Rf 6 0 9_Xychi 2Fit

Likelihood and minimization: setting up a chi^2 fit to an unbinned dataset with X,Y,err(Y) values (and optionally err(X) values)

Author: Wouter Verkerke
This notebook tutorial was automatically generated with ROOTBOOK-izer from the macro found in the ROOT repository on Sunday, July 05, 2020 at 08:33 AM.

In [1]:
%%cpp -d
#include "RooRealVar.h"
#include "RooDataSet.h"
#include "RooPolyVar.h"
#include "RooConstVar.h"
#include "RooChi2Var.h"
#include "TCanvas.h"
#include "TAxis.h"
#include "RooPlot.h"
#include "TRandom.h"
In [2]:
%%cpp -d
// This is a workaround to make sure the namespace is used inside functions
using namespace RooFit;

Create dataset with x and y values

Make weighted xy dataset with asymmetric errors stored The StoreError() argument is essential as it makes the dataset store the error in addition to the values of the observables. If errors on one or more observables are asymmetric, one can store the asymmetric error using the StoreAsymError() argument

In [3]:
RooRealVar x("x", "x", -11, 11);
RooRealVar y("y", "y", -10, 200);
RooDataSet dxy("dxy", "dxy", RooArgSet(x, y), StoreError(RooArgSet(x, y)));
RooFit v3.60 -- Developed by Wouter Verkerke and David Kirkby 
                Copyright (C) 2000-2013 NIKHEF, University of California & Stanford University
                All rights reserved, please read http://roofit.sourceforge.net/license.txt

Fill an example dataset with x,err(x),y,err(y) values

In [4]:
for (int i = 0; i <= 10; i++) {

   // Set X value and error
   x = -10 + 2 * i;
   x.setError(i < 5 ? 0.5 / 1. : 1.0 / 1.);

   // Set Y value and error
   y = x.getVal() * x.getVal() + 4 * fabs(gRandom->Gaus());
   y.setError(sqrt(y.getVal()));

   dxy.add(RooArgSet(x, y));
}

Perform chi2 fit to x+/-dx and y+/-dy values

Make fit function

In [5]:
RooRealVar a("a", "a", 0.0, -10, 10);
RooRealVar b("b", "b", 0.0, -100, 100);
RooPolyVar f("f", "f", x, RooArgList(b, a, RooConst(1)));

Plot dataset in x-y interpretation

In [6]:
RooPlot *frame = x.frame(Title("Chi^2 fit of function set of (X#pmdX,Y#pmdY) values"));
dxy.plotOnXY(frame, YVar(y));

Fit chi^2 using x and y errors

In [7]:
f.chi2FitTo(dxy, YVar(y));
[#1] INFO:Minization -- RooMinimizer::optimizeConst: activating const optimization
 **********
 **    1 **SET PRINT           1
 **********
 **********
 **    2 **SET NOGRAD
 **********
 PARAMETER DEFINITIONS:
    NO.   NAME         VALUE      STEP SIZE      LIMITS
     1 a            0.00000e+00  2.00000e+00   -1.00000e+01  1.00000e+01
     2 b            0.00000e+00  2.00000e+01   -1.00000e+02  1.00000e+02
 **********
 **    3 **SET ERR           1
 **********
 **********
 **    4 **SET PRINT           1
 **********
 **********
 **    5 **SET STR           1
 **********
 NOW USING STRATEGY  1: TRY TO BALANCE SPEED AGAINST RELIABILITY
 **********
 **    6 **MIGRAD        1000           1
 **********
 FIRST CALL TO USER FUNCTION AT NEW START POINT, WITH IFLAG=4.
 START MIGRAD MINIMIZATION.  STRATEGY  1.  CONVERGENCE WHEN EDM .LT. 1.00e-03
 FCN=2.04222 FROM MIGRAD    STATUS=INITIATE        8 CALLS           9 TOTAL
                     EDM= unknown      STRATEGY= 1      NO ERROR MATRIX       
  EXT PARAMETER               CURRENT GUESS       STEP         FIRST   
  NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE 
   1  a            0.00000e+00   2.00000e+00   2.01358e-01   1.35235e+01
   2  b            0.00000e+00   2.00000e+01   2.01358e-01  -3.08578e+02
 MIGRAD MINIMIZATION HAS CONVERGED.
 MIGRAD WILL VERIFY CONVERGENCE AND ERROR MATRIX.
 COVARIANCE MATRIX CALCULATED SUCCESSFULLY
 FCN=1.43197 FROM MIGRAD    STATUS=CONVERGED      32 CALLS          33 TOTAL
                     EDM=3.23329e-05    STRATEGY= 1      ERROR MATRIX ACCURATE 
  EXT PARAMETER                                   STEP         FIRST   
  NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE 
   1  a           -1.60654e-01   5.24023e-01   3.93124e-05   1.09375e-01
   2  b            3.34226e-01   4.91235e-01   3.85614e-06  -1.06196e+00
 EXTERNAL ERROR MATRIX.    NDIM=  25    NPAR=  2    ERR DEF=1
  2.749e-01 -1.964e-02 
 -1.964e-02  2.413e-01 
 PARAMETER  CORRELATION COEFFICIENTS  
       NO.  GLOBAL      1      2
        1  0.07625   1.000 -0.076
        2  0.07625  -0.076  1.000
 **********
 **    7 **SET ERR           1
 **********
 **********
 **    8 **SET PRINT           1
 **********
 **********
 **    9 **HESSE        1000
 **********
 COVARIANCE MATRIX CALCULATED SUCCESSFULLY
 FCN=1.43197 FROM HESSE     STATUS=OK             10 CALLS          43 TOTAL
                     EDM=3.23291e-05    STRATEGY= 1      ERROR MATRIX ACCURATE 
  EXT PARAMETER                                INTERNAL      INTERNAL  
  NO.   NAME      VALUE            ERROR       STEP SIZE       VALUE   
   1  a           -1.60654e-01   5.24019e-01   7.86248e-06  -1.60660e-02
   2  b            3.34226e-01   4.91231e-01   7.71228e-07   3.34226e-03
 EXTERNAL ERROR MATRIX.    NDIM=  25    NPAR=  2    ERR DEF=1
  2.748e-01 -1.961e-02 
 -1.961e-02  2.413e-01 
 PARAMETER  CORRELATION COEFFICIENTS  
       NO.  GLOBAL      1      2
        1  0.07614   1.000 -0.076
        2  0.07614  -0.076  1.000

