| //+------------------------------------------------------------------+
|
| //| Lognormal.mqh |
|
| //| Copyright 2000-2025, MetaQuotes Ltd. |
|
| //| https://www.mql5.com |
|
| //+------------------------------------------------------------------+
|
| #include "Math.mqh"
|
| #include "Normal.mqh"
|
|
|
| //+------------------------------------------------------------------+
|
| //| Lognormal density function (PDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the probability density function |
|
| //| of the Lognormal distribution with parameters mu and sigma. |
|
| //| |
|
| //| f(x,mu,sigma)=[1/(x*sigma*sqrt(2pi)]*exp(-(ln(x)-mu)/(2*sigma^2))|
|
| //| |
|
| //| Arguments: |
|
| //| x : Random variable |
|
| //| mu : Log mean |
|
| //| sigma : Log standard deviation |
|
| //| log_mode : Logarithm mode flag, if true it returns Log values |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The probability density evaluated at x. |
|
| //+------------------------------------------------------------------+
|
| double MathProbabilityDensityLognormal(const double x,const double mu,const double sigma,const bool log_mode,int &error_code)
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(x) || !MathIsValidNumber(mu) || !MathIsValidNumber(sigma))
|
| {
|
| error_code=ERR_ARGUMENTS_NAN;
|
| return QNaN;
|
| }
|
| //--- check sigma
|
| if(sigma<0)
|
| {
|
| error_code=ERR_ARGUMENTS_INVALID;
|
| return QNaN;
|
| }
|
|
|
| error_code=ERR_OK;
|
| //--- check x
|
| if(x<=0.0)
|
| return TailLog0(true,log_mode);
|
| //--- check case sigma==0
|
| if(sigma==0)
|
| {
|
| if(MathLog(MathAbs(x))==mu)
|
| {
|
| error_code=ERR_RESULT_INFINITE;
|
| return QPOSINF;
|
| }
|
| else
|
| return TailLog0(true,log_mode);
|
| }
|
| //--- prepare argument
|
| double y=(MathLog(x)-mu)/sigma;
|
| //--- check argument
|
| if(!MathIsValidNumber(y))
|
| {
|
| error_code=ERR_ARGUMENTS_INVALID;
|
| return QNaN;
|
| }
|
| //--- check overflow
|
| y=MathAbs(y);
|
| if(y>=2*MathSqrt(DBL_MAX))
|
| {
|
| error_code=ERR_ARGUMENTS_INVALID;
|
| return QNaN;
|
| }
|
| //--- return lognormal density
|
| return TailLogValue(M_1_SQRT_2PI*MathExp(-0.5*y*y)/(x*sigma),true,log_mode);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Lognormal density function (PDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the probability density function of |
|
| //| the Lognormal distribution with parameters mu and sigma |
|
| //| for values in x. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Array with random variables |
|
| //| mu : Log mean |
|
| //| sigma : Log standard deviation |
|
| //| log_mode : Logarithm mode flag,if true it calculates Log values|
|
| //| result : Array with calculated values |
|
| //| |
|
| //| Return value: |
|
| //| true if successful, otherwise false. |
|
| //+------------------------------------------------------------------+
|
| bool MathProbabilityDensityLognormal(const double &x[],const double mu,const double sigma,const bool log_mode,double &result[])
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(mu) || !MathIsValidNumber(sigma))
|
| return false;
|
| //--- check sigma
|
| if(sigma<0)
|
| return false;
|
|
|
| int data_count=ArraySize(x);
|
| if(data_count==0)
|
| return false;
|
|
|
| int error_code=0;
|
| ArrayResize(result,data_count);
|
|
|
| //--- check case sigma==0
|
| if(sigma==0)
|
| {
|
| for(int i=0; i<data_count; i++)
|
| {
|
| if(MathLog(MathAbs(x[i]))==mu)
|
| result[i]=QPOSINF;
|
| else
|
| result[i]=TailLog0(true,log_mode);
|
| return true;
|
| }
|
| }
|
|
|
| for(int i=0; i<data_count; i++)
|
| {
|
| double x_arg=x[i];
|
|
|
| if(!MathIsValidNumber(x_arg))
|
| return false;
|
|
|
| //--- check x
|
| if(x_arg<=0.0)
|
| result[i]=TailLog0(true,log_mode);
|
| else
|
| {
|
| //--- prepare argument
|
| double y=(MathLog(x_arg)-mu)/sigma;
|
| //--- check argument
|
