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Parameter uncertainties from 2D chi-squared histogram

Dear rooters,

I have a question that I’ve found a few times before but none of them help me solve this issue.
I am computing the chi2 value between two histograms changing two parameters and I would like to found the values of the parameters that minimize the chi2, and also the 1-sigma uncertainties of these parameters.

I have a two-dimensional histogram, with the first parameter in the x-axis and the second one in the y-axis, and this histogram is filled with chi2 values. In the figure there’s an example of how the histogram looks like, and in the .root file I leave it attached.
chiplot.root (150.4 KB)

With respect to the values that minimize the chi2 I have no problem: I can just simply look for the minimum bin in the x and y direction. But, how can I get the statistical uncertainties of the parameters?

Thank you very much for your time

Best regards,

ROOT Version: 6.22/01
Platform: macOS Catalina and lxplus

@linev could you please take a look or maybe suggest who can help here? Thank you!

I guess, it is question for @moneta

If the histogram represents chi2 values and looking at the contours it seems that the two parameter are weakly correlated, you can just find the errors are the values where the chi2 function = minimum_value + 1 in the separate x and y projections.
If you want I can provide you an example. However, I see the binning of your chi2 is rather large. From the attache plot I see that changing by one bin value the Delta(chi2) is already approximately 1. You might need to use a smaller binning around the minimum if you can, to get more accurate errors.

Hi Lorenzo,
Thank you very much for your answer.

Well in this case the parameters seems to be weekly correlated. In reality, I have more the 1000 of these plots, and they all differ from each other: in ones the correlation is positive, others negative, and in some of them the correlation is almost zero. I am aware of the chi2 = minimum_value + 1 method for only one parameter (I’ve already used that method when using only one parameter). In analogy to that method but in two dimensions, I can find the ellipse for which chi2_function = minimum_value + 2.3 (for 68.3% confidence level) but that is more complicated (I am currently working with it, performing some tests…)

With respect to the binning I am afraid I can only change the binning in the y-direction (stretch in my plots), since shifts represents shifting of one histogram with respect to another, by one bin at the time. I tried working with simulated annealing but, as you have said here this method doesn’t return uncertainties. For this case, as the shift represent steps with fixed size, I can’t use Minuit minimizer to minimize the chi2 function neither.

I was wondering if there is any other method to find the uncertainties for cases like these, as they seem to be very common practices (or am I wrong?)

Thank you very much.

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