| //+------------------------------------------------------------------+
|
| //| T.mqh |
|
| //| Copyright 2000-2025, MetaQuotes Ltd. |
|
| //| https://www.mql5.com |
|
| //+------------------------------------------------------------------+
|
| #include "Math.mqh"
|
| #include "Gamma.mqh"
|
|
|
| //+------------------------------------------------------------------+
|
| //| T probability density function (PDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the probability density function |
|
| //| of the T-distribution with parameter nu. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Random variable |
|
| //| nu : Degrees of freedom |
|
| //| log_mode : Logarithm mode, if true it calculates Log values |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The probability density evaluated at x. |
|
| //+------------------------------------------------------------------+
|
| double MathProbabilityDensityT(const double x,const double nu,const bool log_mode,int &error_code)
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(x) || !MathIsValidNumber(nu))
|
| {
|
| error_code=ERR_ARGUMENTS_NAN;
|
| return QNaN;
|
| }
|
| //--- check nu
|
| if(nu!=MathRound(nu) || nu<=0.0)
|
| {
|
| error_code=ERR_ARGUMENTS_INVALID;
|
| return QNaN;
|
| }
|
|
|
| error_code=ERR_OK;
|
| //--- calculate T density
|
| double pdf=MathExp(MathGammaLog((nu+1.0)*0.5)-MathGammaLog(nu*0.5));
|
| pdf=pdf/(MathSqrt(nu*M_PI)*MathPow(1+x*x/nu,(nu+1.0)*0.5));
|
| //--- return density
|
| return TailLogValue(pdf,true,log_mode);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| T probability density function (PDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the probability density function |
|
| //| of the T-distribution with parameter nu. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Random variable |
|
| //| nu : Degrees of freedom |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The probability density evaluated at x. |
|
| //+------------------------------------------------------------------+
|
| double MathProbabilityDensityT(const double x,const double nu,int &error_code)
|
| {
|
| return MathProbabilityDensityT(x,nu,false,error_code);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| T probability density function (PDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the probability density function of |
|
| //| the T distribution with parameter nu for values in x[] array. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Array with random variables |
|
| //| nu : Degrees of freedom |
|
| //| log_mode : Logarithm mode flag, if true it returns Log values |
|
| //| result : Array with calculated values |
|
| //| |
|
| //| Return value: |
|
| //| true if successful, otherwise false. |
|
| //+------------------------------------------------------------------+
|
| bool MathProbabilityDensityT(const double &x[],const double nu,const bool log_mode,double &result[])
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(nu))
|
| return false;
|
| //--- check nu
|
| if(nu!=MathRound(nu) || nu<=0.0)
|
| return false;
|
|
|
| int data_count=ArraySize(x);
|
| if(data_count==0)
|
| return false;
|
|
|
| int error_code=0;
|
| ArrayResize(result,data_count);
|
| for(int i=0; i<data_count; i++)
|
| {
|
| double x_arg=x[i];
|
|
|
| if(!MathIsValidNumber(x_arg))
|
| return false;
|
|
|
| error_code=ERR_OK;
|
| //--- calculate T density
|
| double pdf=MathExp(MathGammaLog((nu+1.0)*0.5)-MathGammaLog(nu*0.5));
|
| pdf=pdf/(MathSqrt(nu*M_PI)*MathPow(1+x_arg*x_arg/nu,(nu+1.0)*0.5));
|
| //--- return density
|
| result[i]=TailLogValue(pdf,true,log_mode);
|
| }
|
| return true;
|
| }
|
| //+------------------------------------------------------------------+
|
| //| T probability density function (PDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the probability density function of |
|
| //| the T distribution with parameter nu for values in x[] array. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Array with random variables |
|
| //| nu : Degrees of freedom |
|
| //| result : Array with calculated values |
|
| //| |
|
| //| Return value: |
|
| //| true if successful, otherwise false. |
|
| //+------------------------------------------------------------------+
|
| bool MathProbabilityDensityT(const double &x[],const double nu,double &result[])
|
| {
|
| return MathProbabilityDensityT(x,nu,false,result);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| T cumulative distribution function (CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the probability that an observation from |
