437 lines
13 KiB
C
437 lines
13 KiB
C
#include <float.h>
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#include <limits.h>
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#include <math.h>
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#include <stdint.h>
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#include <stdio.h>
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#include <stdlib.h>
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#include <sys/types.h>
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#include <time.h>
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#define MAX_ERROR_LENGTH 500
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#define EXIT_ON_ERROR 0
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#define PROCESS_ERROR(error_msg) process_error(error_msg, EXIT_ON_ERROR, __FILE__, __LINE__)
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const double PI = 3.14159265358979323846; // M_PI in gcc gnu99
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// Pseudo Random number generator
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uint64_t xorshift32(uint32_t* seed)
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{
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// Algorithm "xor" from p. 4 of Marsaglia, "Xorshift RNGs"
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// See <https://stackoverflow.com/questions/53886131/how-does-xorshift64-works>
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// https://en.wikipedia.org/wiki/Xorshift
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// Also some drama: <https://www.pcg-random.org/posts/on-vignas-pcg-critique.html>, <https://prng.di.unimi.it/>
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uint64_t x = *seed;
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x ^= x << 13;
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x ^= x >> 17;
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x ^= x << 5;
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return *seed = x;
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}
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uint64_t xorshift64(uint64_t* seed)
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{
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// Algorithm "xor" from p. 4 of Marsaglia, "Xorshift RNGs"
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// See <https://stackoverflow.com/questions/53886131/how-does-xorshift64-works>
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// https://en.wikipedia.org/wiki/Xorshift
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// Also some drama: <https://www.pcg-random.org/posts/on-vignas-pcg-critique.html>, <https://prng.di.unimi.it/>
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uint64_t x = *seed;
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x ^= x << 13;
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x ^= x >> 7;
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x ^= x << 17;
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return *seed = x;
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}
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// Distribution & sampling functions
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// Unit distributions
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double sample_unit_uniform(uint64_t* seed)
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{
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// samples uniform from [0,1] interval.
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return ((double)xorshift64(seed)) / ((double)UINT64_MAX);
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}
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double sample_unit_normal(uint64_t* seed)
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{
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// See: <https://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform>
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double u1 = sample_unit_uniform(seed);
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double u2 = sample_unit_uniform(seed);
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double z = sqrtf(-2.0 * log(u1)) * sin(2 * PI * u2);
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return z;
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}
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// Composite distributions
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double sample_uniform(double start, double end, uint64_t* seed)
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{
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return sample_unit_uniform(seed) * (end - start) + start;
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}
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double sample_normal(double mean, double sigma, uint64_t* seed)
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{
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return (mean + sigma * sample_unit_normal(seed));
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}
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double sample_lognormal(double logmean, double logstd, uint64_t* seed)
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{
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return exp(sample_normal(logmean, logstd, seed));
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}
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double sample_to(double low, double high, uint64_t* seed)
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{
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// Given a (positive) 90% confidence interval,
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// returns a sample from a lognormal
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// with a matching 90% c.i.
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const double NORMAL95CONFIDENCE = 1.6448536269514722;
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double loglow = logf(low);
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double loghigh = logf(high);
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double logmean = (loglow + loghigh) / 2;
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double logstd = (loghigh - loglow) / (2.0 * NORMAL95CONFIDENCE);
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return sample_lognormal(logmean, logstd, seed);
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}
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double sample_gamma(double alpha, uint64_t* seed)
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{
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// A Simple Method for Generating Gamma Variables, Marsaglia and Wan Tsang, 2001
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// https://dl.acm.org/doi/pdf/10.1145/358407.358414
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// see also the references/ folder
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// Note that the Wikipedia page for the gamma distribution includes a scaling parameter
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// k or beta
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// https://en.wikipedia.org/wiki/Gamma_distribution
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// such that gamma_k(alpha, k) = k * gamma(alpha)
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// or gamma_beta(alpha, beta) = gamma(alpha) / beta
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// So far I have not needed to use this, and thus the second parameter is by default 1.
