analistica/ex-1/main.c

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_histogram.h>
#include <gsl/gsl_statistics_double.h>

#include "landau.h"
#include "tests.h"
#include "bootstrap.h"


/* Here we generate random numbers in a uniform 
 * range and by using the quantile we map them
 * to a Landau distribution. Then we generate an
 * histogram to check the correctness.
 */
int main(int argc, char** argv) {
  // initialize an RNG
  gsl_rng_env_setup();
  gsl_rng *r = gsl_rng_alloc(gsl_rng_default);

  // prepare histogram
  size_t samples = 100000;
  double* sample = calloc(samples, sizeof(double));
  size_t bins = 40;
  double min = -10;
  double max =  10;
  gsl_histogram* hist = gsl_histogram_alloc(bins);
  gsl_histogram_set_ranges_uniform(hist, min, max);


  /* Sample generation
   *
   * Sample points from the Landau
   * distribution and fill the histogram.
   */
  fprintf(stderr, "# Sampling\n");
  fprintf(stderr, "generating %ld points... ", samples);
  double x;
  for(size_t i=0; i<samples; i++) {
    x = gsl_ran_landau(r);
    sample[i] = x;
    gsl_histogram_increment(hist, x);
  }
  fprintf(stderr, "done\n");

  // sort the sample
  qsort(sample, samples, sizeof(double), &cmp_double);


  /* Kolmogorov-Smirnov test
   *
   * Compute the D statistic and its
   * associated probability.
   */
  double D = 0;
  double d;
  for(size_t i=0; i<samples; i++) {
    d = fabs(landau_cdf(sample[i], NULL) - ((double)i+1)/samples);
    if (d > D)
      D = d;
  }
  fprintf(stderr, "\n\n# Kolmogorov-Smirnov test\n");
  double beta = kolmogorov_cdf(D, samples);

  // print the results
  fprintf(stderr, "\n## Results\n");
  fprintf(stderr, "D: %f\n", D);
  fprintf(stderr, "α: %g\n", 1 - beta);


  /* Mode comparison
   *
   * Compute the half-sample mode by bootstrapping
   * and compare the result with the value found by
   * numerical maximisation of the PDF.
   */
  fprintf(stderr, "\n\n# Mode comparison\n");

  /* A structure used by the optimisation
   * routines in numeric_mode and others
   * functions below.
   */
  gsl_function pdf;
  pdf.function = &landau_pdf;
  pdf.params = NULL;

  double mode_e = numeric_mode(min, max, &pdf, 1);
  uncert mode_o = bootstrap_mode(r, sample, samples, 100);

  // print the results
  fprintf(stderr, "\n## Results\n");
  fprintf(stderr, "expected mode: %.7f\n", mode_e);
  fprintf(stderr, "observed mode: %.4f±%.4f\n", mode_o.n, mode_o.s);

  // t-test
  double t = fabs(mode_e - mode_o.n)/mode_o.s;
  double p = 1 - erf(t/sqrt(2));
  fprintf(stderr, "\n## t-test\n");
  fprintf(stderr, "t=%.3f\n", t);
  fprintf(stderr, "p=%.3f\n", p);


  /* FWHM comparison
   *
   * Estimate the FWHM of the sample by constructing
   * an empirical PDF via a KDE method and applying
   * the definition on it (numerical solution of
   * `f(x) = max/2` ⇒ x₁-x₀). This is again bootstrapped
   * to estimate the standard errors and compared against
   * the numerical value of FWHM from the true PDF.
   */
  fprintf(stderr, "\n\n# FWHM comparison\n");

  double fwhm_e = numeric_fwhm(min, max, &pdf, 1);
  uncert fwhm_o = bootstrap_fwhm(r, min, max, sample, samples, 100);

  // print the results
  fprintf(stderr, "\n## Results\n");
  fprintf(stderr, "expected fwhm: %.7f\n", fwhm_e);
  fprintf(stderr, "observed fwhm: %.4f±%.4f\n", fwhm_o.n, fwhm_o.s);

  // t-test
  t = fabs(fwhm_e - fwhm_o.n)/fwhm_o.s;
  p = 1 - erf(t/sqrt(2));
  fprintf(stderr, "\n## t-test\n");
  fprintf(stderr, "t=%.3f\n", t);
  fprintf(stderr, "p=%.3f\n", p);


  /* Median comparison
   *
   * Compute the median of the sample by bootstrapping
   * it and comparing it with the QDF(1/2).
   */
  fprintf(stderr, "\n\n# Median comparison\n");
  double med_e = landau_qdf(0.5);
  uncert med_o = bootstrap_median(r, sample, samples, 100);

  // print the results
  fprintf(stderr, "\n## Results\n");
  fprintf(stderr, "expected median: %.7f\n", med_e);
  fprintf(stderr, "observed median: %.4f±%.4f\n", med_o.n, med_o.s);

  // t-test
  t = fabs(med_e - med_o.n)/med_o.s;
  p = 1 - erf(t/sqrt(2));
  fprintf(stderr, "\n## t-test\n");
  fprintf(stderr, "t=%.3f\n", t);
  fprintf(stderr, "p=%.3f\n", p);


  // clean up and exit
  gsl_histogram_free(hist);
  gsl_rng_free(r);
  free(sample);
  return EXIT_SUCCESS;
}