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//+------------------------------------------------------------------+ //| Beta.mqh | //| Copyright 2000-2026, MetaQuotes Ltd. | //| www.mql5.com | //+------------------------------------------------------------------+ #include "Math.mqh" //+------------------------------------------------------------------+ //| Beta density function (PDF) | //+------------------------------------------------------------------+ //| The function returns the probability density function of | //| the Beta distribution with shape parameters a and b. | //| | //| f(x,a,b)= (1/Beta(a,b))*x^(a-1)*(1-x)^(b-1) | //| Arguments: | //| x : Random variable | //| a : First shape parameter (a>0) | //| b : Second shape parameter (b>0) | //| log_mode : Logarithm mode flag, if true it returns Log values | //| error_code : Variable for error code | //| | //| Return value: | //| The probability density evaluated at x. | //+------------------------------------------------------------------+ double MathProbabilityDensityBeta(const double x,const double a,const double b,const bool log_mode,int &error_code) { //--- check NaN if(!MathIsValidNumber(x) || !MathIsValidNumber(a) || !MathIsValidNumber(b)) { error_code=ERR_ARGUMENTS_NAN; return QNaN; } //--- a and b must be positive if(a<=0.0 || b<=0.0) { error_code=ERR_ARGUMENTS_INVALID; return QNaN; } error_code=ERR_OK; //--- check x range if(x<=0.0 || x>=1.0) return TailLog0(true,log_mode); double log_result=(a-1.0)*MathLog(x)+(b-1.0)*MathLog(1.0-x)-MathBetaLog(a,b); //--- return log beta density if(log_mode==true) return log_result; //--- return beta density return MathExp(log_result); } //+------------------------------------------------------------------+ //| Beta density function (PDF) | //+------------------------------------------------------------------+ //| The function returns the probability density function of | //| the Beta distribution with shape parameters a and b. | //| | //| f(x,a,b)= (1/Beta(a,b))*x^(a-1)*(1-x)^(b-1) | //| Arguments: | //| x : Random variable | //| a : First shape parameter (a>0) | //| b : Second shape parameter (b>0) | //| error_code : Variable for error code | //| | //| Return value: | //| The probability density evaluated at x. | //+------------------------------------------------------------------+ double MathProbabilityDensityBeta(const double x,const double a,const double b,int &error_code) { return MathProbabilityDensityBeta(x,a,b,false,error_code); } //+------------------------------------------------------------------+ //| Beta density function (PDF) | //+------------------------------------------------------------------+ //| The function calculates the probability density function of the | //| Beta distribution with shape parameters a and b for values in | //| x[] array. | //| | //| Arguments: | //| x : Array with random variables | //| a : First shape parameter (a>0) | //| b : Second shape parameter (b>0) | //| log_mode : Logarithm mode flag,if true it calculates Log values| //| result : Array with calculated values | //| | //| Return value: | //| true if successful, otherwise false. | //+------------------------------------------------------------------+ bool MathProbabilityDensityBeta(const double &x[],const double a,const double b,const bool log_mode,double &result[]) { //--- check NaN if(!MathIsValidNumber(a) || !MathIsValidNumber(b)) return false; //--- a and b must be positive if(a<=0.0 || b<=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=1.0) result[i]=TailLog0(true,log_mode); else { double log_result=(a-1.0)*MathLog(x_arg)+(b-1.0)*MathLog(1.0-x_arg)-MathBetaLog(a,b); if(log_mode==true) result[i]=log_result; else result[i]=MathExp(log_result); } } return true; } //+------------------------------------------------------------------+ //| Beta density function (PDF) | //+------------------------------------------------------------------+ //| The function calculates the probability density function of the | //| Beta distribution with shape parameters a and b for values in | //| x[] array. | //| | //| Arguments: | //| x : Array with random variables | //| a : First shape parameter (a>0) | //| b : Second shape parameter (b>0) | //| result : Array with calculated values | //| | //| Return value: | //| true if successful, otherwise false. | //+------------------------------------------------------------------+ bool MathProbabilityDensityBeta(const double &x[],const double a,const double b,double &result[]) { return