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<title>GNU Scientific Library &ndash; Reference Manual: Fitting robust linear regression example</title>

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<a name="Fitting-robust-linear-regression-example"></a>
<div class="header">
<p>
Next: <a href="Fitting-large-linear-systems-example.html#Fitting-large-linear-systems-example" accesskey="n" rel="next">Fitting large linear systems example</a>, Previous: <a href="Fitting-regularized-linear-regression-example-2.html#Fitting-regularized-linear-regression-example-2" accesskey="p" rel="previous">Fitting regularized linear regression example 2</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="Robust-Linear-Regression-Example"></a>
<h4 class="subsection">38.8.5 Robust Linear Regression Example</h4>

<p>The next program demonstrates the advantage of robust least squares on
a dataset with outliers. The program generates linear <em>(x,y)</em>
data pairs on the line <em>y = 1.45 x + 3.88</em>, adds some random
noise, and inserts 3 outliers into the dataset. Both the robust
and ordinary least squares (OLS) coefficients are computed for
comparison.
</p>
<div class="example">
<pre class="verbatim">#include &lt;stdio.h&gt;
#include &lt;gsl/gsl_multifit.h&gt;
#include &lt;gsl/gsl_randist.h&gt;

int
dofit(const gsl_multifit_robust_type *T,
      const gsl_matrix *X, const gsl_vector *y,
      gsl_vector *c, gsl_matrix *cov)
{
  int s;
  gsl_multifit_robust_workspace * work 
    = gsl_multifit_robust_alloc (T, X-&gt;size1, X-&gt;size2);

  s = gsl_multifit_robust (X, y, c, cov, work);
  gsl_multifit_robust_free (work);

  return s;
}

int
main (int argc, char **argv)
{
  size_t i;
  size_t n;
  const size_t p = 2; /* linear fit */
  gsl_matrix *X, *cov;
  gsl_vector *x, *y, *c, *c_ols;
  const double a = 1.45; /* slope */
  const double b = 3.88; /* intercept */
  gsl_rng *r;

  if (argc != 2)
    {
      fprintf (stderr,&quot;usage: robfit n\n&quot;);
      exit (-1);
    }

  n = atoi (argv[1]);

  X = gsl_matrix_alloc (n, p);
  x = gsl_vector_alloc (n);
  y = gsl_vector_alloc (n);

  c = gsl_vector_alloc (p);
  c_ols = gsl_vector_alloc (p);
  cov = gsl_matrix_alloc (p, p);

  r = gsl_rng_alloc(gsl_rng_default);

  /* generate linear dataset */
  for (i = 0; i &lt; n - 3; i++)
    {
      double dx = 10.0 / (n - 1.0);
      double ei = gsl_rng_uniform(r);
      double xi = -5.0 + i * dx;
      double yi = a * xi + b;

      gsl_vector_set (x, i, xi);
      gsl_vector_set (y, i, yi + ei);
    }

  /* add a few outliers */
  gsl_vector_set(x, n - 3, 4.7);
  gsl_vector_set(y, n - 3, -8.3);

  gsl_vector_set(x, n - 2, 3.5);
  gsl_vector_set(y, n - 2, -6.7);

  gsl_vector_set(x, n - 1, 4.1);
  gsl_vector_set(y, n - 1, -6.0);

  /* construct design matrix X for linear fit */
  for (i = 0; i &lt; n; ++i)
    {
      double xi = gsl_vector_get(x, i);

      gsl_matrix_set (X, i, 0, 1.0);
      gsl_matrix_set (X, i, 1, xi);
    }

  /* perform robust and OLS fit */
  dofit(gsl_multifit_robust_ols, X, y, c_ols, cov);
  dofit(gsl_multifit_robust_bisquare, X, y, c, cov);

  /* output data and model */
  for (i = 0; i &lt; n; ++i)
    {
      double xi = gsl_vector_get(x, i);
      double yi = gsl_vector_get(y, i);
      gsl_vector_view v = gsl_matrix_row(X, i);
      double y_ols, y_rob, y_err;

      gsl_multifit_robust_est(&amp;v.vector, c, cov, &amp;y_rob, &amp;y_err);
      gsl_multifit_robust_est(&amp;v.vector, c_ols, cov, &amp;y_ols, &amp;y_err);

      printf(&quot;%g %g %g %g\n&quot;, xi, yi, y_rob, y_ols);
    }

#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\n&quot;, 
            C(0), C(1));

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

  gsl_matrix_free (X);
  gsl_vector_free (x);
  gsl_vector_free (y);
  gsl_vector_free (c);
  gsl_vector_free (c_ols);
  gsl_matrix_free (cov);
  gsl_rng_free(r);

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

<p>The output from the program is shown in the following plot.
</p>

<hr>
<div class="header">
<p>
Next: <a href="Fitting-large-linear-systems-example.html#Fitting-large-linear-systems-example" accesskey="n" rel="next">Fitting large linear systems example</a>, Previous: <a href="Fitting-regularized-linear-regression-example-2.html#Fitting-regularized-linear-regression-example-2" accesskey="p" rel="previous">Fitting regularized linear regression example 2</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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