1192 строки
46 КиБ
MQL5
1192 строки
46 КиБ
MQL5
 | //+------------------------------------------------------------------+
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//| NoncentralT.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 "T.mqh"
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#include "Normal.mqh"
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//+------------------------------------------------------------------+
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//| Noncentral T probability density function (PDF) |
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//+------------------------------------------------------------------+
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//| The function returns the probability density function |
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//| of the Noncentral T distribution with parameters nu and delta. |
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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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//| delta : Noncentrality parameter |
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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 probability density evaluated at x. |
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//+------------------------------------------------------------------+
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double MathProbabilityDensityNoncentralT(const double x,const double nu,const double delta,const bool log_mode,int &error_code)
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{
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//--- return T
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if(delta==0.0)
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return MathProbabilityDensityT(x,nu,error_code);
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//--- check NaN
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if(!MathIsValidNumber(x) || !MathIsValidNumber(nu) || !MathIsValidNumber(delta))
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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 nu
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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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//---
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double nu_1=nu+1.0;
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double nu_1_half=nu_1*0.5;
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double log_nu=MathLog(nu);
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double factor1=MathExp(-0.5*(MathLog(M_PI)+log_nu)-(delta*delta)*0.5-MathGammaLog(nu*0.5)+nu_1_half*log_nu);
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double nu_xx=nu+x*x;
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double log_nu_xx=MathLog(nu_xx);
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double factor2=MathExp(-nu_1_half*log_nu_xx);
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//---
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const int max_terms=500;
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double pwr=1.0;
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double pwr_factor=x*delta*M_SQRT2;
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double pwr_nuxx=1.0;
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double pwr_nuxx_factor=1.0/MathSqrt(nu_xx);
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double pwr_gamma=1.0;
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int j=0;
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double pdf=0.0;
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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_nuxx*=pwr_nuxx_factor;
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pwr_gamma/=j;
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pwr*=pwr_factor;
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}
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double t=pwr*pwr_gamma*pwr_nuxx*MathGamma((nu_1+j)*0.5);
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pdf+=t;
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//--- check precision
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if((t/(pdf+10E-10))<10E-20)
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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(factor1*factor2*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 T probability density function (PDF) |
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//+------------------------------------------------------------------+
| |||
//| The function returns the probability density function |
| |||
//| of the Noncentral T distribution with parameters nu and delta. |
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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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//| delta : 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 MathProbabilityDensityNoncentralT(const double x,const double nu,const double delta,int &error_code)
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{
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return MathProbabilityDensityNoncentralT(x,nu,delta,false,error_code);
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}
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//+------------------------------------------------------------------+
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//| Noncentral T probability density function (PDF) |
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//+------------------------------------------------------------------+
| |||
//| The function calculates the probability density function of |
| |||
//| the Noncentral T distribution with parameters nu and delta |
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//| for values in x[] array. |
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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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//| delta : 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 MathProbabilityDensityNoncentralT(const double &x[],const double nu,const double delta,const bool log_mode,double &result[])
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{
