ex-1: add infinite moments test

ex-1/main.c

@@ -21,7 +21,7 @@ int main(int argc, char** argv) {
   double m_params[2] = {-0.22278298, 1.1191486};
 
   /* Process CLI arguments */
-  for (size_t i = 1; i < argc; i++) {
+  for (size_t i = 1; i < (size_t)argc; i++) {
     if      (!strcmp(argv[i], "-n")) samples = atol(argv[++i]);
     else if (!strcmp(argv[i], "-m")) distr = argv[++i];
     else {
@@ -48,23 +48,20 @@ int main(int argc, char** argv) {
    */
   fprintf(stderr, "# Sampling\n");
   fprintf(stderr, "generating %ld points... ", samples);
-  double x;
+
-  /* Sample points from the Landau distribution using the GSL Landau generator or
+  /* Sample points from the Landau distribution
-   * from the Moyal distribution using inverse sampling.
+   * using the GSL Landau generator or from the
+   * Moyal distribution using inverse sampling.
    */
   for(size_t i=0; i < samples; i++){
-    if (!strcmp(distr, "lan")){
+    if (!strcmp(distr, "lan"))
-      x = gsl_ran_landau(r);
+      sample[i]  = gsl_ran_landau(r);
-      sample[i] = x;
+    if (!strcmp(distr, "moy"))
-    }
+      sample[i] = moyal_qdf(gsl_rng_uniform(r), m_params);
-    if (!strcmp(distr, "moy")){
-      x = gsl_rng_uniform(r);
-      sample[i] = moyal_qdf(x, m_params);
-    }
   }
   fprintf(stderr, "done\n");
 
-  // sort the sample: needed for HSM and ks tests
+  // sort the sample: needed for HSM and KS tests
   qsort(sample, samples, sizeof(double), &cmp_double);
 
 
@@ -81,7 +78,6 @@ int main(int argc, char** argv) {
     if (d > D)
       D = d;
   }
-  
   double beta = kolmogorov_cdf(D, samples);
 
   // print the results
@@ -174,6 +170,24 @@ int main(int argc, char** argv) {
   fprintf(stderr, "p=%.3f\n", p);
 
 
+  /* Infinite moments test
+   *
+   * Apply the Trapani test for infinite moment
+   * to the mean and variance.
+   *
+   * Use r=n^0.75 points in both cases and rescale
+   * the variance by the ψ=2 moment. The mean is not
+   * rescaled.
+   */
+  fprintf(stderr, "\n\n# Infinite moments test\n");
+  double p_mean = trapani(r, 0.75, 0, 1, sample, samples);
+  double p_var  = trapani(r, 0.75, 1, 2, sample, samples);
+
+  // print the results
+  fprintf(stderr, "\n## Results\n");
+  fprintf(stderr, "mean: p=%.3f\n", p_mean);
+  fprintf(stderr, "variance: p=%.3f\n", p_var);
+
   // clean up and exit
   gsl_rng_free(r);
   free(sample);

ex-1/tests.c

@@ -5,11 +5,14 @@
 
 #include <stdio.h>
 #include <gsl/gsl_randist.h>
+#include <gsl/gsl_sf_gamma.h>
+#include <gsl/gsl_cdf.h>
 #include <gsl/gsl_min.h>
 #include <gsl/gsl_roots.h>
 #include <gsl/gsl_sum.h>
 
 #include "landau.h"
+#include "bootstrap.h"
 
 
 /* Kolmogorov distribution CDF
@@ -46,7 +49,6 @@ double kolmogorov_cdf(double D, int n) {
 }
 