MathProbabilityDensityBeta(x,a,b,false,result); } //+------------------------------------------------------------------+ //| Beta cumulative distribution function (CDF) | //+------------------------------------------------------------------+ //| The function returns the cumulative distribution function of | //| the Beta distribution with shape parameters a and b, evaluated | //| at x. | //| | //| Arguments: | //| x : The desired quantile | //| a : First shape parameter (a>0) | //| b : Second shape parameter (b>0) | //| tail : Flag to calculate lower tail | //| log_mode : Logarithm mode flag,if true it calculates Log values| //| error_code : Variable for error code | //| | //| Return value: | //| The value of the Beta cumulative distribution function with | //| shape parameters a and b, evaluated at x. | //+------------------------------------------------------------------+ double MathCumulativeDistributionBeta(const double x,const double a,const double b,const bool tail,const bool log_mode,int &error_code) { //--- check NaN if(!MathIsValidNumber(x) || !MathIsValidNumber(a) || !MathIsValidNumber(b)) { error_code=ERR_ARGUMENTS_NAN; return QNaN; } //--- a and b must be positive if(a<=0.0 || b<=0.0) { error_code=ERR_ARGUMENTS_INVALID; return QNaN; } error_code=ERR_OK; //--- check x range if(x<=0.0) return TailLog0(tail,log_mode); if(x>=1.0) return TailLog1(tail,log_mode); //--- calculate probability and take into account round-off errors double cdf=MathMin(MathBetaIncomplete(x,a,b),1.0); //--- return result depending on arguments return TailLogValue(cdf,tail,log_mode); } //+------------------------------------------------------------------+ //| Beta cumulative distribution function (CDF) | //+------------------------------------------------------------------+ //| The function returns the cumulative distribution function of | //| the Beta distribution with shape parameters a and b, evaluated | //| at x. | //| | //| Arguments: | //| x : The desired quantile | //| a : First shape parameter (a>0) | //| b : Second shape parameter (b>0) | //| error_code : Variable for error code | //| | //| Return value: | //| The value of the Beta cumulative distribution function with | //| shape parameters a and b, evaluated at x. | //+------------------------------------------------------------------+ double MathCumulativeDistributionBeta(const double x,const double a,const double b,int &error_code) { return MathCumulativeDistributionBeta(x,a,b,true,false,error_code); } //+------------------------------------------------------------------+ //| The Beta cumulative distribution function (CDF) | //+------------------------------------------------------------------+ //| The function returns the cumulative distribution function of | //| the Beta distribution with shape parameters a and b for values | //| in x[] array | //| | //| Arguments: | //| x : Array with random variables | //| a : First shape parameter (a>0) | //| b : Second shape parameter (b>0) | //| tail : Flag to calculate lower tail | //| log_mode : Logarithm mode flag,if true it calculates Log values| //| error_code : Variable for error code | //| | //| Return value: | //| true if successful, otherwise false. | //+------------------------------------------------------------------+ bool MathCumulativeDistributionBeta(const double &x[],const double a,const double b,const bool tail,const bool log_mode,double &result[]) { //--- check NaN if(!MathIsValidNumber(a) || !MathIsValidNumber(b)) return false; //--- a and b must be positive if(a<=0.0 || b<=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=1.0) result[i]=TailLog1(tail,log_mode); else { //--- calculate probability and take into account round-off errors double cdf=MathMin(MathBetaIncomplete(x_arg,a,b),1.0); //--- return result depending on arguments result[i]=TailLogValue(cdf,tail,log_mode); } } else return false; } return(true); } //+------------------------------------------------------------------+ //| Beta cumulative distribution function (CDF) | //+------------------------------------------------------------------+ //| The function calculates the cumulative distribution function of | //| the Beta distribution with shape parameters a and b for values | //| in x[] array. | //| | //| Arguments: | //| x : Array with random variables | //| a : First shape parameter (a>0) | //| b : Second shape parameter (b>0) | //| error_code : Variable for error code | //| | //| Return value: | //| true if successful, otherwise