955 lines
39 KiB
MQL5
955 lines
39 KiB
MQL5
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//+------------------------------------------------------------------+
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//| NoncentralBeta.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 "Beta.mqh"
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#include "NoncentralChiSquare.mqh"
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//+------------------------------------------------------------------+
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//| Noncental Beta density function (PDF) |
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//+------------------------------------------------------------------+
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//| The function returns the probability density function |
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//| of the Noncental Beta distribution with parameters a,b,lambda |
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//| Infinity |
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//| f(x,a,b,lambda)=Sum [p(k)*x^(a+k-1)*(1-x)^(b-1)]/Beta(a+k,b) |
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//| k=0 |
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//| |
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//| where p(k)=(1/k!)*exp(-lambda/2)*(lambda/2)^k, |
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//| Beta(a,b)=Gamma(a)*Gamma(b)/Gamma(a+b) |
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//| |
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//| Arguments: |
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//| x : Random variable |
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//| a : First shape parameter |
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//| b : Second shape parameter |
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//| lambda : Noncentrality parameter |
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//| log_mode : Logarithm mode flag, if true it returns 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 MathProbabilityDensityNoncentralBeta(const double x,const double a,const double b,const double lambda,const bool log_mode,int &error_code)
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{
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//--- if lambda==0, return Beta density
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if(lambda==0.0)
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return MathProbabilityDensityBeta(x,a,b,error_code);
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//--- check parameters
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if(!MathIsValidNumber(x) || !MathIsValidNumber(a) || !MathIsValidNumber(b))
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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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//--- a,b,lambda must be positive
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if(a<=0.0 || b<=0.0 || lambda<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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if(x<=0.0 || x>=1.0)
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return TailLog0(true,log_mode);
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//--- factors
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double lambda_half=lambda*0.5;
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double fact_mult=1.0;
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double pwr_lambda_half=1.0;
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double pwr_x=MathExp((a-1.0)*MathLog(x));
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double r_beta=MathBeta(a,b);
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double pdf=0;
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//--- direct sum calculation
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for(int j=0;; j++)
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{
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if(j>0)
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{
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pwr_x*=x;
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pwr_lambda_half*=lambda_half;
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fact_mult/=j;
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double jm1=j-1;
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r_beta*=((a+jm1)/(a+b+jm1));
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}
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double term=pwr_x*fact_mult*pwr_lambda_half/r_beta;
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//---
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if(term<10E-18)
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break;
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pdf+=term;
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}
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//--- calculate density coef
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pdf*=MathExp((b-1.0)*MathLog(1.0-x))*MathExp(-lambda_half);
