772 行
32 KiB
MQL5
772 行
32 KiB
MQL5
 | //+------------------------------------------------------------------+
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//| Beta.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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//| Beta density function (PDF) |
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//+------------------------------------------------------------------+
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//| The function returns the probability density function of |
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//| the Beta distribution with shape parameters a and b. |
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//| |
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//| f(x,a,b)= (1/Beta(a,b))*x^(a-1)*(1-x)^(b-1) |
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//| Arguments: |
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//| x : Random variable |
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//| a : First shape parameter (a>0) |
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//| b : Second shape parameter (b>0) |
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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 MathProbabilityDensityBeta(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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//--- a and b must be positive
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if(a<=0.0 || 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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error_code=ERR_OK;
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//--- check x range
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if(x<=0.0 || x>=1.0)
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return TailLog0(true,log_mode);
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double log_result=(a-1.0)*MathLog(x)+(b-1.0)*MathLog(1.0-x)-MathBetaLog(a,b);
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//--- return log beta density
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if(log_mode==true)
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return log_result;
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//--- return beta density
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return MathExp(log_result);
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}
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//+------------------------------------------------------------------+
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//| Beta density function (PDF) |
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//+------------------------------------------------------------------+
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//| The function returns the probability density function of |
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//| the Beta distribution with shape parameters a and b. |
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//| |
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//| f(x,a,b)= (1/Beta(a,b))*x^(a-1)*(1-x)^(b-1) |
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//| Arguments: |
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//| x : Random variable |
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//| a : First shape parameter (a>0) |
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//| b : Second shape parameter (b>0) |
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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 MathProbabilityDensityBeta(const double x,const double a,const double b,int &error_code)
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{
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return MathProbabilityDensityBeta(x,a,b,false,error_code);
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}
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//+------------------------------------------------------------------+
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//| Beta density function (PDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the probability density function of the |
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//| Beta distribution with shape parameters a and b for values in |
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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 : First shape parameter (a>0) |
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//| b : Second shape parameter (b>0) |
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//| log_mode : Logarithm mode flag,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 MathProbabilityDensityBeta(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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//--- a and b must be positive
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if(a<=0.0 || 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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if(x_arg<=0.0 || x_arg>=1.0)
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result[i]=TailLog0(true,log_mode);
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else
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{
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double log_result=(a-1.0)*MathLog(x_arg)+(b-1.0)*MathLog(1.0-x_arg)-MathBetaLog(a,b);
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if(log_mode==true)
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result[i]=log_result;
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else
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result[i]=MathExp(log_result);
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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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//| Beta density function (PDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the probability density function of the |
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//| Beta distribution with shape parameters a and b for values in |
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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 : First shape parameter (a>0) |
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//| b : Second shape parameter (b>0) |
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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 MathProbabilityDensityBeta(const double &x[],const double a,const double b,double &result[])
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{
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return MathProbabilityDensityBeta(x,a,b,false,result);
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}
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//+------------------------------------------------------------------+
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//| Beta 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 Beta distribution with shape parameters a and b, evaluated |
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//| at x. |
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//| |
