912 lines
36 KiB
MQL5
912 lines
36 KiB
MQL5
//+------------------------------------------------------------------+
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//| NoncentralChiSquare.mqh |
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//| Copyright 2000-2025, MetaQuotes Ltd. |
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//| https://www.mql5.com |
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//+------------------------------------------------------------------+
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#include "Math.mqh"
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#include "Normal.mqh"
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#include "Poisson.mqh"
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#include "ChiSquare.mqh"
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//+------------------------------------------------------------------+
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//| Noncentral Chi-Square distribution density function (PDF) |
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//+------------------------------------------------------------------+
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//| The function returns the probability density function of the |
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//| Noncentral Chi-Square distribution with parameters nu and sigma. |
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//| |
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//| Arguments: |
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//| x : Random variable |
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//| nu : Degrees of freedom |
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//| sigma : Noncentrality parameter |
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//| log_mode : Logarithm mode flag, if true it returns Log values |
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//| error_code : Variable for error code |
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//| |
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//| Return value: |
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//| The probability density evaluated at x. |
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//+------------------------------------------------------------------+
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double MathProbabilityDensityNoncentralChiSquare(const double x,const double nu,const double sigma,const bool log_mode,int &error_code)
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{
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//--- check NaN
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if(!MathIsValidNumber(x) || !MathIsValidNumber(nu) || !MathIsValidNumber(sigma))
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{
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error_code=ERR_ARGUMENTS_NAN;
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return QNaN;
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}
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//--- check arguments
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if(nu!=MathRound(nu) || nu<=0.0)
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{
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error_code=ERR_ARGUMENTS_INVALID;
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return QNaN;
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}
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error_code=ERR_OK;
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if(x<=0.0)
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return TailLog0(true,log_mode);
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//--- prepare parameters
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int err_code=0;
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int max_terms=1000;
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double lambda=sigma*0.5;
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double half_nu=nu*0.5;
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double pwr_lambda=1.0;
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double pwr_two=MathExp(-half_nu*MathLog(2));
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double pwr_x=MathExp((half_nu-1.0)*MathLog(x));
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double fact_mult=1.0;
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double coef_lambda_x=MathExp(-lambda-x*0.5);
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double coef_gamma=1.0/MathGamma(half_nu);
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double inv_factor=1.0;
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//--- calculate density using direct summation
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int j=0;
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double pdf=0;
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while(j<max_terms)
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{
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if(j>0)
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{
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pwr_lambda*=lambda;
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pwr_x*=x;
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pwr_two*=0.5;
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fact_mult*=1.0/j;
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inv_factor*=1.0/(j+half_nu-1);
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}
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double dp=coef_gamma*inv_factor*pwr_lambda*pwr_two*pwr_x*fact_mult*coef_lambda_x;
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pdf=pdf+dp;
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//--- check stop
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if(dp/(pdf+10E-10)<10E-16)
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break;
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j++;
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}
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//--- check convergence
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if(j<max_terms)
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return TailLogValue(pdf,true,log_mode);
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else
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{
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error_code=ERR_NON_CONVERGENCE;
