## File: Fitting-regularized-linear-regression-example-1.html

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 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231232233234235236237238239240241242243244245246247248249250251252253254255256257258259260261262263264265266267268  GNU Scientific Library – Reference Manual: Fitting regularized linear regression example 1

38.8.3 Regularized Linear Regression Example 1

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, &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, "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; }