IDistribution2DSampler.cpp

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//  ************************************************************************************************
//
//  BornAgain: simulate and fit reflection and scattering
//
//! @file      Sample/Correlations/IDistribution2DSampler.cpp
//! @brief     Defines interface class IProfile1D, and children thereof
//!
//! @homepage  http://www.bornagainproject.org
//! @license   GNU General Public License v3 or higher (see COPYING)
//! @copyright Forschungszentrum Jülich GmbH 2018
//! @authors   Scientific Computing Group at MLZ (see CITATION, AUTHORS)
//
//  ************************************************************************************************
 
#include "Sample/Correlations/IDistribution2DSampler.h"
#include "Base/Math/Constants.h"
#include <random>
 
namespace {
 
double sigma_scale = 3.0;
size_t n_boxes = 256; // number of boxes for Ziggurat sampling
 
struct ZigguratBox {
    ZigguratBox(double x_min, double x_max, double y_max, double y_lower)
        : m_x_min(x_min)
        , m_x_max(x_max)
        , m_y_max(y_max)
        , m_y_lower(y_lower)
    {
    }
 
    double m_x_min; // left edge of the box
    double m_x_max; // right edge of the box
    // m_y_min is inherently 0 for every box and hence has not been defined
    double m_y_max;   // height of box
    double m_y_lower; // minimum height of the box for which points below that height
                      // are located below the density function curve in the box
};
 
std::pair<double, double> samplingZiggurat(double r, double x_func_max, double (*func_phi)(double))
{
    // This sampling is based on vertical boxes instead of the conventional
    // Ziggurat sampling that is done with horizontal boxes
 
    std::random_device rd;  // random device class instance
    std::mt19937 gen(rd()); // Standard mersenne_twister_engine seeded with rd()
    std::uniform_real_distribution<double> uniformDist(0.0, 1.0);
 
    double box_width = (x_func_max + r) / n_boxes; // r = rightmost box's right-edge from x_func_max
    std::vector<ZigguratBox> boxes;
    std::vector<double> cum_area_vector;
 
    double x_min = 0, x_max = 0, y_max = 0, y_lower = 0, cum_area_box = 0;
 
    // Establising vectors of boxes and cumulative area (probability of each box) for Ziggurat
    // sampling
    for (size_t i = 0; i < n_boxes; ++i) {
        if (i != 0)
            x_min = x_max;
 
        x_max += box_width;
 
        if (x_func_max >= x_max) {
            y_max = func_phi(x_max);
            y_lower = func_phi(x_min);
        } else if (x_func_max > x_min && x_func_max <= x_max) {
            y_max = func_phi(x_func_max);
            y_lower = std::min(func_phi(x_min), func_phi(x_max));
        } else {
            y_max = func_phi(x_min);
            y_lower = func_phi(x_max);
        }
 
        boxes.emplace_back(ZigguratBox(x_min, x_max, y_max, y_lower));
 
        cum_area_box += box_width * y_max;
        cum_area_vector.emplace_back(cum_area_box);
    }
 
    // Normalizing the cumulative area to 1
    for (size_t i = 0; i < n_boxes; ++i)
        cum_area_vector[i] = cum_area_vector[i] / cum_area_vector.back();
 
    // Sampling a phi value
    double phi = 0;
    bool solnFound(false);
 
    while (!solnFound) {
        double random_cum_area = uniformDist(gen);
        for (size_t i = 0; i < n_boxes; ++i) {
            if (random_cum_area <= cum_area_vector[i]) {
                double random_y = uniformDist(gen) * boxes[i].m_y_max;
 
                std::uniform_real_distribution<double> uniformDistAB(boxes[i].m_x_min,
                                                                     boxes[i].m_x_max);
                double phi_attempt = uniformDistAB(gen);
 
                if (random_y <= boxes[i].m_y_lower) {
                    phi = phi_attempt;
                    solnFound = true;
                } else {
                    if (random_y <= func_phi(phi_attempt)) {
                        phi = phi_attempt;
                        solnFound = true;
                    }
                }
                break;
            }
        }
    }
 
    // Sampling an alpha value
    double alpha = M_TWOPI * uniformDist(gen);
    return std::make_pair(phi, alpha);
}
 
double func_phi_Cauchy(double phi)
{
    // The independent "phi" density function of the 2D Cauchy distribution
    return phi * std::exp(-phi);
}
 
double func_phi_Cone(double phi)
{
    // The independent "phi" density function of the 2D Cone distribution
    return 6 * (1 - phi) * phi;
}
 
} // namespace
 
 
IDistribution2DSampler::~IDistribution2DSampler() = default;
 
std::pair<double, double> Distribution2DCauchySampler::randomSample() const
{
    // Use Ziggurat sampling instead of Inverse Transform Sampling (ITS requires numerical solver)
 
    double phi_max_Cauchy = 1.0;
    // rightmost box's right-edge from phi_max_Cauchy for Ziggurat Sampling
    double r = sigma_scale * std::sqrt(2); // standard dev of func_phi_Cauchy is sqrt(2)
    std::pair<double, double> samples = samplingZiggurat(r, phi_max_Cauchy, func_phi_Cauchy);
    return std::make_pair(m_omega_x * samples.first * std::cos(samples.second),
                          m_omega_y * samples.first * std::sin(samples.second));
}
 
std::pair<double, double> Distribution2DGaussSampler::randomSample() const
{
    std::random_device rd;  // random device class instance
    std::mt19937 gen(rd()); // Standard mersenne_twister_engine seeded with rd()
    std::uniform_real_distribution<double> uniformDist(0.0, 1.0);
 
    double cdf_value_phi = uniformDist(gen);
 
    // Use ITS and solve for phi from the cdf of radial (phi) distribution
    double phi = std::sqrt(-2 * std::log(1 - cdf_value_phi));
    double alpha = M_TWOPI * uniformDist(gen);
    return std::make_pair(m_omega_x * phi * std::cos(alpha), m_omega_y * phi * std::sin(alpha));
}
 
std::pair<double, double> Distribution2DGateSampler::randomSample() const
{
    std::random_device rd;  // random device class instance
    std::mt19937 gen(rd()); // Standard mersenne_twister_engine seeded with rd()
    std::uniform_real_distribution<double> uniformDist(0.0, 1.0);
 
    double cdf_value_phi = uniformDist(gen);
 
    // Use ITS and solve for phi from the cdf of radial (phi) distribution
    double phi = std::sqrt(cdf_value_phi);
    double alpha = M_TWOPI * uniformDist(gen);
    return std::make_pair(m_omega_x * phi * std::cos(alpha), m_omega_y * phi * std::sin(alpha));
}
 
std::pair<double, double> Distribution2DConeSampler::randomSample() const
{
    // Use Ziggurat sampling instead of Inverse Transform Sampling (ITS requires numerical solver)
 
    double phi_max_Cone = 0.5;
    // rightmost box's right-edge from phi_max_Cone for Ziggurat Sampling
    double r = 0.5;
    std::pair<double, double> samples = samplingZiggurat(r, phi_max_Cone, func_phi_Cone);
    return std::make_pair(m_omega_x * samples.first * std::cos(samples.second),
                          m_omega_y * samples.first * std::sin(samples.second));
}