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return QNaN;
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}
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}
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//+------------------------------------------------------------------+
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//| Noncentral Chi-Square distribution density function (PDF) |
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//+------------------------------------------------------------------+
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//| The function returns the probability density function of the |
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//| Noncentral Chi-Square distribution with parameters nu and sigma. |
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//| |
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//| Arguments: |
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//| x : Random variable |
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//| nu : Degrees of freedom |
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//| sigma : Noncentrality parameter |
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//| error_code : Variable for error code |
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//| |
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//| Return value: |
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//| The probability density evaluated at x. |
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//+------------------------------------------------------------------+
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double MathProbabilityDensityNoncentralChiSquare(double x,const double nu,const double sigma,int &error_code)
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{
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return MathProbabilityDensityNoncentralChiSquare(x,nu,sigma,false,error_code);
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}
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//+------------------------------------------------------------------+
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//| Noncentral Chi-Square distribution density function (PDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the probability density function of |
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//| the Chi Square distribution with parameter nu for values in x. |
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//| |
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//| Arguments: |
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//| x : Array with random variables |
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//| nu : Degrees of freedom |
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//| sigma : Noncentrality parameter |
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//| log_mode : Logarithm mode flag, if true it returns Log values |
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//| result : Array with calculated values |
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//| |
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//| Return value: |
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//| true if successful, otherwise false. |
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//+------------------------------------------------------------------+
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bool MathProbabilityDensityNoncentralChiSquare(const double &x[],const double nu,const double sigma,const bool log_mode,double &result[])
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{
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//--- check NaN
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if(!MathIsValidNumber(nu) || !MathIsValidNumber(sigma))
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return false;
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//--- check arguments
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if(nu!=MathRound(nu) || nu<=0.0)
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return false;
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int data_count=ArraySize(x);
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if(data_count==0)
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return false;
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ArrayResize(result,data_count);
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//--- prepare parameters
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int max_terms=1000;
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double lambda=sigma*0.5;
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double half_nu=nu*0.5;
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double coef_gamma=1.0/MathGamma(half_nu);
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double pwr_two2=MathExp(-half_nu*MathLog(2));
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double pwr_half_num1=(half_nu-1.0);
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double coef_exp_lambda=MathExp(-lambda);
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for(int i=0; i<data_count; i++)
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{
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double x_arg=x[i];
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if(x_arg<=0)
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result[i]=TailLog0(true,log_mode);
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else
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{
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int err_code=0;
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//result[i]=MathProbabilityDensityNoncentralChiSquare(x_arg,nu,sigma,false,err_code);
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double pwr_lambda=1.0;
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double pwr_two=pwr_two2;
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double pwr_x=MathPow(x_arg,pwr_half_num1);
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double fact_mult=1.0;
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double coef_lambda_x=coef_exp_lambda*MathExp(-x_arg*0.5);
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double inv_factor=1.0;
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//--- calculate density using direct summation
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int j=0;
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double pdf=0;
