squiggle.c

squiggle.c (7942B)

---

 #include <limits.h>
 #include <math.h>
 #include <stdint.h>
 #include <stdlib.h>
 
 // Defs
 #define PI 3.14159265358979323846 // M_PI in gcc gnu99
 #define NORMAL90CONFIDENCE 1.6448536269514727
 #define UNUSED(x) (void)(x)
 // ^ https://stackoverflow.com/questions/3599160/how-can-i-suppress-unused-parameter-warnings-in-c
 
 // Pseudo Random number generators
 static uint64_t xorshift64(uint64_t* seed)
 {
     // Algorithm "xor" from p. 4 of Marsaglia, "Xorshift RNGs"
     // See:
     //   <https://en.wikipedia.org/wiki/Xorshift>
     //   <https://stackoverflow.com/questions/53886131/how-does-xorshift32-works>,
     // Also some drama:
     //   <https://www.pcg-random.org/posts/on-vignas-pcg-critique.html>,
     //   <https://prng.di.unimi.it/>
     uint64_t x = *seed;
     x ^= x << 13;
     x ^= x >> 7;
     x ^= x << 17;
     return *seed = x;
 
     /* 
     // if one wanted to generate 32 bit ints, 
     // from which to generate floats,
     // one could do the following: 
     uint32_t x = *seed;
     x ^= x << 13;
     x ^= x >> 17;
     x ^= x << 5;
     return *seed = x;
     */
 }
 
 // Distribution & sampling functions
 // Unit distributions
 double sample_unit_uniform(uint64_t* seed)
 {
     // samples uniform from [0,1] interval.
     return ((double)xorshift64(seed)) / ((double)UINT64_MAX);
 }
 
 double sample_unit_normal(uint64_t* seed)
 {
     // // See: <https://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform>
     double u1 = sample_unit_uniform(seed);
     double u2 = sample_unit_uniform(seed);
     double z = sqrt(-2.0 * log(u1)) * sin(2.0 * PI * u2);
     return z;
 }
 
 // Composite distributions
 double sample_uniform(double start, double end, uint64_t* seed)
 {
     return sample_unit_uniform(seed) * (end - start) + start;
 }
 
 double sample_normal(double mean, double sigma, uint64_t* seed)
 {
     return (mean + sigma * sample_unit_normal(seed));
 }
 
 double sample_lognormal(double logmean, double logstd, uint64_t* seed)
 {
     return exp(sample_normal(logmean, logstd, seed));
 }
 
 double sample_normal_from_90_ci(double low, double high, uint64_t* seed)
 {
     // Explanation of key idea:
     // 1. We know that the 90% confidence interval of the unit normal is
     // [-1.6448536269514722, 1.6448536269514722]
     // see e.g.: https://stackoverflow.com/questions/20626994/how-to-calculate-the-inverse-of-the-normal-cumulative-distribution-function-in-p
     // 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
     // 2. So if we take a unit normal and multiply it by
     // L / 1.6448536269514722, its new 90% confidence interval will be
     // [-L, L], i.e., length 2 * L
     // 3. Instead, if we want to get a confidence interval of length L,
     // we should multiply the unit normal by
     // L / (2 * 1.6448536269514722)
     // Meaning that its standard deviation should be multiplied by that amount
     // see: https://en.wikipedia.org/wiki/Normal_distribution?lang=en#Operations_on_a_single_normal_variable
     // 4. So we have learnt that Normal(0, L / (2 * 1.6448536269514722))
     // has a 90% confidence interval of length L
     // 5. If we want a 90% confidence interval from high to low,
     // we can set mean = (high + low)/2; the midpoint, and L = high-low,
     // Normal([high + low]/2, [high - low]/(2 * 1.6448536269514722))
     double mean = (high + low) * 0.5;
     double std = (high - low) / (2.0 * NORMAL90CONFIDENCE);
     return sample_normal(mean, std, seed);
 }
 
 double sample_to(double low, double high, uint64_t* seed)
 {
     // Given a (positive) 90% confidence interval,
     // returns a sample from a lognorma with a matching 90% c.i.
     // Key idea: If we want a lognormal with 90% confidence interval [a, b]
     // we need but get a normal with 90% confidence interval [log(a), log(b)].
     // Then see code for sample_normal_from_90_ci
     double loglow = log(low);
     double loghigh = log(high);
     return exp(sample_normal_from_90_ci(loglow, loghigh, seed));
 }
 
