| //+------------------------------------------------------------------+
|
| //| NegativeBinomial.mqh |
|
| //| Copyright 2000-2025, MetaQuotes Ltd. |
|
| //| https://www.mql5.com |
|
| //+------------------------------------------------------------------+
|
| #include "Math.mqh"
|
| #include "Gamma.mqh"
|
| #include "Poisson.mqh"
|
|
|
| //+------------------------------------------------------------------+
|
| //| Negative Binomial probability mass function (PDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the probability mass function |
|
| //| of the Negative Binomial distribution with parameters r and p. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Random variable |
|
| //| r : Number of successes |
|
| //| p : Probability of success |
|
| //| log_mode : Logarithm mode flag, if true it returns Log values |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The probability mass evaluated at x. |
|
| //+------------------------------------------------------------------+
|
| double MathProbabilityDensityNegativeBinomial(const double x,const double r,const double p,const bool log_mode,int &error_code)
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(x) || !MathIsValidNumber(r) || !MathIsValidNumber(p))
|
| {
|
| error_code=ERR_ARGUMENTS_NAN;
|
| return QNaN;
|
| }
|
| //--- check arguments
|
| if(r!=MathRound(r) || r<1.0 || p<0.0 || p>1.0)
|
| {
|
| error_code=ERR_ARGUMENTS_INVALID;
|
| return QNaN;
|
| }
|
|
|
| error_code=ERR_OK;
|
| if(x<0.0)
|
| return TailLog0(true,log_mode);
|
| //--- calculate gamma factor for the density
|
| double coef=MathRound(MathExp(MathGammaLog(r+x)-MathGammaLog(x+1.0)-MathGammaLog(r)));
|
| //--- return density
|
| return TailLogValue(coef*MathPow(p,r)*MathPow(1.0-p,x),true,log_mode);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Negative Binomial probability mass function (PDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the probability mass function |
|
| //| of the Negative Binomial distribution with parameters r and p. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Random variable |
|
| //| r : Number of successes |
|
| //| p : Probability of success |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The probability mass evaluated at x. |
|
| //+------------------------------------------------------------------+
|
| double MathProbabilityDensityNegativeBinomial(const double x,const double r,const double p,int &error_code)
|
| {
|
| return MathProbabilityDensityNegativeBinomial(x,r,p,false,error_code);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Negative Binomial probability mass function (PDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the probability mass function |
|
| //| of the Negative Binomial distribution with parameters r and p |
|
| //| for values from x[] array. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Array with random variables |
|
| //| r : Number of successes |
|
| //| p : Probability of success |
|
| //| log_mode : Logarithm mode flag, if true it returns Log values |
|
| //| result : Array with calculated values |
|
| //| |
|
| //| Return value: |
|
| //| true if successful, otherwise false. |
|
| //+------------------------------------------------------------------+
|
| bool MathProbabilityDensityNegativeBinomial(const double &x[],const double r,const double p,const bool log_mode,double &result[])
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(r) || !MathIsValidNumber(p))
|
| return false;
|
| //--- check arguments
|
| if(r!=MathRound(r) || r<1.0 || p<0.0 || p>1.0)
|
| return false;
|
|
|
| int data_count=ArraySize(x);
|
| if(data_count==0)
|
| return false;
|
|
|
| double power_p_r=MathPow(p,r);
|
| double log_gamma_r=MathGammaLog(r);
|
| int error_code=0;
|
| ArrayResize(result,data_count);
|
| for(int i=0; i<data_count; i++)
|
| {
|
| double x_arg=x[i];
|
|
|
| if(!MathIsValidNumber(x_arg))
|
| return false;
|
|
|
| if(x_arg<0.0)
|
| result[i]=TailLog0(true,log_mode);
|
| else
|
| {
|
| //--- calculate pdf
|