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while(j<max_terms)
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{
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if(j>0)
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{
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pwr_lambda*=lambda;
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pwr_x*=x_arg;
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pwr_two*=0.5;
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fact_mult*=1.0/j;
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inv_factor*=1.0/(j+half_nu-1);
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}
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double dp=coef_gamma*inv_factor*pwr_lambda*pwr_two*pwr_x*fact_mult*coef_lambda_x;
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pdf=pdf+dp;
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//--- check stop
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if(dp/(pdf+10E-10)<10E-16)
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break;
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j++;
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}
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//--- check convergence
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if(j<max_terms)
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result[i]=TailLogValue(pdf,true,log_mode);
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else
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return false;
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}
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}
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return true;
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}
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//+------------------------------------------------------------------+
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//| Noncentral Chi-Square distribution density function (PDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the probability density function of |
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//| the Chi-Square distribution with parameter nu for values in x[]. |
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//| |
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//| Arguments: |
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//| x : Array with random variables |
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//| nu : Degrees of freedom |
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//| sigma : Noncentrality parameter |
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//| result : Array with calculated values |
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//| |
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//| Return value: |
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//| true if successful, otherwise false. |
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//+------------------------------------------------------------------+
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bool MathProbabilityDensityNoncentralChiSquare(const double &x[],const double nu,const double sigma,double &result[])
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{
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return MathProbabilityDensityNoncentralChiSquare(x,nu,sigma,false,result);
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}
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//+------------------------------------------------------------------+
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//| Noncentral Chi-Square cumulative distribution function (CDF) |
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//+------------------------------------------------------------------+
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//| The function returns the probability that an observation |
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//| from a Noncentral Chi-Square distribution with parameters |
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//| nu and sigma is less than or equal to x. |
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//| |
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//| Arguments: |
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//| x : The desired quantile |
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//| nu : Degrees of freedom |
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//| sigma : Noncentrality parameter |
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//| tail : Flag to calculate lower tail |
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//| log_mode : Logarithm mode, if true it calculates Log values |
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//| error_code : Variable for error code |
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//| |
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//| Return value: |
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//| The value of Noncentral Chi-Square cumulative distribution |
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//| function with parameters nu and sigma, evaluated at x. |
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//+------------------------------------------------------------------+
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double MathCumulativeDistributionNoncentralChiSquare(const double x,const double nu,const double sigma,const bool tail,const bool log_mode,int &error_code)
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{
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//--- check NaN
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if(!MathIsValidNumber(x) || !MathIsValidNumber(nu) || !MathIsValidNumber(sigma))
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{
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error_code=ERR_ARGUMENTS_NAN;
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return QNaN;
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}
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//--- check arguments
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if(nu!=MathRound(nu) || nu<=0.0)
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{
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error_code=ERR_ARGUMENTS_INVALID;
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return QNaN;
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}
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error_code=ERR_OK;