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//--- return T
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if(delta==0.0)
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return MathProbabilityDensityT(x,nu,log_mode,result);
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//--- check NaN
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if(!MathIsValidNumber(nu) || !MathIsValidNumber(delta))
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return false;
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//--- check nu
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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 nu_1=nu+1.0;
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double nu_1_half=nu_1*0.5;
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double log_nu=MathLog(nu);
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double factor1=MathExp(-0.5*(MathLog(M_PI)+log_nu)-(delta*delta)*0.5-MathGammaLog(nu*0.5)+nu_1_half*log_nu);
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double factor_delta=delta*M_SQRT2;
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const int max_terms=500;
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int error_code=0;
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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(!MathIsValidNumber(x_arg))
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return false;
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double nu_xx=nu+x_arg*x_arg;
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double log_nu_xx=MathLog(nu_xx);
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double factor2=MathExp(-nu_1_half*log_nu_xx);
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//---
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double pwr=1.0;
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double pwr_factor=x_arg*factor_delta;
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double pwr_nuxx=1.0;
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double pwr_nuxx_factor=1.0/MathSqrt(nu_xx);
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double pwr_gamma=1.0;
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int j=0;
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double pdf=0.0;
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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_nuxx*=pwr_nuxx_factor;
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pwr_gamma/=j;
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pwr*=pwr_factor;
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}
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double t=pwr*pwr_gamma*pwr_nuxx*MathGamma((nu_1+j)*0.5);
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pdf+=t;
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//--- check precision
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if((t/(pdf+10E-10))<10E-20)
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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(factor1*factor2*pdf,true,log_mode);
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else
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return false;
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}
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return true;
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}
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//+------------------------------------------------------------------+
| |||
//| Noncentral T probability density function (PDF) |
| |||
//+------------------------------------------------------------------+
| |||
//| The function calculates the probability density function of |
| |||
//| the Noncentral T distribution with parameters nu and delta |
| |||
//| for values in x[] array. |
| |||
//| |
| |||
//| Arguments: |
| |||
//| x : Array with random variables |
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//| nu : Degrees of freedom |
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//| delta : 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 MathProbabilityDensityNoncentralT(const double &x[],const double nu,const double delta,double &result[])
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{
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return MathProbabilityDensityNoncentralT(x,nu,delta,false,result);
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}
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//+------------------------------------------------------------------+
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//| Noncentral T 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 the Noncentral T distribution with parameters nu and delta |
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//| 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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//| delta : 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 the Noncentral T cumulative distribution function |
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//| with parameters nu and delta, evaluated at x. |
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//+------------------------------------------------------------------+
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//| The cdf is calculated using Algorithm 7.2 from |
| |||
//| Denise Benton, K. Krishnamoorthy, |
| |||
//| "Computing discrete mixtures of continuous distributions: |
| |||
//| noncentral chisquare, noncentral t and the distribution |
| |||
//| of the square of the sample multiple correlation coeffcient |
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//| Computational Statistics & Data Analysis 43 (2003) 249-267. |
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//+------------------------------------------------------------------+
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double MathCumulativeDistributionNoncentralT(const double x,const double nu,const double delta,const bool tail,const bool log_mode,int &error_code)
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{
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//--- return T
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if(delta==0.0)
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return MathCumulativeDistributionT(x,nu,error_code);
