrix_set (X, i, 2, xi*xi);

          gsl_vector_set (y, i, yi);
          gsl_vector_set (w, i, 1.0/(ei*ei));
        }

      {
        gsl_multifit_linear_workspace * work
          = gsl_multifit_linear_alloc (n, 3);
        gsl_multifit_wlinear (X, w, y, c, cov,
                              &chisq, work);
        gsl_multifit_linear_free (work);
      }

    #define C(i) (gsl_vector_get(c,(i)))
    #define COV(i,j) (gsl_matrix_get(cov,(i),(j)))

      {
        printf ("# best fit: Y = %g + %g X + %g X^2\n",
                C(0), C(1), C(2));

        printf ("# covariance matrix:\n");
        printf ("[ %+.5e, %+.5e, %+.5e  \n",
                   COV(0,0), COV(0,1), COV(0,2));
        printf ("  %+.5e, %+.5e, %+.5e  \n",
                   COV(1,0), COV(1,1), COV(1,2));
        printf ("  %+.5e, %+.5e, %+.5e ]\n",
                   COV(2,0), COV(2,1), COV(2,2));
        printf ("# chisq = %g\n", chisq);
      }

      gsl_matrix_free (X);
      gsl_vector_free (y);
      gsl_vector_free (w);
      gsl_vector_free (c);
      gsl_matrix_free (cov);

      return 0;
    }

A suitable set of data for fitting can be generated using the following program. It outputs a set of points with gaussian errors from the curve y = e^x in the region 0 < x < 2.

    #include <stdio.h>
    #include <math.h>
    #include <gsl/gsl_randist.h>

    int
    main (void)
    {
      double x;
      const gsl_rng_type * T;
      gsl_rng * r;

      gsl_rng_env_setup ();

      T = gsl_rng_default;
      r = gsl_rng_alloc (T);

      for (x = 0.1; x < 2; x+= 0.1)
        {
          double y0 = exp (x);
          double sigma = 0.1 * y0;
          double dy = gsl_ran_gaussian (r, sigma);

          printf ("%g %g %g\n", x, y0 + dy, sigma);
        }

      gsl_rng_free(r);

      return 0;
    }

The data can be prepared by running the resulting executable program,

    $ GSL_RNG_TYPE=mt19937_1999 ./generate > exp.dat
    $ more exp.dat
    0.1 0.97935 0.110517
    0.2 1.3359 0.12214
    0.3 1.52573 0.134986
    0.4 1.60318 0.149182
    0.5 1.81731 0.164872
    0.6 1.92475 0.182212
    ....

To fit the data use the previous program, with the number of data points given as the first argument. In this case there are 19 data points.

    $ ./fit 19 < exp.dat
    0.1 0.97935 +/- 0.110517
    0.2 1.3359 +/- 0.12214
    ...
    # best fit: Y = 1.02318 + 0.956201 X + 0.876796 X^2
    # covariance matrix:
    [ +1.25612e-02, -3.64387e-02, +1.94389e-02
      -3.64387e-02, +1.42339e-01, -8.48761e-02
      +1.94389e-02, -8.48761e-02, +5.60243e-02 ]
    # chisq = 23.0987

The parameters of the quadratic fit match the coefficients of the expansion of e^x, taking into account the errors on the parameters and the O(x^3) difference between the exponential and quadratic functions for the larger values of x. The errors on the parameters are given by the square-root of the corresponding diagonal elements of the covariance matrix. The chi-squared per degree of freedom is 1.4, indicating a reasonable fit to the data.