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if(x<=0.0)
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return TailLog0(true,log_mode);
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//--- prepare parameters
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double cdf=0.0;
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int max_terms=100;
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double lambda=sigma*0.5;
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double coef_lambda=MathExp(-lambda);
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double pwr_lambda=1.0;
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double fact_mult=1.0;
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double half_x=x*0.5;
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double half_nu=nu*0.5;
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//--- direct summation
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int j=0;
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while(j<max_terms)
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{
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if(j>0)
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{
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pwr_lambda*=lambda;
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fact_mult/=j;
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}
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double coef1=coef_lambda*pwr_lambda*fact_mult;
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double coef2=MathMin(MathGammaIncomplete(half_x,half_nu+j),1.0);
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double dp=coef1*coef2;
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cdf=cdf+dp;
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if((dp/(cdf+10E-10))<10E-16)
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break;
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j++;
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}
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//---
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if(j<max_terms)
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{
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//--- take into account round-off errors for probability
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cdf=MathMin(cdf,1.0);
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return TailLogValue(cdf,tail,log_mode);
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}
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else
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{
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error_code=ERR_NON_CONVERGENCE;
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return QNaN;
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}
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}
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//+------------------------------------------------------------------+
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//| Noncentral Chi-Square cumulative distribution function (CDF) |
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//+------------------------------------------------------------------+
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//| The function returns the probability that an observation |
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//| from a Noncentral Chi-Square distribution with parameters |
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//| nu and sigma is less than or equal to x. |
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//| |
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//| Arguments: |
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//| x : The desired quantile |
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//| nu : Degrees of freedom |
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//| sigma : Noncentrality parameter |
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//| error_code : Variable for error code |
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//| |
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//| Return value: |
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//| The value of Noncentral Chi-Square cumulative distribution |
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//| function with parameters nu and sigma, evaluated at x. |
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//+------------------------------------------------------------------+
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double MathCumulativeDistributionNoncentralChiSquare(const double x,const double nu,const double sigma,int &error_code)
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{
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return MathCumulativeDistributionNoncentralChiSquare(x,nu,sigma,true,false,error_code);
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}
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//+------------------------------------------------------------------+
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//| Noncentral Chi-Square cumulative distribution function (CDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the cumulative distribution function of |
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//| the Noncentral Chi-Square distribution with parameters nu and |
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//| sigma for values in x. |
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//| |
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//| Arguments: |
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//| x : Array with random variables |
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//| nu : Degrees of freedom |
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//| sigma : Noncentrality parameter |
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//| tail : Flag to calculate lower tail |
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//| log_mode : Logarithm mode, if true it calculates Log values |
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//| result : Array with calculated values |
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//| |
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//| Return value: |
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//| true if successful, otherwise false. |
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//+------------------------------------------------------------------+