| double pdf=power_p_r*MathPow(1.0-p,x_arg)*MathRound(MathExp(MathGammaLog(r+x_arg)-MathGammaLog(x_arg+1.0)-log_gamma_r));
|
| result[i]=TailLogValue(pdf,true,log_mode);
|
| }
|
| }
|
| return true;
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Negative Binomial probability mass function (PDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the probability mass function |
|
| //| of the Negative Binomial distribution with parameters r and p |
|
| //| for values from x[] array. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Array with random variables |
|
| //| r : Number of successes |
|
| //| p : Probability of success |
|
| //| result : Array with calculated values |
|
| //| |
|
| //| Return value: |
|
| //| true if successful, otherwise false. |
|
| //+------------------------------------------------------------------+
|
| bool MathProbabilityDensityNegativeBinomial(const double &x[],const double r,const double p,double &result[])
|
| {
|
| return MathProbabilityDensityNegativeBinomial(x,r,p,false,result);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Negative Binomial cumulative distribution function (CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the probability that an observation |
|
| //| from the Negative Binomial distribution with parameters r and p |
|
| //| is less than or equal to x. |
|
| //| |
|
| //| Arguments: |
|
| //| x : The desired quantile |
|
| //| r : Number of successes |
|
| //| p : Probability of success |
|
| //| tail : Flag to calculate lower tail |
|
| //| log_mode : Logarithm mode, if true it calculates Log values |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The value of the Negative Binomial cumulative distribution |
|
| //| function with parameters r and p, evaluated at x. |
|
| //+------------------------------------------------------------------+
|
| double MathCumulativeDistributionNegativeBinomial(const double x,const double r,double p,const bool tail,const bool log_mode,int error_code)
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(x) || !MathIsValidNumber(r) || !MathIsValidNumber(p))
|
| {
|
| error_code=ERR_ARGUMENTS_NAN;
|
| return QNaN;
|
| }
|
| //--- check arguments
|
| if(r!=MathRound(r) || r<1.0 || p<0.0 || p>1.0 || x<0)
|
| {
|
| error_code=ERR_ARGUMENTS_INVALID;
|
| return QNaN;
|
| }
|
|
|
| error_code=ERR_OK;
|
| if(x<0.0)
|
| return TailLog0(tail,log_mode);
|
| int err_code=0;
|
| //--- calculate max term of the sum
|
| int max_j=(int)MathFloor(x);
|
| double p1=1.0-p;
|
| //--- initial factors
|
| double factor1=MathFactorial((int)r-1);
|
| double factor2=1.0;
|
| double factor_p=1.0;
|
| double factor_r=1.0/factor1;
|
| double power_p_r=MathPowInt(p,int(r))*factor_r;
|
| double cdf=0.0;
|
| for(int j=0; j<=max_j; j++)
|
| {
|
| if(j>0)
|
| {
|
| factor1*=(j+1);
|
| factor2*=j;
|
| factor_p*=p1;
|
| }
|
| double pdf=power_p_r*factor1*factor_p/factor2;
|
| cdf+=pdf;
|
| }
|
| //--- take into account round-off errors for probability
|
| return TailLogValue(MathMin(cdf,1.0),tail,log_mode);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Negative Binomial cumulative distribution function (CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the probability that an observation |
|
| //| from the Negative Binomial distribution with parameters r and p |
|
| //| is less than or equal to x. |
|
| //| |
|
| //| Arguments: |
|
| //| x : The desired quantile |
|
| //| r : Number of successes |
|
| //| p : Probability of success |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The value of the Negative Binomial cumulative distribution |
|
| //| function with parameters r and p, evaluated at x. |
|
| //+------------------------------------------------------------------+
|
| double MathCumulativeDistributionNegativeBinomial(const double x,const double r,double p,int error_code)
|
| {
|
| return MathCumulativeDistributionNegativeBinomial(x,r,p,true,false,error_code);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Negative Binomial cumulative distribution function (CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the cumulative distribution function |
