distributions.h

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#ifndef THEORETICA_DISTRIBUTIONS
#define THEORETICA_DISTRIBUTIONS
 
#include "../core/real_analysis.h"
#include "../algebra/vec.h"
#include "./statistics.h"
#include "../core/function.h"
#include "../core/special.h"
 
 
namespace theoretica {
 
 
    namespace stats {
 
        inline real likelihood(const vec<real>& X, const vec<real>& theta, stat_function f) {
 
            real res = 1;
 
            for (unsigned int i = 0; i < X.size(); ++i)
                res *= f(X[i], theta);
 
            return res;
        }
 
 
        inline real log_likelihood(const vec<real>& X, const vec<real>& theta, stat_function f) {
 
            real res = 0;
 
            for (unsigned int i = 0; i < X.size(); ++i)
                res += ln(f(X[i], theta));
 
            return res;
        }
    }
 
 
    namespace distribution {
    
 
        inline real gaussian(real x, real X, real sigma) {
 
            return (1.0 / (sigma * SQRT2 * SQRTPI))
                * exp(-square(x - X) / (2 * square(sigma)));
        }
 
 
        inline real gaussian(real x, const vec<real>& theta) {
 
            if(theta.size() != 2) {
                TH_MATH_ERROR("distribution::gaussian", theta.size(), INVALID_ARGUMENT);
                return nan();
            }
 
            return gaussian(x, theta[0], theta[1]);
        }
 
 
        inline real bernoulli(unsigned int k, real p) {
 
            return pow(p, k) * pow(1 - p, 1 - k);
        }
 
 
        inline real bernoulli(real k, const vec<real>& theta) {
 
            if(theta.size() != 1) {
                TH_MATH_ERROR("distribution::bernoulli", theta.size(), INVALID_ARGUMENT);
                return nan();
            }
 
            return bernoulli(static_cast<unsigned int>(k), theta[0]);
 
        }
 
 
        inline real poisson(unsigned int k, real lambda) {
 
            return exp(-lambda) * pow(lambda, k) / (real) fact(k);
        }
 
 
        inline real poisson(real k, const vec<real>& theta) {
 
            if(theta.size() != 1) {
                TH_MATH_ERROR("distribution::poisson", theta.size(), INVALID_ARGUMENT);
                return nan();
            }
 
            return poisson(static_cast<unsigned int>(k), theta[0]);
        }
 
 
        inline real binomial(unsigned int nu, unsigned int n, real p) {
 
            return binomial_coeff(n, nu) *
                pow(p, nu) * pow(1 - p, n - nu);
        }
 
 
        inline real binomial(real nu, const vec<real>& theta) {
 
            if(theta.size() != 2) {
                TH_MATH_ERROR("distribution::binomial", theta.size(), INVALID_ARGUMENT);
                return nan();
            }
 
            return binomial(static_cast<unsigned int>(nu),
                static_cast<unsigned int>(theta[0]), theta[1]);
        }
 
 
        inline real multinomial(
            std::vector<unsigned int> x,
            unsigned int n,
            unsigned int k,
            std::vector<real> p) {
 
            if(x.size() != p.size() || x.size() != k) {
                TH_MATH_ERROR("distribution::multinomial", x.size(), INVALID_ARGUMENT);
                return nan();
            }
 
            real res = fact(n);
 
            for (unsigned int i = 0; i < k; ++i)
                res *= pow(p[i], x[i]) / fact(x[i]);
 
            return res;
        }
 
 
        inline real chi_squared(real x, unsigned int k) {
 
            if(x < 0)
                return 0;
 
            const real coeff = special::half_gamma(k);
 
            if(k % 2 == 0)
                return pow(x, int(k / 2) - 1) * exp(-x / 2.0)
                    / (pow(SQRT2, k) * coeff);
            else
                return pow(sqrt(x), k - 2) * exp(-x / 2.0)
                    / (pow(SQRT2, k) * coeff);
        }
 
 
        inline real chi_squared(real x, unsigned int k, real half_gamma_k) {
 
            if(x < 0)
                return 0;
 
            if(k % 2 == 0)
                return pow(x, int(k / 2) - 1) * exp(-x / 2.0)
                    / (pow(SQRT2, k) * half_gamma_k);
            else
                return pow(sqrt(x), k - 2) * exp(-x / 2.0)
                    / (pow(SQRT2, k) * half_gamma_k);
        }
 
 
        inline real chi_squared(real x, const vec<real>& theta) {
 
            if(theta.size() != 1) {
                TH_MATH_ERROR("distribution::chi_squared", theta.size(), INVALID_ARGUMENT);
                return nan();
            }
 
            return chi_squared(x, static_cast<unsigned int>(theta[0]));
        }
 
 
        inline real gamma(real x, real alpha, real beta) {
 
            return powf(beta, alpha) * powf(x, alpha - 1)
                * exp(-beta * x) / special::gamma(alpha);
        }
 
 
        inline real gamma(real x, const vec<real>& theta) {
 
            if(theta.size() != 2) {
                TH_MATH_ERROR("distribution::gamma", theta.size(), INVALID_ARGUMENT);
                return nan();
            }
 
            return gamma(x, theta[0], theta[1]);
        }
 
 
        inline real beta(real x, real alpha, real beta) {
 
            return powf(x, alpha - 1) * powf(1 - x, beta - 1)
                / special::beta(alpha, beta);
        }
 
 
        inline real beta(real x, const vec<real>& theta) {
 
            if(theta.size() != 2) {
                TH_MATH_ERROR("distribution::beta", theta.size(), INVALID_ARGUMENT);
                return nan();
            }
 