false. | //+------------------------------------------------------------------+ bool MathCumulativeDistributionBeta(const double &x[],const double a,const double b,double &result[]) { return MathCumulativeDistributionBeta(x,a,b,true,false,result); } //+------------------------------------------------------------------+ //| Beta distribution quantile function (inverse CDF) | //+------------------------------------------------------------------+ //| The function returns the inverse cumulative distribution | //| function of the Beta distribution with shape parameters a and b | //| for the desired probability. | //| | //| Arguments: | //| probability : The desired probability | //| a : First shape parameter (a>0) | //| b : Second shape parameter (b>0) | //| tail : Flag to calculate lower tail | //| log_mode : Logarithm mode flag,if true calculates Log values | //| error_code : Variable for error code | //| | //| Return value: | //| The value of the inverse cumulative distribution function of | //| the Beta distribution with shape parameters a and b. | //+------------------------------------------------------------------+ double MathQuantileBeta(const double probability,const double a,const double b,const bool tail,const bool log_mode,int &error_code) { //--- check parameters if(!MathIsValidNumber(probability) || !MathIsValidNumber(a) || !MathIsValidNumber(b)) { error_code=ERR_ARGUMENTS_NAN; return QNaN; } //--- a and b must be positive if(a<=0.0 || b<=0.0) { error_code=ERR_ARGUMENTS_INVALID; return QNaN; } //--- calculate real probability double prob=TailLogProbability(probability,tail,log_mode); //--- check probability range if(prob<0.0 || prob>1.0) { error_code=ERR_ARGUMENTS_INVALID; return QNaN; } error_code=ERR_OK; //--- check probabilty if(prob==0.0) return 0.0; if(prob==1.0) return 1.0; const double eps=10e-16; //--- set h and h_min double h=1.0; double h_min=MathSqrt(eps); //--- initial x value double x=a/(a+b); if(x==0.0) x=h_min; else if(x==1.0) x=1.0-h_min; int err_code=0; const int max_iterations=100; int iterations=0; //--- Newton iterations while(iterationsh_min*MathAbs(x)) && (MathAbs(h)>h_min))==false) break; //--- calculate pdf and cdf double pdf=MathProbabilityDensityBeta(x,a,b,false,err_code); double cdf=MathCumulativeDistributionBeta(x,a,b,true,false,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(iterations0) | //| b : Second shape parameter (b>0) | //| error_code : Variable for error code | //| | //| Return value: | //| The value of the inverse cumulative distribution function | //| of the Beta distribution with shape parameters a and b. | //+------------------------------------------------------------------+ double MathQuantileBeta(const double probability,const double a,const double b,int &error_code) { return MathQuantileBeta(probability,a,b,true,false,error_code); } //+------------------------------------------------------------------+ //| Beta distribution quantile function (inverse CDF) | //+------------------------------------------------------------------+ //| The function calculates the the inverse cumulative distribution | //| function of the Beta distribution with shape parameters a and b | //| for the probability values from probability[] array. | //| | //| Arguments: | //| probability : Array with probability values | //| a : First shape parameter (a>0) | //| b : Second shape parameter (b>0) | //| tail : Lower tail flag (lower tail of probability used) | //| log_mode : Logarithm mode flag (log probability used) | //| result : Output array with quantile values | //| | //| Return value: | //| true if successful, otherwise false. | //+------------------------------------------------------------------+ bool MathQuantileBeta(const double &probability[],const double a,const double b,const bool tail,const bool log_mode,double &result[]) { //--- check parameters if(!MathIsValidNumber(a) || !MathIsValidNumber(b)) return false; //--- a and b must be positive if(a<=0.0 || b<=0.0) return false; int data_count=ArraySize(probability); if(data_count==0) return false; int error_code=0; ArrayResize(result,data_count); const double eps=10e-16; double h_min=MathSqrt(eps); int err_code=0; const int max_iterations=1000; for(int i=0; i1.0) return false; //--- check probabilty if(prob==0.0) result[i]=0.0; else if(prob==1.0) result[i]=1.0; else { //--- initial x value double x=a/(a+b); if(x==0.0) x=h_min; else if(x==1.0) x=1.0-h_min; double h=1.0; int iterations=0; //--- Newton iterations while(iterationsh_min*MathAbs(x)) && (MathAbs(h)>h_min))==false) break; //--- calculate