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if (alpha >= 1) {
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double d, c, x, v, u;
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d = alpha - 1.0 / 3.0;
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c = 1.0 / sqrt(9.0 * d);
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while (1) {
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do {
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x = sample_unit_normal(seed);
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v = 1.0 + c * x;
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} while (v <= 0.0);
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v = v * v * v;
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u = sample_unit_uniform(seed);
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if (u < 1.0 - 0.0331 * (x * x * x * x)) { // Condition 1
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// the 0.0331 doesn't inspire much confidence
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// however, this isn't the whole story
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// by knowing that Condition 1 implies condition 2
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// we realize that this is just a way of making the algorithm faster
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// i.e., of not using the logarithms
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return d * v;
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}
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if (log(u) < 0.5 * (x * x) + d * (1.0 - v + log(v))) { // Condition 2
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return d * v;
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}
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}
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} else {
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return sample_gamma(1 + alpha, seed) * pow(sample_unit_uniform(seed), 1 / alpha);
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// see note in p. 371 of https://dl.acm.org/doi/pdf/10.1145/358407.358414
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}
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}
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double sample_beta(double a, double b, uint64_t* seed)
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{
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double gamma_a = sample_gamma(a, seed);
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double gamma_b = sample_gamma(b, seed);
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return gamma_a / (gamma_a + gamma_b);
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}
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// Array helpers
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double array_sum(double* array, int length)
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{
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double sum = 0.0;
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for (int i = 0; i < length; i++) {
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sum += array[i];
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}
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return sum;
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}
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void array_cumsum(double* array_to_sum, double* array_cumsummed, int length)
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{
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array_cumsummed[0] = array_to_sum[0];
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for (int i = 1; i < length; i++) {
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array_cumsummed[i] = array_cumsummed[i - 1] + array_to_sum[i];
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}
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}
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double array_mean(double* array, int length)
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{
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double sum = array_sum(array, length);
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return sum / length;
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}
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double array_std(double* array, int length)
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{
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double mean = array_mean(array, length);
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double std = 0.0;
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for (int i = 0; i < length; i++) {
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std += (array[i] - mean) * (array[i] - mean);
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}
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std = sqrt(std / length);
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return std;
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}
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// Mixture function
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double sample_mixture(double (*samplers[])(uint64_t*), double* weights, int n_dists, uint64_t* seed)
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{
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// You can see a simpler version of this function in the git history
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// or in C-02-better-algorithm-one-thread/
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double sum_weights = array_sum(weights, n_dists);
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double* cumsummed_normalized_weights = (double*)malloc(n_dists * sizeof(double));
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cumsummed_normalized_weights[0] = weights[0] / sum_weights;
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for (int i = 1; i < n_dists; i++) {
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cumsummed_normalized_weights[i] = cumsummed_normalized_weights[i - 1] + weights[i] / sum_weights;
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}
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double result;
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int result_set_flag = 0;
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double p = sample_uniform(0, 1, seed);
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for (int k = 0; k < n_dists; k++) {
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if (p < cumsummed_normalized_weights[k]) {
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result = samplers[k](seed);
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result_set_flag = 1;
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break;
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}
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}
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if (result_set_flag == 0)
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result = samplers[n_dists - 1](seed);
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free(cumsummed_normalized_weights);
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return result;
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}
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// Sample from an arbitrary cdf
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struct box {
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int empty;
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double content;
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char* error_msg;
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};
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struct box process_error(const char* error_msg, int should_exit, char* file, int line)
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{
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if (should_exit) {
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printf("@, in %s (%d)", file, line);
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exit(1);
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} else {
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char error_msg[MAX_ERROR_LENGTH];
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snprintf(error_msg, MAX_ERROR_LENGTH, "@, in %s (%d)", file, line);
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struct box error = { .empty = 1, .error_msg = error_msg };
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return error;
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}
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}
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// Inverse cdf at point
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// Two versions of this function:
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// - raw, dealing with cdfs that return doubles
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// - input: cdf: double => double, p
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// - output: Box(number|error)
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// - box, dealing with cdfs that return a box.
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// - input: cdf: double => Box(number|error), p
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// - output: Box(number|error)
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struct box inverse_cdf_double(double cdf(double), double p)
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{
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// given a cdf: [-Inf, Inf] => [0,1]
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// returns a box with either
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// x such that cdf(x) = p
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// or an error
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// if EXIT_ON_ERROR is set to 1, it exits instead of providing an error
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double low = -1.0;
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double high = 1.0;
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// 1. Make sure that cdf(low) < p < cdf(high)
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int interval_found = 0;
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while ((!interval_found) && (low > -FLT_MAX / 4) && (high < FLT_MAX / 4)) {
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// ^ Using FLT_MIN and FLT_MAX is overkill
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// but it's also the *correct* thing to do.