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//| Arguments: |
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//| x : The desired quantile |
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//| a : First shape parameter (a>0) |
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//| b : Second shape parameter (b>0) |
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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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//| Return value: |
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//| The value of the Beta cumulative distribution function with |
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//| shape parameters a and b, evaluated at x. |
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//+------------------------------------------------------------------+
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double MathCumulativeDistributionBeta(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 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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//--- a and b must be positive
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if(a<=0.0 || 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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error_code=ERR_OK;
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//--- check x range
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if(x<=0.0)
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return TailLog0(tail,log_mode);
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if(x>=1.0)
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return TailLog1(tail,log_mode);
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//--- calculate probability and take into account round-off errors
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double cdf=MathMin(MathBetaIncomplete(x,a,b),1.0);
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//--- return result depending on arguments
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return TailLogValue(cdf,tail,log_mode);
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}
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//+------------------------------------------------------------------+
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//| Beta 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 Beta distribution with shape parameters a and b, evaluated |
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//| at x. |
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//| |
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//| Arguments: |
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//| x : The desired quantile |
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//| a : First shape parameter (a>0) |
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//| b : Second shape parameter (b>0) |
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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 Beta cumulative distribution function with |
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//| shape parameters a and b, evaluated at x. |
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//+------------------------------------------------------------------+
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double MathCumulativeDistributionBeta(const double x,const double a,const double b,int &error_code)
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{
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return MathCumulativeDistributionBeta(x,a,b,true,false,error_code);
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}
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//+------------------------------------------------------------------+
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//| The Beta 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 Beta distribution with shape 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 : First shape parameter (a>0) |
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//| b : Second shape parameter (b>0) |
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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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//| Return value: |
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//| true if successful, otherwise false. |
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//+------------------------------------------------------------------+
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bool MathCumulativeDistributionBeta(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 NaN
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if(!MathIsValidNumber(a) || !MathIsValidNumber(b))
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return false;
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//--- a and b must be positive
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if(a<=0.0 || 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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{
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if(x_arg<=0.0)
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result[i]=TailLog0(tail,log_mode);
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if(x_arg>=1.0)
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result[i]=TailLog1(tail,log_mode);
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else
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{
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//--- calculate probability and take into account round-off errors
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double cdf=MathMin(MathBetaIncomplete(x_arg,a,b),1.0);
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//--- return result depending on arguments
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result[i]=TailLogValue(cdf,tail,log_mode);
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}
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}
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else
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return false;
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}
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return(true);
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}
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//+------------------------------------------------------------------+
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//| Beta cumulative distribution function (CDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the cumulative distribution function of |
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//| the Beta distribution with shape 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 : First shape parameter (a>0) |
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//| b : Second shape parameter (b>0) |
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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 successful, otherwise false. |
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//+------------------------------------------------------------------+