pdf and cdf double pdf=MathProbabilityDensityBeta(x,a,b,false,err_code); double cdf=MathCumulativeDistributionBeta(x,a,b,true,false,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(iterations0) | //| b : Second shape parameter (b>0) | //| result : Output array with quantile values | //| | //| Return value: | //| true if successful, otherwise false. | //+------------------------------------------------------------------+ bool MathQuantileBeta(const double &probability[],const double a,const double b,double &result[]) { return MathQuantileBeta(probability,a,b,true,false,result); } //+------------------------------------------------------------------+ //| Random variate from the Beta distribution | //+------------------------------------------------------------------+ //| The function returns a single random deviate from the Beta | //| distribution with parameters a and b. | //| | //| Arguments: | //| a : First shape parameter (a>0) | //| b : Second shape parameter (b>0) | //| | //| Return value: | //| The random value with Beta distribution. | //| | //| Reference: | //| Russell Cheng, | //| "Generating Beta Variates with Nonintegral Shape Parameters", | //| Communications of the ACM, | //| Volume 21, Number 4, April 1978, pages 317-322. | //| | //| Original FORTRAN77 version by Barry Brown, James Lovato. | //| C version by John Burkardt. | //+------------------------------------------------------------------+ double MathRandomBeta(const double a,const double b) { const double log4 = MathLog(4); const double log5 = MathLog(5); double a1,b1,alpha,beta,gamma,delta,r,s,u1,u2,v,y,z; double w=0.0; double value=0; //--- if(1.00) | //| b : Second shape parameter (b>0) | //| error_code : Variable for error code | //| | //| Return value: | //| The random value with Beta distribution. | //+------------------------------------------------------------------+ double MathRandomBeta(const double a,const double b,int &error_code) { //--- check parameters if(!MathIsValidNumber(a) || !MathIsValidNumber(b)) { error_code=ERR_ARGUMENTS_NAN; return QNaN; } //--- a and b must be positive if(a<=0.0 || b<=0.0) { error_code=ERR_ARGUMENTS_INVALID; return QNaN; } error_code=ERR_OK; //--- return beta random value return MathRandomBeta(a,b); } //+------------------------------------------------------------------+ //| Random variate from Beta distribution | //+------------------------------------------------------------------+ //| The function generates random variables from Beta distribution | //| with parameters a and b. | //| | //| Arguments: | //| a : First shape parameter (a>0) | //| b : Second shape parameter (b>0) | //| data_count : Number of values needed | //| result : Output array with random values | //| | //| Return value: | //| true if successful, otherwise false. | //+------------------------------------------------------------------+ bool MathRandomBeta(const double a,const double b,const int data_count,double &result[]) { if(data_count<=0) return false; //--- check parameters if(!MathIsValidNumber(a) || !MathIsValidNumber(b)) return false; //--- a and b must be positive if(a<=0.0 || b<=0.0) return false; //--- prepare output array and calculate random values ArrayResize(result,data_count); for(int i=0; i0) | //| b : Second shape parameter (b>0) | //| mean : Variable for mean value (1st moment) | //| variance : Variable for variance value (2nd moment) | //| skewness : Variable for skewness value (3rd moment) | //| kurtosis : Variable for kurtosis value (4th moment) | //| error_code : Variable for error code | //| | //| Return value: | //| true if moments calculated successfully, otherwise false. | //+------------------------------------------------------------------+ bool MathMomentsBeta(const double a,const double b,double &mean,double &variance,double &skewness,double &kurtosis,int &error_code) { //--- initial values mean =QNaN; variance=QNaN; skewness=QNaN; kurtosis=QNaN; //--- check NaN if(!MathIsValidNumber(a) || !MathIsValidNumber(b)) { error_code=ERR_ARGUMENTS_NAN; return false; } //--- a and b must be positive if(a<=0.0 || b<=0.0) { error_code=ERR_ARGUMENTS_INVALID; return false; } error_code=ERR_OK; //--- calculate moments mean =a/(a+b); variance=(a*b)/((a+b)*(a+b)*(a+b+1)); skewness=2*(b-a)*MathSqrt(a+b+1)/(MathSqrt(a*b)*(a+b+2)); kurtosis=6*(a*a*a+a*a*(1-2*b)+b*b*(1+b)-2*a*b*(2+b))/(a*b*(a+b+2)*(a+b+3)); //--- successful return true; } //+------------------------------------------------------------------+