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//--- check NaN
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if(!MathIsValidNumber(x) || !MathIsValidNumber(nu) || !MathIsValidNumber(delta))
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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 nu (must be positive integer)
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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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//--- special case
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if(nu==1.0)
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{
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error_code=ERR_OK;
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return TailLogValue(0.5+MathArctan(x)*M_1_PI,tail,log_mode);
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}
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error_code=ERR_OK;
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//--- error tolerance
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const double errtol=10E-25;
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//--- maximum number of iterations
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const int max_iterations=1000;
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double xx,del;
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double ptermf,qtermf,ptermb,qtermb,error;
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//--- if t<0, then the transformation in (3.2) must be used;
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if(x<0.0)
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{
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xx=-x;
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del=-delta;
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}
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else
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{
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xx=x;
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del=delta;
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}
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int err_code=0;
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//--- compute the normal cdf at (-del):
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double cdf_normal=MathMin(MathCumulativeDistributionNormal(-del,0,1,err_code),1.0);
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if(xx==0.0)
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return TailLogValue(cdf_normal,tail,log_mode);
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double y=xx*xx/(nu+xx*xx);
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double dels=0.5*del*del;
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//--- k = integral part of (dels)
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int k=(int)dels;
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double a=k+0.5;
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double c=k+1.0;
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double b=0.5*nu;
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//--- initialization to compute the Pk's:
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double pkf = MathExp(-dels+k*MathLog(dels)-MathGammaLog(k+1.0));
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double pkb = pkf;
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//--- initialization to compute the Qk's:
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double qkf = MathExp(-dels+k*MathLog(dels)-MathGammaLog(k+1.5));
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double qkb = qkf;
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//--- compute the incomplete beta function associated with the Pk:
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//--- pbetaf = beta distribution at (y; a; b)
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double pbetaf=MathBetaIncomplete(y,a,b);
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double pbetab=pbetaf;
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//--- compute the incomplete beta function associated with the Qk:
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//--- qbetaf = beta distribution at (y; c; b)
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double qbetaf=MathBetaIncomplete(y,c,b);
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double qbetab=qbetaf;
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//--- initialization to compute the incomplete beta functions associated with the Pi's recursively:
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double pgamf=MathExp(MathGammaLog(a+b-1.0)-MathGammaLog(a)-MathGammaLog(b)+(a-1.0)*MathLog(y)+b*MathLog(1.0-y));
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double pgamb=pgamf*y*(a+b-1.0)/a;
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//--- initialization to compute the incomplete beta functions associated with the Qi's recursively:
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double qgamf=MathExp(MathGammaLog(c+b-1.0)-MathGammaLog(c)-MathGammaLog(b)+(c-1.0)*MathLog(y)+b*MathLog(1.0-y));
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double qgamb=qgamf*y*(c+b-1.0)/c;
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//--- compute the remainder of the Poisson probabilities:
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double rempois=1.0-pkf;
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double delosq2=del/M_SQRT2;
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double sum=pkf*pbetaf+delosq2*qkf*qbetaf;
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double cons=0.5*(1.0+0.5*MathAbs(delta));
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int j=0;
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for(;;)
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{
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j++;
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pgamf*=y*(a+b+j-2.0)/(a+j-1.0);
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pbetaf-=pgamf;
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pkf*=dels/(k+j);
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ptermf=pkf*pbetaf;