|
| //| T-distribution with parameter nu is less than or equal to x. |
|
| //| |
|
| //| Arguments: |
|
| //| x : The desired quantile |
|
| //| nu : Degrees of freedom |
|
| //| 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 T cumulative distribution function with |
|
| //| parameter nu, evaluated at x. |
|
| //+------------------------------------------------------------------+
|
| double MathCumulativeDistributionT(const double x,const double nu,const bool tail,const bool log_mode,int &error_code)
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(x) || !MathIsValidNumber(nu))
|
| {
|
| error_code=ERR_ARGUMENTS_NAN;
|
| return QNaN;
|
| }
|
| //--- check nu (must be positive integer)
|
| if(nu!=MathRound(nu) || nu<=0.0)
|
| {
|
| error_code=ERR_ARGUMENTS_INVALID;
|
| return QNaN;
|
| }
|
|
|
| error_code=ERR_OK;
|
| //--- special case
|
| if(nu==1.0)
|
| return TailLogValue(0.5+MathArctan(x)/M_PI,tail,log_mode);
|
| //--- otherwise
|
| if(x==0)
|
| return TailLogValue(0.5,tail,log_mode);
|
| //--- calculate pdf using incomplete Beta
|
| double cdf=1.0-MathBetaIncomplete(nu/(nu+x*x),nu*0.5,0.5);
|
| cdf=(1.0-cdf)*0.5;
|
| //--- check x
|
| if(x>0.0)
|
| cdf=1.0-cdf;
|
| //--- take into account round-off errors for probability
|
| return TailLogValue(MathMin(cdf,1.0),tail,log_mode);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| T cumulative distribution function (CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the probability that an observation from |
|
| //| T-distribution with parameter nu is less than or equal to x. |
|
| //| |
|
| //| Arguments: |
|
| //| x : The desired quantile |
|
| //| nu : Degrees of freedom |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The value of T cumulative distribution function with parameter |
|
| //| nu, evaluated at x. |
|
| //+------------------------------------------------------------------+
|
| double MathCumulativeDistributionT(const double x,const double nu,int &error_code)
|
| {
|
| return MathCumulativeDistributionT(x,nu,true,false,error_code);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| T cumulative distribution function (CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the cumulative distribution function of |
|
| //| the T distribution with parameter nu for values in x. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Array with random variables |
|
| //| nu : Degrees of freedom |
|
| //| 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 MathCumulativeDistributionT(const double &x[],const double nu,const bool tail,const bool log_mode,double &result[])
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(nu))
|
| return false;
|
| //--- check nu (must be positive integer)
|
| if(nu!=MathRound(nu) || nu<=0.0)
|
| return false;
|
|
|
| int data_count=ArraySize(x);
|
| if(data_count==0)
|
| return false;
|
|
|
| int error_code=0;
|
| ArrayResize(result,data_count);
|
| for(int i=0; i<data_count; i++)
|
| {
|
| double x_arg=x[i];
|
|
|
| if(!MathIsValidNumber(x_arg))
|
| return false;
|
|
|
| //--- special case
|
| if(nu==1.0)
|
| result[i]=TailLogValue(0.5+MathArctan(x_arg)/M_PI,tail,log_mode);
|
| else
|
| //--- otherwise
|
| if(x_arg==0)
|
| result[i]=TailLogValue(0.5,tail,log_mode);
|
| else
|
| {
|
| //--- calculate pdf using incomplete Beta
|
| double cdf=1.0-MathBetaIncomplete(nu/(nu+x_arg*x_arg),nu*0.5,0.5);
|
| cdf=(1.0-cdf)*0.5;
|
| //--- check x
|
| if(x_arg>0.0)
|
| cdf=1.0-cdf;
|
| //--- take into account round-off errors for probability
|
| result[i]=TailLogValue(MathMin(cdf,1.0),tail,log_mode);
|
| }
|
| }
|
| return true;
|
| }
|
| //+------------------------------------------------------------------+
|
| //| T cumulative distribution function (CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the cumulative distribution function of |
|
| //| the T distribution with parameter nu for values in x[] array. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Array with random variables |
|
| //| nu : Degrees of freedom |
|
| //| result : Array with calculated values |
|
| //| |
|
| //| Return value: |
|
| //| true if successful, otherwise false. |
|
| //+------------------------------------------------------------------+
|
| bool MathCumulativeDistributionT(const double &x[],const double nu,double &result[])
|
| {
|
| return MathCumulativeDistributionT(x,nu,true,false,result);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| T distribution quantile function (inverse CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the inverse cumulative distribution |
