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<a name="Fitting-regularized-linear-regression-example-2"></a>
<div class="header">
<p>
Next: <a href="Fitting-robust-linear-regression-example.html#Fitting-robust-linear-regression-example" accesskey="n" rel="next">Fitting robust linear regression example</a>, Previous: <a href="Fitting-regularized-linear-regression-example-1.html#Fitting-regularized-linear-regression-example-1" accesskey="p" rel="previous">Fitting regularized linear regression example 1</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="Regularized-Linear-Regression-Example-2"></a>
<h4 class="subsection">38.8.4 Regularized Linear Regression Example 2</h4>

<p>The following example program minimizes the cost function
</p>
<div class="example">
<pre class="example">||y - X c||^2 + \lambda^2 ||x||^2
</pre></div>

<p>where <em>X</em> is the <em>10</em>-by-<em>8</em> Hilbert matrix whose
entries are given by
</p>
<div class="example">
<pre class="example">X_{ij} = 1 / (i + j - 1)
</pre></div>

<p>and the right hand side vector is given by
<em>y = [1,-1,1,-1,1,-1,1,-1,1,-1]^T</em>. Solutions
are computed for <em>\lambda = 0</em> (unregularized) as
well as for optimal parameters <em>\lambda</em> chosen by
analyzing the L-curve and GCV curve.
</p>
<p>Here is the program output:
</p><div class="example">
<pre class="example">matrix condition number = 3.565872e+09
=== Unregularized fit ===
residual norm = 2.15376
solution norm = 2.92217e+09
chisq/dof = 2.31934
=== Regularized fit (L-curve) ===
optimal lambda: 7.11407e-07
residual norm = 2.60386
solution norm = 424507
chisq/dof = 3.43565
=== Regularized fit (GCV) ===
optimal lambda: 1.72278
residual norm = 3.1375
solution norm = 0.139357
chisq/dof = 4.95076
</pre></div>

<p>Here we see the unregularized solution results in a large solution
norm due to the ill-conditioned matrix. The L-curve solution finds
a small value of <em>\lambda = 7.11e-7</em> which still results in
a badly conditioned system and a large solution norm. The GCV method
finds a parameter <em>\lambda = 1.72</em> which results in a well-conditioned
system and small solution norm.
</p>
<p>The L-curve and its computed corner, as well as the GCV curve and its
minimum are plotted below.
</p>

<p>The program is given below.
</p><div class="example">
<pre class="verbatim">#include &lt;gsl/gsl_math.h&gt;
#include &lt;gsl/gsl_vector.h&gt;
#include &lt;gsl/gsl_matrix.h&gt;
#include &lt;gsl/gsl_multifit.h&gt;
#include &lt;gsl/gsl_blas.h&gt;

static int
hilbert_matrix(gsl_matrix * m)
{
  const size_t N = m-&gt;size1;
  const size_t M = m-&gt;size2;
  size_t i, j;

  for (i = 0; i &lt; N; i++)
    {
      for (j = 0; j &lt; M; j++)
        {
          gsl_matrix_set(m, i, j, 1.0/(i+j+1.0));
        }
    }

  return GSL_SUCCESS;
}

int
main()
{
  const size_t n = 10; /* number of observations */
  const size_t p = 8;  /* number of model parameters */
  size_t i;
  gsl_matrix *X = gsl_matrix_alloc(n, p);
  gsl_vector *y = gsl_vector_alloc(n);

  /* construct Hilbert matrix and rhs vector */
  hilbert_matrix(X);

  {
    double val = 1.0;
    for (i = 0; i &lt; n; ++i)
      {
        gsl_vector_set(y, i, val);
        val *= -1.0;
      }
  }

  {
    const size_t npoints = 200;                   /* number of points on L-curve and GCV curve */
    gsl_multifit_linear_workspace *w =
      gsl_multifit_linear_alloc(n, p);
    gsl_vector *c = gsl_vector_alloc(p);          /* OLS solution */
    gsl_vector *c_lcurve = gsl_vector_alloc(p);   /* regularized solution (L-curve) */
    gsl_vector *c_gcv = gsl_vector_alloc(p);      /* regularized solution (GCV) */
    gsl_vector *reg_param = gsl_vector_alloc(npoints);
    gsl_vector *rho = gsl_vector_alloc(npoints);  /* residual norms */
    gsl_vector *eta = gsl_vector_alloc(npoints);  /* solution norms */
    gsl_vector *G = gsl_vector_alloc(npoints);    /* GCV function values */
    double lambda_l;                              /* optimal regularization parameter (L-curve) */
    double lambda_gcv;                            /* optimal regularization parameter (GCV) */
    double G_gcv;                                 /* G(lambda_gcv) */
    size_t reg_idx;                               /* index of optimal lambda */
    double rcond;                                 /* reciprocal condition number of X */
    double chisq, rnorm, snorm;