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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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//| Noncental Beta density function (PDF) |
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//+------------------------------------------------------------------+
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//| The function returns the probability density function |
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//| of the Noncental Beta distribution with parameters a,b,lambda. |
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//| |
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//| Arguments: |
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//| x : Random variable |
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//| a : First shape parameter |
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//| b : Second shape parameter |
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//| lambda : 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 MathProbabilityDensityNoncentralBeta(const double x,const double a,const double b,const double lambda,int &error_code)
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{
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return MathProbabilityDensityNoncentralBeta(x,a,b,lambda,false,error_code);
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}
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//+------------------------------------------------------------------+
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//| Noncental Beta density function (PDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the probability density function of |
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//| the Noncentral Beta distribution with parameters a,b,lambda |
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//| 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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//| a : First shape parameter |
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//| b : Second shape parameter |
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//| lambda : Noncentrality parameter |
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//| tail : Flag to calculate lower tail |
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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 MathProbabilityDensityNoncentralBeta(const double &x[],const double a,const double b,const double lambda,const bool log_mode,double &result[])
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{
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//--- if lambda==0, return Beta density
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if(lambda==0.0)
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return MathProbabilityDensityBeta(x,a,b,log_mode,result);
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//--- check parameters
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if(!MathIsValidNumber(a) || !MathIsValidNumber(b))
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return false;
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//--- a,b,lambda must be positive
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if(a<=0.0 || b<=0.0 || lambda<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 lambda_half=lambda*0.5;
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double exp_lambda_half=MathExp(-lambda_half);
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double r_beta0=MathBeta(a,b);
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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(x_arg<=0.0 || x_arg>=1.0)
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result[i]=TailLog0(true,log_mode);
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else
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{
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double fact_mult=1.0;
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double pwr_lambda_half=1.0;
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double pwr_x=MathExp((a-1.0)*MathLog(x_arg));
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double r_beta=r_beta0;
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double pdf=0;
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for(int j=0;; j++)
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{
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if(j>0)
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{
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pwr_x*=x_arg;
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pwr_lambda_half*=lambda_half;
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fact_mult/=j;
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double jm1=j-1;
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r_beta*=((a+jm1)/(a+b+jm1));
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}
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double term=pwr_x*fact_mult*pwr_lambda_half/r_beta;
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//---
