| //+------------------------------------------------------------------+
|
| //| NoncentralChiSquare.mqh |
|
| //| Copyright 2000-2025, MetaQuotes Ltd. |
|
| //| https://www.mql5.com |
|
| //+------------------------------------------------------------------+
|
| #include "Math.mqh"
|
| #include "Normal.mqh"
|
| #include "Poisson.mqh"
|
| #include "ChiSquare.mqh"
|
|
|
| //+------------------------------------------------------------------+
|
| //| Noncentral Chi-Square distribution density function (PDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the probability density function of the |
|
| //| Noncentral Chi-Square distribution with parameters nu and sigma. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Random variable |
|
| //| nu : Degrees of freedom |
|
| //| sigma : Noncentrality parameter |
|
| //| log_mode : Logarithm mode flag, if true it returns Log values |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The probability density evaluated at x. |
|
| //+------------------------------------------------------------------+
|
| double MathProbabilityDensityNoncentralChiSquare(const double x,const double nu,const double sigma,const bool log_mode,int &error_code)
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(x) || !MathIsValidNumber(nu) || !MathIsValidNumber(sigma))
|
| {
|
| error_code=ERR_ARGUMENTS_NAN;
|
| return QNaN;
|
| }
|
| //--- check arguments
|
| if(nu!=MathRound(nu) || nu<=0.0)
|
| {
|
| error_code=ERR_ARGUMENTS_INVALID;
|
| return QNaN;
|
| }
|
|
|
| error_code=ERR_OK;
|
| if(x<=0.0)
|
| return TailLog0(true,log_mode);
|
| //--- prepare parameters
|
| int err_code=0;
|
| int max_terms=1000;
|
| double lambda=sigma*0.5;
|
| double half_nu=nu*0.5;
|
| double pwr_lambda=1.0;
|
| double pwr_two=MathExp(-half_nu*MathLog(2));
|
| double pwr_x=MathExp((half_nu-1.0)*MathLog(x));
|
| double fact_mult=1.0;
|
| double coef_lambda_x=MathExp(-lambda-x*0.5);
|
| double coef_gamma=1.0/MathGamma(half_nu);
|
| 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=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)
|
| return TailLogValue(pdf,true,log_mode);
|
| else
|
| {
|
| error_code=ERR_NON_CONVERGENCE;
|
| return QNaN;
|
| }
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Noncentral Chi-Square distribution density function (PDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the probability density function of the |
|
| //| Noncentral Chi-Square distribution with parameters nu and sigma. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Random variable |
|
| //| nu : Degrees of freedom |
|
| //| sigma : Noncentrality parameter |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The probability density evaluated at x. |
|
| //+------------------------------------------------------------------+
|
| double MathProbabilityDensityNoncentralChiSquare(double x,const double nu,const double sigma,int &error_code)
|
| {
|
| return MathProbabilityDensityNoncentralChiSquare(x,nu,sigma,false,error_code);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Noncentral Chi-Square distribution density function (PDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the probability density function of |
|
| //| the Chi Square distribution with parameter nu for values in x. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Array with random variables |
|
| //| nu : Degrees of freedom |
|
| //| sigma : Noncentrality parameter |
|
| //| log_mode : Logarithm mode flag, if true it returns Log values |
|
| //| result : Array with calculated values |
|
| //| |
|
| //| Return value: |
|
| //| true if successful, otherwise false. |
|
| //+------------------------------------------------------------------+
|
| bool MathProbabilityDensityNoncentralChiSquare(const double &x[],const double nu,const double sigma,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(x);
|
| if(data_count==0)
|
| return false;
|
|
|
| ArrayResize(result,data_count);
|
| //--- prepare parameters
|
| int max_terms=1000;
|
| double lambda=sigma*0.5;
|
| double half_nu=nu*0.5;
|
| double coef_gamma=1.0/MathGamma(half_nu);
|
| double pwr_two2=MathExp(-half_nu*MathLog(2));
|
| double pwr_half_num1=(half_nu-1.0);
|
| double coef_exp_lambda=MathExp(-lambda);
|
| for(int i=0; i<data_count; i++)
|
| {
|
| double x_arg=x[i];
|
|
|
| if(x_arg<=0)
|
| result[i]=TailLog0(true,log_mode);
|
| else
|
