Confidence Band on TF1 fit

Dear rooters,

I am trying to fit a TH1 (which is the ratio of two histoograms) in a control region in order to extrapolate a correction factor to some yields in a signal region.

I am fitting the ratio of the histograms with an exponenential function. In order to understand the impact of the error on the parameters on the fit function, I am plotting the 68% confidence level on the function using the GetConfidenceInterval function of the TFitter class.

The code is basically readapted from one of the tutorials.

#include "TGraphErrors.h"
#include "TGraph2DErrors.h"
#include "TCanvas.h"
#include "TF2.h"
#include "TH1.h"
#include "TVirtualFitter.h"
#include "TRandom.h"
#include "TFitResultPtr.h"
#include "TMinuit.h"

void ConfidenceIntervals()



    TRandom* random = new TRandom();

    TCanvas *myc = new TCanvas("myc",
      "Confidence intervals on the fitted function",1200, 500);

    TF1 *f = new TF1("fexp", "[0]*exp(x*[1])", 5300, 10000);
    //TF1 *f = new TF1("fexp", "[0]+[1]*x", 5300, 10000); //This looks reasonable
    f->SetParameters(0.77, -0.00004);

   ////2. A histogram
   //Create, fill and fit a histogram
   Int_t nh=5000;
   TH1D *h = new TH1D("h",
      "Fitted gaussian with", 43, 0, 10000);
    h->GetYaxis()->SetRangeUser(0., 2.);
   //h->FillRandom("fgaus", nh);
   for (int i=1; i<44; ++i){
       double value = f->Eval(h->GetBinCenter(i));
       value = value * random->Gaus(1, 0.10);
       if (h->GetBinCenter(i)>5300){
         h->SetBinContent(i, value);
   f->SetParameters(1., 0);
   h->Fit(f, "EV");

   //Create a histogram to hold the confidence intervals
   TH1D *hint = new TH1D("hint",
      "Fitted exponential with", 100, 0, 10000);
   (TVirtualFitter::GetFitter())->GetConfidenceIntervals(hint, 0.68);
   //Now the "hint" histogram has the fitted function values as the
   //bin contents and the confidence intervals as bin errors
   hint->Draw("e3 same");

   TGraph *graph3 =(TGraph*) gMinuit->Contour(80, 0,1);

   TGraph *graph2 =(TGraph*) gMinuit->Contour(80, 0,1);
   graph2->Draw("lf, same");

   TGraph *graph =(TGraph*) gMinuit->Contour(80, 0,1);


As you see from the following plot, the confidence band just explodes at high x. And what is also surprising is that it explodes where the points are located.

At first I thought it was only a problem of a non so parabolic chi2 around the minimum (the plot on the left). But adding more points, and with a more parabolic chi2, the feature remains.

I know that the evaluation of the error band is done using the linear approximation in the error propagation formula. So is what I see only a problem arising from the fact that the function is not linear in the parameters? If I use a linear model, than the error is well behaved, but I would like to fully understand why this is happening with the exponential and why in this strange way.

Thanks a lot for any help

_ROOT Version: 6.14
Platform: Not Provided
Compiler: Not Provided

And if I may add another (statistics related) question: I have tried to dig into the code and I have discovered that the Confidence level is constructed using the Student’s T distribution. i think this is okay if fitting a TGraph with no errors, but in general it should be a Chi2 distribution if fitting a TGraphErrors with the usual Chi2 definition (assuming the data distributes in a gaussian way around the expected value with the observed error). Am I correct?

If yes, how is it possible to evaluate such a confidence band?


In the meanwhile I was also trying to debug this myself, and I have seen that the problem goes away just by rescaling the problem.

Namely I have rescaled the x value to x/1000. Consequently the exponential factor has been rescaled by 1000. (It was 0.00004, now it is 0.04). The two problems are then equivalent.

In this way the confidence band looks much more reasonable.

Is this then pointing towards a numerical issue, rather than a conceptual one?

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