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if(term<10E-18)
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break;
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pdf+=term;
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}
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//--- calculate density coef
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pdf*=MathExp((b-1.0)*MathLog(1.0-x_arg))*exp_lambda_half;
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result[i]=TailLogValue(pdf,true,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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//| Noncental Beta density function (PDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the probability density function of |
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//| the Noncentral Beta distribution with parameters a,b,lambda |
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//| 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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//| a : First shape parameter |
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//| b : Second shape parameter |
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//| lambda : 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 MathProbabilityDensityNoncentralBeta(const double &x[],const double a,const double b,const double lambda,double &result[])
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{
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return MathProbabilityDensityNoncentralBeta(x,a,b,lambda,false,result);
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}
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//+------------------------------------------------------------------+
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//| Noncental Beta 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 Noncental Beta distribution with parameters a,b,lambda |
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//| is less than or equal to x. |
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//| |
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//| Input parameters: |
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//| x : The desired quantile |
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//| a : First shape parameter |
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//| b : Second shape parameter |
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//| lambda : 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 value of the Noncental Beta cumulative distribution function |
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//| with parameters a,b,lambda, evaluated at x. |
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//| |
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//| Infinity |
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//| F(x,a,b,lambda)=Sum p(k)*Ix(a+k,b) |
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//| k=0 |
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//| |
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//| where p(k)=(1/k!)*exp(-lambda/2)*(lambda/2)^k, |
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//| Ix(a,b) - incomplete Beta function |
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//| |
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//| Author: John Burkardt |
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//| |
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//| Reference: |
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//| Harry Posten,"An Effective Algorithm for the Noncentral Beta |
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//| Distribution Function", The American Statistician, |
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//| Volume 47, Number 2, May 1993, pages 129-131. |
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//+------------------------------------------------------------------+
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double MathCumulativeDistributionNoncentralBeta(const double x,const double a,const double b,const double lambda,const bool tail,const bool log_mode,int &error_code)
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{
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//--- if lambda==0, return Beta CDF
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if(lambda==0.0)
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return MathCumulativeDistributionBeta(x,a,b,error_code);
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//--- check parameters
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if(!MathIsValidNumber(x) || !MathIsValidNumber(a) || !MathIsValidNumber(b))
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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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//--- a,b,lambda must be positive
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if(a<=0.0 || b<=0.0 || lambda<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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if(x<=0.0)
