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<title>GNU Scientific Library – Reference Manual: Fitting robust linear regression example</title>
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<a name="Fitting-robust-linear-regression-example"></a>
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<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> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<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 <stdio.h>
#include <gsl/gsl_multifit.h>
#include <gsl/gsl_randist.h>
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->size1, X->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,"usage: robfit n\n");
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 < 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 < 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 < 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(&v.vector, c, cov, &y_rob, &y_err);
gsl_multifit_robust_est(&v.vector, c_ols, cov, &y_ols, &y_err);
printf("%g %g %g %g\n", 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 ("# best fit: Y = %g + %g X\n",
C(0), C(1));
printf ("# covariance matrix:\n");
printf ("# [ %+.5e, %+.5e\n",
COV(0,0), COV(0,1));
printf ("# %+.5e, %+.5e\n",
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>
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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> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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