| if(!MathIsValidNumber(y))
|
| return false;
|
| //--- check overflow
|
| y=MathAbs(y);
|
| if(y>=2*MathSqrt(DBL_MAX))
|
| return false;
|
| //--- return lognormal density
|
| result[i]=TailLogValue(M_1_SQRT_2PI*MathExp(-0.5*y*y)/(x_arg*sigma),true,log_mode);
|
| }
|
| }
|
| return true;
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Lognormal density function (PDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the probability density function of |
|
| //| the Lognormal distribution with parameters mu and sigma |
|
| //| for values in x[] array. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Array with random variables |
|
| //| mu : Log mean |
|
| //| sigma : Log standard deviation |
|
| //| result : Array with calculated values |
|
| //| |
|
| //| Return value: |
|
| //| true if successful, otherwise false. |
|
| //+------------------------------------------------------------------+
|
| bool MathProbabilityDensityLognormal(const double &x[],const double mu,const double sigma,double &result[])
|
| {
|
| return MathProbabilityDensityLognormal(x,mu,sigma,false,result);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Lognormal density function (PDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the probability density function |
|
| //| of the Lognormal distribution with parameters mu and sigma. |
|
| //| |
|
| //| f(x,mu,sigma)=[1/(x*sigma*sqrt(2pi)]*exp(-(ln(x)-mu)/(2*sigma^2))|
|
| //| |
|
| //| Arguments: |
|
| //| x : Random variable |
|
| //| mu : Log mean |
|
| //| sigma : Log standard deviation |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The probability density evaluated at x. |
|
| //+------------------------------------------------------------------+
|
| double MathProbabilityDensityLognormal(const double x,const double mu,const double sigma,int &error_code)
|
| {
|
| return MathProbabilityDensityLognormal(x,mu,sigma,false,error_code);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Lognormal cumulative distribution function (CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the probability that an observation |
|
| //| from the Lognormal distribution with parameters mu and sigma |
|
| //| is less than or equal to x. |
|
| //| |
|
| //| Arguments: |
|
| //| x : The desired quantile |
|
| //| mu : Log mean |
|
| //| sigma : Log standard deviation |
|
| //| 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 Lognormal cumulative distribution function |
|
| //| with parameters mu and sigma, evaluated at x. |
|
| //+------------------------------------------------------------------+
|
| double MathCumulativeDistributionLognormal(const double x,const double mu,const double sigma,const bool tail,const bool log_mode,int &error_code)
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(x) || !MathIsValidNumber(mu) || !MathIsValidNumber(sigma))
|
| {
|
| error_code=ERR_ARGUMENTS_NAN;
|
| return QNaN;
|
| }
|
| //--- check sigma
|
| if(sigma<0)
|
| {
|
| error_code=ERR_ARGUMENTS_INVALID;
|
| return QNaN;
|
| }
|
|
|
| error_code=ERR_OK;
|
| //--- check x
|
| if(x<=0.0)
|
| return TailLog0(tail,log_mode);
|
| //--- return lognormal cdf using Normal cdf
|
| return MathCumulativeDistributionNormal(MathLog(x),mu,sigma,tail,log_mode,error_code);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Lognormal cumulative distribution function (CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the probability that an observation |
|
| //| from the Lognormal distribution with parameters mu and sigma |
|
| //| is less than or equal to x. |
|
| //| |
|
| //| Arguments: |
|
| //| x : The desired quantile |
|
| //| mu : Log mean |
|
| //| sigma : Log standard deviation |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The value of the Lognormal cumulative distribution function |
|
| //| with parameters mu and sigma, evaluated at x. |
|