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qgamf*=y*(c+b+j-2.0)/(c+j-1.0);
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qbetaf-=qgamf;
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qkf*=dels/(k+j+0.5);
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qtermf=qkf*qbetaf;
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double term=ptermf+delosq2*qtermf;
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sum+=term;
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error=rempois*cons*pbetaf;
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rempois-=pkf;
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//--- do forward and backward computations k times or until convergence:
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if(j<=k)
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{
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pgamb*=(a-j+1.0)/(y*(a+b-j));
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pbetab+=pgamb;
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pkb=(k-j+1.0)*pkb/dels;
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ptermb=pkb*pbetab;
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qgamb*=(c-j+1.0)/(y*(c+b-j));
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qbetab+=qgamb;
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qkb=(k-j+1.5)*qkb/dels;
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qtermb=qkb*qbetab;
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term=ptermb+delosq2*qtermb;
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sum+=term;
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rempois-=pkb;
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if(rempois<=errtol || j>=max_iterations)
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break;
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}
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else
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{
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if(error<=errtol || j>=max_iterations)
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break;
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}
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}
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//--- check convergence
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if(j<max_iterations)
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{
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double cdf=0.5*sum+cdf_normal;
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//--- if x is negative
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if(x<0)
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cdf=1.0-cdf;
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//--- take into account round-off errors for probability
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return TailLogValue(MathMin(cdf,1.0),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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//+------------------------------------------------------------------+
| |||
//| Noncentral T cumulative distribution function (CDF) |
| |||
//+------------------------------------------------------------------+
| |||
//| The function returns the probability that an observation |
| |||
//| from the Noncentral T distribution with parameters nu and delta |
| |||
//| is less than or equal to x. |
| |||
//| |
| |||
//| Arguments: |
| |||
//| x : The desired quantile |
| |||
//| nu : Degrees of freedom |
| |||
//| delta : Noncentrality parameter |
| |||
//| error_code : Variable for error code |
| |||
//| |
| |||
//| Return value: |
| |||
//| The value of the Noncentral T cumulative distribution function |
| |||
//| with parameters nu and delta, evaluated at x. |
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//+------------------------------------------------------------------+
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double MathCumulativeDistributionNoncentralT(const double x,const double nu,const double delta,int &error_code)
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{
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return MathCumulativeDistributionNoncentralT(x,nu,delta,true,false,error_code);
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}
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//+------------------------------------------------------------------+
| |||
//| Noncentral T cumulative distribution function (CDF) |
| |||
//+------------------------------------------------------------------+
| |||
//| The function calculates the cumulative distribution function of |
| |||
//| the Noncentral T distribution with parameters nu and delta |
| |||
//| for values in x. |
| |||
//| |
| |||
//| Arguments: |
| |||
//| x : Array with random variables |
| |||
//| nu : Degrees of freedom |
| |||
//| delta : 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. |
| |||
//+------------------------------------------------------------------+
| |||
//| The cdf is calculated using Algorithm 7.2 from |
| |||
//| Denise Benton, K. Krishnamoorthy, |
| |||
//| "Computing discrete mixtures of continuous distributions: |
| |||
//| noncentral chisquare, noncentral t and the distribution |
| |||
//| of the square of the sample multiple correlation coeffcient |
| |||
//| Computational Statistics & Data Analysis 43 (2003) 249-267. |
| |||
//+------------------------------------------------------------------+
| |||
bool MathCumulativeDistributionNoncentralT(const double &x[],const double nu,const double delta,const bool tail,const bool log_mode,double &result[])
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{
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//--- return T
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if(delta==0.0)
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return MathCumulativeDistributionT(x,nu,tail,log_mode,result);
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//--- check NaN
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if(!MathIsValidNumber(nu) || !MathIsValidNumber(delta))
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return false;
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//--- check nu (must be positive integer)