|
| //| function of the T distribution with parameter nu for the desired |
|
| //| probability. |
|
| //| |
|
| //| Arguments: |
|
| //| probability : The desired probability |
|
| //| nu : Degrees of freedom |
|
| //| 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 the T-distribution with parameter nu. |
|
| //+------------------------------------------------------------------+
|
| double MathQuantileT(const double probability,const double nu,const bool tail,const bool log_mode,int &error_code)
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(probability) || !MathIsValidNumber(nu))
|
| {
|
| error_code=ERR_ARGUMENTS_NAN;
|
| return QNaN;
|
| }
|
| //--- check nu
|
| 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;
|
| }
|
|
|
| //--- check cases when probability==0 or 1
|
| if(prob==0.0 || prob==1.0)
|
| {
|
| error_code=ERR_RESULT_INFINITE;
|
| //---
|
| if(prob==0.0)
|
| return(QNEGINF);
|
| else
|
| return(QPOSINF);
|
| }
|
|
|
| error_code=ERR_OK;
|
| //--- special case nu=1
|
| if(nu==1.0)
|
| return MathTan(M_PI*(prob-0.5));
|
| //--- special case
|
| if(prob==0.5)
|
| return 0.0;
|
| //---
|
| int max_iterations=50;
|
| int iterations=0;
|
| //--- initial values
|
| double h=1.0;
|
| double h_min=10E-20;
|
| double x=0.5;
|
| int err_code=0;
|
| //--- Newton iterations
|
| while(iterations<max_iterations)
|
| {
|
| //--- check convegence
|
| if((MathAbs(h)>h_min && MathAbs(h)>MathAbs(h_min*x))==false)
|
| break;
|
| //--- calculate pdf and cdf
|
| double pdf=MathProbabilityDensityT(x,nu,err_code);
|
| double cdf=MathCumulativeDistributionT(x,nu,err_code);
|
| //--- calculate ratio
|
| h=(cdf-prob)/pdf;
|
| //---
|
| double x_new=x-h;
|
| //--- check x
|
| 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;
|
|
|
| x=x_new;
|
|
|
| iterations++;
|
| }
|
| //--- check convergence
|
| if(iterations<max_iterations)
|
| return x;
|
| else
|
| {
|
| error_code=ERR_NON_CONVERGENCE;
|
| return QNaN;
|
| }
|
| }
|
| //+------------------------------------------------------------------+
|
| //| T distribution quantile function (inverse CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the inverse cumulative distribution |
|
| //| function of the T distribution with parameter nu for the desired |
|
| //| probability. |
|
| //| |
|
| //| Arguments: |
|
| //| probability : The desired probability |
|
| //| nu : Degrees of freedom |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The value of the inverse cumulative distribution function |
|
| //| of the T-distribution with parameter nu. |
|
| //+------------------------------------------------------------------+
|
| double MathQuantileT(const double probability,const double nu,int &error_code)
|
| {
|
| return MathQuantileT(probability,nu,true,false,error_code);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| T distribution quantile function (inverse CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the inverse cumulative distribution |
|
| //| function of the T distribution with parameter nu for |
|
| //| values from the probability[] array. |
|
| //| |
|
| //| Arguments: |
|
| //| probability : Array with probabilities |
|
| //| nu : Degrees of freedom |
|
| //| 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 MathQuantileT(const double &probability[],const double nu,const bool tail,const bool log_mode,double &result[])
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(nu))
|
| return false;
|
| //--- check nu
|
| if(nu!=MathRound(nu) || nu<0.0)
|
| return false;
|
|
|
| int data_count=ArraySize(probability);
|
| if(data_count==0)
|
| return false;
|
|
|
| int error_code=0;
|
| 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;
|
|
|
| //--- special case p=0.5
|
| if(prob==0.5)
|
| result[i]=0.0;
|
| else
|
| if(prob==0.0)
|
| result[i]=QNEGINF;
|
| else
|
| if(prob==1.0)
|
| result[i]=QPOSINF;
|
| else
|
| {
|
| //--- special case nu=1
|
| if(nu==1.0)
|
| result[i]=MathTan(M_PI*(prob-0.5));
|
| else
|
| {
|
| int max_iterations=50;
|
| int iterations=0;
|
| //--- initial values
|
| double h=1.0;
|
| double h_min=10E-18;
|
| double x=0.5;
|
| int err_code=0;
|
| //--- Newton iterations
|
| while(iterations<max_iterations)
|
| {
|
| //--- check convegence
|
| if((MathAbs(h)>h_min && MathAbs(h)>MathAbs(h_min*x))==false)
|
| break;
|
| //--- calculate pdf and cdf
|
| double pdf=MathProbabilityDensityT(x,nu,err_code);
|
| double cdf=MathCumulativeDistributionT(x,nu,err_code);
|
| //--- calculate ratio
|
| h=(cdf-prob)/pdf;
|
| //---
|
| double x_new=x-h;