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bool MathCumulativeDistributionNoncentralChiSquare(const double &x[],const double nu,const double sigma,const bool tail,const bool log_mode,double &result[])
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{
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//--- check NaN
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if(!MathIsValidNumber(nu) || !MathIsValidNumber(sigma))
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return false;
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//--- check arguments
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if(nu!=MathRound(nu) || nu<=0.0)
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return false;
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int data_count=ArraySize(x);
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if(data_count==0)
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return false;
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//--- common factors
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double lambda=sigma*0.5;
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double coef_lambda=MathExp(-lambda);
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double half_nu=nu*0.5;
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const int max_terms=100;
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ArrayResize(result,data_count);
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for(int i=0; i<data_count; i++)
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{
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double x_arg=x[i];
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if(x_arg<=0.0)
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result[i]=TailLog0(tail,log_mode);
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else
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{
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double pwr_lambda=1.0;
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double fact_mult=1.0;
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double half_x=x_arg*0.5;
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double cdf=0.0;
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int j=0;
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//--- direct summation
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while(j<max_terms)
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{
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if(j>0)
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{
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pwr_lambda*=lambda;
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fact_mult/=j;
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}
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double coef1=coef_lambda*pwr_lambda*fact_mult;
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double coef2=MathMin(MathGammaIncomplete(half_x,half_nu+j),1.0);
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double dp=coef1*coef2;
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cdf=cdf+dp;
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if((dp/(cdf+10E-10))<10E-16)
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break;
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j++;
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}
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//---
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if(j<max_terms)
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{
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//--- take into account round-off errors for probability
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cdf=MathMin(cdf,1.0);
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result[i]=TailLogValue(cdf,tail,log_mode);
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}
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else
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return false;
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}
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}
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return true;
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}
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//+------------------------------------------------------------------+
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//| Noncentral Chi-Square cumulative distribution function (CDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the cumulative distribution function of |
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//| the Noncentral Chi-Square distribution with parameters nu and |
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//| sigma for values in x. |
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//| |
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//| Arguments: |
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//| x : Array with random variables |
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//| nu : Degrees of freedom |
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//| sigma : Noncentrality parameter |
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//| result : Array with calculated values |
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//| |
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//| Return value: |
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//| true if successful, otherwise false. |
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//+------------------------------------------------------------------+
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bool MathCumulativeDistributionNoncentralChiSquare(const double &x[],const double nu,const double sigma,double &result[])
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{
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return MathCumulativeDistributionNoncentralChiSquare(x,nu,sigma,true,false,result);
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}
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//+------------------------------------------------------------------+
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//| Noncentral Chi-Square distribution quantile function(inverse CDF)|
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//+------------------------------------------------------------------+
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//| The function returns the inverse cumulative distribution |