|
| //| of the Negative Binomial distribution with parameters r and p |
|
| //| for values from x[] array. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Array with random variables |
|
| //| r : Number of successes |
|
| //| p : Probability of success |
|
| //| 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 MathCumulativeDistributionNegativeBinomial(const double &x[],const double r,double p,const bool tail,const bool log_mode,double &result[])
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(r) || !MathIsValidNumber(p))
|
| return false;
|
| //--- check arguments
|
| if(r!=MathRound(r) || r<1.0 || p<0.0 || p>1.0)
|
| return false;
|
|
|
| int data_count=ArraySize(x);
|
| if(data_count==0)
|
| return false;
|
| //--- common factors
|
| double fact1=MathFactorial((int)r-1);
|
| double factor_r=1.0/fact1;
|
| double power_p_r=MathPowInt(p,int(r))*factor_r;
|
| double p1=1.0-p;
|
| int error_code=0;
|
| ArrayResize(result,data_count);
|
| for(int i=0; i<data_count; i++)
|
| {
|
| double x_arg=x[i];
|
|
|
| if(!MathIsValidNumber(x_arg))
|
| return false;
|
|
|
| if(x_arg<0.0)
|
| result[i]=TailLog0(tail,log_mode);
|
| else
|
| {
|
| int err_code=0;
|
| //--- calculate max term of the sum
|
| int max_j=(int)MathFloor(x_arg);
|
| //--- initial factors
|
| double factor1=fact1;
|
| double factor2=1.0;
|
| double factor_p=1.0;
|
| double cdf=0.0;
|
| for(int j=0; j<=max_j; j++)
|
| {
|
| if(j>0)
|
| {
|
| factor1*=(j+1);
|
| factor2*=j;
|
| factor_p*=p1;
|
| }
|
| double pdf=power_p_r*factor1*factor_p/factor2;
|
| cdf+=pdf;
|
| }
|
| //--- take into account round-off errors for probability
|
| result[i]=TailLogValue(MathMin(cdf,1.0),tail,log_mode);
|
| }
|
| }
|
| return true;
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Negative Binomial cumulative distribution function (CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the cumulative distribution function |
|
| //| of the Negative Binomial distribution with parameters r and p |
|
| //| for values from x[] array. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Array with random variables |
|
| //| r : Number of successes |
|
| //| p : Probability of success |
|
| //| result : Array with calculated values |
|
| //| |
|
| //| Return value: |
|
| //| true if successful, otherwise false. |
|
| //+------------------------------------------------------------------+
|
| bool MathCumulativeDistributionNegativeBinomial(const double &x[],const double r,double p,double &result[])
|
| {
|
| return MathCumulativeDistributionNegativeBinomial(x,r,p,true,false,result);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Negative Binomial distribution quantile function (inverse CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the inverse cumulative distribution |
|
| //| function of the Negative Binomial distribution with parameters |
|
| //| r and p for the desired probability. |
|
| //| |
|
| //| Arguments: |
|
| //| probability : The desired probability |
|
| //| r : Number of successes |
|
| //| p : Probability of success |
|
| //| tail : Flag to calculate lower tail |
|
| //| log_mode : Logarithm mode, if true it calculates Log values |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The value of the inverse cumulative distribution function |
|
| //| of the Negative Binomial distribution with parameters r and p. |
|
| //+------------------------------------------------------------------+
|
| double MathQuantileNegativeBinomial(const double probability,const double r,const double p,const bool tail,const bool log_mode,int &error_code)
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(probability) || !MathIsValidNumber(r) || !MathIsValidNumber(p))
|
| {
|
| error_code=ERR_ARGUMENTS_NAN;
|
| return QNaN;
|
| }
|
| //--- check arguments
|
| if(r!=MathRound(r) || r<1.0 || p<0.0 || p>1.0)
|
| {
|
| error_code=ERR_ARGUMENTS_INVALID;
|
| return QNaN;
|
| }
|
| //--- calculate real probability
|
| double prob=TailLogProbability(probability,tail,log_mode);
|
| //--- check probability range
|
| if(prob<0.0 || prob>1.0)