| {
|
| int err_code=0;
|
| //result[i]=MathProbabilityDensityNoncentralChiSquare(x_arg,nu,sigma,false,err_code);
|
| double pwr_lambda=1.0;
|
| double pwr_two=pwr_two2;
|
| double pwr_x=MathPow(x_arg,pwr_half_num1);
|
| double fact_mult=1.0;
|
| double coef_lambda_x=coef_exp_lambda*MathExp(-x_arg*0.5);
|
| 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_arg;
|
| 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)
|
| result[i]=TailLogValue(pdf,true,log_mode);
|
| else
|
| return false;
|
| }
|
| }
|
| return true;
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Noncentral Chi-Square distribution density function (PDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the probability density function of |
|
| //| the Chi-Square distribution with parameter nu for values in x[]. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Array with random variables |
|
| //| nu : Degrees of freedom |
|
| //| sigma : Noncentrality parameter |
|
| //| result : Array with calculated values |
|
| //| |
|
| //| Return value: |
|
| //| true if successful, otherwise false. |
|
| //+------------------------------------------------------------------+
|
| bool MathProbabilityDensityNoncentralChiSquare(const double &x[],const double nu,const double sigma,double &result[])
|
| {
|
| return MathProbabilityDensityNoncentralChiSquare(x,nu,sigma,false,result);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Noncentral Chi-Square cumulative distribution function (CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the probability that an observation |
|
| //| from a Noncentral Chi-Square distribution with parameters |
|
| //| nu and sigma is less than or equal to x. |
|
| //| |
|
| //| Arguments: |
|
| //| x : The desired quantile |
|
| //| nu : Degrees of freedom |
|
| //| sigma : Noncentrality parameter |
|
| //| tail : Flag to calculate lower tail |
|
| //| log_mode : Logarithm mode, if true it calculates Log values |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The value of Noncentral Chi-Square cumulative distribution |
|
| //| function with parameters nu and sigma, evaluated at x. |
|
| //+------------------------------------------------------------------+
|
| double MathCumulativeDistributionNoncentralChiSquare(const double x,const double nu,const double sigma,const bool tail,const bool log_mode,int &error_code)
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(x) || !MathIsValidNumber(nu) || !MathIsValidNumber(sigma))
|
| {
|
| error_code=ERR_ARGUMENTS_NAN;
|
| return QNaN;
|
| }
|
| //--- check arguments
|
| if(nu!=MathRound(nu) || nu<=0.0)
|
| {
|
| error_code=ERR_ARGUMENTS_INVALID;
|
| return QNaN;
|
| }
|
|
|
| error_code=ERR_OK;
|
| if(x<=0.0)
|
| return TailLog0(true,log_mode);
|
|
|
| //--- prepare parameters
|
| double cdf=0.0;
|
| int max_terms=100;
|
| double lambda=sigma*0.5;
|
| double coef_lambda=MathExp(-lambda);
|
| double pwr_lambda=1.0;
|
| double fact_mult=1.0;
|
| double half_x=x*0.5;
|
| double half_nu=nu*0.5;
|
| //--- direct summation
|
| int j=0;
|
| 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)
|
| {
|
| //--- take into account round-off errors for probability
|
| cdf=MathMin(cdf,1.0);
|
| return TailLogValue(cdf,tail,log_mode);
|
| }
|
| else
|
| {
|
| error_code=ERR_NON_CONVERGENCE;
|
| return QNaN;
|
| }
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Noncentral Chi-Square cumulative distribution function (CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the probability that an observation |
|
| //| from a Noncentral Chi-Square distribution with parameters |
|
| //| nu and sigma is less than or equal to x. |
|
| //| |
|
| //| Arguments: |
|
| //| x : The desired quantile |
|
| //| nu : Degrees of freedom |
|
| //| sigma : Noncentrality parameter |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The value of Noncentral Chi-Square cumulative distribution |
|
| //| function with parameters nu and sigma, evaluated at x. |
|
| //+------------------------------------------------------------------+
|
| double MathCumulativeDistributionNoncentralChiSquare(const double x,const double nu,const double sigma,int &error_code)
|
| {
|
| return MathCumulativeDistributionNoncentralChiSquare(x,nu,sigma,true,false,error_code);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Noncentral Chi-Square cumulative distribution function (CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the cumulative distribution function of |