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//| function of Noncentral Chi-Square distribution with parameters |
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//| nu and sigma for the desired probability. |
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//| |
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//| Arguments: |
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//| probability : The desired probability |
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//| nu : Degrees of freedom |
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//| sigma : Noncentrality parameter |
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//| error_code : Variable for error code |
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//| |
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//| Return value: |
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//| The value of the inverse cumulative distribution function of |
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//| Noncentral Chi-Square distribution with parameters nu and sigma. |
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//+------------------------------------------------------------------+
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double MathQuantileNoncentralChiSquare(const double probability,const double nu,const double sigma,const bool tail,const bool log_mode,int &error_code)
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{
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if(log_mode==true)
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{
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if(probability==QNEGINF)
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return 0.0;
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}
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//--- check NaN
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if(!MathIsValidNumber(probability) || !MathIsValidNumber(nu) || !MathIsValidNumber(sigma))
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{
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error_code=ERR_ARGUMENTS_NAN;
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return QNaN;
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}
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//--- check arguments
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if(nu!=MathRound(nu) || nu<=0.0)
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{
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error_code=ERR_ARGUMENTS_INVALID;
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return QNaN;
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}
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//--- calculate real probability
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double prob=TailLogProbability(probability,tail,log_mode);
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//--- check probability range
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if(prob<0.0 || prob>1.0)
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{
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error_code=ERR_ARGUMENTS_INVALID;
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return QNaN;
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}
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error_code=ERR_OK;
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if(prob==0.0)
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return 0.0;
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if(prob==1.0)
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return QPOSINF;
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error_code=ERR_OK;
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//--- common factors for pdf and cdf calculation
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const int max_terms=1000;
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double lambda=sigma*0.5;
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double half_nu=nu*0.5;
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double coef_lambda=MathExp(-lambda);
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double half_nu_m1=half_nu-1.0;
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double coef_gamma=1.0/MathGamma(half_nu);
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double pwr_two2=MathExp(-half_nu*MathLog(2));
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double pwr_half_num1=(half_nu-1.0);
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//--- prepare values for initial x estimation
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double x=0.5;
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double h=1.0;
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double h_min=10E-10;
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//--- Newton iterations
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const int max_iterations=50;
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int iterations=0;
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// int err_code=0;
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while(iterations<max_iterations)
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{
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//--- check convergence
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if((MathAbs(h)>h_min && MathAbs(h)>MathAbs(h_min*x))==false)
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break;
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//double pdf=MathProbabilityDensityNoncentralChiSquare(x,nu,sigma,false,err_code);
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double half_x=x*0.5;
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double pwr_lambda=1.0;
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double pwr_two=pwr_two2;
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double pwr_x=MathPow(x,pwr_half_num1);
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double fact_mult=1.0;
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double coef_lambda_x=coef_lambda*MathExp(-half_x);
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double inv_factor=1.0;
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//--- calculate density using direct summation
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int j=0;
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double pdf=0;
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while(j<max_terms)
