squiggle.c/squiggle.c

#include <float.h>
#include <limits.h>
#include <math.h>
#include <stdint.h>
#include <stdio.h>
#include <stdlib.h>
#include <sys/types.h>
#include <time.h>

// Some error niceties; these won't be used until later
#define MAX_ERROR_LENGTH 500
#define EXIT_ON_ERROR 0
#define PROCESS_ERROR(error_msg) process_error(error_msg, EXIT_ON_ERROR, __FILE__, __LINE__)

#define PI 3.14159265358979323846 // M_PI in gcc gnu99
#define NORMAL90CONFIDENCE 1.6448536269514722

// # Key functionality
// Define the minimum number of functions needed to do simple estimation
// Starts here, ends until the end of the mixture function

// Pseudo Random number generator
uint64_t xorshift32(uint32_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 >> 17;
    x ^= x << 5;
    return *seed = x;
}

uint64_t xorshift64(uint64_t* seed)
{
    // same as above, but for generating doubles instead of floats
    uint64_t x = *seed;
    x ^= x << 13;
    x ^= x >> 7;
    x ^= x << 17;
    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 = sqrtf(-2.0 * log(u1)) * sin(2 * 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));
}

inline double sample_normal_from_90_confidence_interval(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
    // 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) / 2.0;
    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_95_confidence_interval
    double loglow = logf(low);
    double loghigh = logf(high);
    return exp(sample_normal_from_90_confidence_interval(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(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;
}

// # More cool stuff
// This is no longer necessary to do basic estimation,
// but is still cool

// ## Sample from an arbitrary cdf
struct box {
    int empty;
    double content;
    char* error_msg;
};

struct box process_error(const char* error_msg, int should_exit, char* file, int line)
{
    if (should_exit) {
        printf("@, in %s (%d)", file, line);
        exit(1);
    } else {
        char error_msg[MAX_ERROR_LENGTH];
        snprintf(error_msg, MAX_ERROR_LENGTH, "@, in %s (%d)", file, line); // NOLINT: We are being carefull here by considering MAX_ERROR_LENGTH explicitly.
        struct box error = { .empty = 1, .error_msg = error_msg };
        return error;
    }
}

// Inverse cdf at point
// Two versions of this function:
//   - raw, dealing with cdfs that return doubles
//     - input: cdf: double => double, p
//     - output: Box(number|error)
//   - box, dealing with cdfs that return a box.
//     - input: cdf: double => Box(number|error), p
//     - output: Box(number|error)
struct box inverse_cdf_double(double cdf(double), double p)
{
    // given a cdf: [-Inf, Inf] => [0,1]
    // returns a box with either
    // x such that cdf(x) = p
    // or an error
    // if EXIT_ON_ERROR is set to 1, it exits instead of providing an error

    double low = -1.0;
    double high = 1.0;

    // 1. Make sure that cdf(low) < p < cdf(high)
    int interval_found = 0;
    while ((!interval_found) && (low > -FLT_MAX / 4) && (high < FLT_MAX / 4)) {
        // ^ Using FLT_MIN and FLT_MAX is overkill
        // but it's also the *correct* thing to do.

        int low_condition = (cdf(low) < p);
        int high_condition = (p < cdf(high));
        if (low_condition && high_condition) {
            interval_found = 1;
        } else if (!low_condition) {
            low = low * 2;
        } else if (!high_condition) {
            high = high * 2;
        }
    }

    if (!interval_found) {
        return PROCESS_ERROR("Interval containing the target value not found, in function inverse_cdf");
    } else {

        int convergence_condition = 0;
        int count = 0;
        while (!convergence_condition && (count < (INT_MAX / 2))) {
            double mid = (high + low) / 2;
            int mid_not_new = (mid == low) || (mid == high);
            // double width = high - low;
            // if ((width < 1e-8) || mid_not_new){
            if (mid_not_new) {
                convergence_condition = 1;
            } else {
                double mid_sign = cdf(mid) - p;
                if (mid_sign < 0) {
                    low = mid;
                } else if (mid_sign > 0) {
                    high = mid;
                } else if (mid_sign == 0) {
                    low = mid;
                    high = mid;
                }
            }
        }

        if (convergence_condition) {
            struct box result = { .empty = 0, .content = low };
            return result;
        } else {
            return PROCESS_ERROR("Search process did not converge, in function inverse_cdf");
        }
    }
}

struct box inverse_cdf_box(struct box cdf_box(double), double p)
{
    // given a cdf: [-Inf, Inf] => Box([0,1])
    // returns a box with either
    // x such that cdf(x) = p
    // or an error
    // if EXIT_ON_ERROR is set to 1, it exits instead of providing an error

    double low = -1.0;
    double high = 1.0;

