791 lines
31 KiB
MQL5
791 lines
31 KiB
MQL5
//+------------------------------------------------------------------+
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//| Poisson.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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//| Poisson probability mass function (PDF) |
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//+------------------------------------------------------------------+
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//| The function returns the probability mass function |
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//| of the Poisson distribution with parameter lambda. |
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//| |
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//| Arguments: |
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//| x : Random variable |
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//| lambda : Mean |
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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 mass evaluated at x. |
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//+------------------------------------------------------------------+
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double MathProbabilityDensityPoisson(const double x,const double lambda,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(lambda))
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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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//--- lambda must be positive, x must be integer
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if(lambda<=0.0 || x!=MathRound(x))
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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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//--- check x
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if(x<0.0)
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return TailLog0(true,log_mode);
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//--- calculate log pdf using LogGamma
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double log_pdf=-lambda+x*MathLog(lambda)-MathGammaLog(x+1.0);
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if(log_mode)
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return log_pdf;
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//--- return density
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return MathExp(log_pdf);
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}
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//+------------------------------------------------------------------+
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//| Poisson probability mass function (PDF) |
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//+------------------------------------------------------------------+
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//| The function returns the probability mass function |
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//| of the Poisson distribution with parameter lambda. |
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//| |
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//| Arguments: |
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//| x : Random variable |
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//| lambda : Mean |
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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 mass evaluated at x. |
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//+------------------------------------------------------------------+
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double MathProbabilityDensityPoisson(const double x,const double lambda,int &error_code)
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{
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return MathProbabilityDensityPoisson(x,lambda,false,error_code);
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}
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//+------------------------------------------------------------------+
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//| Poisson probability mass function (PDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the probability density function of |
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//| the Poisson distribution with parameter lambda 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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//| lambda : Mean |
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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 MathProbabilityDensityPoisson(const double &x[],const double lambda,const bool log_mode,double &result[])
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{
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//--- check NaN
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if(!MathIsValidNumber(lambda))
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return false;
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//--- lambda must be positive
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if(lambda<=0.0)
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return false;
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int data_count=ArraySize(x);
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if(data_count==0)
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return false;
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ArrayResize(result,data_count);
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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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if(x_arg!=MathRound(x_arg))
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return false;
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//--- check x
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if(x_arg<0.0)
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result[i]=TailLog0(true,log_mode);
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else
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{
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//--- calculate log pdf using LogGamma
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double log_pdf=-lambda+x_arg*MathLog(lambda)-MathGammaLog(x_arg+1.0);
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if(log_mode)
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result[i]=log_pdf;
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else
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result[i]=MathExp(log_pdf);
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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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//| Poisson probability mass function (PDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the probability density function of |