|
| //--- check x
|
| 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;
|
| }
|
| //+------------------------------------------------------------------+
|
| //| T distribution quantile function (inverse CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the inverse cumulative distribution |
|
| //| function of the T distribution with parameter nu for |
|
| //| values from the probability[] array. |
|
| //| |
|
| //| Arguments: |
|
| //| probability : Array with probabilities |
|
| //| nu : Degrees of freedom |
|
| //| result : Array with calculated values |
|
| //| |
|
| //| Return value: |
|
| //| true if successful, otherwise false. |
|
| //+------------------------------------------------------------------+
|
| bool MathQuantileT(const double &probability[],const double nu,double &result[])
|
| {
|
| return MathQuantileT(probability,nu,true,false,result);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Random variate from the T distribution |
|
| //+------------------------------------------------------------------+
|
| //| Computes the random variable from the T distribution |
|
| //| with parameter nu. |
|
| //| |
|
| //| Arguments: |
|
| //| nu : Degrees of freedom |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The random value with T distribution. |
|
| //+------------------------------------------------------------------+
|
| double MathRandomT(const double nu,int error_code)
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(nu))
|
| {
|
| 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;
|
| //--- calculate normal random variable using Box-Muller transform
|
| double x1,x2,r2;
|
| do
|
| {
|
| x1=2.0*MathRandomNonZero()-1.0;
|
| x2=2.0*MathRandomNonZero()-1.0;
|
| r2=x1*x1+x2*x2;
|
| }
|
| while(r2>=1.0 || r2==0.0);
|
| //--- generate normal and gamma random variables
|
| double rnd_normal=x2*MathSqrt(-2.0*MathLog(r2)/r2);
|
| double rnd_gamma=MathRandomGamma(nu*0.5,1,error_code);
|
| //--- calculate ratio
|
| double result=0;
|
| if(rnd_gamma!=0)
|
| result=MathSqrt(nu*0.5)*rnd_normal/MathSqrt(rnd_gamma);
|
| return(result);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Random variate from the T distribution |
|
| //+------------------------------------------------------------------+
|
| //| Generates random variables from the T distribution with |
|
| //| parameter nu. |
|
| //| |
|
| //| Arguments: |
|
| //| nu : Degrees of freedom |
|
| //| data_count : Number of values needed |
|
| //| result : Output array with random values |
|
| //| |
|
| //| Return value: |
|
| //| true if successful, otherwise false. |
|
| //+------------------------------------------------------------------+
|
| bool MathRandomT(const double nu,const int data_count,double &result[])
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(nu))
|
| return false;
|
| //--- check arguments
|
| if(nu!=MathRound(nu) || nu<=0.0)
|
| return false;
|
|
|
| int error_code=0;
|
| //--- prepare output array and calculate random values
|
| ArrayResize(result,data_count);
|
| for(int i=0; i<data_count; i++)
|
| {
|
| //--- calculate normal random variable using Box-Muller transform
|
| double x1,x2,r2;
|
| do
|
| {
|
| x1=2.0*MathRandomNonZero()-1.0;
|
| x2=2.0*MathRandomNonZero()-1.0;
|
| r2=x1*x1+x2*x2;
|
| }
|
| while(r2>=1.0 || r2==0.0);
|
| //--- generate normal and gamma random variables
|
| double rnd_normal=x2*MathSqrt(-2.0*MathLog(r2)/r2);
|
| double rnd_gamma=MathRandomGamma(nu*0.5,1,error_code);
|
| //--- calculate ratio
|
| double rnd=0;
|
| if(rnd_gamma!=0)
|
| rnd=MathSqrt(nu*0.5)*rnd_normal/MathSqrt(rnd_gamma);
|
| result[i]=rnd;
|
| }
|
| return true;
|
| }
|
| //+------------------------------------------------------------------+
|
| //| T distribution moments |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates 4 first moments of the T distribution |
|
| //| with parameter nu. |
|
| //| |
|
| //| Arguments: |
|
| //| nu : Degrees of freedom |
|
| //| 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 MathMomentsT(const double nu,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))
|
| {
|
| error_code=ERR_ARGUMENTS_NAN;
|
| return QNaN;
|
| }
|
| //--- check nu
|
| if(nu!=MathRound(nu) || nu<0.0)
|
| {
|
| error_code=ERR_ARGUMENTS_INVALID;
|
| return QNaN;
|
| }
|
|
|
| error_code=ERR_OK;
|
| //--- calculate moments
|
| mean=0;
|
| if(nu>2)
|
| variance=nu/(nu-2);
|
| skewness=0;
|
| if(nu>4)
|
| kurtosis=6/(nu-4);
|
| //--- successful
|
| return true;
|
| }
|
| //+------------------------------------------------------------------+
|