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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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//--- special case
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if(nu==1.0)
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{
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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(!MathIsValidNumber(x_arg))
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return false;
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result[i]=TailLogValue(0.5+MathArctan(x_arg)*M_1_PI,tail,log_mode);
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}
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return true;
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}
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int err_code=0;
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//--- compute the normal cdf at (-del):
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double cdf_normal=MathMin(MathCumulativeDistributionNormal(-delta,0,1,err_code),1.0);
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//--- error tolerance
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const double errtol=10E-25;
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//--- maximum number of iterations
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const int max_iterations=1000;
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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(!MathIsValidNumber(x_arg))
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return false;
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double xx,del;
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double ptermf,qtermf,ptermb,qtermb,error;
| |||
//--- if t<0, then the transformation in (3.2) must be used;
| |||
if(x_arg<0.0)
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{
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xx=-x_arg;
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del=-delta;
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}
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else
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{
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xx=x_arg;
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del=delta;
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}
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if(xx==0.0)
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result[i]=TailLogValue(cdf_normal,tail,log_mode);
| |||
else
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{
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double y=xx*xx/(nu+xx*xx);
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double dels=0.5*del*del;
| |||
//--- k = integral part of (dels)
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int k=(int)dels;
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double a=k+0.5;
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double c=k+1.0;
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double b=0.5*nu;
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//--- initialization to compute the Pk's:
| |||
double pkf = MathExp(-dels+k*MathLog(dels)-MathGammaLog(k+1.0));
| |||
double pkb = pkf;
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//--- initialization to compute the Qk's:
| |||
double qkf = MathExp(-dels+k*MathLog(dels)-MathGammaLog(k+1.5));
| |||
double qkb = qkf;
| |||
//--- compute the incomplete beta function associated with the Pk:
| |||
//--- pbetaf = beta distribution at (y; a; b)
| |||
double pbetaf=MathBetaIncomplete(y,a,b);
| |||
double pbetab=pbetaf;
| |||
//--- compute the incomplete beta function associated with the Qk:
| |||
//--- qbetaf = beta distribution at (y; c; b)
| |||
double qbetaf=MathBetaIncomplete(y,c,b);
| |||
double qbetab=qbetaf;
| |||
//--- initialization to compute the incomplete beta functions associated with the Pi's recursively:
| |||
double pgamf=MathExp(MathGammaLog(a+b-1.0)-MathGammaLog(a)-MathGammaLog(b)+(a-1.0)*MathLog(y)+b*MathLog(1.0-y));
| |||
double pgamb=pgamf*y*(a+b-1.0)/a;
| |||
//--- initialization to compute the incomplete beta functions associated with the Qi's recursively:
| |||
double qgamf=MathExp(MathGammaLog(c+b-1.0)-MathGammaLog(c)-MathGammaLog(b)+(c-1.0)*MathLog(y)+b*MathLog(1.0-y));
| |||
double qgamb=qgamf*y*(c+b-1.0)/c;
| |||
//--- compute the remainder of the Poisson probabilities:
| |||
double rempois=1.0-pkf;
| |||
double delosq2=del/M_SQRT2;
| |||
double sum=pkf*pbetaf+delosq2*qkf*qbetaf;
| |||
double cons=0.5*(1.0+0.5*MathAbs(delta));
| |||
int j=0;
| |||
for(;;)
| |||
{
| |||
j++;
| |||
pgamf*=y*(a+b+j-2.0)/(a+j-1.0);
| |||
pbetaf-=pgamf;
| |||
pkf*=dels/(k+j);
| |||
ptermf=pkf*pbetaf;
| |||
qgamf*=y*(c+b+j-2.0)/(c+j-1.0);
| |||
qbetaf-=qgamf;
| |||
qkf*=dels/(k+j+0.5);
| |||
qtermf=qkf*qbetaf;
| |||
double term=ptermf+delosq2*qtermf;
| |||
sum+=term;
| |||
error=rempois*cons*pbetaf;
| |||
rempois-=pkf;
| |||
//--- do forward and backward computations k times or until convergence:
| |||
if(j<=k)
| |||
{
| |||
pgamb*=(a-j+1.0)/(y*(a+b-j));
| |||
pbetab+=pgamb;
| |||
pkb=(k-j+1.0)*pkb/dels;
| |||
ptermb=pkb*pbetab;
| |||
qgamb*=(c-j+1.0)/(y*(c+b-j));
| |||
qbetab+=qgamb;
| |||
qkb=(k-j+1.5)*qkb/dels;
| |||
qtermb=qkb*qbetab;
| |||
term=ptermb+delosq2*qtermb;
| |||
sum+=term;
| |||
rempois-=pkb;
| |||
if(rempois<=errtol || j>=max_iterations)
| |||
break;
| |||
}
| |||
else
| |||
{
| |||
if(error<=errtol || j>=max_iterations)
| |||
break;
| |||
}
| |||
}
| |||
//--- check convergence
| |||
if(j<max_iterations)
| |||
{
| |||
double cdf=0.5*sum+cdf_normal;
| |||
//--- if x is negative
| |||
if(x_arg<0)
| |||
cdf=1.0-cdf;
| |||
//--- take into account round-off errors for probability
| |||
result[i]=TailLogValue(MathMin(cdf,1.0),tail,log_mode);
| |||
}
| |||
else
| |||
return false;
| |||
}
| |||
}
| |||
return true;
| |||
}
| |||
//+------------------------------------------------------------------+
| |||
//| Noncentral T cumulative distribution function (CDF) |
| |||
//+------------------------------------------------------------------+
| |||
//| The function calculates the cumulative distribution function of |
| |||
//| the Noncentral T distribution with parameters nu and delta |
| |||
//| for values in x[] array. |