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//| the Poisson distribution with parameter lambda 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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//| lambda : Mean |
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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 MathProbabilityDensityPoisson(const double &x[],const double lambda,double &result[])
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{
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return MathProbabilityDensityPoisson(x,lambda,false,result);
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}
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//+------------------------------------------------------------------+
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//| Poisson 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 Poisson distribution with parameter lambda |
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//| is less than or equal to x. |
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//| |
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//| Arguments: |
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//| x : The desired quantile |
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//| lambda : Mean |
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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 Poisson cumulative distribution function with |
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//| parameter lambda, evaluated at x. |
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//+------------------------------------------------------------------+
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double MathCumulativeDistributionPoisson(const double x,const double lambda,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(lambda))
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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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//--- lambda must be positive, x must be integer
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if(lambda<=0.0 || x!=MathRound(x))
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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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//--- check x
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if(x<0.0)
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return TailLog0(tail,log_mode);
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int err_code=0;
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int t=(int)MathFloor(x+10e-10);
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double cdf=MathCumulativeDistributionGamma(lambda,t+1,1,false,false,err_code);
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return TailLogValue(cdf,tail,log_mode);
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}
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//+------------------------------------------------------------------+
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//| Poisson 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 Poisson distribution with parameter lambda |
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//| is less than or equal to x. |
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//| |
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//| Arguments: |
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//| x : The desired quantile |
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//| lambda : Mean |
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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 Poisson cumulative distribution function with |
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//| parameter lambda, evaluated at x. |
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//+------------------------------------------------------------------+
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double MathCumulativeDistributionPoisson(const double x,const double lambda,int &error_code)
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{
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return MathCumulativeDistributionPoisson(x,lambda,true,false,error_code);
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}
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//+------------------------------------------------------------------+
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//| Poisson 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 Poisson distribution with parameter lambda 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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//| lambda : Mean |
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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 MathCumulativeDistributionPoisson(const double &x[],const double lambda,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(lambda))
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return false;
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//--- lambda must be positive
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if(lambda<=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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double coef_lambda=MathExp(-lambda);
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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!=MathRound(x_arg))
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return false;
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if(x_arg<0.0)
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result[i]=TailLog0(tail,log_mode);
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else
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{
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int err_code=0;
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int t=(int)MathFloor(x_arg+10e-10);
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double cdf=MathCumulativeDistributionGamma(lambda,t+1,1,false,false,err_code);
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result[i]=TailLogValue(cdf,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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//| Poisson 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 Poisson distribution with parameter lambda 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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//| lambda : Mean |
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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 MathCumulativeDistributionPoisson(const double &x[],const double lambda,double &result[])
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{
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return MathCumulativeDistributionPoisson(x,lambda,true,false,result);
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}
