654 行
27 KiB
MQL5
654 行
27 KiB
MQL5
 | //+------------------------------------------------------------------+
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//| T.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 "Gamma.mqh"
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//+------------------------------------------------------------------+
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//| 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 T-distribution with parameter nu. |
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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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//| 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 MathProbabilityDensityT(const double x,const double nu,const bool log_mode,int &error_code)
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{
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//--- check NaN
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if(!MathIsValidNumber(x) || !MathIsValidNumber(nu))
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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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//--- calculate T density
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double pdf=MathExp(MathGammaLog((nu+1.0)*0.5)-MathGammaLog(nu*0.5));
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pdf=pdf/(MathSqrt(nu*M_PI)*MathPow(1+x*x/nu,(nu+1.0)*0.5));
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//--- return density
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return TailLogValue(pdf,true,log_mode);
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}
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//+------------------------------------------------------------------+
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//| 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 T-distribution with parameter nu. |
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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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//| 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 MathProbabilityDensityT(const double x,const double nu,int &error_code)
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{
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return MathProbabilityDensityT(x,nu,false,error_code);
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}
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//+------------------------------------------------------------------+
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//| T probability density function (PDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the probability density function of |
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//| the T distribution with parameter nu 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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//| 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 MathProbabilityDensityT(const double &x[],const double nu,const bool log_mode,double &result[])
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{
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//--- check NaN
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if(!MathIsValidNumber(nu))
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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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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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error_code=ERR_OK;
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//--- calculate T density
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double pdf=MathExp(MathGammaLog((nu+1.0)*0.5)-MathGammaLog(nu*0.5));
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pdf=pdf/(MathSqrt(nu*M_PI)*MathPow(1+x_arg*x_arg/nu,(nu+1.0)*0.5));
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//--- return density
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result[i]=TailLogValue(pdf,true,log_mode);
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}
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return true;
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}
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//+------------------------------------------------------------------+
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//| T probability density function (PDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the probability density function of |
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//| the T distribution with parameter nu 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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//| 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 MathProbabilityDensityT(const double &x[],const double nu,double &result[])
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{
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return MathProbabilityDensityT(x,nu,false,result);
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}
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//+------------------------------------------------------------------+
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//| T cumulative distribution function (CDF) |
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//+------------------------------------------------------------------+
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//| The function returns the probability that an observation from |
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//| T-distribution with parameter nu 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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//| 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 T cumulative distribution function with |
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//| parameter nu, evaluated at x. |
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//+------------------------------------------------------------------+
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double MathCumulativeDistributionT(const double x,const double nu,const bool tail,const bool log_mode,int &error_code)
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{
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//--- check NaN
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if(!MathIsValidNumber(x) || !MathIsValidNumber(nu))
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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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error_code=ERR_OK;
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//--- special case
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if(nu==1.0)
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return TailLogValue(0.5+MathArctan(x)/M_PI,tail,log_mode);
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//--- otherwise
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if(x==0)
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return TailLogValue(0.5,tail,log_mode);
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//--- calculate pdf using incomplete Beta
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double cdf=1.0-MathBetaIncomplete(nu/(nu+x*x),nu*0.5,0.5);
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cdf=(1.0-cdf)*0.5;
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//--- check x
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if(x>0.0)
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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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//+------------------------------------------------------------------+
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//| T cumulative distribution function (CDF) |
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//+------------------------------------------------------------------+
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//| The function returns the probability that an observation from |
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//| T-distribution with parameter nu 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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//| error_code : Variable for error code |
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//| |
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//| Return value: |
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//| The value of T cumulative distribution function with parameter |
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//| nu, evaluated at x. |
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//+------------------------------------------------------------------+
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double MathCumulativeDistributionT(const double x,const double nu,int &error_code)
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{
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return MathCumulativeDistributionT(x,nu,true,false,error_code);
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}
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//+------------------------------------------------------------------+
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//| T cumulative distribution function (CDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the cumulative distribution function of |
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//| the T distribution with parameter nu for values in x. |
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//| |
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//| Arguments: |
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//| x : Array with random variables |
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//| nu : Degrees of freedom |
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//| tail : Flag to calculate lower tail |
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//| log_mode : Logarithm mode, if true it calculates Log values |
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//| result : Array with calculated values |
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//| |
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//| Return value: |
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//| true if successful, otherwise false. |
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//+------------------------------------------------------------------+
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bool MathCumulativeDistributionT(const double &x[],const double nu,const bool tail,const bool log_mode,double &result[])
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{