| //+------------------------------------------------------------------+
|
| double MathCumulativeDistributionLognormal(const double x,const double mu,const double sigma,int &error_code)
|
| {
|
| return MathCumulativeDistributionLognormal(x,mu,sigma,true,false,error_code);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Lognormal cumulative distribution function (CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the cumulative distribution function of |
|
| //| the Lognormal distribution with parameters mu and sigma |
|
| //| for values in x[] array. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Array with random variables |
|
| //| mu : Log mean |
|
| //| sigma : Log standard deviation |
|
| //| 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 MathCumulativeDistributionLognormal(const double &x[],const double mu,const double sigma,const bool tail,const bool log_mode,double &result[])
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(mu) || !MathIsValidNumber(sigma))
|
| return false;
|
| //--- check sigma
|
| if(sigma<0)
|
| return false;
|
|
|
| int data_count=ArraySize(x);
|
| if(data_count==0)
|
| return false;
|
|
|
| int error_code=0;
|
| ArrayResize(result,data_count);
|
| for(int i=0; i<data_count; i++)
|
| {
|
| double x_arg=x[i];
|
|
|
| if(!MathIsValidNumber(x_arg))
|
| return false;
|
|
|
| //--- check x
|
| if(x_arg<=0.0)
|
| result[i]=TailLog0(tail,log_mode);
|
| else
|
| //--- return lognormal cdf using Normal cdf
|
| result[i]=MathCumulativeDistributionNormal(MathLog(x_arg),mu,sigma,tail,log_mode,error_code);
|
| }
|
| return true;
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Lognormal cumulative distribution function (CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the cumulative distribution function of |
|
| //| the Lognormal distribution with parameters mu and sigma |
|
| //| for values in x[] array. |
|
| //| |
|
| //| Arguments: |
|
| //| x : Array with random variables |
|
| //| mu : Log mean |
|
| //| sigma : Log standard deviation |
|
| //| result : Array with calculated values |
|
| //| |
|
| //| Return value: |
|
| //| true if successful, otherwise false. |
|
| //+------------------------------------------------------------------+
|
| bool MathCumulativeDistributionLognormal(const double &x[],const double mu,const double sigma,double &result[])
|
| {
|
| return MathCumulativeDistributionLognormal(x,mu,sigma,true,false,result);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Lognormal distribution quantile function (inverse CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the inverse cumulative distribution |
|
| //| function of the Lognormal distribution with parameters mu |
|
| //| and sigma for the desired probability. |
|
| //| |
|
| //| Arguments: |
|
| //| probability : The desired probability |
|
| //| mu : Log mean |
|
| //| sigma : Log standard deviation |
|
| //| tail : Flag to calculate lower tail |
|
| //| log_mode : Logarithm mode,if true it calculates for Log values|
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The quantile value of the Lognormal distribution. |
|
| //+------------------------------------------------------------------+
|
| double MathQuantileLognormal(const double probability,const double mu,const double sigma,const bool tail,const bool log_mode,int &error_code)
|
| {
|
| if(log_mode==true && probability==QNEGINF)
|
| {
|
| error_code=ERR_OK;
|
| return 0.0;
|
| }
|
| //--- check NaN
|
| if(!MathIsValidNumber(probability) || !MathIsValidNumber(mu) || !MathIsValidNumber(sigma))
|
| {
|
| error_code=ERR_ARGUMENTS_NAN;
|
| return QNaN;
|
| }
|
| //--- check sigma
|
| if(sigma<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;
|
| }
|
|
|
| //--- special cases exp(a+b-+infinity)
|
| if(prob==0.0 || prob==1.0)
|
| {
|
| if(sigma==0.0)
|
| {
|
| error_code=ERR_OK;
|
| return MathExp(mu);
