From 56fcdd5fbbe8efb9f9a16145b1605d35d12763ff Mon Sep 17 00:00:00 2001
From: rnhmjoj <rnhmjoj@inventati.org>
Date: Fri, 3 Apr 2020 01:16:47 +0000
Subject: [PATCH] ex-1: implement bootstrapping for the median

---
 ex-1/bootstrap.c | 47 +++++++++++++++++++++++++++++++++++++++++++++++
 ex-1/bootstrap.h | 24 ++++++++++++++++++++++++
 ex-1/main.c      | 20 ++++++++++++--------
 makefile         |  2 +-
 4 files changed, 84 insertions(+), 9 deletions(-)
 create mode 100644 ex-1/bootstrap.c
 create mode 100644 ex-1/bootstrap.h

diff --git a/ex-1/bootstrap.c b/ex-1/bootstrap.c
new file mode 100644
index 0000000..0f10809
--- /dev/null
+++ b/ex-1/bootstrap.c
@@ -0,0 +1,47 @@
+#include <stdlib.h>
+#include <gsl/gsl_rstat.h>
+#include <gsl/gsl_statistics_double.h>
+
+#include "bootstrap.h"
+
+
+/* Computes an approximation to the asymptotic median
+ * and its standard deviation by bootstrapping (ie
+ * repeated resampling) the original `sample`, `boots`
+ * times.
+ *
+ * The functions returns an `uncert` pair of mean and
+ * stdev of the medians computed on each sample.
+ */
+uncert bootstrap_median(
+    const gsl_rng *r,
+    double *sample, size_t n,
+    int boots) {
+
+  /* We use a running statistics to not 
+   * store the full resampled array.
+   */
+  gsl_rstat_workspace* w = gsl_rstat_alloc();
+
+  double *values = calloc(boots, sizeof(double));
+
+  for (size_t i = 0; i < boots; i++) {
+    /* The sampling is simply done by generating
+     * an array index uniformely in [0, n-1].
+     */
+    for (size_t j = 0; j < n; j++) {
+      size_t choice = gsl_rng_uniform_int(r, n);
+      gsl_rstat_add(sample[choice], w);
+    }
+    values[i] = gsl_rstat_median(w);
+  }
+
+  /* Compute mean and stdev of the medians
+   * of each newly bootstrapped sample.
+   */
+  uncert median;
+  median.n = gsl_stats_mean(values, 1, boots);
+  median.s = gsl_stats_sd(values, 1, boots);
+
+  return median;
+}
diff --git a/ex-1/bootstrap.h b/ex-1/bootstrap.h
new file mode 100644
index 0000000..453bf06
--- /dev/null
+++ b/ex-1/bootstrap.h
@@ -0,0 +1,24 @@
+#include <gsl/gsl_rng.h>
+
+
+/* A pair structure that represent
+ * a value with an uncertainty
+ */
+typedef struct {
+  double n;  // nominal value
+  double s;  // uncertainty
+} uncert;
+
+
+/* Computes an approximation to the asymptotic median
+ * and its standard deviation by bootstrapping (ie
+ * repeated resampling) the original `sample`, `boots`
+ * times.
+ *
+ * The functions returns an `uncert` pair of mean and
+ * stdev of the medians computed on each sample.
+ */
+uncert bootstrap_median(
+    const gsl_rng *r,
+    double *sample, size_t n,
+    int boots);
diff --git a/ex-1/main.c b/ex-1/main.c
index 2d9af4d..fe47307 100644
--- a/ex-1/main.c
+++ b/ex-1/main.c
@@ -7,6 +7,7 @@
 
 #include "landau.h"
 #include "tests.h"
+#include "bootstrap.h"
 
 
 /* Function that compare doubles for sorting:
@@ -149,21 +150,24 @@ int main(int argc, char** argv) {
 
   /* Median comparison
    *
-   * Compute the median of the sample
-   * and compare it with QDF(1/2).
+   * 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);
-  double med_o = gsl_stats_median_from_sorted_data(
-    sample,   // sorted data
-    1,        // array stride
-    samples); // number of elements
+  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: %.7f\n", med_o);
-  
+  fprintf(stderr, "observed median: %.4f±%.4f\n", med_o.n, med_o.s);
+
+  double t = (med_e - med_o.n)/med_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);
+
 
   // clean up and exit
   gsl_histogram_free(hist);
diff --git a/makefile b/makefile
index 9ce2042..c17acce 100644
--- a/makefile
+++ b/makefile
@@ -6,7 +6,7 @@ CCOMPILE = \
 	mkdir -p $(@D); \
 	$(CC) $(CFLAGS) $^ -o $@
 
-ex-1/bin/main: ex-1/main.c ex-1/landau.c ex-1/tests.c
+ex-1/bin/main: ex-1/main.c ex-1/landau.c ex-1/tests.c ex-1/bootstrap.c
 	$(CCOMPILE)
 ex-1/bin/pdf: ex-1/pdf.c
 	$(CCOMPILE)