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#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_bspline.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_linalg.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
double
f(double x)
{
return (1.0 / (1.0 + 25.0*x*x));
}
int
main (void)
{
const size_t n = 500; /* number of data points to fit */
const size_t k = 10; /* spline order */
const double a = -1.0; /* data interval [a,b] */
const double b = 1.0;
const double sigma = 0.03; /* noise */
const double lambda_sq = 10.0; /* regularization parameter */
const size_t nderiv = 1; /* derivative order to regularize at endpoints */
gsl_bspline_workspace *work = gsl_bspline_alloc(k, 20); /* 20 breakpoints */
const size_t p = gsl_bspline_ncontrol(work); /* number of control points */
gsl_vector *c = gsl_vector_alloc(p); /* control points for unregularized model */
gsl_vector *creg = gsl_vector_alloc(p); /* control points for regularized model */
gsl_vector *x = gsl_vector_alloc(n); /* x data */
gsl_vector *y = gsl_vector_alloc(n); /* y data */
gsl_vector *w = gsl_vector_alloc(n); /* data weights */
gsl_matrix *XTX = gsl_matrix_alloc(p, k); /* normal equations matrix */
gsl_vector *XTy = gsl_vector_alloc(p); /* normal equations rhs */
gsl_matrix *cov = gsl_matrix_alloc(p, p); /* covariance matrix */
gsl_matrix *cov_reg = gsl_matrix_alloc(p, p); /* covariance matrix for regularized model */
gsl_matrix *A1 = gsl_matrix_alloc(p, k); /* left regularization matrix */
gsl_matrix *A2 = gsl_matrix_alloc(p, k); /* right regularization matrix */
size_t i;
gsl_rng *r;
gsl_rng_env_setup();
r = gsl_rng_alloc(gsl_rng_default);
/* this is the data to be fitted */
i = 0;
while (i < n)
{
double ui = gsl_rng_uniform(r);
double xi = (b - a) * ui + a;
double yi, dyi;
dyi = gsl_ran_gaussian(r, sigma);
yi = f(xi) + dyi;
gsl_vector_set(x, i, xi);
gsl_vector_set(y, i, yi);
gsl_vector_set(w, i, 1.0 / (sigma * sigma));
printf("%f %f\n", xi, yi);
++i;
}
printf("\n\n");
/* use uniform breakpoints on [a, b] */
gsl_bspline_init_uniform(a, b, work);
gsl_bspline_lsnormal(x, y, w, XTy, XTX, work); /* form normal equations matrix */
gsl_linalg_cholesky_band_decomp(XTX); /* banded Cholesky decomposition */
gsl_linalg_cholesky_band_solve(XTX, XTy, c); /* solve for unregularized solution */
gsl_bspline_covariance(XTX, cov, work); /* compute covariance matrix */
/* compute regularization matrices */
gsl_bspline_oprod(nderiv, a, A1, work);
gsl_bspline_oprod(nderiv, b, A2, work);
/* multiply by lambda^2 */
gsl_matrix_scale(A1, lambda_sq);
gsl_matrix_scale(A2, lambda_sq);
gsl_bspline_lsnormal(x, y, w, XTy, XTX, work); /* form normal equations matrix */
gsl_matrix_add(XTX, A1); /* add regularization terms */
gsl_matrix_add(XTX, A2);
gsl_linalg_cholesky_band_decomp(XTX); /* banded Cholesky decomposition */
gsl_linalg_cholesky_band_solve(XTX, XTy, creg); /* solve for regularized solution */
gsl_bspline_covariance(XTX, cov_reg, work); /* compute covariance matrix */
/* output the spline curves */
{
double xi;
for (xi = a; xi <= b; xi += 0.001)
{
double result_unreg, result_reg;
double err_unreg, err_reg;
/* compute unregularized spline value and error at xi */
gsl_bspline_calc(xi, c, &result_unreg, work);
gsl_bspline_err(xi, 0, cov, &err_unreg, work);
/* compute regularized spline value and error at xi */
gsl_bspline_calc(xi, creg, &result_reg, work);
gsl_bspline_err(xi, 0, cov_reg, &err_reg, work);
printf("%f %e %e %e %e %e\n",
xi,
f(xi),
result_unreg,
err_unreg,
result_reg,
err_reg);
}
}
{
double result0, result1;
gsl_bspline_calc_deriv(a, c, nderiv, &result0, work);
gsl_bspline_calc_deriv(b, c, nderiv, &result1, work);
fprintf(stderr, "unregularized endpoint deriv %zu: [%14.6e, %14.6e]\n", nderiv, result0, result1);
gsl_bspline_calc_deriv(a, creg, nderiv, &result0, work);
gsl_bspline_calc_deriv(b, creg, nderiv, &result1, work);
fprintf(stderr, " regularized endpoint deriv %zu: [%14.6e, %14.6e]\n", nderiv, result0, result1);
}
gsl_rng_free(r);
gsl_vector_free(x);
gsl_vector_free(y);
gsl_vector_free(w);
gsl_vector_free(c);
gsl_vector_free(creg);
gsl_bspline_free(work);
gsl_matrix_free(XTX);
gsl_vector_free(XTy);
gsl_matrix_free(cov);
gsl_matrix_free(cov_reg);
gsl_matrix_free(A1);
gsl_matrix_free(A2);
return 0;
}
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