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//+------------------------------------------------------------------+
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//| Poisson distribution quantile function (inverse CDF) |
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//+------------------------------------------------------------------+
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//| Computes the inverse cumulative distribution function of the |
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//| Poisson distribution with parameter lambda 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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//| lambda : Mean |
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//| tail : Flag to calculate lower tail |
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//| log_mode : Logarithm mode,if true it calculates for 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 Poisson distribution with parameter lambda. |
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//+------------------------------------------------------------------+
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double MathQuantilePoisson(const double probability,const double lambda,const bool tail,const bool log_mode,int &error_code)
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{
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//--- check parameters
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if(!MathIsValidNumber(probability) || !MathIsValidNumber(lambda))
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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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//--- lambda must be positive
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if(lambda<=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
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if(prob==1.0)
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{
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error_code=ERR_RESULT_INFINITE;
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return QPOSINF;
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}
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error_code=ERR_OK;
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if(prob==0.0)
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return 0.0;
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prob*=1-1000*DBL_EPSILON;
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int err_code=0;
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int j=0;
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const int max_terms=500;
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double coef_lambda=MathExp(-lambda);
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double pwr_lambda=1.0;
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double inverse_fact=1.0;
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double sum=0;
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//--- direct calculation of the quantile
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while(sum<prob && j<max_terms)
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{
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if(j>0)
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{
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pwr_lambda*=lambda;
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inverse_fact/=j;
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}
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sum+=coef_lambda*pwr_lambda*inverse_fact;
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j++;
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}
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//--- check convergence
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if(j<max_terms)
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{
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if(j==0)
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return 0;
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else
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return j-1;
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}
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else
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{
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error_code=ERR_RESULT_INFINITE;
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return QPOSINF;
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}
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}
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//+------------------------------------------------------------------+
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//| Poisson distribution quantile function (inverse CDF) |
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//+------------------------------------------------------------------+
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//| Computes the inverse cumulative distribution function of the |
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//| Poisson distribution with parameter lambda 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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//| lambda : Mean |
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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 Poisson distribution with parameter lambda. |
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//+------------------------------------------------------------------+
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double MathQuantilePoisson(const double probability,const double lambda,int &error_code)
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{
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return MathQuantilePoisson(probability,lambda,true,false,error_code);
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}
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//+------------------------------------------------------------------+
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//| Poisson 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 Poisson distribution with parameter lambda |
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//| for 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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//| lambda : Mean |
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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 MathQuantilePoisson(const double &probability[],const double lambda,const bool tail,const bool log_mode,double &result[])
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{
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//--- NaN
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if(!MathIsValidNumber(lambda))
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return false;
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//--- lambda must be positive