 
-
 /* This is a high-order function (ie a function that operates
  * on functions) in disguise. It takes a function f and produces
  * a function that computes -f. In lambda calculus it would be
@@ -219,3 +221,126 @@ double numeric_fwhm(double min, double max,
   gsl_root_fsolver_free(s);
   return x_upp - x_low;
 }
+
+
+/* The absolute moment of given `order`
+ * of the normal distribution N(0, 1).
+ *
+ *  μ_k = E[|X|^k] = 2^(k/2) Γ((k+1)/2) / √π
+ */
+double normal_moment(double order)  {
+  return pow(2, order/2) * gsl_sf_gamma((order+1)/2) / sqrt(M_PI);
+}
+
+
+/* Trapani infinite moment test
+ *
+ * Tests whether the `order`-th moment of the
+ * distribution of `sample` is infinite.
+ * The test generates an artificial sample of
+ * r points, where r = n^θ and returns a p-value.
+ */
+double trapani(gsl_rng *rng, double theta, double psi,
+               int order, double *sample, size_t n) {
+  fprintf(stderr, "\n## Trapani test (k = %d)\n", order);
+
+  /* Compute the moment μ_k and rescale it
+   * to make it scale-free. This is done by
+   * also computing the moment μ_ψ and taking
+   *
+   *   m_k = μ_k / μ_ψ^(k/ψ)
+   *
+   * where:
+   *   1. ψ ∈ (0, k)
+   *   2. k is the `order` of the moment
+   */
+
+  double moment = 0;
+  double moment_psi = 0;
+
+  for (size_t i = 0; i < n; i++) {
+    moment += pow(fabs(sample[i]), order);
+    moment_psi += pow(fabs(sample[i]), psi);
+  }
+  moment /= n;
+  moment_psi /= n;
+  fprintf(stderr, "%d-th sample moment: %.2g\n", order, moment);
+
+  /* Rescale the moment
+   * Note: k/ψ = 2 becase ψ=k/2
+   */
+  if (psi > 0) {
+    fprintf(stderr, "%g-th sample moment: %.2g\n", psi, moment_psi);
+    moment /= pow(moment_psi, order/psi);
+
+    /* Rescale further by the standard gaussian
+     * moments.
+     *
+     * m_k' = m_k (μ_ψ)^(k/ψ) / μ_k
+     */
+    moment *= pow(normal_moment(psi), order/psi) / normal_moment(order);
+
+    fprintf(stderr, "rescaled moment: %.2g\n", moment);
+  }
+
+  /* Generate r points from a standard
+   * normal distribution and multiply
+   * them by √(e^m_k).
+   */
+  size_t r = round(pow(n, theta));
+
+  fprintf(stderr, "n: %ld\n", n);
+  fprintf(stderr, "r: %ld\n", r);
+  double factor = sqrt(exp(moment));
+  double *noise = calloc(r, sizeof(double));
+  for (size_t i = 0; i < r; i++) {
+    noise[i] = factor * gsl_ran_gaussian(rng, 1); // σ = 1
+  }
+
+  /* Compute the intel of the squared
+   * sum of the residues:
+   *
+   * Θ = ∫[-L, L] sum(u) φ(u)du
+   *
+   * where sum(u) = 2/√r (Σ_j=0 ^r θ(u - noise[j]) - 1/2)
+   *
+   * L = 1 and φ(u)=1/2L is a good enough choice.
+   */
+
+  // sort the noise sample
+  qsort(noise, r, sizeof(double), &cmp_double);
+
+  double integral = 0;
+  double L = 1;
+  for (size_t i = 0; i < r; i++) {
+    if (noise[i] < -L) {
+      if (i+1 < r && noise[i+1] < -L)
+        factor = 0;
+      else if (i+1 < r && noise[i+1] < L)
+        factor = noise[i+1] + L;
+      else
+        factor = 2*L;
+    }
+    else if (noise[i] < L) {
+      if (i+1 < r && noise[i+1] < L)
+        factor = noise[i+1] - noise[i];
+      else
+        factor = L - noise[i];
+    }
+    else
+      factor = 0;
+    integral += (i + 1)*((i + 1)/(double)r - 1) * factor;
+  }
+  integral = r + 2/L * integral;
+  fprintf(stderr, "Θ=%g\n", integral);
+
+  // free memory
+  free(noise);
+
+  /* Assuming Θ is distributed as a χ² with
+   * 1 DoF, compute the the p-value:
+   *
+   *  p = P(χ² > Θ ; ν=1)
+   */
+  return gsl_cdf_chisq_Q(integral, 1);
+}

ex-1/tests.h

@@ -3,6 +3,7 @@
  * from a Landau distribution.
  */
 
+#include <gsl/gsl_rng.h>
 #include <gsl/gsl_roots.h>
 
 #pragma once
@@ -38,3 +39,14 @@ double numeric_mode(double min, double max,
 double numeric_fwhm(double min, double max,
                     gsl_function *pdf,
                     int err);
+
+
+/* Trapani infinite moment test
+ *
+ * Tests whether the `order`-th moment of the
+ * distribution of `sample` is infinite.
+ * The test generates an artificial sample of
+ * r points, where r = n^θ and returns a p-value.
+ */
+double trapani(gsl_rng *rng, double theta, double psi,
+               int order, double *sample, size_t n);