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return TailLog0(tail,log_mode);
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if(x>=1.0)
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return TailLog1(tail,log_mode);
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const int max_terms=100;
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double c=lambda*0.5;
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double x0 = int(MathMax(c - 5*MathSqrt(c), 0));
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double a0 = a + x0;
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double beta = MathGammaLog(a0) + MathGammaLog(b) - MathGammaLog(a0+b);
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double temp = MathBetaIncomplete(x, a0, b);
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double gx=MathExp(a0*MathLog(x)+b*MathLog(1-x)-beta-MathLog(a0));
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double q=0;
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if(a0>a)
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q=MathExp(-c+x0*MathLog(c)-MathGammaLog(x0+1));
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else
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q=MathExp(-c);
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double sumq=1-q;
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double betanc=q*temp;
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double ab=a+b;
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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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temp-=gx;
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gx*=x*(ab+j-1)/(a+j);
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q*=c/j;
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sumq-=q;
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betanc+=temp*q;
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double err=(temp-gx)*sumq;
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if(j>max_terms || err<1E-18)
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break;
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}
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double cdf=MathMin(betanc,1.0);
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return TailLogValue(cdf,tail,log_mode);
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}
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//+------------------------------------------------------------------+
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//| Noncental Beta 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 Noncental Beta distribution with parameters a,b,lambda |
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//| is less than or equal to x. |
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//| |
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//| Input parameters: |
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//| x : The desired quantile |
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//| a : First shape parameter |
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//| b : Second shape parameter |
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//| lambda : 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 value of the Noncental Beta cumulative distribution function |
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//| with parameters a,b,lambda, evaluated at x. |
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//+------------------------------------------------------------------+
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double MathCumulativeDistributionNoncentralBeta(const double x,const double a,const double b,const double lambda,int &error_code)
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{
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return MathCumulativeDistributionNoncentralBeta(x,a,b,lambda,true,false,error_code);
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}
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//+------------------------------------------------------------------+
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//| Noncental Beta 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 Noncentral Beta distribution with parameters a,b,lambda |
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//| 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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//| a : First shape parameter |
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//| b : Second shape parameter |
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//| lambda : Noncentrality parameter |
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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 MathCumulativeDistributionNoncentralBeta(const double &x[],const double a,const double b,const double lambda,const bool tail,const bool log_mode,double &result[])
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{
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//--- if lambda==0, return Beta CDF
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if(lambda==0.0)