 double sample_gamma(double alpha, uint64_t* seed)
 {
 
     // A Simple Method for Generating Gamma Variables, Marsaglia and Wan Tsang, 2001
     // https://dl.acm.org/doi/pdf/10.1145/358407.358414
     // see also the references/ folder
     // Note that the Wikipedia page for the gamma distribution includes a scaling parameter
     // k or beta
     // https://en.wikipedia.org/wiki/Gamma_distribution
     // such that gamma_k(alpha, k) = k * gamma(alpha)
     // or gamma_beta(alpha, beta) = gamma(alpha) / beta
     // So far I have not needed to use this, and thus the second parameter is by default 1.
     if (alpha >= 1) {
         double d, c, x, v, u;
         d = alpha - 1.0 / 3.0;
         c = 1.0 / sqrt(9.0 * d);
         while (1) {
 
             do {
                 x = sample_unit_normal(seed);
                 v = 1.0 + c * x;
             } while (v <= 0.0);
 
             v = v * v * v;
             u = sample_unit_uniform(seed);
             if (u < 1.0 - 0.0331 * (x * x * x * x)) { // Condition 1
                 // the 0.0331 doesn't inspire much confidence
                 // however, this isn't the whole story
                 // by knowing that Condition 1 implies condition 2
                 // we realize that this is just a way of making the algorithm faster
                 // i.e., of not using the logarithms
                 return d * v;
             }
             if (log(u) < 0.5 * (x * x) + d * (1.0 - v + log(v))) { // Condition 2
                 return d * v;
             }
         }
     } else {
         return sample_gamma(1 + alpha, seed) * pow(sample_unit_uniform(seed), 1 / alpha);
         // see note in p. 371 of https://dl.acm.org/doi/pdf/10.1145/358407.358414
     }
 }
 
 double sample_beta(double a, double b, uint64_t* seed)
 {
     // See: https://en.wikipedia.org/wiki/Gamma_distribution#Related_distributions
     double gamma_a = sample_gamma(a, seed);
     double gamma_b = sample_gamma(b, seed);
     return gamma_a / (gamma_a + gamma_b);
 }
 
 double sample_laplace(double successes, double failures, uint64_t* seed)
 {
     // see <https://en.wikipedia.org/wiki/Beta_distribution?lang=en#Rule_of_succession>
     return sample_beta(successes + 1, failures + 1, seed);
 }
 
 // Array helpers
 double array_sum(double* array, int length)
 {
     double sum = 0.0;
     for (int i = 0; i < length; i++) {
         sum += array[i];
     }
     return sum;
 }
 
 void array_cumsum(double* array_to_sum, double* array_cumsummed, int length)
 {
     array_cumsummed[0] = array_to_sum[0];
     for (int i = 1; i < length; i++) {
         array_cumsummed[i] = array_cumsummed[i - 1] + array_to_sum[i];
     }
 }
 
 double array_mean(double* array, int length)
 {
     double sum = array_sum(array, length);
     return sum / length;
 }
 
 double array_std(double* array, int length)
 {
     double mean = array_mean(array, length);
     double std = 0.0;
     for (int i = 0; i < length; i++) {
         std += (array[i] - mean) * (array[i] - mean);
     }
     std = sqrt(std / length);
     return std;
 }
 
 // Mixture function
 double sample_mixture(double (*samplers[])(uint64_t*), double* weights, int n_dists, uint64_t* seed)
 {
     // Sample from samples with frequency proportional to their weights.
     double sum_weights = array_sum(weights, n_dists);
     double* cumsummed_normalized_weights = (double*)malloc((size_t)n_dists * sizeof(double));
     cumsummed_normalized_weights[0] = weights[0] / sum_weights;
     for (int i = 1; i < n_dists; i++) {
         cumsummed_normalized_weights[i] = cumsummed_normalized_weights[i - 1] + weights[i] / sum_weights;
     }
 
     double result;
     int result_set_flag = 0;
     double p = sample_uniform(0, 1, seed);
     for (int k = 0; k < n_dists; k++) {
         if (p < cumsummed_normalized_weights[k]) {
             result = samplers[k](seed);
             result_set_flag = 1;
             break;
         }
     }
     if (result_set_flag == 0)
         result = samplers[n_dists - 1](seed);
 
     free(cumsummed_normalized_weights);
     return result;
 }