Hi Alexander
This is because in a chi-square fit the errors are estimated
by the square root of the bin contents. Thus, the bins with
low contents get smaller errors assigned than the bins with
high contents, even if the differences in contents come from
statistical fluctuations. Thus, the low points are being taken
more seriously than the high points. Thus, the fitted line
is always somewhat below the points. It only becomes visible
when the statistics is low.
In the maximum likelihood method the errors are taken from
the fit function and thus are independent on statistical
fluctuations of the points.
Best regards
Dariusz Miskowiec
Alexander Dietz wrote:
>
> here,
>
> I have a problem when fitting with the chi2-square method!
>
> With the following program I create several random-histograms consisting
> of two parts: a flat poisson-background (with mean 2) and a Gaussian line
> centered at channel 50, with sigma 2.0 and with 100 counts in it.
> Then I fit this histogram with the function 'function' (assuming a flat
> backgriund and a gaussian line), using the Chi-square method or the
> Maximum likelihood method (adding a 'L' to the fitting options).
> Each results of each fit are then given into a second histogram 'sum'.
>
> When I do the fit with Maximum likelihood, the results in histogram 'sum'
> are really good, fluctuating around the true value of 100. But when I use
> the chi-square method, the results are centered around 95 which is quite
> wrong! Only a few percent of the results give values larger than 100!
>
> So what is wrong here? Maybe the chi-square method itself gives confusing
> results? Or is there a bug in the fitting procedure?
>
> Thanks for help
>
> A. Dietz
>
> P.S. When doing the same but with 10000 counts in the line, BOTH methods
> give very good results.
>
> Program:
>
> start()
> {
> gROOT->Reset();
>
> c1 = new TCanvas("c1","The fitting problem",200,10,600,400);
> c1->SetGrid();
>
> // Create one histograms.
> main = new TH1F("main","Test spectrum",100,0,100);
> sum = new TH1F("sum" ,"Fittig results",100,80,120);
> sum->SetStats(kFALSE);
> sum->SetFillColor(47);
> main->SetStats(kFALSE);
> main->SetFillColor(46);
>
> for (Int_t k=0;k<=100;k++) {
>
> // setting random seeds
> gRandom->SetSeed(k);
>
> Float_t value;
> // filling spectrum with Poisson background
> for (Int_t i=0;i<=100;i++) {
> value=gRandom->Poisson(1);
> main->SetBinContent(i,value);
> }
>
> // adding gaussian line at channel 50, sigma 2.0=channel with 100
> counts in it
> Int_t counts=100;
> for ( Int_t i=0; i<counts; i++) {
> value = gRandom->Gaus(50,2.0);
> main->Fill(value);
> }
>
> main->Draw("");
> c1->Update();
> c1->Modified();
>
> // fiting the spectrum
> TF1* func=new TF1("func",function,0,100,4);
> func->SetParameters(1,100,50,2);
> main->Fit("func","QWEMR"); // or with likelihood: "LQWEMR"
>
> // printing out size of line
> Float_t size=func->GetParameter(1);
> cout << "Size of line: " << size << endl;
>
> // store result to histogram 'sum'
> sum->Fill(size);
> }
>
> // showing the results of the fit
> sum->Draw();
> }
>
> // fitting function
> Double_t function(Double_t* x, Double_t* p)
> {
> Double_t dummy=(x[0]-p[2])/p[3];
> Double_t result=p[0] + p[1]*exp(-0.5*dummy*dummy)/ (2.506 * p[3]);
> return result;
> }
--
+-----------------------------------------------------------------+
+ Dariusz Miskowiec E-mail: D.Miskowiec@gsi.de +
+ GSI, Planckstr. 1 Phone: 0049-6159-712-133 +
+ D-64291 Darmstadt Fax: 0049-6159-712-989 +
+-----------------------------------------------------------------+
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