|
| //| the Noncentral Chi-Square distribution with parameters nu and |
|
| //| sigma for values in x. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Array with random variables |
|
| //| 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 MathCumulativeDistributionNoncentralChiSquare(const double &x[],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(x);
|
| if(data_count==0)
|
| return false;
|
|
|
| //--- common factors
|
| double lambda=sigma*0.5;
|
| double coef_lambda=MathExp(-lambda);
|
| double half_nu=nu*0.5;
|
| const int max_terms=100;
|
| ArrayResize(result,data_count);
|
| for(int i=0; i<data_count; i++)
|
| {
|
| double x_arg=x[i];
|
|
|
| if(x_arg<=0.0)
|
| result[i]=TailLog0(tail,log_mode);
|
| else
|
| {
|
| double pwr_lambda=1.0;
|
| double fact_mult=1.0;
|
| double half_x=x_arg*0.5;
|
| double cdf=0.0;
|
| int 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)
|
| {
|
| //--- take into account round-off errors for probability
|
| cdf=MathMin(cdf,1.0);
|
| result[i]=TailLogValue(cdf,tail,log_mode);
|
| }
|
| else
|
| return false;
|
| }
|
| }
|
| return true;
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Noncentral Chi-Square cumulative distribution function (CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the cumulative distribution function of |
|
| //| the Noncentral Chi-Square distribution with parameters nu and |
|
| //| sigma for values in x. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Array with random variables |
|
| //| nu : Degrees of freedom |
|
| //| sigma : Noncentrality parameter |
|
| //| result : Array with calculated values |
|
| //| |
|
| //| Return value: |
|
| //| true if successful, otherwise false. |
|
| //+------------------------------------------------------------------+
|
| bool MathCumulativeDistributionNoncentralChiSquare(const double &x[],const double nu,const double sigma,double &result[])
|
| {
|
| return MathCumulativeDistributionNoncentralChiSquare(x,nu,sigma,true,false,result);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Noncentral Chi-Square distribution quantile function(inverse CDF)|
|
| //+------------------------------------------------------------------+
|
| //| The function returns the inverse cumulative distribution |
|
| //| function of Noncentral Chi-Square distribution with parameters |
|
| //| nu 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 nu and sigma. |
|
| //+------------------------------------------------------------------+
|
| double MathQuantileNoncentralChiSquare(const double probability,const double nu,const double sigma,const bool tail,const bool log_mode,int &error_code)
|
| {
|
| if(log_mode==true)
|
| {
|
| if(probability==QNEGINF)
|
| return 0.0;
|
| }
|
| //--- check NaN
|
| if(!MathIsValidNumber(probability) || !MathIsValidNumber(nu) || !MathIsValidNumber(sigma))
|
| {
|
| error_code=ERR_ARGUMENTS_NAN;
|
| return QNaN;
|
| }
|
| //--- check arguments
|
| if(nu!=MathRound(nu) || nu<=0.0)
|
| {
|
| error_code=ERR_ARGUMENTS_INVALID;
|
| return QNaN;
|
| }
|
|
|
| //--- calculate real probability
|
| double prob=TailLogProbability(probability,tail,log_mode);
|
| //--- check probability range
|
| if(prob<0.0 || prob>1.0)
|
| {
|
| error_code=ERR_ARGUMENTS_INVALID;
|
| return QNaN;
|
| }
|
|
|
| error_code=ERR_OK;
|
| if(prob==0.0)
|
| return 0.0;
|
|
|
| if(prob==1.0)
|
| return QPOSINF;
|
|
|
| error_code=ERR_OK;
|
| //--- common factors for pdf and cdf calculation
|
| const int max_terms=1000;
|
| double lambda=sigma*0.5;
|
| double half_nu=nu*0.5;
|
| double coef_lambda=MathExp(-lambda);
|
| double half_nu_m1=half_nu-1.0;
|
| double coef_gamma=1.0/MathGamma(half_nu);
|
| double pwr_two2=MathExp(-half_nu*MathLog(2));
|
| double pwr_half_num1=(half_nu-1.0);
|
| //--- prepare values for initial x estimation
|
| double x=0.5;
|
| double h=1.0;
|
| double h_min=10E-10;
|
| //--- Newton iterations
|
| const int max_iterations=50;
|
| int iterations=0;
|
| // int err_code=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_two2;
|
| double pwr_x=MathPow(x,pwr_half_num1);
|
| 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=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;
|
| }
|
| //+------------------------------------------------------------------+
|