|
| {
|
| error_code=ERR_ARGUMENTS_INVALID;
|
| return QNaN;
|
| }
|
| //--- check cases p=0 and p=1
|
| if(prob==1.0)
|
| {
|
| error_code=ERR_RESULT_INFINITE;
|
| return QPOSINF;
|
| }
|
| error_code=ERR_OK;
|
| if(prob==0.0)
|
| return 0.0;
|
|
|
| int max_terms=1000;
|
| int err_code=0;
|
| //--- factors
|
| double fact1=MathFactorial((int)r-1);
|
| double factor_r=1.0/fact1;
|
| double power_p_r=MathPowInt(p,int(r))*factor_r;
|
| double p1=1.0-p;
|
| //--- initial factors
|
| double factor1=fact1;
|
| double factor2=1.0;
|
| double factor_p=1.0;
|
| double cdf=0.0;
|
| int j=0;
|
| while(cdf<prob && j<max_terms)
|
| {
|
| if(j>0)
|
| {
|
| factor1*=(j+1);
|
| factor2*=j;
|
| factor_p*=p1;
|
| }
|
| double pdf=power_p_r*factor1*factor_p/factor2;
|
| cdf+=pdf;
|
| j++;
|
| }
|
| //--- check convergence
|
| if(j<max_terms)
|
| {
|
| if(j==0)
|
| return 0;
|
| else
|
| return j-1;
|
| }
|
| else
|
| {
|
| error_code=ERR_NON_CONVERGENCE;
|
| return 0;
|
| }
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Negative Binomial distribution quantile function (inverse CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the inverse cumulative distribution |
|
| //| function of the Negative Binomial distribution with parameters |
|
| //| r and p for the desired probability. |
|
| //| |
|
| //| Arguments: |
|
| //| probability : The desired probability |
|
| //| r : Number of successes |
|
| //| p : Probability of success |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The value of the inverse cumulative distribution function |
|
| //| of the Negative Binomial distribution with parameters r and p. |
|
| //+------------------------------------------------------------------+
|
| double MathQuantileNegativeBinomial(const double probability,const double r,const double p,int &error_code)
|
| {
|
| return MathQuantileNegativeBinomial(probability,r,p,true,false,error_code);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Negative Binomial distribution quantile function (inverse CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the inverse cumulative distribution |
|
| //| function of the Negative Binomial distribution with parameters |
|
| //| r and p for values form the probability[] array. |
|
| //| |
|
| //| Arguments: |
|
| //| probability : Array with probabilities |
|
| //| r : Number of successes |
|
| //| p : Probability of success |
|
| //| 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 MathQuantileNegativeBinomial(const double &probability[],const double r,const double p,const bool tail,const bool log_mode,double &result[])
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(r) || !MathIsValidNumber(p))
|
| return false;
|
| //--- check arguments
|
| if(r!=MathRound(r) || r<1.0 || p<0.0 || p>1.0)
|
| return false;
|
|
|
| int data_count=ArraySize(probability);
|
| if(data_count==0)
|
| return false;
|
| //--- common factors
|
| double fact1=MathFactorial((int)r-1);
|
| double factor_r=1.0/fact1;
|
| double power_p_r=MathPowInt(p,int(r))*factor_r;
|
| double p1=1.0-p;
|
| int max_terms=500;
|
| ArrayResize(result,data_count);
|
| for(int i=0; i<data_count; i++)
|
| {
|
| //--- calculate real probability
|
| double prob=TailLogProbability(probability[i],tail,log_mode);
|
|
|
| //--- check probability range
|
| if(prob<0.0 || prob>1.0)
|
| return false;
|
|
|
| if(prob==0.0)
|
| result[i]=0.0;
|
| else
|
| if(prob==1.0)
|
| result[i]=QPOSINF;
|
| else
|
| {
|
| double factor1=fact1;
|
| double factor2=1.0;
|
| double factor_p=1.0;
|
| double cdf=0.0;
|
| int j=0;
|
| while(cdf<prob && j<max_terms)
|
| {
|
| if(j>0)
|
| {
|
| factor1*=(j+1);
|
| factor2*=j;
|
| factor_p*=p1;
|
| }
|
| double pdf=power_p_r*factor1*factor_p/factor2;
|
| cdf+=pdf;
|
| j++;
|
| }
|
| if(j<max_terms)
|
| {
|
| if(j==0)
|
| result[i]=0;
|
| else
|
| result[i]=j-1;
|
| }
|
| else
|
| return false;
|
| }
|
| }
|
| return true;
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Negative Binomial distribution quantile function (inverse CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the inverse cumulative distribution |
