Mustaqil ish s in C++ C++ da statistik taqsimotlar

Standart normal taqsimotni amalga oshirish
Birinchidan, standart normal taqsimlash uchun amalga oshirishimiz kerak bo'lgan turli usullarning formulasini qisqacha ko'rib chiqamiz. Standart normal taqsimotning ehtimollik zichligi funksiyasi quyidagicha beriladi:

Yig’indi zichlik funktsiyasi quyidagilar tomonidan beriladi:

Standart normal taqsimlashning (ehtimollik funktsiyasi deb ham ataladi) invers kümülatif zichligi funktsiyasi biroz ko'proq ishtirok etadi. Bu muayyan funktsiya uchun hech qanday analitik formula mavjud emas va shuning uchun u raqamli usullar bilan taxmin qilinishi kerak. Korn[3] da uchraydigan Beasley-Springer-Moro algoritmidan foydalanamiz. Men bu erda batafsil ta'riflamayman, chunki tafsilotlar Korn[3] ga tegishli.
Biz standart normal taqsimot bilan shug'ullanayotganimizni hisobga olsak, o'rtacha oddiygina μ=0, variance=1 va standart sapma, =1. Bosh fayl uchun amalga oshirilishi (bu yuqorida statistika.h ning davomi ) quyidagicha:
class StandardNormalDistribution : public StatisticalDistribution {
public:
StandardNormalDistribution();
virtual ~StandardNormalDistribution();


// Distribution functions
virtual double pdf(const double& x) const;
virtual double cdf(const double& x) const;


// Inverse cumulative distribution function (aka the probit function)
virtual double inv_cdf(const double& quantile) const;

// Descriptive stats
virtual double mean() const; // equal to 0
virtual double var() const; // equal to 1
virtual double stdev() const; // equal to 1


// Obtain a sequence of random draws from the standard normal distribution
virtual void random_draws(const std::vector& uniform_draws,
std::vector& dist_draws);
};
Manba fayl quyida koʻrsatilgan:
#ifndef __STATISTICS_CPP
#define __STATISTICS_CPP


#define _USE_MATH_DEFINES


#include "statistics.h"
#include


StatisticalDistribution::StatisticalDistribution() {}
StatisticalDistribution::~StatisticalDistribution() {}


// Constructor/destructor
StandardNormalDistribution::StandardNormalDistribution() {}
StandardNormalDistribution::~StandardNormalDistribution() {}


// Probability density function
double StandardNormalDistribution::pdf(const double& x) const {
return (1.0/sqrt(2.0 * M_PI)) * exp(-0.5*x*x);
}


// Cumulative density function
double StandardNormalDistribution::cdf(const double& x) const {
double k = 1.0/(1.0 + 0.2316419*x);
double k_sum = k*(0.319381530 + k*(-0.356563782 + k*(1.781477937 + k*(-1.821255978 + 1.330274429*k))));


if (x >= 0.0) {
return (1.0 - (1.0/(pow(2*M_PI,0.5)))*exp(-0.5*x*x) * k_sum);
} else {
return 1.0 - cdf(-x);
}
}


// Inverse cumulative distribution function (aka the probit function)
double StandardNormalDistribution::inv_cdf(const double& quantile) const {
// This is the Beasley-Springer-Moro algorithm which can
// be found in Glasserman [2004]. We won't go into the
// details here, so have a look at the reference for more info
static double a[4] = { 2.50662823884,
-18.61500062529,
41.39119773534,
-25.44106049637};


static double b[4] = { -8.47351093090,
23.08336743743,
-21.06224101826,
3.13082909833};


static double c[9] = {0.3374754822726147,
0.9761690190917186,
0.1607979714918209,
0.0276438810333863,
0.0038405729373609,
0.0003951896511919,
0.0000321767881768,
0.0000002888167364,
0.0000003960315187};


if (quantile >= 0.5 && quantile <= 0.92) {
double num = 0.0;
double denom = 1.0;


for (int i=0; i<4; i++) {
num += a[i] * pow((quantile - 0.5), 2*i + 1);
denom += b[i] * pow((quantile - 0.5), 2*i);
}
return num/denom;


} else if (quantile > 0.92 && quantile < 1) {
double num = 0.0;


for (int i=0; i<9; i++) {
num += c[i] * pow((log(-log(1-quantile))), i);
}
return num;


} else {
return -1.0*inv_cdf(1-quantile);
}
}

// Expectation/mean
double StandardNormalDistribution::mean() const { return 0.0; }


// Variance
double StandardNormalDistribution::var() const { return 1.0; }


// Standard Deviation
double StandardNormalDistribution::stdev() const { return 1.0; }


// Obtain a sequence of random draws from this distribution
void StandardNormalDistribution::random_draws(const std::vector& uniform_draws,
std::vector& dist_draws) {
// The simplest method to calculate this is with the Box-Muller method,
// which has been used procedurally in many other articles on QuantStart.com

// Check that the uniform draws and dist_draws are the same size and
// have an even number of elements (necessary for B-M)
if (uniform_draws.size() != dist_draws.size()) {
std::cout << "Draws vectors are of unequal size in standard normal dist." << std::endl;
return;
}

// Check that uniform draws have an even number of elements (necessary for B-M)
if (uniform_draws.size() % 2 != 0) {
std::cout << "Uniform draw vector size not an even number." << std::endl;
return;
}


// Slow, but easy to implement
for (int i=0; i dist_draws[2*i] = sqrt(-2.0*log(uniform_draws[2*i])) * sin(2*M_PI*uniform_draws[2*i+1]);
dist_draws[2*i+1] = sqrt(-2.0*log(uniform_draws[2*i])) * cos(2*M_PI*uniform_draws[2*i+1]);
}


return;
}
#endif
Bu erda ba'zi amalga oshirishlarni qisqacha muhokama qilaman. Kümülatif taqsimlanish funksiyasi (cdf) Joshi[1] dan ko'rsatilgan. Bu yopiq shakldagi eritma emas, balki yaqinlashishdir. Inverse CDF (inv_cdf) Korndagi amalga oshirishdan to'g'ridan-to'g'ri kodlagan Beasley-Springer-Moro algoritmidan foydalanadi[3]. Shunga oʻxshash usulni Joshi[2] da uchratsa boʻladi. Algoritm yana yopiq shakl yechimiga emas, balki haqiqiy funktsiyaga yaqinlashishdir. Sonli usul random_draws. Bu holda biz Box-Muller algoritmidan foydalanamiz. Biroq, buning o'rniga biz yanada samarali Ziggurat algoritmidan foydalanishimiz mumkin edi, ammo bu keyingi maqola uchun potentsial muhokama bo'ladi!