539 lines
24 KiB
MQL5
539 lines
24 KiB
MQL5
//+------------------------------------------------------------------+
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//| Cauchy.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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//+------------------------------------------------------------------+
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//| Cauchy density function (PDF) |
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//+------------------------------------------------------------------+
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//| Computes the value of the Cauchy probability density function |
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//| with parameters a and b at the desired quantile x. |
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//| |
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//| f(x,a,b)= 1/(pi*b*(1.0+((x-a)/b)^2) |
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//| Arguments: |
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//| x : Random variable |
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//| a : Mean |
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//| b : Scale |
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//| log_mode : Logarithm mode flag, if true it returns Log values |
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//| error_code : Variable for error code |
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//| |
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//| Return value: |
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//| The probability density evaluated at x. |
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//+------------------------------------------------------------------+
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double MathProbabilityDensityCauchy(const double x,const double a,const double b,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(a) || !MathIsValidNumber(b))
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{
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error_code=ERR_ARGUMENTS_NAN;
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return QNaN;
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}
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//--- check scale
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if(b<=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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//--- prepare argument
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double y=(x-a)/b;
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//--- check result
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if(!MathIsValidNumber(y))
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{
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error_code=ERR_ARGUMENTS_INVALID;
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return QNaN;
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}
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error_code=ERR_OK;
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if(log_mode==true)
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return -MathLog(M_PI*b*(1.0+y*y));
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//--- return Cauchy density
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return 1.0/(M_PI*b*(1.0+y*y));
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}
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//+------------------------------------------------------------------+
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//| Cauchy density function (PDF) |
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//+------------------------------------------------------------------+
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//| Computes the value of the Cauchy probability density function |
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//| with parameters a and b at the desired quantile x. |
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//| |
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//| f(x,a,b)= 1/(pi*b*(1.0+((x-a)/b)^2) |
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//| Arguments: |
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//| x : Random variable |
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//| a : Mean |
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//| b : Scale |
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//| error_code : Variable for error code |
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//| |
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//| Return value: |
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//| The probability density evaluated at x. |
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//+------------------------------------------------------------------+
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double MathProbabilityDensityCauchy(const double x,const double a,const double b,int &error_code)
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{
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return MathProbabilityDensityCauchy(x,a,b,false,error_code);
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}
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//+------------------------------------------------------------------+
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//| Cauchy density function (PDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the probability density function of |
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//| Cauchy distribution with parameters a and b for values |
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//| from x[] array. |
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//| |
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//| Arguments: |
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//| x : Array with random variables |
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//| a : Mean |
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//| b : Scale |
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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 MathProbabilityDensityCauchy(const double &x[],const double a,const double b,const bool log_mode,double &result[])
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{
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//--- check NaN
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if(!MathIsValidNumber(a) || !MathIsValidNumber(b))
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return false;
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//--- check scale
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if(b<=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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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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//--- prepare argument
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double y=(x_arg-a)/b;
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if(log_mode==true)
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result[i]=-MathLog(M_PI*b*(1.0+y*y));
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else
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result[i]=(1.0/(M_PI*b*(1.0+y*y)));
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}
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return true;
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}
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//+------------------------------------------------------------------+
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//| Cauchy density function (PDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the probability density function of |
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//| Cauchy distribution with parameters a and b for values |
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//| in x[] array. |
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//| |
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//| Arguments: |
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//| x : Array with random variables |
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//| a : Mean |
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//| b : Scale |
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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 MathProbabilityDensityCauchy(const double &x[],const double a,const double b,double &result[])
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{
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return MathProbabilityDensityCauchy(x,a,b,false,result);
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}
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//+------------------------------------------------------------------+
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//| Cauchy cumulative distribution function (CDF) |
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//+------------------------------------------------------------------+
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//| The function returns the probability that an observation |
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//| from the Cauchy distribution with parameters a and b |
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//| is less than or equal to x. |
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//| F(x,a,b)=(1/2)+(1/pi)*arctan((x-a)/b) |
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//| Arguments: |
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//| x : The desired quantile |
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//| a : Mean |
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//| b : Scale |
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//| tail : Flag to calculate lower tail |
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//| log_mode : Logarithm mode flag,if true it calculates Log values|
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//| error_code : Variable for error code |
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//| |
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//| The value of the Cauchy cumulative distribution function with |
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//| parameters a and b, evaluated at x. |
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//+------------------------------------------------------------------+
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double MathCumulativeDistributionCauchy(const double x,const double a,const double b,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(x) || !MathIsValidNumber(a) || !MathIsValidNumber(b))
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{
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error_code=ERR_ARGUMENTS_NAN;
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return QNaN;
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}
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//--- check scale
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if(b<=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 argument
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double y=(x-a)/b;
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//--- check result
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if(!MathIsValidNumber(y))
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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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//--- calculate probability and take into account round-off errors
