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MQL5 ReferenceStandard LibraryMathematicsStatisticsNoncentral chi-squared distribution 

Noncentral chi-squared distribution

This section contains functions for working with noncentral chi-squared distribution. They allow to calculate density, probability, quantiles and to generate pseudo-random numbers distributed according to the corresponding law. The noncentral chi-squared distribution is defined by the following formula:

pdf_noncentral_chi_square_distribution

where:

  • x – value of the random variable
  • ν – number of degrees of freedom
  • σ – noncentrality parameter

DemoNoncentralChiSquare

In addition to the calculation of the individual random variables, the library also implements the ability to work with arrays of random variables.

Function

Description

MathProbabilityDensityNoncentralChiSquare

Calculates the probability density function of the noncentral chi-squared distribution

MathCumulativeDistributionNoncentralChiSquare

Calculates the value of the noncentral chi-squared probability distribution function

MathQuantileNoncentralChiSquare

Calculates the value of the inverse noncentral chi-squared distribution function for the specified probability

MathRandomNoncentralChiSquare

Generates a pseudorandom variable/array of pseudorandom variables distributed according to the noncentral chi-squared distribution law

MathMomentsNoncentralChiSquare

Calculates the theoretical numerical values of the first 4 moments of the noncentral chi-squared distribution

Example:

#include <Graphics\Graphic.mqh>
#include <Math\Stat\NoncentralChiSquare.mqh>
#include <Math\Stat\Math.mqh>
#property script_show_inputs
//--- input parameters
input double nu_par=8;    // the number of degrees of freedom
input double si_par=1;    // noncentrality parameter
//+------------------------------------------------------------------+
//| Script program start function                                    |
//+------------------------------------------------------------------+
void OnStart()
  {
//--- hide the price chart
   ChartSetInteger(0,CHART_SHOW,false);
//--- initialize the random number generator  
   MathSrand(GetTickCount());
//--- generate a sample of the random variable
   long chart=0;
   string name="GraphicNormal";
   int n=1000000;       // the number of values in the sample
   int ncells=51;       // the number of intervals in the histogram
   double x[];          // centers of the histogram intervals
   double y[];          // the number of values from the sample falling within the interval
   double data[];       // sample of random values
   double max,min;      // the maximum and minimum values in the sample
//--- obtain a sample from the noncentral chi-squared distribution
   MathRandomNoncentralChiSquare(nu_par,si_par,n,data);
//--- calculate the data to plot the histogram
   CalculateHistogramArray(data,x,y,max,min,ncells);
//--- obtain the sequence boundaries and the step for plotting the theoretical curve
   double step;
   GetMaxMinStepValues(max,min,step);
   step=MathMin(step,(max-min)/ncells);
//--- obtain the theoretically calculated data at the interval of [min,max]
   double x2[];
   double y2[];
   MathSequence(min,max,step,x2);
   MathProbabilityDensityNoncentralChiSquare(x2,nu_par,si_par,false,y2);
//--- set the scale
   double theor_max=y2[ArrayMaximum(y2)];
   double sample_max=y[ArrayMaximum(y)];
   double k=sample_max/theor_max;
   for(int i=0; i<ncells; i++)
      y[i]/=k;
//--- output charts
   CGraphic graphic;
   if(ObjectFind(chart,name)<0)
      graphic.Create(chart,name,0,0,0,780,380);
   else
      graphic.Attach(chart,name);
   graphic.BackgroundMain(StringFormat("Noncentral ChiSquare distribution nu=%G sigma=%G",nu_par,si_par));
   graphic.BackgroundMainSize(16);
//--- disable automatic scaling of the X axis
   graphic.XAxis().AutoScale(false);
   graphic.XAxis().Max(NormalizeDouble(max,0));
   graphic.XAxis().Min(min);
//--- plot all curves
   graphic.CurveAdd(x,y,CURVE_HISTOGRAM,"Sample").HistogramWidth(6);
//--- and now plot the theoretical curve of the distribution density
   graphic.CurveAdd(x2,y2,CURVE_LINES,"Theory");
   graphic.CurvePlotAll();
//--- plot all curves
   graphic.Update();
  }
//+------------------------------------------------------------------+
//|  Calculate frequencies for data set                              |
//+------------------------------------------------------------------+
bool CalculateHistogramArray(const double &data[],double &intervals[],double &frequency[],
                             double &maxv,double &minv,const int cells=10)
  {
   if(cells<=1) return (false);
   int size=ArraySize(data);
   if(size<cells*10) return (false);
   minv=data[ArrayMinimum(data)];
   maxv=data[ArrayMaximum(data)];
   double range=maxv-minv;
   double width=range/cells;
   if(width==0) return false;
   ArrayResize(intervals,cells);
   ArrayResize(frequency,cells);
//--- define the interval centers
   for(int i=0; i<cells; i++)
     {
      intervals[i]=minv+(i+0.5)*width;
      frequency[i]=0;
     }
//--- fill the frequencies of falling within the interval
   for(int i=0; i<size; i++)
     {
      int ind=int((data[i]-minv)/width);
      if(ind>=cells) ind=cells-1;
      frequency[ind]++;
     }
   return (true);
  }
//+------------------------------------------------------------------+
//|  Calculates values for sequence generation                       |
//+------------------------------------------------------------------+
void GetMaxMinStepValues(double &maxv,double &minv,double &stepv)
  {
//--- calculate the absolute range of the sequence to obtain the precision of normalization
   double range=MathAbs(maxv-minv);
   int degree=(int)MathRound(MathLog10(range));
//--- normalize the maximum and minimum values to the specified precision
   maxv=NormalizeDouble(maxv,degree);
   minv=NormalizeDouble(minv,degree);
//--- sequence generation step is also set based on the specified precision
   stepv=NormalizeDouble(MathPow(10,-degree),degree);
   if((maxv-minv)/stepv<10)
      stepv/=10.;
  }


Updated: 2017.02.06