Overlay fitted function

In [8]:
f.plotOn(frame);

Alternative: fit chi^2 integrating f(x) over ranges defined by x errors, rather than taking point at center of bin

In [9]:
f.chi2FitTo(dxy, YVar(y), Integrate(kTRUE));
[#1] INFO:Minization -- RooMinimizer::optimizeConst: activating const optimization
 **********
 **   10 **SET PRINT           1
 **********
 **********
 **   11 **SET NOGRAD
 **********
 PARAMETER DEFINITIONS:
    NO.   NAME         VALUE      STEP SIZE      LIMITS
     1 a           -1.60654e-01  5.24019e-01   -1.00000e+01  1.00000e+01
     2 b            3.34226e-01  4.91231e-01   -1.00000e+02  1.00000e+02
 **********
 **   12 **SET ERR           1
 **********
 **********
 **   13 **SET PRINT           1
 **********
 **********
 **   14 **SET STR           1
 **********
 NOW USING STRATEGY  1: TRY TO BALANCE SPEED AGAINST RELIABILITY
 **********
 **   15 **MIGRAD        1000           1
 **********
 FIRST CALL TO USER FUNCTION AT NEW START POINT, WITH IFLAG=4.
 START MIGRAD MINIMIZATION.  STRATEGY  1.  CONVERGENCE WHEN EDM .LT. 1.00e-03
 FCN=176984 FROM MIGRAD    STATUS=INITIATE        8 CALLS           9 TOTAL
                     EDM= unknown      STRATEGY= 1      NO ERROR MATRIX       
  EXT PARAMETER               CURRENT GUESS       STEP         FIRST   
  NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE 
   1  a           -1.60654e-01   5.24019e-01   5.24327e-02  -8.81282e+03
   2  b            3.34226e-01   4.91231e-01   4.91235e-03   2.93731e+07
 MIGRAD MINIMIZATION HAS CONVERGED.
 MIGRAD WILL VERIFY CONVERGENCE AND ERROR MATRIX.
 COVARIANCE MATRIX CALCULATED SUCCESSFULLY
 FCN=76478.7 FROM MIGRAD    STATUS=CONVERGED      30 CALLS          31 TOTAL
                     EDM=2.56889e-08    STRATEGY= 1      ERROR MATRIX ACCURATE 
  EXT PARAMETER                                   STEP         FIRST   
  NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE 
   1  a           -5.61220e-02   1.57588e-02   2.12800e-04  -1.43819e-01
   2  b           -3.49788e-01   2.15812e-03   2.91421e-06  -1.32588e-01
 EXTERNAL ERROR MATRIX.    NDIM=  25    NPAR=  2    ERR DEF=1
  2.483e-04  3.343e-08 
  3.343e-08  4.657e-06 
 PARAMETER  CORRELATION COEFFICIENTS  
       NO.  GLOBAL      1      2
        1  0.00098   1.000  0.001
        2  0.00098   0.001  1.000
 **********
 **   16 **SET ERR           1
 **********
 **********
 **   17 **SET PRINT           1
 **********
 **********
 **   18 **HESSE        1000
 **********
 COVARIANCE MATRIX CALCULATED SUCCESSFULLY
 FCN=76478.7 FROM HESSE     STATUS=OK             10 CALLS          41 TOTAL
                     EDM=2.56903e-08    STRATEGY= 1      ERROR MATRIX ACCURATE 
  EXT PARAMETER                                INTERNAL      INTERNAL  
  NO.   NAME      VALUE            ERROR       STEP SIZE       VALUE   
   1  a           -5.61220e-02   1.57588e-02   4.25601e-05  -5.61223e-03
   2  b           -3.49788e-01   2.15812e-03   5.82843e-07  -3.49789e-03
 EXTERNAL ERROR MATRIX.    NDIM=  25    NPAR=  2    ERR DEF=1
  2.483e-04  3.343e-08 
  3.343e-08  4.657e-06 
 PARAMETER  CORRELATION COEFFICIENTS  
       NO.  GLOBAL      1      2
        1  0.00098   1.000  0.001
        2  0.00098   0.001  1.000

Overlay alternate fit result

In [10]:
f.plotOn(frame, LineStyle(kDashed), LineColor(kRed));

Draw the plot on a canvas

In [11]:
new TCanvas("rf609_xychi2fit", "rf609_xychi2fit", 600, 600);
gPad->SetLeftMargin(0.15);
frame->GetYaxis()->SetTitleOffset(1.4);
frame->Draw();

Draw all canvases

In [12]:
%jsroot on
gROOT->GetListOfCanvases()->Draw()