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double cdf=0;
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if(y>-1.0)
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cdf=MathMin(0.5+M_1_PI*MathArctan(y),1.0);
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else
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cdf=MathMin(M_1_PI*MathArctan(-1/y),1.0);
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return TailLogValue(cdf,tail,log_mode);
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}
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//+------------------------------------------------------------------+
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//| Cauchy cumulative distribution function (CDF) |
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//+------------------------------------------------------------------+
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//| The function returns the cumulative distribution function of |
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//| the Cauchy distribution with parameters a and b, evaluated at x. |
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//| |
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//| Arguments: |
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//| x : The desired quantile |
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//| a : Mean |
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//| b : Scale |
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//| error_code : Variable for error code |
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//| |
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//| The value of the Cauchy cumulative distribution function with |
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//| parameters a and b, evaluated at x. |
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//+------------------------------------------------------------------+
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double MathCumulativeDistributionCauchy(const double x,const double a,const double b,int &error_code)
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{
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return MathCumulativeDistributionCauchy(x,a,b,true,false,error_code);
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}
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//+------------------------------------------------------------------+
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//| Cauchy 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 Cauchy distribution with parameters a and b for values from |
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//| x[] array. |
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//| |
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//| Arguments: |
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//| x : Array with random variables |
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//| a : Mean |
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//| b : Scale |
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//| tail : Flag to calculate lower tail |
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//| log_mode : Logarithm mode flag,if true it calculates Log values|
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//| error_code : Variable for error code |
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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 MathCumulativeDistributionCauchy(const double &x[],const double a,const double b,const bool tail,const bool log_mode,double &result[])
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{
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//--- check parameters
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if(!MathIsValidNumber(a) || !MathIsValidNumber(b))
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return false;
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//--- check scale
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if(b<=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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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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//--- calculate argument
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double y=(x_arg-a)/b;
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//--- check result
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if(!MathIsValidNumber(y))
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return false;
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//--- calculate probability and take into account round-off errors
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double cdf=0;
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if(y>-1.0)
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cdf=MathMin(0.5+M_1_PI*MathArctan(y),1.0);
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else
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cdf=MathMin(M_1_PI*MathArctan(-1/y),1.0);
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result[i]=TailLogValue(cdf,tail,log_mode);
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}
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return true;
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}
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//+------------------------------------------------------------------+
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//| Cauchy cumulative distribution function (CDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the cumulative distribution function |
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//| of the Cauchy distribution with parameters a and b for values |
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//| from x[] array. |
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//| |
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//| Arguments: |
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//| x : Array with random variables |
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//| a : Mean |
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//| b : Scale |
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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 MathCumulativeDistributionCauchy(const double &x[],const double a,const double b,double &result[])
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{
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return MathCumulativeDistributionCauchy(x,a,b,true,false,result);
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}
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//+------------------------------------------------------------------+
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//| Cauchy distribution quantile function (inverse CDF) |
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//+------------------------------------------------------------------+
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//| The function returns the inverse cumulative distribution |
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//| function of the Cauchy distribution with parameters a and b |
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//| for the desired probability. |
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//| Q(p,a,b)=a+b*tan*(pi*(p-1/2)) |
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//| Arguments: |
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//| probability : The desired probability |
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//| a : Mean |
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//| b : Scale |
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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 inverse cumulative distribution function |
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//| of the Cauchy distribution with parameters a and b. |
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//+------------------------------------------------------------------+
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double MathQuantileCauchy(const double probability,const double a,const double b,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(probability) || !MathIsValidNumber(a) || !MathIsValidNumber(b))
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{
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error_code=ERR_ARGUMENTS_NAN;
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return QNaN;
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}
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//--- check scale
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if(b<=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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//--- f(1)= + infinity
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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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//--- f(0)= - infinity
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if(prob==0.0)
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{
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error_code=ERR_RESULT_INFINITE;
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return QNEGINF;
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}
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error_code=ERR_OK;
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//--- return quantile
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return a+b*MathTan(M_PI*(prob-0.5));
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}
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//+------------------------------------------------------------------+
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//| Cauchy distribution quantile function (inverse CDF) |
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//+------------------------------------------------------------------+
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//| The function returns the inverse cumulative distribution |
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//| function of the Cauchy distribution with parameters a and b |
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//| for the desired probability. |
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//| |
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//| Arguments: |
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//| probability : The desired probability |
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//| a : Mean |
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//| b : Scale |
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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 Cauchy distribution with parameters a and b. |
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//+------------------------------------------------------------------+
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double MathQuantileCauchy(const double probability,const double a,const double b,int &error_code)
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{
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return MathQuantileCauchy(probability,a,b,true,false,error_code);
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}
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//+------------------------------------------------------------------+
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//| Cauchy distribution quantile function (inverse CDF) |