|
| //| function of the Negative Binomial distribution with parameters |
|
| //| r and p for values from the probability[] array. |
|
| //| |
|
| //| Arguments: |
|
| //| probability : Array with probabilities |
|
| //| r : Number of successes |
|
| //| p : Probability of success |
|
| //| result : Array with calculated values |
|
| //| |
|
| //| Return value: |
|
| //| true if successful, otherwise false. |
|
| //+------------------------------------------------------------------+
|
| bool MathQuantileNegativeBinomial(const double &probability[],const double r,const double p,double &result[])
|
| {
|
| return MathQuantileNegativeBinomial(probability,r,p,true,false,result);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Random variate from the Negative Binomial distribution |
|
| //+------------------------------------------------------------------+
|
| //| Computes the random variable from the Negative Binomial |
|
| //| distribution with parameters r and p. |
|
| //| |
|
| //| Arguments: |
|
| //| r : Number of successes |
|
| //| p : Probability of success |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The random value with Negative Binomial distribution. |
|
| //+------------------------------------------------------------------+
|
| double MathRandomNegativeBinomial(const double r,const double p,int error_code)
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(r) || !MathIsValidNumber(p))
|
| {
|
| error_code=ERR_ARGUMENTS_NAN;
|
| return QNaN;
|
| }
|
| //--- check arguments
|
| if(r<=0.0 || p<=0.0 || p>=1.0)
|
| {
|
| error_code=ERR_ARGUMENTS_NAN;
|
| return QNaN;
|
| }
|
| double r_gamma=MathRandomGamma(r,(1-p)/p);
|
| return MathRandomPoisson(r_gamma,error_code);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Random variate from the Negative Binomial distribution |
|
| //+------------------------------------------------------------------+
|
| //| Generates random variables from the Negative Binomial |
|
| //| distribution with parameters r and p. |
|
| //| |
|
| //| Arguments: |
|
| //| r : Number of successes |
|
| //| p : Probability of success |
|
| //| data_count : Number of values needed |
|
| //| result : Output array with random values |
|
| //| |
|
| //| Return value: |
|
| //| true if successful, otherwise false. |
|
| //+------------------------------------------------------------------+
|
| bool MathRandomNegativeBinomial(const double r,const double p,const int data_count,double &result[])
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(r) || !MathIsValidNumber(p))
|
| return false;
|
| //--- check arguments
|
| if(r<=0.0 || p<=0.0 || p>=1.0)
|
| return false;
|
|
|
| double p_coef=(1-p)/p;
|
| int error_code=0;
|
| //--- prepare output array and calculate random values
|
| ArrayResize(result,data_count);
|
| for(int i=0; i<data_count; i++)
|
| {
|
| double r_gamma=MathRandomGamma(r,p_coef);
|
| result[i]=MathRandomPoisson(r_gamma,error_code);
|
| }
|
| return true;
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Negative Binomial distribution moments |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates 4 first moments of Negative Binomial |
|
| //| distribution with parameters r and p. |
|
| //| |
|
| //| Arguments: |
|
| //| r : Number of successes |
|
| //| p : Probability of success |
|
| //| 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 MathMomentsNegativeBinomial(const double r,double p,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(r) || !MathIsValidNumber(p))
|
| {
|
| error_code=ERR_ARGUMENTS_NAN;
|
| return false;
|
| }
|
| //--- check arguments
|
| if(r!=MathRound(r) || r<1.0 || p<=0.0 || p>=1.0)
|
| {
|
| error_code=ERR_ARGUMENTS_INVALID;
|
| return false;
|
| }
|
|
|
| error_code=ERR_OK;
|
| //--- calculate moments
|
| mean =r*(1.0-p)/p;
|
| variance=mean/p;
|
| skewness=(2.0-p)/MathSqrt((r*(1.0-p)));
|
| kurtosis=(p*p-6*p+6)/(r*(1.0-p));
|
| //--- successful
|
| return true;
|
| }
|
| //+------------------------------------------------------------------+
|