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if(lambda<=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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double coef_lambda=MathExp(-lambda);
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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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else
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if(prob==1.0)
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result[i]=QPOSINF;
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if(prob==0.0)
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result[i]=0;
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else
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{
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prob*=1-1000*DBL_EPSILON;
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int err_code=0;
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int j=0;
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double sum=0.0;
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const int max_terms=500;
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double pwr_lambda=1.0;
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double inverse_fact=1.0;
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//--- direct calculation
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while(sum<prob && j<max_terms)
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{
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if(j>0)
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{
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pwr_lambda*=lambda;
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inverse_fact/=j;
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}
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sum+=coef_lambda*pwr_lambda*inverse_fact;
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j++;
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}
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//--- check convergence
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if(j<max_terms)
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{
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if(j==0)
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result[i]=0;
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else
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result[i]=j-1;
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}
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else
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return false;
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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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//| Poisson 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 Poisson distribution with parameter lambda |
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//| for 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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//| lambda : Mean |
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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 MathQuantilePoisson(const double &probability[],const double lambda,double &result[])
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{
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return MathQuantilePoisson(probability,lambda,true,false,result);
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}
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//+------------------------------------------------------------------+
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//| Random variate from the Poisson distribution |
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//+------------------------------------------------------------------+
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//| Compute the random variable from the Poisson distribution |
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//| with parameter lambda. |
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//| |
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//| Arguments |
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//| lambda : Mean |
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//| |
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//| Return value: |
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//| The random value with Poisson distribution. |
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//+------------------------------------------------------------------+
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//| Original FORTRAN77 version by Barry Brown, James Lovato. |
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//| C version by John Burkardt. |
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//| |
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//| Reference: |
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//| Joachim Ahrens, Ulrich Dieter, "Computer Generation of Poisson |
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//| "Deviates From Modified Normal Distributions", |
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//| ACM Transactions on Mathematical Software, |
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//| Volume 8, Number 2, June 1982, pages 163-179. |
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//+------------------------------------------------------------------+
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double MathRandomPoisson(const double lambda)
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{
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const double a0 = -0.5;
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const double a1 = 0.3333333;
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const double a2 = -0.2500068;
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const double a3 = 0.2000118;
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const double a4 = -0.1661269;
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const double a5 = 0.1421878;
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const double a6 = -0.1384794;
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const double a7 = 0.1250060;
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int kflag;
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double fk=0,difmuk=0;
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double e=0,fx,fy,g,p0,px,py,p,q,s,t,u=0,v,x,xx;
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int value=0;
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//--- start new table and calculate P0
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if(lambda<10.0)
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{
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int m=MathMax(1,(int)(lambda));
