TGraphQQ.cxx

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// @(#)root/graf:$Id$
// Author: Anna Kreshuk 18/11/2005

/*************************************************************************
 * Copyright (C) 1995-2005, Rene Brun and Fons Rademakers.               *
 * All rights reserved.                                                  *
 *                                                                       *
 * For the licensing terms see $ROOTSYS/LICENSE.                         *
 * For the list of contributors see $ROOTSYS/README/CREDITS.             *
 *************************************************************************/

#include "TGraphQQ.h"
#include "TAxis.h"
#include "TF1.h"
#include "TMath.h"
#include "TVirtualPad.h"
#include "TLine.h"

ClassImp(TGraphQQ)

/** \class TGraphQQ
\ingroup BasicGraphics

This class allows to draw quantile-quantile plots

Plots can be drawn for 2 datasets or for a dataset and a theoretical
distribution function

## 2 datasets:
  Quantile-quantile plots are used to determine whether 2 samples come from
  the same distribution.
  A qq-plot draws the quantiles of one dataset against the quantile of the
  the other. The quantiles of the dataset with fewer entries are on Y axis,
  with more entries - on X axis.
  A straight line, going through 0.25 and 0.75 quantiles is also plotted
  for reference. It represents a robust linear fit, not sensitive to the
  extremes of the datasets.
  If the datasets come from the same distribution, points of the plot should
  fall approximately on the 45 degrees line. If they have the same
  distribution function, but location or scale different parameters,
  they should still fall on the straight line, but not the 45 degrees one.
  The greater their departure from the straight line, the more evidence there
  is, that the datasets come from different distributions.
  The advantage of qq-plot is that it not only shows that the underlying
  distributions are different, but, unlike the analytical methods, it also
  gives information on the nature of this difference: heavier tails,
  different location/scale, different shape, etc.

  Some examples of qqplots of 2 datasets:

\image html graf_graphqq1.png

## 1 dataset:
  Quantile-quantile plots are used to determine if the dataset comes from the
  specified theoretical distribution, such as normal.
  A qq-plot draws quantiles of the dataset against quantiles of the specified
  theoretical distribution.
  (NOTE, that density, not CDF should be specified)
  A straight line, going through 0.25 and 0.75 quantiles can also be plotted
  for reference. It represents a robust linear fit, not sensitive to the
  extremes of the dataset.
  As in the 2 datasets case, departures from straight line indicate departures
  from the specified distribution.

  "The correlation coefficient associated with the linear fit to the data
   in the probability plot (qq plot in our case) is a measure of the
   goodness of the fit.
   Estimates of the location and scale parameters  of the distribution
   are given by the intercept and slope. Probability plots can be generated
   for several competing distributions to see which provides the best fit,
   and the probability plot generating the highest correlation coefficient
   is the best choice since it generates the straightest probability plot."

  From "Engineering statistic handbook",

  http://www.itl.nist.gov/div898/handbook/eda/section3/probplot.htm

  Example of a qq-plot of a dataset from N(3, 2) distribution and
          TMath::Gaus(0, 1) theoretical function. Fitting parameters
          are estimates of the distribution mean and sigma.

\image html graf_graphqq2.png

References:

http://www.itl.nist.gov/div898/handbook/eda/section3/qqplot.htm

http://www.itl.nist.gov/div898/handbook/eda/section3/probplot.htm
*/

////////////////////////////////////////////////////////////////////////////////
/// default constructor

TGraphQQ::TGraphQQ()
{
   fF   = 0;
   fY0  = 0;
   fNy0 = 0;
   fXq1 = 0.;
   fXq2 = 0.;
   fYq1 = 0.;
   fYq2 = 0.;

}

////////////////////////////////////////////////////////////////////////////////
/// Creates a quantile-quantile plot of dataset x.
/// Theoretical distribution function can be defined later by SetFunction method

TGraphQQ::TGraphQQ(Int_t n, Double_t *x)
   : TGraph(n)
{
   fNy0 = 0;
   fXq1 = 0.;
   fXq2 = 0.;
   fYq1 = 0.;
   fYq2 = 0.;

   Int_t *index = new Int_t[n];
   TMath::Sort(n, x, index, kFALSE);
   for (Int_t i=0; i<fNpoints; i++)
      fY[i] = x[index[i]];
   fF=0;
   fY0=0;
   delete [] index;
}

////////////////////////////////////////////////////////////////////////////////
/// Creates a quantile-quantile plot of dataset x against function f

TGraphQQ::TGraphQQ(Int_t n, Double_t *x, TF1 *f)
   : TGraph(n)
{
   fNy0 = 0;

   Int_t *index = new Int_t[n];
   TMath::Sort(n, x, index, kFALSE);
   for (Int_t i=0; i<fNpoints; i++)
      fY[i] = x[index[i]];
   delete [] index;
   fF = f;
   fY0=0;
   MakeFunctionQuantiles();
}