| |||
//| |
| |||
//| Arguments: |
| |||
//| x : Array with random variables |
| |||
//| nu : Degrees of freedom |
| |||
//| delta : Noncentrality parameter |
| |||
//| result : Array with calculated values |
| |||
//| |
| |||
//| Return value: |
| |||
//| true if successful, otherwise false. |
| |||
//+------------------------------------------------------------------+
| |||
bool MathCumulativeDistributionNoncentralT(const double &x[],const double nu,const double delta,double &result[])
| |||
{
| |||
return MathCumulativeDistributionNoncentralT(x,nu,delta,true,false,result);
| |||
}
| |||
//+------------------------------------------------------------------+
| |||
//| The Noncentral T distribution quantile function (inverse CDF) |
| |||
//+------------------------------------------------------------------+
| |||
//| The function returns the inverse cumulative distribution |
| |||
//| function of Noncentral T distribution with parameters nu and |
| |||
//| delta for the desired probability. |
| |||
//| |
| |||
//| Arguments: |
| |||
//| probability : The desired probability |
| |||
//| nu : Degrees of freedom |
| |||
//| delta : 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 the inverse cumulative distribution function |
| |||
//| of Noncentral T-distribution with parameters nu and delta. |
| |||
//+------------------------------------------------------------------+
| |||
double MathQuantileNoncentralT(const double probability,const double nu,const double delta,const bool tail,const bool log_mode,int &error_code)
| |||
{
| |||
//--- return T if delta==0
| |||
if(delta==0.0)
| |||
return MathQuantileT(probability,nu,error_code);
| |||
//--- check NaN
| |||
if(!MathIsValidNumber(nu) || !MathIsValidNumber(delta))
| |||
{
| |||
error_code=ERR_ARGUMENTS_NAN;
| |||
return QNaN;
| |||
}
| |||
//--- check nu
| |||
if(nu!=MathRound(nu) || nu<0.0)
| |||
{
| |||
error_code=ERR_ARGUMENTS_INVALID;
| |||
return QNaN;
| |||
}
| |||
//--- check delta
| |||
if(delta<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;
| |||
}
| |||
| |||
if(prob==0.0 || prob==1.0)
| |||
{
| |||
error_code=ERR_RESULT_INFINITE;
| |||
if(prob==0.0)
| |||
return QNEGINF;
| |||
else
| |||
return QPOSINF;
| |||
}
| |||
//--- coefficients for pdf and cdf
| |||
double sqr_delta_half=delta*delta*0.5;
| |||
double nu_half=nu*0.5;
| |||
double log_sqr_delta_half=MathLog(sqr_delta_half);
| |||
double exp_sqr_delta_half=MathExp(-sqr_delta_half);
| |||
double sqrt_sqr_delta_half=MathSqrt(sqr_delta_half);
| |||
double nu_1=nu+1.0;
| |||
double nu_1_half=nu_1*0.5;
| |||
double log_nu=MathLog(nu);
| |||
double factor1=MathExp(-0.5*(MathLog(M_PI)+log_nu)-sqr_delta_half-MathGammaLog(nu*0.5)+nu_1_half*log_nu);
| |||
| |||
//--- probability for -delta
| |||
int err_code=0;
| |||
double p0=MathCumulativeDistributionNormal(-delta,0.0,1.0,err_code);
| |||
//---
| |||
error_code=ERR_OK;
| |||
double precision=10E-20;
| |||
const int max_iterations=150;
| |||
int iterations=0;
| |||
double x=0.5;
| |||
double h=1.0;
| |||
double h_min=precision;
| |||
const int max_terms=500;
| |||
//--- Newton iterations
| |||
while(iterations<max_iterations)
| |||
{
| |||
//--- check convergence
| |||
if((MathAbs(h)>h_min && MathAbs(h)>MathAbs(h_min*x))==false)
| |||
break;
| |||
| |||
//--- calculate pdf
| |||
double x_arg_sqr=x*x;
| |||
double nu_xx=nu+x_arg_sqr;
| |||
double log_nu_xx=MathLog(nu_xx);
| |||
double factor2=MathExp(-nu_1_half*log_nu_xx);
| |||
double pwr=1.0;
| |||
double pwr_factor=x*delta*M_SQRT2;
| |||
double pwr_nuxx=1.0;
| |||
double pwr_nuxx_factor=1.0/MathSqrt(nu_xx);
| |||
double pwr_gamma=1.0;
| |||
int j=0;
| |||
double pdf=0.0;
| |||
while(j<max_terms)
| |||
{
| |||
if(j>0)
| |||
{
| |||
pwr_nuxx*=pwr_nuxx_factor;
| |||
pwr_gamma/=j;
| |||
pwr*=pwr_factor;
| |||
}
| |||
double t=pwr*pwr_gamma*pwr_nuxx*MathGamma((nu_1+j)*0.5);
| |||
pdf+=t;
| |||
//--- check precision
| |||
if((t/(pdf+10E-10))<10E-20)
| |||
break;
| |||
j++;
| |||
}
| |||
//--- check convergence
| |||
if(j>max_terms)
| |||
return false;
| |||
pdf=factor1*factor2*pdf;
| |||
| |||
//--- calculate cdf
| |||
double t=(x*x)/(nu+x*x);
| |||
double sum1 = 0.0;
| |||
double sum2 = 0.0;
| |||
//---
| |||
double pwr_coef1=1.0;
| |||
double pwr_coef2=1.0;
| |||
double fact=1.0;
| |||
j=0;
| |||
if(x!=0)
| |||
{
| |||
while(j<max_terms)
| |||
{
| |||
if(j>0)
| |||
{
| |||
pwr_coef1*=sqrt_sqr_delta_half;
| |||
pwr_coef2*=sqr_delta_half;
| |||
fact/=j;
| |||
}
| |||
double coef=1.0/MathGamma(j*0.5+1.0);
| |||
//--- term1: t between 0 and x
| |||
double t1=pwr_coef1*coef*MathBetaIncomplete(t,(j+1.0)*0.5,nu_half);
| |||
sum1+=t1;
| |||
//--- term2: t between x and -x
| |||
double t2=pwr_coef2*fact*MathBetaIncomplete(t,(j+0.5),nu_half);
| |||
sum2+=t2;
| |||
//--- check precision
| |||
if((MathAbs(t1/(sum1+10E-10))<precision) && (MathAbs(t2/(sum2+10E-10))<precision))
| |||
break;
| |||
j++;
| |||
}
| |||
}
| |||
//--- check convergence
| |||
if(j>max_terms)
| |||
return false;
| |||
| |||
//--- compute probability for positive x
| |||
double cdf=p0+exp_sqr_delta_half*sum1*0.5;
| |||
//--- compute probability for negative x
| |||
if(x<0)
| |||
cdf=cdf-exp_sqr_delta_half*sum2;
| |||
//--- take into account round-off errors for probability
| |||
cdf=MathMin(cdf,1.0);
| |||
| |||
//--- 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-(1.0-x)*0.1;
| |||
| |||
if(MathAbs(x_new-x)<10E-15)
| |||
break;
| |||
| |||
x=x_new;
| |||
iterations++;
| |||
}
| |||
//--- check convergence
| |||
if(iterations<max_iterations)
| |||
return x;
| |||
else
| |||
{
| |||
error_code=ERR_NON_CONVERGENCE;
| |||
return QNaN;
| |||
}
| |||
}
| |||
//+------------------------------------------------------------------+
| |||
//| Noncentral T distribution quantile function (inverse CDF) |
| |||
//+------------------------------------------------------------------+
| |||
//| The function returns the inverse cumulative distribution |
| |||
//| function of the Noncentral T distribution with parameters nu |
| |||
//| and delta for the desired probability. |
| |||
//| |
| |||
//| Arguments: |
| |||
//| probability : The desired probability |
| |||
//| nu : Degrees of freedom |
| |||
//| delta : Noncentrality parameter |
| |||
//| error_code : Variable for error code |
| |||
//| |
| |||
//| Return value: |
| |||