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{
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if(j>0)
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{
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pwr_lambda*=lambda;
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pwr_x*=x;
|
|
pwr_two*=0.5;
|
|
fact_mult*=1.0/j;
|
|
inv_factor*=1.0/(j+half_nu-1);
|
|
}
|
|
double dp=coef_gamma*inv_factor*pwr_lambda*pwr_two*pwr_x*fact_mult*coef_lambda_x;
|
|
pdf=pdf+dp;
|
|
//--- check stop
|
|
if(dp/(pdf+10E-10)<10E-16)
|
|
break;
|
|
j++;
|
|
}
|
|
//--- check convergence
|
|
if(j>max_terms)
|
|
{
|
|
error_code=ERR_NON_CONVERGENCE;
|
|
return QNaN;
|
|
}
|
|
|
|
//--- calculate cdf
|
|
pwr_lambda=1.0;
|
|
fact_mult=1.0;
|
|
double cdf=0.0;
|
|
j=0;
|
|
//--- direct summation
|
|
while(j<max_terms)
|
|
{
|
|
if(j>0)
|
|
{
|
|
pwr_lambda*=lambda;
|
|
fact_mult/=j;
|
|
}
|
|
double coef1=coef_lambda*pwr_lambda*fact_mult;
|
|
double coef2=MathMin(MathGammaIncomplete(half_x,half_nu+j),1.0);
|
|
double dp=coef1*coef2;
|
|
cdf=cdf+dp;
|
|
if((dp/(cdf+10E-10))<10E-16)
|
|
break;
|
|
j++;
|
|
}
|
|
//---
|
|
if(j>max_terms)
|
|
{
|
|
error_code=ERR_NON_CONVERGENCE;
|
|
return QNaN;
|
|
}
|
|
|
|
//--- calculate ratio
|
|
h=(cdf-prob)/pdf;
|
|
|
|
double x_new=x-h;
|
|
if(x_new<0.0)
|
|
x_new=x*0.1;
|
|
else
|
|
if(x_new>1.0)
|
|
x_new=1.0-(1-x)*0.1;
|
|
x=x_new;
|
|
|
|
iterations++;
|
|
}
|
|
//--- check convergence
|
|
if(iterations<max_iterations)
|
|
return x;
|
|
else
|
|
{
|
|
error_code=ERR_NON_CONVERGENCE;
|
|
return QNaN;
|
|
}
|
|
}
|
|
//+------------------------------------------------------------------+
|
|
//| Noncentral Chi-Square distribution quantile function(inverse CDF)|
|
|
//+------------------------------------------------------------------+
|
|
//| The function returns the inverse cumulative distribution |
|
|
//| function of the Noncentral Chi-Square distribution |
|
|
//| with parameters mu and sigma for the desired probability. |
|
|
//| |
|
|
//| Arguments: |
|
|
//| probability : The desired probability |
|
|
//| nu : Degrees of freedom |
|
|
//| sigma : Noncentrality parameter |
|
|
//| error_code : Variable for error code |
|
|
//| |
|
|
//| Return value: |
|
|
//| The value of the inverse cumulative distribution function of |
|
|
//| Noncentral Chi-Square distribution with parameters mu and sigma. |
|
|
//+------------------------------------------------------------------+
|
|
double MathQuantileNoncentralChiSquare(const double probability,const double nu,const double sigma,int &error_code)
|
|
{
|
|
return MathQuantileNoncentralChiSquare(probability,nu,sigma,true,false,error_code);
|
|
}
|
|
//+------------------------------------------------------------------+
|
|
//| Noncentral Chi-Square distribution quantile function(inverse CDF)|
|
|
//+------------------------------------------------------------------+
|
|
//| The function calculates the inverse cumulative distribution |
|
|
//| function of the Noncentral Chi-Square distribution with |
|
|
//| parameters nu and sigma for values from the probability[] array. |
|
|
//| |
|
|
//| Arguments: |
|
|
//| probability : Array with probabilities |
|
|
//| nu : Degrees of freedom |
|
|
//| sigma : Noncentrality parameter |
|
|
//| tail : Flag to calculate lower tail |
|
|
//| log_mode : Logarithm mode, if true it calculates Log values |
|
|
//| result : Array with calculated values |
|
|
//| |
|
|
//| Return value: |
|
|
//| true if successful, otherwise false. |
|
|
//+------------------------------------------------------------------+
|
|
bool MathQuantileNoncentralChiSquare(const double &probability[],const double nu,const double sigma,const bool tail,const bool log_mode,double &result[])
|
|
{
|
|
//--- check NaN
|
|
if(!MathIsValidNumber(nu) || !MathIsValidNumber(sigma))
|
|
return false;
|
|
//--- check arguments
|
|
if(nu!=MathRound(nu) || nu<=0.0)
|
|
return false;
|
|
|
|
int data_count=ArraySize(probability);
|
|
if(data_count==0)
|
|
return false;
|
|
|
|
//--- common factors for pdf and cdf calculation
|
|
double lambda=sigma*0.5;
|
|
double half_nu=nu*0.5;
|
|
double pwr_two0=MathExp(-half_nu*MathLog(2));
|
|
double pwr_gamma0=1.0/MathGamma(half_nu);
|
|
double coef_lambda=MathExp(-lambda);
|
|
double half_nu_m1=half_nu-1.0;
|
|
const int max_terms=1000;
|
|
const int max_iterations=50;
|
|
double h_min=10E-10;
|
|
ArrayResize(result,data_count);
|
|
for(int i=0; i<data_count; i++)
|
|
{
|
|
//--- calculate real probability
|
|
double prob=TailLogProbability(probability[i],tail,log_mode);
|
|
|
|
if(prob==0.0)
|
|
result[i]=0.0;
|
|
else
|
|
if(prob==1.0)
|
|
result[i]=QPOSINF;
|
|
else
|
|
{
|
|
//--- check probability range
|
|
if(prob<0.0 || prob>1.0)
|
|
return false;
|
|
|
|
//--- prepare values for initial x estimation
|
|
int err_code=0;
|
|
double x=0.5;
|
|
double h=1.0;
|
|
//--- Newton iterations
|
|
int iterations=0;
|
|
while(iterations<max_iterations)
|
|
{
|
|
//--- check convergence
|
|
if((MathAbs(h)>h_min && MathAbs(h)>MathAbs(h_min*x))==false)
|
|
break;
|
|
|
|
//double pdf=MathProbabilityDensityNoncentralChiSquare(x,nu,sigma,false,err_code);
|
|
double half_x=x*0.5;
|
|
double pwr_lambda=1.0;
|
|
double pwr_two=pwr_two0;
|
|
double pwr_x=MathPow(x,half_nu_m1);
|
|
double fact_mult=1.0;
|
|
double coef_lambda_x=coef_lambda*MathExp(-half_x);
|
|
double inv_factor=1.0;
|
|
//--- calculate density using direct summation
|
|
int j=0;
|
|
double pdf=0;
|
|
while(j<max_terms)
|
|
{
|
|
if(j>0)
|
|
{
|
|
pwr_lambda*=lambda;
|
|
pwr_x*=x;
|
|
pwr_two*=0.5;
|
|
fact_mult*=1.0/j;
|
|
inv_factor*=1.0/(j+half_nu-1);
|
|
}
|
|
double dp=pwr_gamma0*inv_factor*pwr_lambda*pwr_two*pwr_x*fact_mult*coef_lambda_x;
|
|
pdf=pdf+dp;
|
|
//--- check stop
|
|
if(dp/(pdf+10E-10)<10E-16)
|
|
break;
|
|
j++;
|
|
}
|
|
//--- check convergence
|
|
if(j>max_terms)
|
|
return false;
|
|
|
|
//--- calculate cdf
|
|
pwr_lambda=1.0;
|
|
fact_mult=1.0;
|
|
pwr_lambda=1.0;
|
|
fact_mult=1.0;
|
|
double cdf=0.0;
|
|
j=0;
|
|
//--- direct summation
|
|
while(j<max_terms)
|
|
{
|
|
if(j>0)
|
|
{
|
|
pwr_lambda*=lambda;
|
|
fact_mult/=j;
|
|
}
|
|
double coef1=coef_lambda*pwr_lambda*fact_mult;
|
|
double coef2=MathMin(MathGammaIncomplete(half_x,half_nu+j),1.0);
|
|
double dp=coef1*coef2;
|
|
cdf=cdf+dp;
|
|
if((dp/(cdf+10E-10))<10E-16)
|
|
break;
|
|
j++;
|
|
}
|
|
//---
|
|
if(j>max_terms)
|
|
return false;
|
|
|
|
//--- calculate ratio
|
|
h=(cdf-prob)/pdf;
|
|
|
|
double x_new=x-h;
|
|
if(x_new<0.0)
|
|
x_new=x*0.1;
|
|
else
|
|