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bool MathCumulativeDistributionBeta(const double &x[],const double a,const double b,double &result[])
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{
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return MathCumulativeDistributionBeta(x,a,b,true,false,result);
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}
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//+------------------------------------------------------------------+
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//| Beta 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 Beta distribution with shape 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 : First shape parameter (a>0) |
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//| b : Second shape parameter (b>0) |
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//| tail : Flag to calculate lower tail |
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//| log_mode : Logarithm mode flag,if true 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 of |
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//| the Beta distribution with shape parameters a and b. |
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//+------------------------------------------------------------------+
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double MathQuantileBeta(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 parameters
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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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//--- a and b must be positive
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if(a<=0.0 || 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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error_code=ERR_OK;
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//--- check probabilty
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if(prob==0.0)
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return 0.0;
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if(prob==1.0)
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return 1.0;
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const double eps=10e-16;
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//--- set h and h_min
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double h=1.0;
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double h_min=MathSqrt(eps);
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//--- initial x value
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double x=a/(a+b);
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if(x==0.0)
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x=h_min;
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else
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if(x==1.0)
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x=1.0-h_min;
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int err_code=0;
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const int max_iterations=100;
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int iterations=0;
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//--- Newton iterations
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while(iterations<max_iterations)
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{
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//--- check convergence
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if(((MathAbs(h)>h_min*MathAbs(x)) && (MathAbs(h)>h_min))==false)
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break;
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//--- calculate pdf and cdf
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double pdf=MathProbabilityDensityBeta(x,a,b,false,err_code);
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double cdf=MathCumulativeDistributionBeta(x,a,b,true,false,err_code);
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//--- calculate ratio
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h=(cdf-prob)/pdf;
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double x_new=x-h;
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//--- check x
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if(x_new<0.0)
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x_new=x*0.1;
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else
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if(x_new>1.0)
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x_new=1.0-(1.0-x)*0.1;
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x=x_new;
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iterations++;
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}
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//--- check convergence
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if(iterations<max_iterations)
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return x;
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else
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{
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error_code=ERR_NON_CONVERGENCE;
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return QNaN;
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}
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}
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//+------------------------------------------------------------------+
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//| Beta 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 Beta distribution with shape 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 : First shape parameter (a>0) |
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//| b : Second shape parameter (b>0) |
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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 Beta distribution with shape parameters a and b. |
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//+------------------------------------------------------------------+
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double MathQuantileBeta(const double probability,const double a,const double b,int &error_code)
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{
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return MathQuantileBeta(probability,a,b,true,false,error_code);
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}
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//+------------------------------------------------------------------+
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//| Beta distribution quantile function (inverse CDF) |
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//+------------------------------------------------------------------+
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//| The function calculates the the inverse cumulative distribution |
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//| function of the Beta distribution with shape parameters a and b |