//| The value of the inverse cumulative distribution function |
| |||
//| of Noncentral T-distribution with parameters nu and delta. |
| |||
//+------------------------------------------------------------------+
| |||
double MathQuantileNoncentralT(const double probability,const double nu,const double delta,int &error_code)
| |||
{
| |||
return MathQuantileNoncentralT(probability,nu,delta,true,false,error_code);
| |||
}
| |||
//+------------------------------------------------------------------+
| |||
//| Noncentral T distribution quantile function (inverse CDF) |
| |||
//+------------------------------------------------------------------+
| |||
//| The function calculates the inverse cumulative distribution |
| |||
//| function of the Noncentral T distribution with parameters nu |
| |||
//| and delta for values from the probability[] array. |
| |||
//| |
| |||
//| Arguments: |
| |||
//| probability : Array with probabilities |
| |||
//| nu : Degrees of freedom |
| |||
//| delta : Noncentrality parameter |
| |||
//| tail : Flag to calculate lower tail |
| |||
//| log_mode : Logarithm mode,if true it calculates for Log values|
| |||
//| result : Array with calculated values |
| |||
//| |
| |||
//| Return value: |
| |||
//| true if successful, otherwise false. |
| |||
//+------------------------------------------------------------------+
| |||
bool MathQuantileNoncentralT(const double &probability[],const double nu,const double delta,const bool tail,const bool log_mode,double &result[])
| |||
{
| |||
//--- return T if delta==0
| |||
if(delta==0.0)
| |||
return MathQuantileT(probability,nu,tail,log_mode,result);
| |||
//--- check NaN
| |||
if(!MathIsValidNumber(nu) || !MathIsValidNumber(delta))
| |||
return false;
| |||
//--- check nu
| |||
if(nu!=MathRound(nu) || nu<0.0)
| |||
return false;
| |||
//--- check delta
| |||
if(delta<0.0)
| |||
return false;
| |||
| |||
int data_count=ArraySize(probability);
| |||
if(data_count==0)
| |||
return false;
| |||
| |||
//--- coefficients for pdf and cdf
| |||
double sqr_delta_half=delta*delta*0.5;
| |||
double nu_half=nu*0.5;
| |||
double log_sqr_delta_half=MathLog(sqr_delta_half);
| |||
double exp_sqr_delta_half=MathExp(-sqr_delta_half);
| |||
double sqrt_sqr_delta_half=MathSqrt(sqr_delta_half);
| |||
double nu_1=nu+1.0;
| |||
double nu_1_half=nu_1*0.5;
| |||
double log_nu=MathLog(nu);
| |||
double factor1=MathExp(-0.5*(MathLog(M_PI)+log_nu)-sqr_delta_half-MathGammaLog(nu*0.5)+nu_1_half*log_nu);
| |||
| |||
//--- probability for -delta
| |||
int err_code=0;
| |||
double p0=MathCumulativeDistributionNormal(-delta,0.0,1.0,err_code);
| |||
//---
| |||
double precision=10E-20;
| |||
double h_min=precision;
| |||
const int max_iterations=50;
| |||
const int max_terms=500;
| |||
| |||
ArrayResize(result,data_count);
| |||
for(int i=0; i<data_count; i++)
| |||
{
| |||
//--- calculate real probability
| |||
double prob=TailLogProbability(probability[i],tail,log_mode);
| |||
| |||
//--- check probability range
| |||
if(prob<0.0 || prob>1.0)
| |||
return false;
| |||
| |||
if(prob==0.0 || prob==1.0)
| |||
{
| |||
if(prob==0.0)
| |||
result[i]=QNEGINF;
| |||
else
| |||
result[i]=QPOSINF;
| |||
}
| |||
else
| |||
{
| |||
double x=0.5;
| |||
double h=1.0;
| |||
int iterations=0;
| |||
//--- Newton iterations
| |||
while(iterations<max_iterations)
| |||
{
| |||
//--- check convergence
| |||
if((MathAbs(h)>h_min && MathAbs(h)>MathAbs(h_min*x))==false)
| |||
break;
| |||
| |||
//--- calculate pdf and cdf
| |||
//double cdf=MathCumulativeDistributionNoncentralT(x,nu,delta,err_code);
| |||
//double pdf=MathProbabilityDensityNoncentralT(x,nu,delta,err_code);
| |||
| |||
//--- calculate pdf
| |||
double x_arg_sqr=x*x;
| |||
double nu_xx=nu+x_arg_sqr;
| |||
double log_nu_xx=MathLog(nu_xx);
| |||
double factor2=MathExp(-nu_1_half*log_nu_xx);
| |||
double pwr=1.0;
| |||
double pwr_factor=x*delta*M_SQRT2;
| |||
double pwr_nuxx=1.0;
| |||
double pwr_nuxx_factor=1.0/MathSqrt(nu_xx);
| |||
double pwr_gamma=1.0;
| |||
int j=0;
| |||
double pdf=0.0;
| |||
while(j<max_terms)
| |||
{
| |||
if(j>0)
| |||
{
| |||
pwr_nuxx*=pwr_nuxx_factor;
| |||
pwr_gamma/=j;
| |||
pwr*=pwr_factor;
| |||
}
| |||
double t=pwr*pwr_gamma*pwr_nuxx*MathGamma((nu_1+j)*0.5);
| |||
pdf+=t;
| |||
//--- check precision
| |||
if((t/(pdf+10E-10))<10E-20)
| |||
break;
| |||
j++;
| |||
}
| |||
//--- check convergence
| |||
if(j>max_terms)
| |||
return false;
| |||
| |||
pdf=factor1*factor2*pdf;
| |||
| |||
//--- calculate cdf
| |||
double t=(x*x)/(nu+x*x);
| |||
double sum1 = 0.0;
| |||
double sum2 = 0.0;
| |||
//---
| |||
double pwr_coef1=1.0;
| |||
double pwr_coef2=1.0;
| |||
double fact=1.0;
| |||
j=0;
| |||
if(x!=0)
| |||
{
| |||
while(j<max_terms)
| |||
{
| |||
if(j>0)
| |||
{
| |||
pwr_coef1*=sqrt_sqr_delta_half;
| |||
pwr_coef2*=sqr_delta_half;
| |||
fact/=j;
| |||
}
| |||
double coef=1.0/MathGamma(j*0.5+1.0);
| |||
//--- term1: t between 0 and x
| |||
double t1=pwr_coef1*coef*MathBetaIncomplete(t,(j+1.0)*0.5,nu_half);
| |||
sum1+=t1;
| |||
//--- term2: t between x and -x
| |||
double t2=pwr_coef2*fact*MathBetaIncomplete(t,(j+0.5),nu_half);
| |||
sum2+=t2;
| |||
//--- check precision
| |||
if((MathAbs(t1/(sum1+10E-10))<precision) && (MathAbs(t2/(sum2+10E-10))<precision))
| |||
break;
| |||
j++;
| |||
}
| |||
}
| |||
//--- check convergence
| |||
if(j>max_terms)
| |||
return false;
| |||
| |||
//--- compute probability for positive x
| |||
double cdf=p0+exp_sqr_delta_half*sum1*0.5;
| |||
//--- compute probability for negative x
| |||
if(x<0)
| |||
cdf=cdf-exp_sqr_delta_half*sum2;
| |||
//--- take into account round-off errors for probability
| |||
cdf=MathMin(cdf,1.0);
| |||
| |||
//--- 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-(1.0-x)*0.1;
| |||
| |||
if(MathAbs(x_new-x)<10E-15)
| |||
break;
| |||
| |||
x=x_new;
| |||
iterations++;
| |||
}
| |||
//--- check convergence
| |||
if(iterations<max_iterations)
| |||
result[i]=x;
| |||
else
| |||
return false;
| |||
}
| |||
}
| |||
return true;
| |||
}
| |||
//+------------------------------------------------------------------+
| |||
//| Noncentral T distribution quantile function (inverse CDF) |
| |||
//+------------------------------------------------------------------+
| |||
//| The function calculates the inverse cumulative distribution |
| |||
//| function of the Noncentral T distribution with parameters nu |