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p = MathExp(-lambda);
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q = p;
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p0= p;
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//--- uniform sample for inversion method
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for(;;)
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{
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u=MathRandomNonZero();
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value=0;
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if(u<=p0)
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return value;
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//--- creation of new Poisson probabilities
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for(int k=1; k<=35; k++)
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{
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p=p*lambda/double(k);
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q=q+p;
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if(u<=q)
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{
|
|
value=k;
|
|
return value;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
else
|
|
{
|
|
s=MathSqrt(lambda);
|
|
double d=6.0*lambda*lambda;
|
|
int l=(int)(lambda-1.1484);
|
|
//--- generate normal deviate
|
|
double f,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);
|
|
//--- Box-Muller transform
|
|
f=MathSqrt(-2.0*MathLog(r2)/r2);
|
|
double snorm=f*x2;
|
|
//--- normal sample
|
|
g=lambda+s*snorm;
|
|
|
|
if(0.0<=g)
|
|
{
|
|
value=(int)(g);
|
|
//--- immediate acceptance if large enough
|
|
if(l<=value)
|
|
return value;
|
|
//--- squeeze acceptance
|
|
fk=(double)(value);
|
|
difmuk=lambda-fk;
|
|
u=MathRandomNonZero();
|
|
//---
|
|
if(difmuk*difmuk*difmuk<=d*u)
|
|
return value;
|
|
}
|
|
//--- preparation for steps P and Q
|
|
double omega=0.3989423/s;
|
|
double b1 = 0.04166667/lambda;
|
|
double b2 = 0.3*b1*b1;
|
|
double c3 = 0.1428571*b1*b2;
|
|
double c2 = b2 - 15.0*c3;
|
|
double c1 = b1 - 6.0*b2 + 45.0*c3;
|
|
double c0 = 1.0 - b1 + 3.0*b2 - 15.0*c3;
|
|
double c=0.1069/lambda;
|
|
double del=0;
|
|
|
|
if(0.0<=g)
|
|
{
|
|
kflag=0;
|
|
|
|
if(value<10)
|
|
{
|
|
px = -lambda;
|
|
py = MathPow(lambda,value)/MathFactorial(value);
|
|
}
|
|
else
|
|
{
|
|
del = 0.8333333E-01/fk;
|
|
del = del - 4.8*del*del*del;
|
|
v=difmuk/fk;
|
|
|
|
if(0.25<MathAbs(v))
|
|
{
|
|
px=fk*MathLog(1.0+v)-difmuk-del;
|
|
}
|
|
else
|
|
{
|
|
px=fk*v*v*(((((((a7*v+a6)*v+a5)*v+a4)*v+a3)*v+a2)*v+a1)*v+a0)-del;
|
|
}
|
|
py=0.3989423/MathSqrt(fk);
|
|
}
|
|
x=(0.5-difmuk)/s;
|
|
xx = x * x;
|
|
fx = -0.5 * xx;
|
|
fy = omega*(((c3*xx+c2)*xx+c1)*xx+c0);
|
|
|
|
if(kflag<=0)
|
|
{
|
|
if(fy-u*fy<=py*MathExp(px-fx))
|
|
return value;
|
|
}
|
|
else
|
|
{
|
|
if(c*MathAbs(u)<=py*MathExp(px+e)-fy*MathExp(fx+e))
|
|
return value;
|
|
}
|
|
}
|
|
//--- exponential sample
|
|
for(;;)
|
|
{
|
|
double rnd=MathRandomNonZero();
|
|
e=-MathLog(1.0-rnd);
|
|
|
|
u=2.0*MathRandomNonZero()-1.0;
|
|
if(u<0.0)
|
|
t=1.8-MathAbs(e);
|
|
else
|
|
t=1.8+MathAbs(e);
|
|
|
|
if(t<=-0.6744)
|
|
continue;
|
|
|
|
value=(int)(lambda+s*t);
|
|
fk=(double)(value);
|
|
difmuk=lambda-fk;
|
|
|
|
kflag=1;
|
|
//--- calculation of PX, PY, FX, FY
|
|
if(value<10)
|
|
{
|
|
px = -lambda;
|
|
py = MathPow(lambda,value)/MathFactorial(value);
|
|
}
|
|
else
|
|
{
|
|
del = 0.8333333E-01/fk;
|
|
del = del - 4.8*del*del*del;
|
|
v=difmuk/fk;
|
|
|
|
if(0.25<MathAbs(v))
|
|
px=fk*MathLog(1.0+v)-difmuk-del;
|
|
else
|
|
px=fk*v*v*(((((((a7*v+a6)*v+a5)*v+a4)*v+a3)*v+a2)*v+a1)*v+a0)-del;
|
|
|
|
py=0.3989423/MathSqrt(fk);
|
|
}
|
|
|
|
x=(0.5-difmuk)/s;
|
|
xx = x*x;
|
|
fx = -0.5*xx;
|
|
fy = omega*(((c3*xx+c2)*xx+c1)*xx+c0);
|
|
|
|
if(kflag<=0)
|
|
{
|
|
if(fy-u*fy<=py*MathExp(px-fx))
|
|
return value;
|
|
}
|
|
else
|
|
{
|
|
if(c*MathAbs(u)<=py*MathExp(px+e)-fy*MathExp(fx+e))
|
|
return value;
|
|
}
|
|
}
|
|
}
|
|
return value;
|
|
}
|
|
//+------------------------------------------------------------------+
|
|
//| Random variate from the Poisson distribution |
|
|
//+------------------------------------------------------------------+
|
|
//| Compute the random variable from the Poisson distribution |
|
|
//| with parameter lambda. |
|
|
//| |
|
|
//| Arguments |
|
|
//| lambda : Mean |
|
|
//| error_code : Variable for error code |
|
|
//| |
|
|
//| Return value: |
|
|
//| The random value with Poisson distribution. |
|
|
//+------------------------------------------------------------------+
|
|
double MathRandomPoisson(const double lambda,int &error_code)
|
|
{
|
|
//--- check parameters
|
|
if(!MathIsValidNumber(lambda))
|
|
{
|
|
error_code=ERR_ARGUMENTS_NAN;
|
|
return QNaN;
|
|
}
|
|
//--- lambda must be positive
|
|
if(lambda<=0.0)
|
|
{
|
|
error_code=ERR_ARGUMENTS_INVALID;
|
|
return QNaN;
|
|
}
|
|
error_code=ERR_OK;
|
|
return MathRandomPoisson(lambda);
|
|
}
|
|
//+------------------------------------------------------------------+
|
|
//| Random variate from the Poisson distribution |
|
|
//+------------------------------------------------------------------+
|
|
//| Generates random variables from the Poisson distribution |
|
|
//| with parameter lambda. |
|
|
//| |
|
|
//| Arguments: |
|
|
//| lambda : Mean |
|
|
//| data_count : Number of values needed |
|
|
//| result : Output array with random values |
|
|
//| |
|
|
//| Return value: |
|
|
//| true if successful, otherwise false. |
|
|
//+------------------------------------------------------------------+
|
|
bool MathRandomPoisson(const double lambda,const int data_count,double &result[])
|
|
{
|
|
//--- check NaN
|
|
if(!MathIsValidNumber(lambda))
|
|
return false;
|
|
//--- lambda must be positive
|
|
if(lambda<=0.0)
|
|
return false;
|
|
//--- prepare output array and calculate random values
|
|
ArrayResize(result,data_count);
|
|
for(int i=0; i<data_count; i++)
|
|
{
|
|
result[i]=MathRandomPoisson(lambda);
|
|
}
|
|
return true;
|
|
}
|
|
//+------------------------------------------------------------------+
|
|
//| Poisson distribution moments |
|
|
//+------------------------------------------------------------------+
|
|
//| The function calculates 4 first moments of the Poisson |
|
|
//| distribution with parameter lambda. |
|
|
//| |
|
|
//| Arguments: |
|
|
//| lambda : Mean |
|
|
//| 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 MathMomentsPoisson(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(lambda))
|
|
{
|
|
error_code=ERR_ARGUMENTS_NAN;
|
|
return false;
|
|
}
|
|
//--- lambda must be positive
|
|
if(lambda<=0.0)
|
|
{
|
|
error_code=ERR_ARGUMENTS_INVALID;
|
|
return false;
|
|
}
|
|
|
|
error_code=ERR_OK;
|
|
//--- calculate moments
|
|
mean =lambda;
|
|
variance=lambda;
|
|
skewness=MathPow(lambda,-0.5);
|
|
kurtosis=1.0/lambda;
|
|
//--- successful
|
|
return true;
|
|
}
|
|
//+------------------------------------------------------------------+
|