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The next program demonstrates the difference between ordinary and regularized least squares when the design matrix is near-singular. In this program, we generate two random normally distributed variables u and v, with v = u + noise so that u and v are nearly colinear. We then set a third dependent variable y = u + v + noise and solve for the coefficients c_1,c_2 of the model Y(c_1,c_2) = c_1 u + c_2 v. Since u \approx v, the design matrix X is nearly singular, leading to unstable ordinary least squares solutions.
Here is the program output:
matrix condition number = 1.025113e+04 === Unregularized fit === best fit: y = -43.6588 u + 45.6636 v residual norm = 31.6248 solution norm = 63.1764 chisq/dof = 1.00213 === Regularized fit (L-curve) === optimal lambda: 4.51103 best fit: y = 1.00113 u + 1.0032 v residual norm = 31.6547 solution norm = 1.41728 chisq/dof = 1.04499 === Regularized fit (GCV) === optimal lambda: 0.0232029 best fit: y = -19.8367 u + 21.8417 v residual norm = 31.6332 solution norm = 29.5051 chisq/dof = 1.00314
We see that the ordinary least squares solution is completely wrong, while the L-curve regularized method with the optimal \lambda = 4.51103 finds the correct solution c_1 \approx c_2 \approx 1. The GCV regularized method finds a regularization parameter \lambda = 0.0232029 which is too small to give an accurate solution, although it performs better than OLS. 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_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_multifit.h>
int
main()
{
const size_t n = 1000; /* number of observations */
const size_t p = 2; /* number of model parameters */
size_t i;
gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
gsl_matrix *X = gsl_matrix_alloc(n, p);
gsl_vector *y = gsl_vector_alloc(n);
for (i = 0; i < n; ++i)
{
/* generate first random variable u */
double ui = 5.0 * gsl_ran_gaussian(r, 1.0);
/* set v = u + noise */
double vi = ui + gsl_ran_gaussian(r, 0.001);
/* set y = u + v + noise */
double yi = ui + vi + gsl_ran_gaussian(r, 1.0);
/* since u =~ v, the matrix X is ill-conditioned */
gsl_matrix_set(X, i, 0, ui);
gsl_matrix_set(X, i, 1, vi);
/* rhs vector */
gsl_vector_set(y, i, yi);
}
{
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, "best fit: y = %g u + %g v\n",
gsl_vector_get(c, 0), gsl_vector_get(c, 1));
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, ®_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, "best fit: y = %g u + %g v\n",
gsl_vector_get(c_lcurve, 0), gsl_vector_get(c_lcurve, 1));
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, "best fit: y = %g u + %g v\n",
gsl_vector_get(c_gcv, 0), gsl_vector_get(c_gcv, 1));
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_rng_free(r);
gsl_matrix_free(X);
gsl_vector_free(y);
return 0;
}