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return MathCumulativeDistributionBeta(x,a,b,tail,log_mode,result);
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//--- check parameters
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if(!MathIsValidNumber(a) || !MathIsValidNumber(b))
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return false;
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//--- a,b,lambda must be positive
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if(a<=0.0 || b<=0.0 || lambda<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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const int max_terms=100;
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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(x_arg<=0.0)
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result[i]=TailLog0(tail,log_mode);
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if(x_arg>=1.0)
|
||
|
|
result[i]=TailLog1(tail,log_mode);
|
||
|
|
else
|
||
|
|
{
|
||
|
|
double c=lambda*0.5;
|
||
|
|
double x0 = int(MathMax(c - 5*MathSqrt(c), 0));
|
||
|
|
double a0 = a + x0;
|
||
|
|
double beta = MathGammaLog(a0) + MathGammaLog(b) - MathGammaLog(a0+b);
|
||
|
|
double temp = MathBetaIncomplete(x_arg, a0, b);
|
||
|
|
double gx=MathExp(a0*MathLog(x_arg)+b*MathLog(1-x_arg)-beta-MathLog(a0));
|
||
|
|
|
||
|
|
double q=0;
|
||
|
|
if(a0>a)
|
||
|
|
q=MathExp(-c+x0*MathLog(c)-MathGammaLog(x0+1));
|
||
|
|
else
|
||
|
|
q=MathExp(-c);
|
||
|
|
|
||
|
|
double sumq=1-q;
|
||
|
|
double betanc=q*temp;
|
||
|
|
int j=0;
|
||
|
|
double ab=a+b;
|
||
|
|
for(;;)
|
||
|
|
{
|
||
|
|
j++;
|
||
|
|
temp-=gx;
|
||
|
|
gx*=x_arg*(ab+j-1)/(a+j);
|
||
|
|
q*=c/j;
|
||
|
|
sumq-=q;
|
||
|
|
betanc+=temp*q;
|
||
|
|
double err=(temp-gx)*sumq;
|
||
|
|
if(j>max_terms || err<1E-18)
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
double cdf=MathMin(betanc,1.0);
|
||
|
|
result[i]=TailLogValue(cdf,tail,log_mode);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return true;
|
||
|
|
}
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| Noncental Beta cumulative distribution function (CDF) |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| The function calculates the cumulative distribution function of |
|
||
|
|
//| the Noncentral Beta distribution with parameters a,b,lambda |
|
||
|
|
//| for values in x[] array. |
|
||
|
|
//| |
|
||
|
|
//| Arguments: |
|
||
|
|
//| x : Array with random variables |
|
||
|
|
//| a : First shape parameter |
|
||
|
|
//| b : Second shape parameter |
|
||
|
|
//| lambda : Noncentrality parameter |
|
||
|
|
//| result : Array with calculated values |
|
||
|
|
//| |
|
||
|
|
//| Return value: |
|
||
|
|
//| true if successful, otherwise false. |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
bool MathCumulativeDistributionNoncentralBeta(const double &x[],const double a,const double b,const double lambda,double &result[])
|
||
|
|
{
|
||
|
|
return MathCumulativeDistributionNoncentralBeta(x,a,b,lambda,true,false,result);
|
||
|
|
}
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| Noncental Beta distribution quantile function (inverse CDF) |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| The function returns the inverse cumulative distribution |
|
||
|
|
//| function of the Noncental Beta distribution with parameters a,b |
|
||
|
|
//| and lambda for the desired probability. |
|
||
|
|
//| |
|
||
|
|
//| Arguments: |
|
||
|
|
//| probability : The desired probability |
|
||
|
|
//| a : First shape parameter |
|
||
|
|
//| b : Second shape parameter |
|
||
|
|
//| lambda : Noncentrality parameter |
|
||
|
|
//| error_code : Variable for error code |
|
||
|
|
//| |
|
||
|
|
//| Return value: |
|
||
|
|
//| The value of the inverse cumulative distribution function of |
|
||
|
|
//| of Noncental Beta distribution with parameters a,b and lambda. |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
double MathQuantileNoncentralBeta(const double probability,const double a,const double b,const double lambda,const bool tail,const bool log_mode,int &error_code)
|
||
|
|
{
|
||
|
|
if(log_mode==true && probability==QNEGINF)
|
||
|
|
return 0.0;
|
||
|
|
if(log_mode==false && probability==0)
|
||
|
|
return 0.0;
|
||
|
|
//--- if lambda==0, return beta quantile
|
||
|
|
if(lambda==0.0)
|
||
|
|
return MathQuantileBeta(probability,a,b,error_code);
|
||
|
|
//--- check parameters
|
||
|
|
if(!MathIsValidNumber(probability) || !MathIsValidNumber(a) || !MathIsValidNumber(b) || !MathIsValidNumber(lambda))
|
||
|
|
{
|
||
|
|
error_code=ERR_ARGUMENTS_NAN;
|
||
|
|
return QNaN;
|
||
|
|
}
|
||
|
|
//--- a,b,lambda must be positive
|
||
|
|
if(a<=0.0 || b<=0.0 || lambda<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;
|
||
|
|
|
||
|
|
double lambda_half=lambda*0.5;
|
||
|
|
double lambda_half_log=MathLog(lambda_half);
|
||
|
|
double lambda_half_sqrt=MathSqrt(lambda_half);
|
||
|
|
double lambda_half_exp=MathExp(-lambda_half);
|
||
|
|
|
||
|
|
double x0=int(MathMax(lambda_half-5*lambda_half_sqrt,0));
|
||
|
|
double b_gamma_log=MathGammaLog(b);
|
||
|
|
double eps=10E-18;
