Detailed Description

Estimate the error in the integral of a fitted function taking into account the errors in the parameters resulting from the fit.

The error is estimated also using the correlations values obtained from the fit

run the macro doing:

.x ErrorIntegral.C

After having computed the integral and its error using the integral and the integral error using the generic functions TF1::Integral and TF1::IntegralError, we compute the integrals and its error analytically using the fact that the fitting function is \( f(x) = p[1]* sin(p[0]*x) \).

Therefore we have:

integral in [0,1] : ic = p[1]* (1-std::cos(p[0]) )/p[0]
derivative of integral with respect to p0: c0c = p[1] * (std::cos(p[0]) + p[0]*std::sin(p[0]) -1.)/p[0]/p[0]
derivative of integral with respect to p1: c1c = (1-std::cos(p[0]) )/p[0]

and then we can compute the integral error using error propagation and the covariance matrix for the parameters p obtained from the fit.

integral error : sic = std::sqrt( c0c*c0c * covMatrix(0,0) + c1c*c1c * covMatrix(1,1) + 2.* c0c*c1c * covMatrix(0,1))

Note that, if possible, one should fit directly the function integral, which are the number of events of the different components (e.g. signal and background). In this way one obtains a better and more correct estimate of the integrals uncertainties, since they are obtained directly from the fit without using the approximation of error propagation. This is possible in ROOT. when using the TF1NormSum class, see the tutorial fitNormSum.C

 
 FCN=49.5952 FROM MIGRAD    STATUS=CONVERGED      52 CALLS          53 TOTAL
                     EDM=1.22682e-09    STRATEGY= 1  ERROR MATRIX UNCERTAINTY   2.5 per cent
  EXT PARAMETER                                   STEP         FIRST   
  NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE 
   1  p0           3.13201e+00   3.12699e-02  -3.64656e-05   2.15221e-03
   2  p1           2.97626e+01   1.00773e+00   6.67621e-05  -4.02033e-06
Covariance matrix from the fit 
2x2 matrix is as follows
 
     |      0    |      1    |
-------------------------------
   0 |  0.0009778    0.009128 
   1 |   0.009128       1.016 
 
Integral = 19.005 +/- 0.6159

 
#include "TF1.h"
#include "TH1D.h"
#include "TFitResult.h"
#include "TMath.h"
#include <assert.h>
#include <iostream>
#include <cmath>
 
TF1 * fitFunc;  // fit function pointer
 
const int NPAR = 2; // number of function parameters;
 
//____________________________________________________________________
double f(double * x, double * p) {
   // function used to fit the data
   return p[1]*TMath::Sin( p[0] * x[0] );
}
 
//____________________________________________________________________
void ErrorIntegral() {
   fitFunc = new TF1("f",f,0,1,NPAR);
   TH1D * h1     = new TH1D("h1","h1",50,0,1);
 
   double  par[NPAR] = { 3.14, 1.};
   fitFunc->SetParameters(par);
 
   h1->FillRandom("f",1000); // fill histogram sampling fitFunc
   fitFunc->SetParameter(0,3.);  // vary a little the parameters
   auto fitResult = h1->Fit(fitFunc,"S");             // fit the histogram and get fit result pointer
 
   h1->Draw();
 
   /* calculate the integral*/
   double integral = fitFunc->Integral(0,1);
 
   auto covMatrix = fitResult->GetCovarianceMatrix();
   std::cout << "Covariance matrix from the fit ";
   covMatrix.Print();
 
   // need to pass covariance matrix to fit result.
   // Parameters values are are stored inside the function but we can also retrieve from TFitResult
   double sigma_integral = fitFunc->IntegralError(0,1, fitResult->GetParams() , covMatrix.GetMatrixArray());
 
   std::cout << "Integral = " << integral << " +/- " << sigma_integral
             << std::endl;
 
   // estimated integral  and error analytically
 
   double * p = fitFunc->GetParameters();
   double ic  = p[1]* (1-std::cos(p[0]) )/p[0];
   double c0c = p[1] * (std::cos(p[0]) + p[0]*std::sin(p[0]) -1.)/p[0]/p[0];
   double c1c = (1-std::cos(p[0]) )/p[0];
 
   // estimated error with correlations
   double sic = std::sqrt( c0c*c0c * covMatrix(0,0) + c1c*c1c * covMatrix(1,1)
      + 2.* c0c*c1c * covMatrix(0,1));
 
   if ( std::fabs(sigma_integral-sic) > 1.E-6*sic )
      std::cout << " ERROR: test failed : different analytical  integral : "
                << ic << " +/- " << sic << std::endl;
}

Author: Lorenzo Moneta

Definition in file ErrorIntegral.C.