    // 1. Make sure that cdf(low) < p < cdf(high)
    int interval_found = 0;
    while ((!interval_found) && (low > -FLT_MAX / 4) && (high < FLT_MAX / 4)) {
        // ^ Using FLT_MIN and FLT_MAX is overkill
        // but it's also the *correct* thing to do.
        struct box cdf_low = cdf_box(low);
        if (cdf_low.empty) {
            return PROCESS_ERROR(cdf_low.error_msg);
        }

        struct box cdf_high = cdf_box(high);
        if (cdf_high.empty) {
            return PROCESS_ERROR(cdf_low.error_msg);
        }

        int low_condition = (cdf_low.content < p);
        int high_condition = (p < cdf_high.content);
        if (low_condition && high_condition) {
            interval_found = 1;
        } else if (!low_condition) {
            low = low * 2;
        } else if (!high_condition) {
            high = high * 2;
        }
    }

    if (!interval_found) {
        return PROCESS_ERROR("Interval containing the target value not found, in function inverse_cdf");
    } else {

        int convergence_condition = 0;
        int count = 0;
        while (!convergence_condition && (count < (INT_MAX / 2))) {
            double mid = (high + low) / 2;
            int mid_not_new = (mid == low) || (mid == high);
            // double width = high - low;
            if (mid_not_new) {
                // if ((width < 1e-8) || mid_not_new){
                convergence_condition = 1;
            } else {
                struct box cdf_mid = cdf_box(mid);
                if (cdf_mid.empty) {
                    return PROCESS_ERROR(cdf_mid.error_msg);
                }
                double mid_sign = cdf_mid.content - p;
                if (mid_sign < 0) {
                    low = mid;
                } else if (mid_sign > 0) {
                    high = mid;
                } else if (mid_sign == 0) {
                    low = mid;
                    high = mid;
                }
            }
        }

        if (convergence_condition) {
            struct box result = { .empty = 0, .content = low };
            return result;
        } else {
            return PROCESS_ERROR("Search process did not converge, in function inverse_cdf");
        }
    }
}

// Sampler based on inverse cdf and randomness function
struct box sampler_cdf_box(struct box cdf(double), uint64_t* seed)
{
    double p = sample_unit_uniform(seed);
    struct box result = inverse_cdf_box(cdf, p);
    return result;
}
struct box sampler_cdf_double(double cdf(double), uint64_t* seed)
{
    double p = sample_unit_uniform(seed);
    struct box result = inverse_cdf_double(cdf, p);
    return result;
}

/* Could also define other variations, e.g.,
double sampler_danger(struct box cdf(double), uint64_t* seed)
{
    double p = sample_unit_uniform(seed);
    struct box result = inverse_cdf_box(cdf, p);
		if(result.empty){
			exit(1);
		}else{
			return result.content;
		}
}
*/

// Get confidence intervals, given a sampler

typedef struct ci_t {
    float low;
    float high;
} ci;
int compare_doubles(const void* p, const void* q)
{
    // https://wikiless.esmailelbob.xyz/wiki/Qsort?lang=en
    double x = *(const double*)p;
    double y = *(const double*)q;

    /* Avoid return x - y, which can cause undefined behaviour
       because of signed integer overflow. */
    if (x < y)
        return -1; // Return -1 if you want ascending, 1 if you want descending order.
    else if (x > y)
        return 1; // Return 1 if you want ascending, -1 if you want descending order.

    return 0;
}
ci get_90_confidence_interval(double (*sampler)(uint64_t*), uint64_t* seed)
{
    int n = 100 * 1000;
    double* samples_array = malloc(n * sizeof(double));
    for (int i = 0; i < n; i++) {
        samples_array[i] = sampler(seed);
    }
    qsort(samples_array, n, sizeof(double), compare_doubles);

    ci result = {
        .low = samples_array[5000],
        .high = samples_array[94999],
    };
    free(samples_array);

    return result;
}

// # Small algebra manipulations

// here I discover named structs,
// which mean that I don't have to be typing
// struct blah all the time.
typedef struct normal_params_t {
    double mean;
    double std;
} normal_params;

normal_params algebra_sum_normals(normal_params a, normal_params b)
{
    normal_params result = {
        .mean = a.mean + b.mean,
        .std = sqrt((a.std * a.std) + (b.std * b.std)),
    };
    return result;
}

typedef struct lognormal_params_t {
    double logmean;
    double logstd;
} lognormal_params;

lognormal_params algebra_product_lognormals(lognormal_params a, lognormal_params b)
{
    lognormal_params result = {
        .logmean = a.logmean + b.logmean,
        .logstd = sqrt((a.logstd * a.logstd) + (b.logstd * b.logstd)),
    };
    return result;
}

lognormal_params convert_ci_to_lognormal_params(ci x)
{
    double loghigh = logf(x.high);
    double loglow = logf(x.low);
    double logmean = (loghigh + loglow) / 2.0;
    double logstd = (loghigh - loglow) / (2.0 * NORMAL90CONFIDENCE);
    lognormal_params result = { .logmean = logmean, .logstd = logstd}; 
    return result;
}

ci convert_lognormal_params_to_ci(lognormal_params y)
{
    double h = y.logstd * NORMAL90CONFIDENCE;
    double loghigh = y.logmean + h;
    double loglow = y.logmean - h;
    ci result = { .low=exp(loglow), .high=exp(loghigh)};
    return result;

}