|
||
|
|
double h_min=MathSqrt(eps);
|
||
|
|
|
||
|
|
//double lambda_half=lambda*0.5;
|
||
|
|
double r_beta0=MathBeta(a,b);
|
||
|
|
|
||
|
|
int err_code=0;
|
||
|
|
double x=0.5;
|
||
|
|
double h=1.0;
|
||
|
|
const int max_terms=100;
|
||
|
|
//--- Newton iterations
|
||
|
|
const int max_iterations=50;
|
||
|
|
int iterations=0;
|
||
|
|
while(iterations<max_iterations)
|
||
|
|
{
|
||
|
|
//--- check convergence
|
||
|
|
if((MathAbs(h)>h_min*MathAbs(x) && MathAbs(h)>h_min)==false)
|
||
|
|
break;
|
||
|
|
|
||
|
|
//--- calculate PDF
|
||
|
|
double pdf=0;
|
||
|
|
if(x<=0.0 || x>=1.0)
|
||
|
|
pdf=0;
|
||
|
|
else
|
||
|
|
{
|
||
|
|
double fact_mult=1.0;
|
||
|
|
double pwr_lambda_half=1.0;
|
||
|
|
double pwr_x=MathExp((a-1.0)*MathLog(x));
|
||
|
|
double r_beta=r_beta0;
|
||
|
|
//--- direct sum calculation
|
||
|
|
for(int j=0;; j++)
|
||
|
|
{
|
||
|
|
if(j>0)
|
||
|
|
{
|
||
|
|
pwr_x*=x;
|
||
|
|
pwr_lambda_half*=lambda_half;
|
||
|
|
fact_mult/=j;
|
||
|
|
double jm1=j-1;
|
||
|
|
r_beta*=((a+jm1)/(a+b+jm1));
|
||
|
|
}
|
||
|
|
double term=pwr_x*fact_mult*pwr_lambda_half/r_beta;
|
||
|
|
//---
|
||
|
|
if(term<10E-18)
|
||
|
|
break;
|
||
|
|
pdf+=term;
|
||
|
|
}
|
||
|
|
//--- calculate density coef
|
||
|
|
pdf*=MathExp((b-1.0)*MathLog(1.0-x))*lambda_half_exp;
|
||
|
|
}
|
||
|
|
|
||
|
|
//--- calculate CDF
|
||
|
|
double cdf=0;
|
||
|
|
if(x<=0.0)
|
||
|
|
cdf=0;
|
||
|
|
if(x>=1.0)
|
||
|
|
cdf=1;
|
||
|
|
else
|
||
|
|
{
|
||
|
|
double a0=a+x0;
|
||
|
|
double beta = MathGammaLog(a0) + b_gamma_log - MathGammaLog(a0+b);
|
||
|
|
double temp = MathBetaIncomplete(x, a0, b);
|
||
|
|
double gx=MathExp(a0*MathLog(x)+b*MathLog(1-x)-beta-MathLog(a0));
|
||
|
|
|
||
|
|
double q=0;
|
||
|
|
if(a0>a)
|
||
|
|
q=MathExp(-lambda_half+x0*lambda_half_log-MathGammaLog(x0+1));
|
||
|
|
else
|
||
|
|
q=lambda_half_exp;
|
||
|
|
|
||
|
|
double sumq=1-q;
|
||
|
|
double betanc=q*temp;
|
||
|
|
int j=0;
|
||
|
|
double ab=a+b;
|
||
|
|
for(;;)
|
||
|
|
{
|
||
|
|
j++;
|
||
|
|
temp-=gx;
|
||
|
|
gx*=x*(ab+j-1)/(a+j);
|
||
|
|
q*=lambda_half/j;
|
||
|
|
sumq-=q;
|
||
|
|
betanc+=temp*q;
|
||
|
|
double err=(temp-gx)*sumq;
|
||
|
|
if(j>max_terms || err<1E-18)
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
cdf=MathMin(betanc,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-x)*0.1;
|
||
|
|
|
||
|
|
if(MathAbs(x_new-x)<10E-16)
|
||
|
|
break;
|
||
|
|
x=x_new;
|
||
|
|
|
||
|
|
iterations++;
|
||
|
|
}
|
||
|
|
//--- check convergence
|
||
|
|
if(iterations<max_iterations)
|
||
|
|
return x;
|
||
|
|
else
|
||
|
|
{
|
||
|
|
error_code=ERR_NON_CONVERGENCE;
|
||
|
|
return QNaN;
|
||
|
|
}
|
||
|
|
return x;
|
||
|
|
}
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| Noncental Beta distribution quantile function (inverse CDF) |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| The function returns the inverse cumulative distribution |
|
||
|
|
//| function of the Noncental Beta distribution with parameters a, b |
|
||
|
|
//| and lambda for the desired probability. |
|
||
|
|
//| |
|
||
|
|
//| Arguments: |
|
||
|
|
//| probability : The desired probability |
|
||
|
|
//| a : First shape parameter |
|
||
|
|
//| b : Second shape parameter |
|
||
|
|
//| lambda : Noncentrality parameter |
|
||
|
|
//| error_code : Variable for error code |
|
||
|
|
//| |
|
||
|
|
//| Return value: |
|
||
|
|
//| The value of the inverse cumulative distribution function of |
|
||
|
|
//| of Noncental Beta distribution with parameters a,b and lambda. |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
double MathQuantileNoncentralBeta(const double probability,const double a,const double b,const double lambda,int &error_code)
|
||
|
|
{
|
||
|
|
return MathQuantileNoncentralBeta(probability,a,b,lambda,true,false,error_code);
|
||
|
|
}
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| Noncental Beta distribution quantile function (inverse CDF) |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| The function calculates the inverse cumulative distribution |
|
||
|
|
//| function of the Noncentral Beta distribution with parameter a,b |
|
||
|
|
//| lambda for the probability values from array. |
|
||
|
|
//| |
|
||
|
|
//| Arguments: |
|
||
|
|
//| probability : Array with probabilities |
|
||
|
|
//| a : First shape parameter |
|
||
|
|
//| b : Second shape parameter |
|
||
|
|
//| lambda : Noncentrality parameter |
|
||
|
|
//| tail : Flag to calculate lower tail |
|
||
|
|
//| log_mode : Logarithm mode, if true it calculates Log values |
|
||
|
|
//| result : Array with calculated values |
|
||
|
|
//| |
|
||
|
|
//| Return value: |
|
||
|
|
//| true if successful, otherwise false. |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
bool MathQuantileNoncentralBeta(const double &probability[],const double a,const double b,const double lambda,const bool tail,const bool log_mode,double &result[])
|
||
|
|
{
|
||
|
|
//--- if lambda==0, return beta quantile
|
||
|
|
if(lambda==0.0)
|
||
|
|
return MathQuantileBeta(probability,a,b,tail,log_mode,result);
|
||
|
|
//--- check parameters
|
||
|
|
if(!MathIsValidNumber(a) || !MathIsValidNumber(b) || !MathIsValidNumber(lambda))