if(x_new>1.0)
|
|
x_new=1.0-(1-x)*0.1;
|
|
x=x_new;
|
|
|
|
iterations++;
|
|
}
|
|
//--- check convergence
|
|
if(iterations<max_iterations)
|
|
result[i]=x;
|
|
else
|
|
return false;
|
|
}
|
|
}
|
|
return true;
|
|
}
|
|
//+------------------------------------------------------------------+
|
|
//| Noncentral Chi-Square distribution quantile function(inverse CDF)|
|
|
//+------------------------------------------------------------------+
|
|
//| The function calculates the inverse cumulative distribution |
|
|
//| function of the Noncentral Chi-Square distribution with |
|
|
//| parameters nu and sigma for values from the probability[] array. |
|
|
//| |
|
|
//| Arguments: |
|
|
//| probability : Array with probabilities |
|
|
//| nu : Degrees of freedom |
|
|
//| sigma : Noncentrality parameter |
|
|
//| result : Array with calculated values |
|
|
//| |
|
|
//| Return value: |
|
|
//| true if successful, otherwise false. |
|
|
//+------------------------------------------------------------------+
|
|
bool MathQuantileNoncentralChiSquare(const double &probability[],const double nu,const double sigma,double &result[])
|
|
{
|
|
return MathQuantileNoncentralChiSquare(probability,nu,sigma,true,false,result);
|
|
}
|
|
//+------------------------------------------------------------------+
|
|
//| Random variate from the Noncentral Chi-Square distribution |
|
|
//+------------------------------------------------------------------+
|
|
//| Compute the random variable from the Noncentral Chi-Square |
|
|
//| distribution with parameters nu and sigma. |
|
|
//| |
|
|
//| Arguments: |
|
|
//| nu : Degrees of freedom |
|
|
//| sigma : Noncentrality parameter |
|
|
//| error_code : Variable for error code |
|
|
//| |
|
|
//| Return value: |
|
|
//| The random value with Noncentral Chi-Square distribution. |
|
|
//+------------------------------------------------------------------+
|
|
//| Author: Robert Kern |
|
|
//+------------------------------------------------------------------+
|
|
double MathRandomNoncentralChiSquare(const double nu,const double sigma,int &error_code)
|
|
{
|
|
//--- return ChiSquare if sigma==0
|
|
if(sigma==0.0)
|
|
{
|
|
return MathRandomChiSquare(nu,error_code);
|
|
}
|
|
//--- check NaN
|
|
if(!MathIsValidNumber(nu) || !MathIsValidNumber(sigma))
|
|
{
|
|
error_code=ERR_ARGUMENTS_NAN;
|
|
return QNaN;
|
|
}
|
|
//--- check nu
|
|
if(nu!=MathRound(nu) || nu<=0)
|
|
{
|
|
error_code=ERR_ARGUMENTS_INVALID;
|
|
return QNaN;
|
|
}
|
|
//--- check sigma
|
|
if(sigma<0.0)
|
|
{
|
|
error_code=ERR_ARGUMENTS_INVALID;
|
|
return QNaN;
|
|
}
|
|
|
|
error_code=ERR_OK;
|
|
int err_code=0;
|
|
if(nu>1.0)
|
|
{
|
|
double rnd_chisquare=MathRandomGamma((nu-1)*0.5,2.0,err_code);
|
|
double rnd_normal=MathSqrt(sigma)+MathRandomNormal(0,1,err_code);
|
|
return rnd_chisquare+rnd_normal*rnd_normal;
|
|
}
|
|
else
|
|
{
|
|
int rnd_poisson=(int)MathRandomPoisson(sigma*0.5);
|
|
return MathRandomChiSquare(nu+2*rnd_poisson,err_code);
|
|
}
|
|
}
|
|
//+------------------------------------------------------------------+
|
|
//| Random variate from the Noncentral Chi-Square distribution |
|
|
//+------------------------------------------------------------------+
|
|
//| Generates random variables from the Noncentral Chi-Square |
|
|
//| distribution with parameters nu and sigma. |
|
|
//| |
|
|
//| Arguments: |
|
|
//| nu : Degrees of freedom |
|
|
//| sigma : Noncentrality parameter |
|
|
//| data_count : Number of values needed |
|
|
//| result : Output array with random values |
|
|
//| |
|
|
//| Return value: |
|
|
//| true if successful, otherwise false. |
|
|
//+------------------------------------------------------------------+
|
|
//| Author: Robert Kern |
|
|
//+------------------------------------------------------------------+
|
|
bool MathRandomNoncentralChiSquare(const double nu,const double sigma,const int data_count,double &result[])
|
|
{
|
|
//--- return ChiSquare if sigma==0
|
|
if(sigma==0.0)
|
|
return MathRandomChiSquare(nu,data_count,result);
|
|
//--- check NaN
|
|
if(!MathIsValidNumber(nu) || !MathIsValidNumber(sigma))
|
|
return false;
|
|
//--- check nu
|
|
if(nu!=MathRound(nu) || nu<=0)
|
|
return false;
|
|
//--- check sigma
|
|
if(sigma<0.0)
|
|
return false;
|
|
|
|
int err_code=0;
|
|
//--- prepare output array and calculate random values
|
|
ArrayResize(result,data_count);
|
|
for(int i=0; i<data_count; i++)
|
|
{
|
|
if(nu>1.0)
|
|
{
|
|
double rnd_chisquare=MathRandomGamma((nu-1)*0.5,2.0,err_code);
|
|
double rnd_normal=MathSqrt(sigma)+MathRandomNormal(0,1,err_code);
|
|
result[i]=rnd_chisquare+rnd_normal*rnd_normal;
|
|
}
|
|
else
|
|
{
|
|
int rnd_poisson=(int)MathRandomPoisson(sigma*0.5);
|
|
result[i]=MathRandomChiSquare(nu+2*rnd_poisson,err_code);
|
|
}
|
|
}
|
|
return true;
|
|
}
|
|
//+------------------------------------------------------------------+
|
|
//| Noncentral Chi-Square distribution moments |
|
|
//+------------------------------------------------------------------+
|
|
//| The function calculates 4 first moments of Noncental Chi-Square |
|
|
//| distribution with parameters nu and sigma. |
|
|
//| |
|
|
//| Arguments: |
|
|
//| nu : Degrees of freedom |
|
|
//| sigma : Noncentrality parameter |
|
|
//| mean : Variable for mean value (1st moment) |
|
|
//| variance : Variable for variance value (2nd moment) |
|
|
//| skewness : Variable for skewness value (3rd moment) |
|
|
//| kurtosis : Variable for kurtosis value (4th moment) |
|
|
//| error_code : Variable for error code |
|
|
//| |
|
|
//| Return value: |
|
|
//| true if moments calculated successfully, otherwise false. |
|
|
//+------------------------------------------------------------------+
|
|
bool MathMomentsNoncentralChiSquare(const double nu,const double sigma,double &mean,double &variance,double &skewness,double &kurtosis,int &error_code)
|
|
{
|
|
//--- default values
|
|
mean =QNaN;
|
|
variance=QNaN;
|
|
skewness=QNaN;
|
|
kurtosis=QNaN;
|
|
//--- check NaN
|
|
if(!MathIsValidNumber(nu) || !MathIsValidNumber(sigma))
|
|
{
|
|
error_code=ERR_ARGUMENTS_NAN;
|
|
return false;
|
|
}
|
|
//--- check nu
|
|
if(nu!=MathRound(nu) || nu<=0.0)
|
|
{
|
|
error_code=ERR_ARGUMENTS_INVALID;
|
|
return false;
|
|
}
|
|
|
|
error_code=ERR_OK;
|
|
//--- calculate moments
|
|
mean =nu+sigma;
|
|
variance=2*nu+4*sigma;
|
|
skewness=2*M_SQRT2*(nu+3*sigma)*MathPow(nu+2*sigma,-1.5);
|
|
kurtosis=12*(nu+4*sigma)*MathPow(nu+2*sigma,-2);
|
|
//--- successful
|
|
return true;
|
|
}
|
|
//+------------------------------------------------------------------+
|