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<a name="Fitting-multi_002dparameter-linear-regression-example"></a>
<div class="header">
<p>
Next: <a href="Fitting-regularized-linear-regression-example-1.html#Fitting-regularized-linear-regression-example-1" accesskey="n" rel="next">Fitting regularized linear regression example 1</a>, Previous: <a href="Fitting-linear-regression-example.html#Fitting-linear-regression-example" accesskey="p" rel="previous">Fitting linear regression example</a>, Up: <a href="Fitting-Examples.html#Fitting-Examples" accesskey="u" rel="up">Fitting Examples</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Multi_002dparameter-Linear-Regression-Example"></a>
<h4 class="subsection">38.8.2 Multi-parameter Linear Regression Example</h4>

<p>The following program performs a quadratic fit <em>y = c_0 + c_1 x + c_2
x^2</em> to a weighted dataset using the generalised linear fitting function
<code>gsl_multifit_wlinear</code>.  The model matrix <em>X</em> for a quadratic
fit is given by,
</p>
<div class="example">
<pre class="example">X = [ 1   , x_0  , x_0^2 ;
      1   , x_1  , x_1^2 ;
      1   , x_2  , x_2^2 ;
      ... , ...  , ...   ]
</pre></div>

<p>where the column of ones corresponds to the constant term <em>c_0</em>.
The two remaining columns corresponds to the terms <em>c_1 x</em> and
<em>c_2 x^2</em>.
</p>
<p>The program reads <var>n</var> lines of data in the format (<var>x</var>, <var>y</var>,
<var>err</var>) where <var>err</var> is the error (standard deviation) in the
value <var>y</var>.
</p>
<div class="example">
<pre class="verbatim">#include &lt;stdio.h&gt;
#include &lt;gsl/gsl_multifit.h&gt;

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,&quot;usage: fit n &lt; data\n&quot;);
      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 &lt; n; i++)
    {
      int count = fscanf (stdin, &quot;%lg %lg %lg&quot;,
                          &amp;xi, &amp;yi, &amp;ei);

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

      printf (&quot;%g %g +/- %g\n&quot;, 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,
                          &amp;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 (&quot;# best fit: Y = %g + %g X + %g X^2\n&quot;, 
            C(0), C(1), C(2));

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

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

  return 0;
}
</pre></div>

<p>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
<em>y = e^x</em> in the region <em>0 &lt; x &lt; 2</em>.
</p>
<div class="example">
<pre class="verbatim">#include &lt;stdio.h&gt;
#include &lt;math.h&gt;
#include &lt;gsl/gsl_randist.h&gt;

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

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

  gsl_rng_free(r);

  return 0;
}
</pre></div>

<p>The data can be prepared by running the resulting executable program,
</p>
<div class="example">
<pre class="example">$ GSL_RNG_TYPE=mt19937_1999 ./generate &gt; 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
....
</pre></div>

<p>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.
</p>
<div class="example">
<pre class="example">$ ./fit 19 &lt; 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
</pre></div>

<p>The parameters of the quadratic fit match the coefficients of the
expansion of <em>e^x</em>, taking into account the errors on the
parameters and the <em>O(x^3)</em> difference between the exponential and
quadratic functions for the larger values of <em>x</em>.  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.
</p>

<hr>
<div class="header">
<p>
Next: <a href="Fitting-regularized-linear-regression-example-1.html#Fitting-regularized-linear-regression-example-1" accesskey="n" rel="next">Fitting regularized linear regression example 1</a>, Previous: <a href="Fitting-linear-regression-example.html#Fitting-linear-regression-example" accesskey="p" rel="previous">Fitting linear regression example</a>, Up: <a href="Fitting-Examples.html#Fitting-Examples" accesskey="u" rel="up">Fitting Examples</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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