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//| for the probability values from probability[] array. |
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//| |
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//| Arguments: |
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//| probability : Array with probability values |
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//| a : First shape parameter (a>0) |
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//| b : Second shape parameter (b>0) |
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//| tail : Lower tail flag (lower tail of probability used) |
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//| log_mode : Logarithm mode flag (log probability used) |
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//| result : Output array with quantile 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 MathQuantileBeta(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 parameters
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if(!MathIsValidNumber(a) || !MathIsValidNumber(b))
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return false;
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//--- a and b must be positive
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if(a<=0.0 || 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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const double eps=10e-16;
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double h_min=MathSqrt(eps);
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int err_code=0;
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const int max_iterations=1000;
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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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if(MathIsValidNumber(prob))
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{
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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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//--- check probabilty
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if(prob==0.0)
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result[i]=0.0;
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else
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if(prob==1.0)
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result[i]=1.0;
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else
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{
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//--- initial x value
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double x=a/(a+b);
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if(x==0.0)
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x=h_min;
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else
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if(x==1.0)
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x=1.0-h_min;
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double h=1.0;
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int iterations=0;
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//--- Newton iterations
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while(iterations<max_iterations)
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{
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//--- check convergence
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if(((MathAbs(h)>h_min*MathAbs(x)) && (MathAbs(h)>h_min))==false)
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break;
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//--- calculate pdf and cdf
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double pdf=MathProbabilityDensityBeta(x,a,b,false,err_code);
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double cdf=MathCumulativeDistributionBeta(x,a,b,true,false,err_code);
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//--- calculate ratio
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h=(cdf-prob)/pdf;
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double x_new=x-h;
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//--- check x
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if(x_new<0.0)
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x_new=x*0.1;
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else
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if(x_new>1.0)
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x_new=1.0-(1.0-x)*0.1;
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x=x_new;
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iterations++;
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}
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//--- check convergence
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if(iterations<max_iterations)
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result[i]=x;
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else
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return false;
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}
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}
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else
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return false;
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}
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return true;
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}
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//+------------------------------------------------------------------+
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//| Beta distribution quantile function (inverse CDF) |
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//+------------------------------------------------------------------+
| |||
//| The function calculates the the inverse cumulative distribution |
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//| function of the Beta distribution with shape parameters a and b |
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//| for the probability values from probability[] array. |
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//| |
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//| Arguments: |
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//| probability : Array with probability values |
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//| a : First shape parameter (a>0) |
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//| b : Second shape parameter (b>0) |
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//| result : Output array with quantile 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 MathQuantileBeta(const double &probability[],const double a,const double b,double &result[])
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{
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return MathQuantileBeta(probability,a,b,true,false,result);
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}
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//+------------------------------------------------------------------+
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//| Random variate from the Beta distribution |