|
| }
|
| else
|
| if(prob==0.0)
|
| {
|
| if(sigma>0)
|
| {
|
| error_code=ERR_OK;
|
| return 0.0;
|
| }
|
| else
|
| if(sigma<0)
|
| {
|
| error_code=ERR_RESULT_INFINITE;
|
| return QPOSINF;
|
| }
|
| }
|
| else
|
| {
|
| if(sigma<0)
|
| {
|
| error_code=ERR_OK;
|
| return 0.0;
|
| }
|
| else
|
| if(sigma>0)
|
| {
|
| error_code=ERR_RESULT_INFINITE;
|
| return QPOSINF;
|
| }
|
| }
|
| }
|
| //--- return lognormal quantile using Normal distribution
|
| return MathExp(MathQuantileNormal(prob,mu,sigma,error_code));
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Lognormal distribution quantile function (inverse CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function returns the inverse cumulative distribution |
|
| //| function of Lognormal distribution with parameters mu and sigma |
|
| //| for the desired probability. |
|
| //| |
|
| //| Arguments: |
|
| //| probability : The desired probability |
|
| //| mu : Log mean |
|
| //| sigma : Log standard deviation |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The quantile value of the Lognormal distribution. |
|
| //+------------------------------------------------------------------+
|
| double MathQuantileLognormal(const double probability,const double mu,const double sigma,int &error_code)
|
| {
|
| return MathQuantileLognormal(probability,mu,sigma,true,false,error_code);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Lognormal distribution quantile function (inverse CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the inverse cumulative distribution |
|
| //| function of the Lognormal distribution with parameters mu and |
|
| //| sigma for values from the probability[] array. |
|
| //| |
|
| //| Arguments: |
|
| //| probability : Array with probabilities |
|
| //| mu : Log mean |
|
| //| sigma : Log standard deviation |
|
| //| tail : Flag to calculate lower tail |
|
| //| log_mode : Logarithm mode,if true it calculates for Log values|
|
| //| result : Array with calculated values |
|
| //| |
|
| //| Return value: |
|
| //| true if successful, otherwise false. |
|
| //+------------------------------------------------------------------+
|
| bool MathQuantileLognormal(const double &probability[],const double mu,const double sigma,const bool tail,const bool log_mode,double &result[])
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(mu) || !MathIsValidNumber(sigma))
|
| return false;
|
| //--- check sigma
|
| if(sigma<0)
|
| return false;
|
|
|
| int data_count=ArraySize(probability);
|
| if(data_count==0)
|
| return false;
|
|
|
| int error_code=0;
|
| 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;
|
|
|
| //--- special cases exp(a+b-+infinity)
|
| if(prob==0.0 || prob==1.0)
|
| {
|
| if(sigma==0.0)
|
| result[i]=MathExp(mu);
|
| else
|
| if(prob==0.0)
|
| {
|
| if(sigma>0)
|
| result[i]=0.0;
|
| else
|
| if(sigma<0)
|
| result[i]=QPOSINF;
|
| }
|
| else
|
| {
|
| if(sigma<0)
|
| result[i]=0.0;
|
| else
|
| if(sigma>0)
|
| result[i]=QPOSINF;
|
| }
|
| }
|
| else
|
| //--- calculate lognormal quantile using Normal distribution
|
| result[i]=MathExp(MathQuantileNormal(prob,mu,sigma,error_code));
|
| }
|
| return true;
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Lognormal distribution quantile function (inverse CDF) |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates the inverse cumulative distribution |
|
| //| function of the Lognormal distribution with parameters mu and |
|
| //| sigma for values from the probability[] array. |
|
| //| |
|
| //| Arguments: |
|
| //| probability : Array with probabilities |
|
| //| mu : Log mean |
|
| //| sigma : Log standard deviation |
|
| //| result : Array with calculated values |
|
| //| |
|
| //| Return value: |
|
| //| true if successful, otherwise false. |
|
| //+------------------------------------------------------------------+
|
