#### 38.8.4 Regularized Linear Regression Example 2

The following example program minimizes the cost function

    ||y - X c||^2 + \lambda^2 ||x||^2

where X is the 10-by-8 Hilbert matrix whose entries are given by

    X_{ij} = 1 / (i + j - 1)

and the right hand side vector is given by y = [1,-1,1,-1,1,-1,1,-1,1,-1]^T. Solutions are computed for \lambda = 0 (unregularized) as well as for optimal parameters \lambda chosen by analyzing the L-curve and GCV curve.

Here is the program output:

    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

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 \lambda = 7.11e-7 which still results in a badly conditioned system and a large solution norm. The GCV method finds a parameter \lambda = 1.72 which results in a well-conditioned system and small solution norm.

The L-curve and its computed corner, as well as the GCV curve and its minimum are plotted below.

The program is given below.

    #include <gsl/gsl_math.h>
    #include <gsl/gsl_vector.h>
    #include <gsl/gsl_matrix.h>
    #include <gsl/gsl_multifit.h>
    #include <gsl/gsl_blas.h>

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

      for (i = 0; i < N; i++)
        {
          for (j = 0; j < 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 < 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, "matrix condition number = %e\n", 1.0 / rcond);

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

        fprintf(stderr, "=== Unregularized fit ===\n");
        fprintf(stderr, "residual norm = %g\n", rnorm);
        fprintf(stderr, "solution norm = %g\n", snorm);
        fprintf(stderr, "chisq/dof = %g\n", 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, &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, &rnorm, &snorm, w);
        chisq = pow(rnorm, 2.0) + pow(lambda_l * snorm, 2.0);

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

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

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

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

        /* output L-curve and GCV curve */
        for (i = 0; i < npoints; ++i)
          {
            printf("%e %e %e %e\n",
                   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("\n\n%f %f\n",
               gsl_vector_get(rho, reg_idx),
               gsl_vector_get(eta, reg_idx));

        /* output GCV curve corner minimum */
        printf("\n\n%e %e\n",
               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;
    }