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//--- check NaN
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if(!MathIsValidNumber(nu))
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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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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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//--- special case
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if(nu==1.0)
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result[i]=TailLogValue(0.5+MathArctan(x_arg)/M_PI,tail,log_mode);
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else
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//--- otherwise
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if(x_arg==0)
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result[i]=TailLogValue(0.5,tail,log_mode);
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else
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{
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//--- calculate pdf using incomplete Beta
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double cdf=1.0-MathBetaIncomplete(nu/(nu+x_arg*x_arg),nu*0.5,0.5);
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cdf=(1.0-cdf)*0.5;
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//--- check x
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if(x_arg>0.0)
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cdf=1.0-cdf;
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//--- take into account round-off errors for probability
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result[i]=TailLogValue(MathMin(cdf,1.0),tail,log_mode);
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}
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}
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return true;
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}
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//+------------------------------------------------------------------+
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//| T cumulative distribution function (CDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the cumulative distribution function of |
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//| the T distribution with parameter nu 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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//| 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 MathCumulativeDistributionT(const double &x[],const double nu,double &result[])
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{
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return MathCumulativeDistributionT(x,nu,true,false,result);
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}
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//+------------------------------------------------------------------+
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//| T distribution quantile function (inverse CDF) |
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//+------------------------------------------------------------------+
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//| The function returns the inverse cumulative distribution |
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//| function of the T distribution with parameter nu for the desired |
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//| probability. |
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//| |
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//| Arguments: |
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//| probability : The desired probability |
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//| nu : Degrees of freedom |
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//| 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 inverse cumulative distribution function |
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//| of the T-distribution with parameter nu. |
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//+------------------------------------------------------------------+
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double MathQuantileT(const double probability,const double nu,const bool tail,const bool log_mode,int &error_code)
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{
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//--- check NaN
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if(!MathIsValidNumber(probability) || !MathIsValidNumber(nu))
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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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//--- calculate real probability
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double prob=TailLogProbability(probability,tail,log_mode);
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//--- check probability range
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if(prob<0.0 || prob>1.0)
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{
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error_code=ERR_ARGUMENTS_INVALID;
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return QNaN;
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}
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//--- check cases when probability==0 or 1
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if(prob==0.0 || prob==1.0)
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{
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error_code=ERR_RESULT_INFINITE;
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//---
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if(prob==0.0)
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return(QNEGINF);
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else
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return(QPOSINF);
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}
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error_code=ERR_OK;
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//--- special case nu=1
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if(nu==1.0)
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return MathTan(M_PI*(prob-0.5));
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//--- special case
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if(prob==0.5)
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return 0.0;
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//---
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int max_iterations=50;
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int iterations=0;
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//--- initial values
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double h=1.0;
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double h_min=10E-20;
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double x=0.5;
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int err_code=0;
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//--- Newton iterations
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while(iterations<max_iterations)
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{
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//--- check convegence
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if((MathAbs(h)>h_min && MathAbs(h)>MathAbs(h_min*x))==false)
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break;
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//--- calculate pdf and cdf
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double pdf=MathProbabilityDensityT(x,nu,err_code);
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double cdf=MathCumulativeDistributionT(x,nu,err_code);
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//--- calculate ratio
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h=(cdf-prob)/pdf;
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//---
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double x_new=x-h;
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//--- check x
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if(x_new<0.0)
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x_new=x*0.1;
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else
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if(x_new>1.0)
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x_new=1.0-(1.0-x)*0.1;
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x=x_new;
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iterations++;
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}
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//--- check convergence
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if(iterations<max_iterations)
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return x;
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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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//| T distribution quantile function (inverse CDF) |
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//+------------------------------------------------------------------+
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//| The function returns the inverse cumulative distribution |
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//| function of the T distribution with parameter nu for the desired |
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//| probability. |
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//| |
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//| Arguments: |
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//| probability : The desired probability |
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//| nu : Degrees of freedom |
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//| error_code : Variable for error code |
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//| |
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//| Return value: |
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//| The value of the inverse cumulative distribution function |
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//| of the T-distribution with parameter nu. |
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//+------------------------------------------------------------------+
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double MathQuantileT(const double probability,const double nu,int &error_code)
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{
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return MathQuantileT(probability,nu,true,false,error_code);
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}
