Error on Integral of function not Resulting From Fit

Roofit and RooStats
1

Hi All,

I’m trying to compute the error of an integral which has values in its integrand with known uncertainties. In particular, this is an integral of a function which is not the result of a fit. Consider, for example, a Gaussian defined in the code below with an amplitude, mean, and width with uncertainties.

I’ve attempted to do this with TF1::IntegralError() which seems like it should be capable of this, but it complains:
“Error in TF1Helper::IntegralError: No existing fitter can be used for computing the integral error”.

Am I doing something wrong here?

Thanks,
Chris

//How to find the error on an integral of a function
//that isn't the result of a fit

void integralError(){

  TF1 *f1 = new TF1("f1","gaus(0)",-10,10);

  f1->SetParameters(100,0,2);
  f1->SetParError(0,10);
  f1->SetParError(1,.6);
  f1->SetParError(2, .3);

  cout <<"Integral: " <<f1->Integral(-10,10) <<endl
       <<"Integral Error: " <<f1->IntegralError(-10,10);

}
2

Read the notes about the “covmat” parameter in the TF1::IntegralError method description.

3

Okay…thanks for making me read that again. When I was reading that late last night somehow I concluded that the parameter errors were automatically used and that entering a covariance matrix was optional. Now I see that if you don’t supply your own covariance matrix it defaults to the last fit that the function was used in. Since I’m not using this function to perform a fit, there was no covariance matrix…and thus the error message.

In the code below I have supplied the parameter errors and a null covariance matrix. I get a number for the integral error rather than a error message, but the number is not what I expect it to be - it returns an integral error of 0. I expect this gaussian to have an error of 36.05 by old school pencil and paper propagation of errors.

[code]//How to find the error on an integral of a function
//that isn’t the result of a fit

void integralError(){

TF1 *f1 = new TF1(“f1”,“gaus(0)”,-10,10);

f1->SetParameters(100,0,2);
f1->SetParError(0,10);
f1->SetParError(1,.6);
f1->SetParError(2, .3);

cout <<"Number of Parameters: " <GetNpar();

Double_t *parErrors = f1->GetParErrors();

TMatrixD covMatrix(3,3);
for (Int_t i=0; i<3; i++){
for (Int_t j=0; j<3; j++){
covMatrix[i][j] = 0.0;
}
}

covMatrix.Print();

cout <<"Integral: " <Integral(-10,10) <<endl
<<"Integral Error: " <IntegralError(-10,10,parErrors,covMatrix.GetMatrixArray())
<<endl;

}[/code]

4

I’m not sure now what TF1::GetParError really returns, but assuming that it’s simply the “sigma” of the distribution (if it’s not the “sigma”, then you need to “scale” it to “sigma” in the equation below), I would try something like: covMatrix[i][j] = (i == j ? (parErrors[i] * parErrors[i]) : 0.0);
Note that for a “gaussian” you should expect a STRONG (anti-)correlation between its fitted “amplitude” and its fitted “width” (which means that some off-diagonal “covMatrix” elements should be BIG, too).

See:
Wolfram MathWorld - Covariance
Wolfram MathWorld - Variance