|
||
|
|
return false;
|
||
|
|
//--- a,b,lambda must be positive
|
||
|
|
if(a<=0.0 || b<=0.0 || lambda<0)
|
||
|
|
return false;
|
||
|
|
|
||
|
|
int data_count=ArraySize(probability);
|
||
|
|
if(data_count==0)
|
||
|
|
return false;
|
||
|
|
|
||
|
|
int err_code=0;
|
||
|
|
ArrayResize(result,data_count);
|
||
|
|
|
||
|
|
double lambda_half=lambda*0.5;
|
||
|
|
double lambda_half_log=MathLog(lambda_half);
|
||
|
|
double lambda_half_sqrt=MathSqrt(lambda_half);
|
||
|
|
double lambda_half_exp=MathExp(-lambda_half);
|
||
|
|
double r_beta0=MathBeta(a,b);
|
||
|
|
|
||
|
|
double x0=int(MathMax(lambda_half-5*lambda_half_sqrt,0));
|
||
|
|
double b_gamma_log=MathGammaLog(b);
|
||
|
|
const double eps=10E-18;
|
||
|
|
double h_min=MathSqrt(eps);
|
||
|
|
const int max_terms=100;
|
||
|
|
|
||
|
|
for(int i=0; i<data_count; i++)
|
||
|
|
{
|
||
|
|
//--- calculate real probability
|
||
|
|
double prob=TailLogProbability(probability[i],tail,log_mode);
|
||
|
|
|
||
|
|
if(!MathIsValidNumber(prob))
|
||
|
|
return false;
|
||
|
|
|
||
|
|
if(prob==0.0)
|
||
|
|
result[i]=0.0;
|
||
|
|
else
|
||
|
|
if(prob==1.0)
|
||
|
|
result[i]=1.0;
|
||
|
|
else
|
||
|
|
{
|
||
|
|
double x=0.5;
|
||
|
|
double h=1.0;
|
||
|
|
//--- Newton iterations
|
||
|
|
const int max_iterations=50;
|
||
|
|
int iterations=0;
|
||
|
|
while(iterations<max_iterations)
|
||
|
|
{
|
||
|
|
//--- check convergence
|
||
|
|
if((MathAbs(h)>h_min*MathAbs(x) && MathAbs(h)>h_min)==false)
|
||
|
|
break;
|
||
|
|
|
||
|
|
//--- calculate PDF
|
||
|
|
double pdf=0;
|
||
|
|
if(x<=0.0 || x>=1.0)
|
||
|
|
pdf=0;
|
||
|
|
else
|
||
|
|
{
|
||
|
|
double fact_mult=1.0;
|
||
|
|
double pwr_lambda_half=1.0;
|
||
|
|
double pwr_x=MathExp((a-1.0)*MathLog(x));
|
||
|
|
double r_beta=r_beta0;
|
||
|
|
//--- direct sum calculation
|
||
|
|
for(int j=0;; j++)
|
||
|
|
{
|
||
|
|
if(j>0)
|
||
|
|
{
|
||
|
|
pwr_x*=x;
|
||
|
|
pwr_lambda_half*=lambda_half;
|
||
|
|
fact_mult/=j;
|
||
|
|
double jm1=j-1;
|
||
|
|
r_beta*=((a+jm1)/(a+b+jm1));
|
||
|
|
}
|
||
|
|
double term=pwr_x*fact_mult*pwr_lambda_half/r_beta;
|
||
|
|
//---
|
||
|
|
if(term<10E-18)
|
||
|
|
break;
|
||
|
|
pdf+=term;
|
||
|
|
}
|
||
|
|
//--- calculate density coef
|
||
|
|
pdf*=MathExp((b-1.0)*MathLog(1.0-x))*lambda_half_exp;
|
||
|
|
}
|
||
|
|
|
||
|
|
//--- calculate CDF
|
||
|
|
double cdf=0;
|
||
|
|
if(x<=0.0)
|
||
|
|
cdf=0;
|
||
|
|
if(x>=1.0)
|
||
|
|
cdf=1;
|
||
|
|
else
|
||
|
|
{
|
||
|
|
double a0=a+x0;
|
||
|
|
double beta = MathGammaLog(a0) + b_gamma_log - MathGammaLog(a0+b);
|
||
|
|
double temp = MathBetaIncomplete(x, a0, b);
|
||
|
|
double gx=MathExp(a0*MathLog(x)+b*MathLog(1-x)-beta-MathLog(a0));
|
||
|
|
|
||
|
|
double q=0;
|
||
|
|
if(a0>a)
|
||
|
|
q=MathExp(-lambda_half+x0*lambda_half_log-MathGammaLog(x0+1));
|
||
|
|
else
|
||
|
|
q=lambda_half_exp;
|
||
|
|
|
||
|
|
double sumq=1-q;
|
||
|
|
double betanc=q*temp;
|
||
|
|
int j=0;
|
||
|
|
double ab=a+b;
|
||
|
|
for(;;)
|
||
|
|
{
|
||
|
|
j++;
|
||
|
|
temp-=gx;
|
||
|
|
gx*=x*(ab+j-1)/(a+j);
|
||
|
|
q*=lambda_half/j;
|
||
|
|
sumq-=q;
|
||
|
|
betanc+=temp*q;
|
||
|
|
double err=(temp-gx)*sumq;
|
||
|
|
if(j>max_terms || err<1E-18)
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
cdf=MathMin(betanc,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-x)*0.1;
|
||
|
|
|
||
|
|
if(MathAbs(x_new-x)<10E-16)
|
||
|
|
break;
|
||
|
|
x=x_new;
|
||
|
|
|
||
|
|
iterations++;
|
||
|
|
}
|
||
|
|
//--- check convergence
|
||
|
|
if(iterations<max_iterations)
|
||
|
|
result[i]=x;
|
||
|
|
else
|
||
|
|
return false;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return true;
|
||
|
|
}
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| Noncental Beta distribution quantile function (inverse CDF) |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| The function calculates the inverse cumulative distribution |
|
||
|
|
//| function of the Noncentral Beta distribution with parameter a,b |
|
||
|
|
//| lambda for the probability values from array. |
|
||
|
|
//| |
|
||
|
|
//| Arguments: |
|
||
|
|
//| probability : Array with probabilities |
|
||
|
|
//| a : First shape parameter |
|
||
|
|
//| b : Second shape parameter |
|
||
|
|
//| lambda : Noncentrality parameter |
|
||
|
|
//| result : Array with calculated values |
|
||
|
|
//| |
|
||
|
|
//| Return value: |
|
||
|
|
//| true if successful, otherwise false. |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
bool MathQuantileNoncentralBeta(const double &probability[],const double a,const double b,const double lambda,double &result[])
|
||
|
|
{
|
||
|
|
return MathQuantileNoncentralBeta(probability,a,b,lambda,true,false,result);
|
||
|
|
}
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| Random variate from the Noncentral Beta distribution |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| Compute the random variable from the Noncentral Beta |
|
||
|
|
//| distribution with parameters a,b and lambda. |
|
||
|
|
//| |
|
||
|
|
//| Arguments: |
|
||
|
|
//| a : First shape parameter |
|
||
|
|
//| b : Second shape parameter |
|
||
|
|