| bool MathQuantileLognormal(const double &probability[],const double mu,const double sigma,double &result[])
|
| {
|
| return MathQuantileLognormal(probability,mu,sigma,true,false,result);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Random variate from the Lognormal distribution |
|
| //+------------------------------------------------------------------+
|
| //| Computes the random variable from the Lognormal distribution |
|
| //| with parameters mu and sigma. |
|
| //| |
|
| //| Arguments: |
|
| //| mu : Log mean |
|
| //| sigma : Log standard deviation |
|
| //| error_code : Variable for error code |
|
| //| |
|
| //| Return value: |
|
| //| The random value with Lognormal distribution. |
|
| //+------------------------------------------------------------------+
|
| double MathRandomLognormal(const double mu,const double sigma,int &error_code)
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(mu) || !MathIsValidNumber(sigma))
|
| {
|
| error_code=ERR_ARGUMENTS_NAN;
|
| return QNaN;
|
| }
|
| //--- check sigma
|
| if(sigma<0)
|
| {
|
| error_code=ERR_ARGUMENTS_INVALID;
|
| return QNaN;
|
| }
|
| error_code=ERR_OK;
|
| //--- generate random number
|
| double rnd=MathRandomNonZero();
|
| //---
|
| rnd=MathQuantileNormal(rnd,mu,sigma,true,false,error_code);
|
| return MathExp(rnd);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Random variate from the Lognormal distribution |
|
| //+------------------------------------------------------------------+
|
| //| Generates random variables from the Lognormal distribution |
|
| //| with parameters mu and sigma. |
|
| //| |
|
| //| Arguments: |
|
| //| mu : Log mean |
|
| //| sigma : Log standard deviation |
|
| //| data_count : Number of values needed |
|
| //| result : Output array with random values |
|
| //| |
|
| //| Return value: |
|
| //| true if successful, otherwise false. |
|
| //+------------------------------------------------------------------+
|
| bool MathRandomLognormal(const double mu,const double sigma,const int data_count,double &result[])
|
| {
|
| //--- check NaN
|
| if(!MathIsValidNumber(mu) || !MathIsValidNumber(sigma))
|
| return false;
|
| //--- check sigma
|
| if(sigma<0)
|
| return false;
|
| //--- prepare output array and calculate random values
|
| ArrayResize(result,data_count);
|
| int err_code=0;
|
| for(int i=0; i<data_count; i++)
|
| result[i]=MathRandomNonZero();
|
| //--- return normal random array using quantile
|
| MathQuantileNormal(result,mu,sigma,result);
|
| return MathExp(result);
|
| }
|
| //+------------------------------------------------------------------+
|
| //| Lognormal distribution moments |
|
| //+------------------------------------------------------------------+
|
| //| The function calculates 4 first moments of the Lognormal |
|
| //| distribution with parameters mu and sigma. |
|
| //| |
|
| //| Arguments: |
|
| //| mu : Log mean |
|
| //| sigma : Log standard deviation |
|
| //| 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 MathMomentsLognormal(const double mu,const double sigma,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(mu) || !MathIsValidNumber(sigma))
|
| {
|
| error_code=ERR_ARGUMENTS_NAN;
|
| return false;
|
| }
|
| //--- check sigma
|
| if(sigma<0)
|
| {
|
| error_code=ERR_ARGUMENTS_INVALID;
|
| return false;
|
| }
|
|
|
| error_code=ERR_OK;
|
| //--- sigma squared
|
| double sigma_sqr=sigma*sigma;
|
| double exp_sigma_sqr=MathExp(sigma_sqr);
|
| //--- calculate moments
|
| mean =MathExp(mu+sigma_sqr*0.5);
|
| variance=(exp_sigma_sqr-1.0)*MathExp(2*mu+sigma_sqr);
|
| skewness=MathSqrt(exp_sigma_sqr-1.0)*(exp_sigma_sqr+2.0);
|
| kurtosis=3*MathPowInt(exp_sigma_sqr,2)+2*MathPowInt(exp_sigma_sqr,3)+MathPowInt(exp_sigma_sqr,4)-3-3;
|
| //--- successful
|
| return true;
|
| }
|
| //+------------------------------------------------------------------+
|