////////////////////////////////////////////////////////////////////////////////
/// Creates a quantile-quantile plot of dataset x against dataset y
/// Parameters nx and ny are respective array sizes

TGraphQQ::TGraphQQ(Int_t nx, Double_t *x, Int_t ny, Double_t *y)
{
   fNy0 = 0;
   fXq1 = 0.;
   fXq2 = 0.;
   fYq1 = 0.;
   fYq2 = 0.;

   nx<=ny ? fNpoints=nx : fNpoints=ny;

   if (!CtorAllocate()) return;
   fF=0;
   Int_t *index = new Int_t[TMath::Max(nx, ny)];
   TMath::Sort(nx, x, index, kFALSE);
   if (nx <=ny){
      for (Int_t i=0; i<fNpoints; i++)
         fY[i] = x[index[i]];
      TMath::Sort(ny, y, index, kFALSE);
      if (nx==ny){
         for (Int_t i=0; i<fNpoints; i++)
            fX[i] = y[index[i]];
         fY0 = 0;
         Quartiles();
      } else {
         fNy0 = ny;
         fY0 = new Double_t[ny];
         for (Int_t i=0; i<ny; i++)
            fY0[i] = y[i];
         MakeQuantiles();
      }
   } else {
      fNy0 = nx;
      fY0 = new Double_t[nx];
      for (Int_t i=0; i<nx; i++)
         fY0[i] = x[index[i]];
      TMath::Sort(ny, y, index, kFALSE);
      for (Int_t i=0; i<ny; i++)
         fY[i] = y[index[i]];
      MakeQuantiles();
   }


   delete [] index;
}

////////////////////////////////////////////////////////////////////////////////
/// Destroys a TGraphQQ

TGraphQQ::~TGraphQQ()
{
   if (fY0)
      delete [] fY0;
   if (fF)
      fF = 0;
}

////////////////////////////////////////////////////////////////////////////////
/// Computes quantiles of theoretical distribution function

void TGraphQQ::MakeFunctionQuantiles()
{
   if (!fF) return;
   TString s = fF->GetTitle();
   Double_t pk;
   if (s.Contains("TMath::Gaus") || s.Contains("gaus")){
      //use plotting positions optimal for normal distribution
      for (Int_t k=1; k<=fNpoints; k++){
         pk = (k-0.375)/(fNpoints+0.25);
         fX[k-1]=TMath::NormQuantile(pk);
      }
   } else {
      Double_t *prob = new Double_t[fNpoints];
      if (fNpoints > 10){
         for (Int_t k=1; k<=fNpoints; k++)
            prob[k-1] = (k-0.5)/fNpoints;
      } else {
         for (Int_t k=1; k<=fNpoints; k++)
            prob[k-1] = (k-0.375)/(fNpoints+0.25);
      }
      //fF->GetQuantiles(fNpoints, prob, fX);
      fF->GetQuantiles(fNpoints, fX, prob);
      delete [] prob;
   }

   Quartiles();
}

////////////////////////////////////////////////////////////////////////////////
/// When sample sizes are not equal, computes quantiles of the bigger sample
/// by linear interpolation

void TGraphQQ::MakeQuantiles()
{


   if (!fY0) return;

   Double_t pi, pfrac;
   Int_t pint;
   for (Int_t i=0; i<fNpoints-1; i++){
      pi = (fNy0-1)*Double_t(i)/Double_t(fNpoints-1);
      pint = TMath::FloorNint(pi);
      pfrac = pi - pint;
      fX[i] = (1-pfrac)*fY0[pint]+pfrac*fY0[pint+1];
   }
   fX[fNpoints-1]=fY0[fNy0-1];

   Quartiles();
}

////////////////////////////////////////////////////////////////////////////////
/// compute quartiles
/// a quartile is a 25 per cent or 75 per cent quantile

void TGraphQQ::Quartiles()
{
   Double_t prob[]={0.25, 0.75};
   Double_t x[2];
   Double_t y[2];
   TMath::Quantiles(fNpoints, 2, fY, y, prob, kTRUE);
   if (fY0)
      TMath::Quantiles(fNy0, 2, fY0, x, prob, kTRUE);
   else if (fF) {
      TString s = fF->GetTitle();
      if (s.Contains("TMath::Gaus") || s.Contains("gaus")){
         x[0] = TMath::NormQuantile(0.25);
         x[1] = TMath::NormQuantile(0.75);
      } else
         fF->GetQuantiles(2, x, prob);
   }
   else
      TMath::Quantiles(fNpoints, 2, fX, x, prob, kTRUE);

   fXq1=x[0]; fXq2=x[1]; fYq1=y[0]; fYq2=y[1];
}

////////////////////////////////////////////////////////////////////////////////
///Sets the theoretical distribution function (density!)
///and computes its quantiles

void TGraphQQ::SetFunction(TF1 *f)
{
   fF = f;
   MakeFunctionQuantiles();
}