            return beta(x, theta[0], theta[1]);
        }
 
 
        inline real student(real x, unsigned int nu) {
 
            const real a = 1 + (x * x / nu);
 
            return (special::half_gamma(nu + 1) / special::half_gamma(nu))
                    * pow(sqrt(a), -nu - 1) / (sqrt(nu) * SQRTPI);
        }
 
 
        inline real student(real x, const vec<real>& theta) {
 
            if(theta.size() != 1) {
                TH_MATH_ERROR("distribution::student", theta.size(), INVALID_ARGUMENT);
                return nan();
            }
 
            return student(x, static_cast<unsigned int>(theta[0]));
        }
 
 
        inline real log_normal(real x, real mu, real sigma) {
 
            return 1.0 / (2.50662827463 * sigma * x) *
                exp(-square(ln(x) - mu) / (2 * square(sigma)));
        }
 
 
        inline real log_normal(real x, const vec<real>& theta) {
 
            if(theta.size() != 2) {
                TH_MATH_ERROR("distribution::log_normal", theta.size(), INVALID_ARGUMENT);
                return nan();
            }
 
            return log_normal(x, theta[0], theta[1]);
        }
 
 
        inline real exponential(real x, real lambda) {
 
            if(x < 0)
                return 0;
 
            return lambda * exp(-lambda * x);
        }
 
 
        inline real exponential(real x, const vec<real>& theta) {
 
            if(theta.size() != 1) {
                TH_MATH_ERROR("distribution::exponential", theta.size(), INVALID_ARGUMENT);
                return nan();
            }
 
            return exponential(x, theta[0]);
        }
 
 
        inline real rayleigh(real x, real sigma) {
 
            if(x < 0)
                return 0;
 
            if(sigma < MACH_EPSILON) {
                TH_MATH_ERROR("distribution::rayleigh", sigma, DIV_BY_ZERO);
                return nan();
            }
 
            return x * exp(-square(x / sigma) / 2) / square(sigma);
        }
 
 
        inline real rayleigh(real x, const vec<real>& theta) {
 
            if(theta.size() != 1) {
                TH_MATH_ERROR("distribution::rayleigh", theta.size(), INVALID_ARGUMENT);
                return nan();
            }
 
            return rayleigh(x, theta[0]);
        }
 
 
        inline real cauchy(real x, real mu, real alpha) {
 
            return 1.0 / (PI * alpha * (1 + (square(x - mu) / square(alpha))));
        }
 
 
        inline real cauchy(real x, const vec<real>& theta) {
 
            if(theta.size() != 2) {
                TH_MATH_ERROR("distribution::cauchy", theta.size(), INVALID_ARGUMENT);
                return nan();
            }
 
            return cauchy(x, theta[0], theta[1]);
        }
 
 
        inline real breit_wigner(real x, real M, real Gamma) {
 
            return Gamma / (2 * PI * (square(x - M) + square(Gamma / 2.0)));
        }
 
 
        inline real breit_wigner(real x, const vec<real>& theta) {
 
            if(theta.size() != 2) {
                TH_MATH_ERROR("distribution::breit_wigner", theta.size(), INVALID_ARGUMENT);
                return nan();
            }
 
            return breit_wigner(x, theta[0], theta[1]);
        }
 
 
        inline real maxwell(real x, real a) {
 
            return (SQRT2 / SQRTPI) * square(x) * exp(-square(x / a) / 2.0) / cube(a);
        }
 
 
        inline real maxwell(real x, const vec<real>& theta) {
 
            if(theta.size() != 1) {
                TH_MATH_ERROR("distribution::maxwell", theta.size(), INVALID_ARGUMENT);
                return nan();
            }
 
            return maxwell(x, theta[0]);
        }
 
 
        inline real laplace(real x, real mu, real b) {
 
            return (1.0 / (2.0 * b)) * exp(-abs(x - mu) / b);
        }
 
 
        inline real laplace(real x, const vec<real>& theta) {
 
            if(theta.size() != 2) {
                TH_MATH_ERROR("distribution::laplace", theta.size(), INVALID_ARGUMENT);
                return nan();
            }
 
            return (1.0 / (2.0 * theta[1])) * exp(-abs(x - theta[0]) / theta[1]);
        }
 
 
        inline real pareto(real x, real x_m, real alpha) {
 
            if(alpha <= 0) {
                TH_MATH_ERROR("distribution::pareto", alpha, OUT_OF_DOMAIN);
                return nan();
            }
 
            if(x < x_m)
                return 0;
 
            return alpha * powf(x_m, alpha) / powf(x, alpha + 1);
        }
 
 
        inline real pareto(real x, const vec<real>& theta) {
 
            if(theta.size() != 2) {
                TH_MATH_ERROR("distribution::pareto", theta.size(), INVALID_ARGUMENT);
                return nan();
            }
 
            return pareto(x, theta[0], theta[1]);
        }
 
 
        inline real erlang(real x, unsigned int k, real lambda) {
 
            if(!k) {
                TH_MATH_ERROR("distribution::erlang", k, INVALID_ARGUMENT);
                return nan();
            }
 
            return pow(lambda, k) * pow(x, k - 1)
                    * exp(-lambda * x) / fact(k - 1);
        }
 
 
        inline real erlang(real x, const vec<real>& theta) {
 
            if(theta.size() != 2) {
                TH_MATH_ERROR("distribution::erlang", theta.size(), INVALID_ARGUMENT);
                return nan();
            }
 
            return erlang(x, static_cast<unsigned int>(theta[0]), theta[1]);
        }
 
    }
}
 
#endif