| |||
//| and delta for values from the probability[] array. |
| |||
//| |
| |||
//| Arguments: |
| |||
//| probability : Array with probabilities |
| |||
//| nu : Degrees of freedom |
| |||
//| delta : Noncentrality parameter |
| |||
//| result : Array with calculated values |
| |||
//| |
| |||
//| Return value: |
| |||
//| true if successful, otherwise false. |
| |||
//+------------------------------------------------------------------+
| |||
bool MathQuantileNoncentralT(const double &probability[],const double nu,const double delta,double &result[])
| |||
{
| |||
return MathQuantileNoncentralT(probability,nu,delta,true,false,result);
| |||
}
| |||
//+------------------------------------------------------------------+
| |||
//| Random variate from the Noncentral T distribution |
| |||
//+------------------------------------------------------------------+
| |||
//| Computes the random variable from the Noncentral T distribution |
| |||
//| with parameters nu and delta. |
| |||
//| |
| |||
//| Arguments: |
| |||
//| nu : Degrees of freedom |
| |||
//| delta : Noncentrality parameter |
| |||
//| error_code : Variable for error code |
| |||
//| |
| |||
//| Return value: |
| |||
//| The random value with Noncentral T distribution. |
| |||
//+------------------------------------------------------------------+
| |||
double MathRandomNoncentralT(const double nu,const double delta,int &error_code)
| |||
{
| |||
//--- return T if delta==0
| |||
if(delta==0.0)
| |||
return MathRandomT(nu,error_code);
| |||
//--- check NaN
| |||
if(!MathIsValidNumber(nu) || !MathIsValidNumber(delta))
| |||
{
| |||
error_code=ERR_ARGUMENTS_NAN;
| |||
return QNaN;
| |||
}
| |||
//--- check arguments
| |||
if(nu!=MathRound(nu) || nu<=0.0)
| |||
{
| |||
error_code=ERR_ARGUMENTS_INVALID;
| |||
return QNaN;
| |||
}
| |||
//--- check delta
| |||
if(delta<0)
| |||
{
| |||
error_code=ERR_ARGUMENTS_INVALID;
| |||
return QNaN;
| |||
}
| |||
| |||
error_code=ERR_OK;
| |||
int err_code=0;
| |||
//--- generate normal and chisquare random variables
| |||
double rnd_normal=delta+MathRandomNormal(0,1,err_code);
| |||
double rnd_nchi=MathRandomGamma(nu*0.5,2.0,err_code);
| |||
//--- calculate ratio
| |||
double rnd_value=0;
| |||
if(rnd_nchi!=0)
| |||
rnd_value=rnd_normal/MathSqrt(rnd_nchi/nu);
| |||
return rnd_value;
| |||
}
| |||
//+------------------------------------------------------------------+
| |||
//| Random variate from the Noncentral T distribution |
| |||
//+------------------------------------------------------------------+
| |||
//| Generates random variables from the Noncentral T distribution |
| |||
//| with parameters nu and delta. |
| |||
//| |
| |||
//| Arguments: |
| |||
//| nu : Degrees of freedom |
| |||
//| delta : Noncentrality parameter |
| |||
//| data_count : Number of values needed |
| |||
//| result : Output array with random values |
| |||
//| |
| |||
//| Return value: |
| |||
//| true if successful, otherwise false. |
| |||
//+------------------------------------------------------------------+
| |||
bool MathRandomNoncentralT(const double nu,const double delta,const int data_count,double &result[])
| |||
{
| |||
//--- return T if delta==0
| |||
if(delta==0.0)
| |||
return MathRandomT(nu,data_count,result);
| |||
//--- check NaN
| |||
if(!MathIsValidNumber(nu) || !MathIsValidNumber(delta))
| |||
return false;
| |||
//--- check arguments
| |||
if(nu!=MathRound(nu) || nu<=0.0)
| |||
return false;
| |||
//--- check delta
| |||
if(delta<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++)
| |||
{
| |||
//--- generate normal and chisquare random variables
| |||
double rnd_normal=delta+MathRandomNormal(0,1,err_code);
| |||
double rnd_nchi=MathRandomGamma(nu*0.5,2.0,err_code);
| |||
//--- calculate ratio
| |||
double rnd_value=0;
| |||
if(rnd_nchi!=0)
| |||
{
| |||
rnd_value=rnd_normal/MathSqrt(rnd_nchi/nu);
| |||
result[i]=rnd_value;
| |||
}
| |||
else
| |||
return false;
| |||
}
| |||
return true;
| |||
}
| |||
//+------------------------------------------------------------------+
| |||
//| Noncentral T distribution moments |
| |||
//+------------------------------------------------------------------+
| |||
//| The function calculates 4 first moments of the Noncentral T |
| |||
//| distribution with parameters nu and delta. |
| |||
//| |
| |||
//| Arguments: |
| |||
//| nu : Degrees of freedom |
| |||
//| delta : 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. |
| |||
//+------------------------------------------------------------------+
| |||
double MathMomentsNoncentralT(const double nu,const double delta,double &mean,double &variance,double &skewness,double &kurtosis,int &error_code)
| |||
{
| |||
//--- if delta==0, calc moments for T
| |||
if(delta==0)
| |||
return(MathMomentsT(nu,mean,variance,skewness,kurtosis,error_code));
| |||
//--- default values
| |||
mean =QNaN;
| |||
variance=QNaN;
| |||
skewness=QNaN;
| |||
kurtosis=QNaN;
| |||
//--- check NaN
| |||
if(!MathIsValidNumber(nu) || !MathIsValidNumber(delta))
| |||
{
| |||
error_code=ERR_ARGUMENTS_NAN;
| |||
return false;
| |||
}
| |||
//--- check nu
| |||
if(nu!=MathRound(nu) || nu<0.0)
| |||
{
| |||
error_code=ERR_ARGUMENTS_INVALID;
| |||
return false;
| |||
}
| |||
//--- check delta
| |||
if(delta<0.0)
| |||
{
| |||
error_code=ERR_ARGUMENTS_INVALID;
| |||
return false;
| |||
}
| |||
| |||
error_code=ERR_OK;
| |||
//--- calculate moments
| |||
//--- mean
| |||
if(nu>1)
| |||
mean=delta*MathSqrt(nu)*MathGamma((nu-1)*0.5)/(MathSqrt(2)*MathGamma(nu*0.5));
| |||
//--- delta^2
| |||
double delta_sqr=delta*delta;
| |||
//--- 1/((nu-3)*(nu-2))
| |||
double nu32=1/((nu-3)*(nu-2));
| |||
if(nu>2)
| |||
variance=((delta_sqr+1)*nu)/(nu-2)-MathPow(mean,2);
| |||
//--- skewness
| |||
if(nu>3)
| |||
{
| |||
skewness=-2*variance;
| |||
skewness+= nu*(delta_sqr+2*nu-3)*nu32;
| |||
skewness*= mean*MathPow(variance,-1.5);
| |||
}
| |||
//--- kurtosis
| |||
if(nu>4)
| |||
{
| |||
kurtosis=-3*variance;
| |||
kurtosis+= nu*(delta_sqr*(nu+1)+3*(3*nu-5))*nu32;
| |||
kurtosis*= -MathPow(mean,2);
| |||
kurtosis+= MathPow(nu,2)*(MathPow(delta,4)+6*delta_sqr+3)/((nu-4)*(nu-2));
| |||
kurtosis*= MathPow(variance,-2);
| |||
kurtosis-=3;
| |||
}
| |||
//--- successful
| |||
return true;
| |||
}
| |||
//+------------------------------------------------------------------+
|