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
Therefore we have:
ic = p[1]* (1-std::cos(p[0]) )/p[0]c0c = p[1] * (std::cos(p[0]) + p[0]*std::sin(p[0]) -1.)/p[0]/p[0]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
Author: Lorenzo Moneta
This notebook tutorial was automatically generated with ROOTBOOK-izer from the macro found in the ROOT repository on Monday, March 27, 2023 at 09:46 AM.
%%cpp -d
#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;
%%cpp -d
double f(double * x, double * p) {
// function used to fit the data
return p[1]*TMath::Sin( p[0] * x[0] );
}
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();
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
Info in <TCanvas::MakeDefCanvas>: created default TCanvas with name c1
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;
Integral = 19.005 +/- 0.6159
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;
Draw all canvases
%jsroot on
gROOT->GetListOfCanvases()->Draw()