    /* compute SVD of X */
    gsl_multifit_linear_svd(X, w);

    rcond = gsl_multifit_linear_rcond(w);
    fprintf(stderr, &quot;matrix condition number = %e\n&quot;, 1.0 / rcond);

    /* unregularized (standard) least squares fit, lambda = 0 */
    gsl_multifit_linear_solve(0.0, X, y, c, &amp;rnorm, &amp;snorm, w);
    chisq = pow(rnorm, 2.0);

    fprintf(stderr, &quot;=== Unregularized fit ===\n&quot;);
    fprintf(stderr, &quot;residual norm = %g\n&quot;, rnorm);
    fprintf(stderr, &quot;solution norm = %g\n&quot;, snorm);
    fprintf(stderr, &quot;chisq/dof = %g\n&quot;, chisq / (n - p));

    /* calculate L-curve and find its corner */
    gsl_multifit_linear_lcurve(y, reg_param, rho, eta, w);
    gsl_multifit_linear_lcorner(rho, eta, &amp;reg_idx);

    /* store optimal regularization parameter */
    lambda_l = gsl_vector_get(reg_param, reg_idx);

    /* regularize with lambda_l */
    gsl_multifit_linear_solve(lambda_l, X, y, c_lcurve, &amp;rnorm, &amp;snorm, w);
    chisq = pow(rnorm, 2.0) + pow(lambda_l * snorm, 2.0);

    fprintf(stderr, &quot;=== Regularized fit (L-curve) ===\n&quot;);
    fprintf(stderr, &quot;optimal lambda: %g\n&quot;, lambda_l);
    fprintf(stderr, &quot;residual norm = %g\n&quot;, rnorm);
    fprintf(stderr, &quot;solution norm = %g\n&quot;, snorm);
    fprintf(stderr, &quot;chisq/dof = %g\n&quot;, chisq / (n - p));

    /* calculate GCV curve and find its minimum */
    gsl_multifit_linear_gcv(y, reg_param, G, &amp;lambda_gcv, &amp;G_gcv, w);

    /* regularize with lambda_gcv */
    gsl_multifit_linear_solve(lambda_gcv, X, y, c_gcv, &amp;rnorm, &amp;snorm, w);
    chisq = pow(rnorm, 2.0) + pow(lambda_gcv * snorm, 2.0);

    fprintf(stderr, &quot;=== Regularized fit (GCV) ===\n&quot;);
    fprintf(stderr, &quot;optimal lambda: %g\n&quot;, lambda_gcv);
    fprintf(stderr, &quot;residual norm = %g\n&quot;, rnorm);
    fprintf(stderr, &quot;solution norm = %g\n&quot;, snorm);
    fprintf(stderr, &quot;chisq/dof = %g\n&quot;, chisq / (n - p));

    /* output L-curve and GCV curve */
    for (i = 0; i &lt; npoints; ++i)
      {
        printf(&quot;%e %e %e %e\n&quot;,
               gsl_vector_get(reg_param, i),
               gsl_vector_get(rho, i),
               gsl_vector_get(eta, i),
               gsl_vector_get(G, i));
      }

    /* output L-curve corner point */
    printf(&quot;\n\n%f %f\n&quot;,
           gsl_vector_get(rho, reg_idx),
           gsl_vector_get(eta, reg_idx));

    /* output GCV curve corner minimum */
    printf(&quot;\n\n%e %e\n&quot;,
           lambda_gcv,
           G_gcv);

    gsl_multifit_linear_free(w);
    gsl_vector_free(c);
    gsl_vector_free(c_lcurve);
    gsl_vector_free(reg_param);
    gsl_vector_free(rho);
    gsl_vector_free(eta);
    gsl_vector_free(G);
  }

  gsl_matrix_free(X);
  gsl_vector_free(y);

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

<hr>
<div class="header">
<p>
Next: <a href="Fitting-robust-linear-regression-example.html#Fitting-robust-linear-regression-example" accesskey="n" rel="next">Fitting robust linear regression example</a>, Previous: <a href="Fitting-regularized-linear-regression-example-1.html#Fitting-regularized-linear-regression-example-1" accesskey="p" rel="previous">Fitting regularized linear regression example 1</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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