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//+------------------------------------------------------------------+
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//| T distribution quantile function (inverse CDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the inverse cumulative distribution |
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//| function of the T distribution with parameter nu for |
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//| values from the probability[] array. |
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//| |
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//| Arguments: |
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//| probability : Array with probabilities |
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//| nu : Degrees of freedom |
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//| tail : Flag to calculate lower tail |
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//| log_mode : Logarithm mode, if true it calculates Log values |
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//| result : Array with calculated values |
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//| |
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//| Return value: |
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//| true if successful, otherwise false. |
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//+------------------------------------------------------------------+
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bool MathQuantileT(const double &probability[],const double nu,const bool tail,const bool log_mode,double &result[])
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{
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//--- check NaN
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if(!MathIsValidNumber(nu))
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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(probability);
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if(data_count==0)
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return false;
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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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//--- calculate real probability
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double prob=TailLogProbability(probability[i],tail,log_mode);
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//--- check probability range
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if(prob<0.0 || prob>1.0)
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return false;
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//--- special case p=0.5
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if(prob==0.5)
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result[i]=0.0;
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else
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if(prob==0.0)
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result[i]=QNEGINF;
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else
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if(prob==1.0)
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result[i]=QPOSINF;
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else
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{
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//--- special case nu=1
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if(nu==1.0)
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result[i]=MathTan(M_PI*(prob-0.5));
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else
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{
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int max_iterations=50;
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int iterations=0;
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//--- initial values
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double h=1.0;
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double h_min=10E-18;
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double x=0.5;
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int err_code=0;
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//--- Newton iterations
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while(iterations<max_iterations)
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{
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//--- check convegence
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if((MathAbs(h)>h_min && MathAbs(h)>MathAbs(h_min*x))==false)
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break;
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//--- calculate pdf and cdf
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double pdf=MathProbabilityDensityT(x,nu,err_code);
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double cdf=MathCumulativeDistributionT(x,nu,err_code);
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//--- calculate ratio
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h=(cdf-prob)/pdf;
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//---
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double x_new=x-h;
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//--- check x
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if(x_new<0.0)
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x_new=x*0.1;
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else
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if(x_new>1.0)
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x_new=1.0-(1.0-x)*0.1;
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if (MathAbs(x_new-x)<10E-15)
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break;
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x=x_new;
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iterations++;
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}
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//--- check convergence
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if(iterations<max_iterations)
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result[i]=x;
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else
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return false;
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}
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}
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}
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return true;
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}
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//+------------------------------------------------------------------+
| |||
//| T distribution quantile function (inverse CDF) |
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//+------------------------------------------------------------------+
| |||
//| The function calculates the inverse cumulative distribution |
| |||
//| function of the T distribution with parameter nu for |
| |||
//| values from the probability[] array. |
| |||
//| |
| |||
//| Arguments: |
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//| probability : Array with probabilities |
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//| nu : Degrees of freedom |
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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 MathQuantileT(const double &probability[],const double nu,double &result[])
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{
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return MathQuantileT(probability,nu,true,false,result);
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}
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//+------------------------------------------------------------------+
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//| Random variate from the T distribution |
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//+------------------------------------------------------------------+
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//| Computes the random variable from the T distribution |
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//| with parameter nu. |
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//| |
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//| Arguments: |
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//| nu : Degrees of freedom |
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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 random value with T distribution. |
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//+------------------------------------------------------------------+
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double MathRandomT(const double nu,int error_code)
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{
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//--- check NaN
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if(!MathIsValidNumber(nu))
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{
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error_code=ERR_ARGUMENTS_NAN;
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return QNaN;
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}
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//--- check arguments
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if(nu!=MathRound(nu) || nu<=0.0)
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{
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error_code=ERR_ARGUMENTS_INVALID;
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return QNaN;
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}
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error_code=ERR_OK;
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//--- calculate normal random variable using Box-Muller transform
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double x1,x2,r2;
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do
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{
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x1=2.0*MathRandomNonZero()-1.0;
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x2=2.0*MathRandomNonZero()-1.0;
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r2=x1*x1+x2*x2;
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}
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while(r2>=1.0 || r2==0.0);
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//--- generate normal and gamma random variables
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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;
| |||
}
| |||
//+------------------------------------------------------------------+
|