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//+------------------------------------------------------------------+
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//| The function returns a single random deviate from the Beta |
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//| distribution with parameters a and b. |
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//| |
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//| Arguments: |
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//| a : First shape parameter (a>0) |
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//| b : Second shape parameter (b>0) |
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//| |
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//| Return value: |
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//| The random value with Beta distribution. |
| |||
//| |
| |||
//| Reference: |
| |||
//| Russell Cheng, |
| |||
//| "Generating Beta Variates with Nonintegral Shape Parameters", |
| |||
//| Communications of the ACM, |
| |||
//| Volume 21, Number 4, April 1978, pages 317-322. |
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//| |
| |||
//| Original FORTRAN77 version by Barry Brown, James Lovato. |
| |||
//| C version by John Burkardt. |
| |||
//+------------------------------------------------------------------+
| |||
double MathRandomBeta(const double a,const double b)
| |||
{
| |||
const double log4 = MathLog(4);
| |||
const double log5 = MathLog(5);
| |||
double a1,b1,alpha,beta,gamma,delta,r,s,u1,u2,v,y,z;
| |||
double w=0.0;
| |||
double value=0;
| |||
//---
| |||
if(1.0<a && 1.0<b)
| |||
{
| |||
//--- algorithm BB
| |||
a1 = MathMin(a,b);
| |||
b1 = MathMax(a,b);
| |||
alpha= a1+b1;
| |||
beta = MathSqrt((alpha-2.0)/(2.0*a1*b1-alpha));
| |||
gamma= a1+1.0/beta;
| |||
//---
| |||
for(;;)
| |||
{
| |||
u1 = MathRandomNonZero();
| |||
u2 = MathRandomNonZero();
| |||
| |||
if(u1!=1.0)
| |||
v=beta*MathLog(u1/(1.0-u1));
| |||
else
| |||
v=0.0;
| |||
| |||
w=a1*MathExp(v);
| |||
| |||
z = u1*u1*u2;
| |||
r = gamma*v - log4;
| |||
s = a1+r-w;
| |||
| |||
if(5.0*z<=s+1.0+log5)
| |||
break;
| |||
| |||
double t=MathLog(z);
| |||
if(t<=s)
| |||
break;
| |||
| |||
if(t<=(r+alpha*MathLog(alpha/(b1+w))))
| |||
break;
| |||
}
| |||
}
| |||
else
| |||
{
| |||
//--- algorithm BC
| |||
a1 = MathMax(a,b);
| |||
b1 = MathMin(a,b);
| |||
alpha=a1+b1;
| |||
beta =1.0/b1;
| |||
delta=1.0+a1-b1;
| |||
double k1=delta*(1.0/72.0+b1/24.0)/(a1/b1-7.0/9.0);
| |||
double k2=0.25+(0.5+0.25/delta)*b1;
| |||
| |||
for(;;)
| |||
{
| |||
u1 = MathRandomNonZero();
| |||
u2 = MathRandomNonZero();
| |||
| |||
if(u1<0.5)
| |||
{
| |||
y = u1*u2;
| |||
z = u1*y;
| |||
| |||
if(k1<=0.25*u2+z-y)
| |||
continue;
| |||
}
| |||
else
| |||
{
| |||
z=u1*u1*u2;
| |||
| |||
if(z<=0.25)
| |||
{
| |||
if(u1!=1.0)
| |||
v=beta*MathLog(u1/(1.0-u1));
| |||
else
| |||
v=0.0;
| |||
| |||
w=a1*MathExp(v);
| |||
| |||
if(a==a1)
| |||
value=w/(b1+w);
| |||
else
| |||
value=b1/(b1+w);
| |||
| |||
return value;
| |||
}
| |||
| |||
if(k2<z)
| |||
continue;
| |||
}
| |||
| |||
if(u1!=1.0)
| |||
v=beta*MathLog(u1/(1.0-u1));
| |||
else
| |||
v=0.0;
| |||
w=a1*MathExp(v);
| |||
| |||
if(MathLog(z)<=alpha*(MathLog(alpha/(b1+w))+v)-log4)
| |||
break;
| |||
}
| |||
}
| |||
| |||
if(a==a1)
| |||
value=w/(b1+w);
| |||
else
| |||
value=b1/(b1+w);
| |||
//---
| |||
return value;
| |||
}
| |||
//+------------------------------------------------------------------+
| |||
//| Random variate from the Beta distribution |
| |||
//+------------------------------------------------------------------+
| |||
//| The function returns a single random deviate from the Beta |
| |||
//| distribution with parameters a and b. |
| |||
//| |
| |||
//| Arguments: |
| |||
//| a : First shape parameter (a>0) |
| |||
//| b : Second shape parameter (b>0) |
| |||
//| error_code : Variable for error code |
| |||
//| |
| |||
//| Return value: |
| |||
//| The random value with Beta distribution. |
| |||
//+------------------------------------------------------------------+
| |||
double MathRandomBeta(const double a,const double b,int &error_code)
| |||
{
| |||
//--- check parameters
| |||
if(!MathIsValidNumber(a) || !MathIsValidNumber(b))
| |||
{
| |||
error_code=ERR_ARGUMENTS_NAN;
| |||
return QNaN;
| |||
}
| |||
//--- a and b must be positive
| |||
if(a<=0.0 || b<=0.0)
| |||
{
| |||
error_code=ERR_ARGUMENTS_INVALID;
| |||
return QNaN;
| |||
}
| |||
| |||
error_code=ERR_OK;
| |||
//--- return beta random value
| |||
return MathRandomBeta(a,b);
| |||
}
| |||
//+------------------------------------------------------------------+
| |||
//| Random variate from Beta distribution |
| |||
//+------------------------------------------------------------------+
| |||
//| The function generates random variables from Beta distribution |
| |||
//| with parameters a and b. |
| |||
//| |
| |||
//| Arguments: |
| |||
//| a : First shape parameter (a>0) |
| |||
//| b : Second shape parameter (b>0) |
| |||
//| data_count : Number of values needed |
| |||
//| result : Output array with random values |
| |||
//| |
| |||
//| Return value: |
| |||
//| true if successful, otherwise false. |
| |||
//+------------------------------------------------------------------+
| |||
bool MathRandomBeta(const double a,const double b,const int data_count,double &result[])
| |||
{
| |||
if(data_count<=0)
| |||
return false;
| |||
//--- check parameters
| |||
if(!MathIsValidNumber(a) || !MathIsValidNumber(b))
| |||
return false;
| |||
//--- a and b must be positive
| |||
if(a<=0.0 || b<=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]=MathRandomBeta(a,b);
| |||
| |||
return true;
| |||
}
| |||
//+------------------------------------------------------------------+
| |||
//| Beta distriburion moments |
| |||
//+------------------------------------------------------------------+
| |||
//| The function calculates 4 first moments of the Beta distribution |
| |||
//| with parameters a and b. |
| |||
//| |
| |||
//| Arguments: |
| |||
//| a : First shape parameter (a>0) |
| |||
//| b : Second shape parameter (b>0) |
| |||
//| 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 MathMomentsBeta(const double a,const double b,double &mean,double &variance,double &skewness,double &kurtosis,int &error_code)
| |||
{
| |||
//--- initial values
| |||
mean =QNaN;
| |||
variance=QNaN;
| |||
skewness=QNaN;
| |||
kurtosis=QNaN;
| |||
//--- check NaN
| |||
if(!MathIsValidNumber(a) || !MathIsValidNumber(b))
| |||
{
| |||
error_code=ERR_ARGUMENTS_NAN;
| |||
return false;
| |||
}
| |||
//--- a and b must be positive
| |||
if(a<=0.0 || b<=0.0)
| |||
{
| |||
error_code=ERR_ARGUMENTS_INVALID;
| |||
return false;
| |||
}
| |||
| |||
error_code=ERR_OK;
| |||
//--- calculate moments
| |||
mean =a/(a+b);
| |||
variance=(a*b)/((a+b)*(a+b)*(a+b+1));
| |||
skewness=2*(b-a)*MathSqrt(a+b+1)/(MathSqrt(a*b)*(a+b+2));
| |||
kurtosis=6*(a*a*a+a*a*(1-2*b)+b*b*(1+b)-2*a*b*(2+b))/(a*b*(a+b+2)*(a+b+3));
| |||
//--- successful
| |||
return true;
| |||
}
| |||
//+------------------------------------------------------------------+
|