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//+------------------------------------------------------------------+
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//| The function returns the inverse cumulative distribution |
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//| function of the Cauchy distribution with parameters a and b |
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//| for the probability values from array. |
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//| |
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//| Arguments: |
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//| probability : Array with probabilities |
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//| a : Mean |
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//| b : Scale |
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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 MathQuantileCauchy(const double &probability[],const double a,const double b,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(a) || !MathIsValidNumber(b))
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return false;
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//--- check scale
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if(b<=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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for(int i=0; i<data_count; i++)
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{
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if(!MathIsValidNumber(probability[i]))
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return false;
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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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//--- f(1)= + infinity
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if(prob==1.0)
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result[i]=QPOSINF;
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else
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//--- f(0)= - infinity
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if(prob==0.0)
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result[i]=QNEGINF;
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else
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result[i]=a+b*MathTan(M_PI*(prob-0.5));
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}
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return true;
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}
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//+------------------------------------------------------------------+
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//| Cauchy distribution quantile function (inverse CDF) |
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//+------------------------------------------------------------------+
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bool MathQuantileCauchy(const double &probability[],const double a,const double b,double &result[])
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{
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return MathQuantileCauchy(probability,a,b,true,false,result);
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}
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//+------------------------------------------------------------------+
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//| Random variate from the Cauchy distribution |
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//+------------------------------------------------------------------+
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//| Compute the random variable from the Cauchy distribution |
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//| with parameters a and b. |
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//| |
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//| Arguments: |
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//| a : Mean |
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//| b : Scale |
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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 random value with Cauchy distribution. |
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//+------------------------------------------------------------------+
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double MathRandomCauchy(const double a,const double b,int &error_code)
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{
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//--- check parameters
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if(!MathIsValidNumber(a) || !MathIsValidNumber(b))
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{
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error_code=ERR_ARGUMENTS_NAN;
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return QNaN;
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}
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//--- check scale
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if(b<0)
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{
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error_code=ERR_ARGUMENTS_INVALID;
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return QNaN;
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}
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error_code=ERR_OK;
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//--- check scale=0
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if(b==0.0)
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return a;
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//--- generate random number
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double rnd=MathRandomNonZero();
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//--- return result
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return a+b*MathTan(M_PI*(rnd-0.5));
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}
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//+------------------------------------------------------------------+
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//| Random variate from the Cauchy distribution |
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//+------------------------------------------------------------------+
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//| Generates random variables from the Cauchy distribution with |
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//| parameters a and b. |
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//| |
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//| Arguments: |
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//| a : Mean |
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//| b : Scale |
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//| data_count : Number of values needed |
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//| result : Output array with random 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 MathRandomCauchy(const double a,const double b,const int data_count,double &result[])
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{
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//--- check parameters
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if(!MathIsValidNumber(a) || !MathIsValidNumber(b))
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return false;
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//--- check scale
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if(b<0)
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return false;
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//--- prepare output array
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ArrayResize(result,data_count);
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//--- check scale=0
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if(b==0.0)
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{
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for(int i=0; i<data_count; i++)
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result[i]=a;
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}
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else
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//--- calculate random values
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for(int i=0; i<data_count; i++)
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{
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//--- generate random number
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double rnd=MathRandomNonZero();
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result[i]=a+b*MathTan(M_PI*(rnd-0.5));
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}
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return true;
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}
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//+------------------------------------------------------------------+
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//| Cauchy distribution moments |
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//+------------------------------------------------------------------+
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//| The function calculates 4 first moments of Cauchy distribution |
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//| with parameters a and b. |
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//| |
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//| Arguments: |
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//| a : Mean |
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//| b : Scale |
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//| mean : Variable for mean value (1st moment) |
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//| variance : Variable for variance value (2nd moment) |
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//| skewness : Variable for skewness value (3rd moment) |
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//| kurtosis : Variable for kurtosis value (4th moment) |
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//| error_code : Variable for error code |
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//| |
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//| Return value: |
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//| true if moments calculated successfully, otherwise false. |
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//+------------------------------------------------------------------+
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bool MathMomentsCauchy(const double a,const double b,double &mean,double &variance,double &skewness,double &kurtosis,int &error_code)
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{
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error_code=ERR_OK;
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//--- set theoretical values for moments (undefined)
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mean =QNaN;
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variance=QNaN;
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skewness=QNaN;
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kurtosis=QNaN;
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//--- successful
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return true;
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}
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//+------------------------------------------------------------------+
|