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int low_condition = (cdf(low) < p);
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int high_condition = (p < cdf(high));
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if (low_condition && high_condition) {
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interval_found = 1;
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} else if (!low_condition) {
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low = low * 2;
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} else if (!high_condition) {
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high = high * 2;
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}
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}
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if (!interval_found) {
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return PROCESS_ERROR("Interval containing the target value not found, in function inverse_cdf");
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} else {
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int convergence_condition = 0;
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int count = 0;
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while (!convergence_condition && (count < (INT_MAX / 2))) {
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double mid = (high + low) / 2;
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int mid_not_new = (mid == low) || (mid == high);
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// double width = high - low;
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// if ((width < 1e-8) || mid_not_new){
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if (mid_not_new) {
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convergence_condition = 1;
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} else {
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double mid_sign = cdf(mid) - p;
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if (mid_sign < 0) {
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low = mid;
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} else if (mid_sign > 0) {
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high = mid;
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} else if (mid_sign == 0) {
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low = mid;
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high = mid;
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}
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}
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}
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if (convergence_condition) {
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struct box result = { .empty = 0, .content = low };
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return result;
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} else {
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return PROCESS_ERROR("Search process did not converge, in function inverse_cdf");
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}
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}
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}
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struct box inverse_cdf_box(struct box cdf_box(double), double p)
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{
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// given a cdf: [-Inf, Inf] => Box([0,1])
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// returns a box with either
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// x such that cdf(x) = p
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// or an error
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// if EXIT_ON_ERROR is set to 1, it exits instead of providing an error
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double low = -1.0;
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double high = 1.0;
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// 1. Make sure that cdf(low) < p < cdf(high)
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int interval_found = 0;
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while ((!interval_found) && (low > -FLT_MAX / 4) && (high < FLT_MAX / 4)) {
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// ^ Using FLT_MIN and FLT_MAX is overkill
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// but it's also the *correct* thing to do.
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struct box cdf_low = cdf_box(low);
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if (cdf_low.empty) {
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return PROCESS_ERROR(cdf_low.error_msg);
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}
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struct box cdf_high = cdf_box(high);
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if (cdf_high.empty) {
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return PROCESS_ERROR(cdf_low.error_msg);
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}
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int low_condition = (cdf_low.content < p);
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int high_condition = (p < cdf_high.content);
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if (low_condition && high_condition) {
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interval_found = 1;
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} else if (!low_condition) {
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low = low * 2;
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} else if (!high_condition) {
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high = high * 2;
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}
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}
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if (!interval_found) {
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return PROCESS_ERROR("Interval containing the target value not found, in function inverse_cdf");
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} else {
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int convergence_condition = 0;
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int count = 0;
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while (!convergence_condition && (count < (INT_MAX / 2))) {
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double mid = (high + low) / 2;
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int mid_not_new = (mid == low) || (mid == high);
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// double width = high - low;
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if (mid_not_new) {
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// if ((width < 1e-8) || mid_not_new){
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convergence_condition = 1;
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} else {
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struct box cdf_mid = cdf_box(mid);
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if (cdf_mid.empty) {
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return PROCESS_ERROR(cdf_mid.error_msg);
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}
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double mid_sign = cdf_mid.content - p;
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if (mid_sign < 0) {
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low = mid;
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} else if (mid_sign > 0) {
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high = mid;
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} else if (mid_sign == 0) {
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low = mid;
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high = mid;
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}
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}
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}
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if (convergence_condition) {
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struct box result = { .empty = 0, .content = low };
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return result;
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} else {
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return PROCESS_ERROR("Search process did not converge, in function inverse_cdf");
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}
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}
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}
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// Sampler based on inverse cdf and randomness function
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struct box sampler_cdf_box(struct box cdf(double), uint64_t* seed)
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{
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double p = sample_unit_uniform(seed);
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struct box result = inverse_cdf_box(cdf, p);
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return result;
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}
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struct box sampler_cdf_double(double cdf(double), uint64_t* seed)
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{
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double p = sample_unit_uniform(seed);
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struct box result = inverse_cdf_double(cdf, p);
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return result;
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}
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// Get confidence intervals, given a sampler
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struct c_i {
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float low;
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float high;
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};
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int compare_doubles(const void* p, const void* q)
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{
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// https://wikiless.esmailelbob.xyz/wiki/Qsort?lang=en
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double x = *(const double*)p;
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double y = *(const double*)q;
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/* Avoid return x - y, which can cause undefined behaviour
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because of signed integer overflow. */
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if (x < y)
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return -1; // Return -1 if you want ascending, 1 if you want descending order.
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else if (x > y)
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return 1; // Return 1 if you want ascending, -1 if you want descending order.
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return 0;
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}
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struct c_i get_90_confidence_interval(double (*sampler)(uint64_t*), uint64_t* seed)
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{
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int n = 100 * 1000;
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double* samples_array = malloc(n * sizeof(double));
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for (int i = 0; i < n; i++) {
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samples_array[i] = sampler(seed);
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}
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qsort(samples_array, n, sizeof(double), compare_doubles);
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struct c_i result = {
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.low = samples_array[5000],
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.high = samples_array[94999],
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};
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free(samples_array);
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return result;
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}
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/* Could also define other variations, e.g.,
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double sampler_danger(struct box cdf(double), uint64_t* seed)
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{
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double p = sample_unit_uniform(seed);
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struct box result = inverse_cdf_box(cdf, p);
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if(result.empty){
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exit(1);
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}else{
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return result.content;
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}
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}
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*/
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