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229 lines
7.8 KiB
229 lines
7.8 KiB
#include <limits.h>
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#include <math.h>
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#include <stdint.h>
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#include <stdlib.h>
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// Defs
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#define PI 3.14159265358979323846 // M_PI in gcc gnu99
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#define NORMAL90CONFIDENCE 1.6448536269514727
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#define UNUSED(x) (void)(x)
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// ^ https://stackoverflow.com/questions/3599160/how-can-i-suppress-unused-parameter-warnings-in-c
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// Pseudo Random number generators
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static 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:
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// <https://en.wikipedia.org/wiki/Xorshift>
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// <https://stackoverflow.com/questions/53886131/how-does-xorshift32-works>,
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// Also some drama:
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// <https://www.pcg-random.org/posts/on-vignas-pcg-critique.html>,
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// <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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// if one wanted to generate 32 bit ints,
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// from which to generate floats,
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// one could do the following:
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uint32_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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}
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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 = sqrt(-2.0 * log(u1)) * sin(2.0 * 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_normal_from_90_ci(double low, double high, uint64_t* seed)
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{
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// Explanation of key idea:
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// 1. We know that the 90% confidence interval of the unit normal is
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// [-1.6448536269514722, 1.6448536269514722]
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// see e.g.: https://stackoverflow.com/questions/20626994/how-to-calculate-the-inverse-of-the-normal-cumulative-distribution-function-in-p
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// or https://www.wolframalpha.com/input?i=N%5BInverseCDF%28normal%280%2C1%29%2C+0.05%29%2C%7B%E2%88%9E%2C100%7D%5D
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// 2. So if we take a unit normal and multiply it by
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// L / 1.6448536269514722, its new 90% confidence interval will be
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// [-L, L], i.e., length 2 * L
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// 3. Instead, if we want to get a confidence interval of length L,
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// we should multiply the unit normal by
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// L / (2 * 1.6448536269514722)
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// Meaning that its standard deviation should be multiplied by that amount
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// see: https://en.wikipedia.org/wiki/Normal_distribution?lang=en#Operations_on_a_single_normal_variable
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// 4. So we have learnt that Normal(0, L / (2 * 1.6448536269514722))
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// has a 90% confidence interval of length L
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// 5. If we want a 90% confidence interval from high to low,
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// we can set mean = (high + low)/2; the midpoint, and L = high-low,
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// Normal([high + low]/2, [high - low]/(2 * 1.6448536269514722))
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double mean = (high + low) * 0.5;
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double std = (high - low) / (2.0 * NORMAL90CONFIDENCE);
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return sample_normal(mean, std, 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 lognorma with a matching 90% c.i.
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// Key idea: If we want a lognormal with 90% confidence interval [a, b]
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// we need but get a normal with 90% confidence interval [log(a), log(b)].
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// Then see code for sample_normal_from_90_ci
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double loglow = log(low);
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double loghigh = log(high);
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return exp(sample_normal_from_90_ci(loglow, loghigh, 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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// See: https://en.wikipedia.org/wiki/Gamma_distribution#Related_distributions
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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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double sample_laplace(double successes, double failures, uint64_t* seed)
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{
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// see <https://en.wikipedia.org/wiki/Beta_distribution?lang=en#Rule_of_succession>
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return sample_beta(successes + 1, failures + 1, seed);
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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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// Sample from samples with frequency proportional to their weights.
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double sum_weights = array_sum(weights, n_dists);
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double* cumsummed_normalized_weights = (double*)malloc((size_t)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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