//| lambda : Noncentrality parameter |
|
||
|
|
//| error_code : Variable for error code |
|
||
|
|
//| |
|
||
|
|
//| Return value: |
|
||
|
|
//| The random value with Noncentral Beta distribution. |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
double MathRandomNoncentralBeta(const double a,const double b,const double lambda,int &error_code)
|
||
|
|
{
|
||
|
|
//--- if lambda==0, return beta random variate
|
||
|
|
if(lambda==0.0)
|
||
|
|
return MathRandomBeta(a,b,error_code);
|
||
|
|
//--- check parameters
|
||
|
|
if(!MathIsValidNumber(a) || !MathIsValidNumber(b) || !MathIsValidNumber(lambda))
|
||
|
|
{
|
||
|
|
error_code=ERR_ARGUMENTS_NAN;
|
||
|
|
return QNaN;
|
||
|
|
}
|
||
|
|
//--- a,b,lambda must be positive
|
||
|
|
if(a<=0.0 || b<=0.0 || lambda<0)
|
||
|
|
{
|
||
|
|
error_code=ERR_ARGUMENTS_INVALID;
|
||
|
|
return QNaN;
|
||
|
|
}
|
||
|
|
|
||
|
|
error_code=ERR_OK;
|
||
|
|
//--- generate random number using Noncentral ChiSquare
|
||
|
|
double chi1=MathRandomNoncentralChiSquare(2*a,lambda,error_code);
|
||
|
|
double chi2=MathRandomChiSquare(2*b,error_code);
|
||
|
|
return chi1/(chi1+chi2);
|
||
|
|
}
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| Random variate from the Noncentral Beta distribution |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| Generates random variables from the Noncentral Beta distribution |
|
||
|
|
//| with parameters a,b, lambda. |
|
||
|
|
//| |
|
||
|
|
//| Arguments: |
|
||
|
|
//| a : First shape parameter |
|
||
|
|
//| b : Second shape parameter |
|
||
|
|
//| lambda : Noncentrality parameter |
|
||
|
|
//| data_count : Number of values needed |
|
||
|
|
//| result : Output array with random values |
|
||
|
|
//| |
|
||
|
|
//| Return value: |
|
||
|
|
//| true if successful, otherwise false. |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
bool MathRandomNoncentralBeta(const double a,const double b,const double lambda,const int data_count,double &result[])
|
||
|
|
{
|
||
|
|
//--- if lambda==0, return beta random variate
|
||
|
|
if(lambda==0.0)
|
||
|
|
return MathRandomBeta(a,b,data_count,result);
|
||
|
|
//--- check parameters
|
||
|
|
if(!MathIsValidNumber(a) || !MathIsValidNumber(b) || !MathIsValidNumber(lambda))
|
||
|
|
return false;
|
||
|
|
//--- a,b,lambda must be positive
|
||
|
|
if(a<=0.0 || b<=0.0 || lambda<0)
|
||
|
|
return false;
|
||
|
|
|
||
|
|
int error_code=0;
|
||
|
|
//--- prepare output array and calculate random values
|
||
|
|
ArrayResize(result,data_count);
|
||
|
|
double a2=a*2;
|
||
|
|
double b2=b*2;
|
||
|
|
for(int i=0; i<data_count; i++)
|
||
|
|
{
|
||
|
|
//--- generate random number using Noncentral ChiSquare
|
||
|
|
double chi1=MathRandomNoncentralChiSquare(a2,lambda,error_code);
|
||
|
|
double chi2=MathRandomChiSquare(b2,error_code);
|
||
|
|
result[i]=chi1/(chi1+chi2);
|
||
|
|
}
|
||
|
|
return true;
|
||
|
|
}
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| Noncental Beta distribution moments |
|
||
|
|
//+------------------------------------------------------------------+
|
||
|
|
//| The function calculates 4 first moments of the Noncental Beta |
|
||
|
|
//| distribution with parameters a,b and lambda. |
|
||
|
|
//| |
|
||
|
|
//| Arguments: |
|
||
|
|
//| a : First shape parameter |
|
||
|
|
//| b : Second shape parameter |
|
||
|
|
//| lambda : 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 MathMomentsNoncentralBeta(const double a,const double b,const double lambda,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(a) || !MathIsValidNumber(b) || !MathIsValidNumber(lambda))
|
||
|
|
{
|
||
|
|
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;
|
||
|
|
}
|
||
|
|
//--- check lambda
|
||
|
|
if(lambda<0.0)
|
||
|
|
{
|
||
|
|
error_code=ERR_ARGUMENTS_INVALID;
|
||
|
|
return false;
|
||
|
|
}
|
||
|
|
|
||
|
|
error_code=ERR_OK;
|
||
|
|
//--- prepare coefficients
|
||
|
|
double lambda_half=lambda*0.5;
|
||
|
|
//--- hypergeometric function values
|
||
|
|
double f1=MathHypergeometric2F2(a+1,a+b,a,a+b+1,lambda_half);
|
||
|
|
double f2=MathHypergeometric2F2(a+2,a+b,a,a+b+2,lambda_half);
|
||
|
|
double f3=MathHypergeometric2F2(a+3,a+b,a,a+b+3,lambda_half);
|
||
|
|
double f4=MathHypergeometric2F2(a+4,a+b,a,a+b+4,lambda_half);
|
||
|
|
//--- exponents
|
||
|
|
double exp_lambda_half=MathExp(-lambda_half);
|
||
|
|
double exp_lambda=MathPow(exp_lambda_half,2);
|
||
|
|
//--- factors
|
||
|
|
double aab=a/(a+b);
|
||
|
|
double aab2=MathPow(aab,2);
|
||
|
|
double ab1=(a+1)/(a+b+1);
|
||
|
|
double ab2=(a+2)/(a+b+2);
|
||
|
|
double ab3=(a+3)/(a+b+3);
|
||
|
|
//--- calculate moments
|
||
|
|
mean=aab*exp_lambda_half*f1;
|
||
|
|
double mean2=MathPow(mean,2);
|
||
|
|
variance=aab*ab1*exp_lambda_half*f2-mean2;
|
||
|
|
skewness=(2*MathPow(mean,3)+exp_lambda_half*aab*ab1*(-3*mean*f2+ab2*f3))*MathPow(variance,-1.5);
|
||
|
|
kurtosis=-3+(-3*MathPow(mean,4)+exp_lambda*f1*aab2*(6*mean*ab1*f2-4*ab1*ab2*f3)+aab*ab1*ab2*ab3*exp_lambda_half*f4)*MathPow(aab*ab1*exp_lambda_half*f2-mean2,-2);
|
||
|
|
//--- successful
|
||
|
|
return true;
|
||
|
|
}
|
||
|
|
//+------------------------------------------------------------------+
|