In this twiki we are going to show the results for the 3D fit on the RMS distributions as part of the chi2-like variable modelization. The main reason this twiki was created is because the old one (that is referencing it) contains too many plots and the speed of loading material is too low. Therefore we created the new on to increase the speed of sharing plots and also choosing the correct information.

Signal weight RMS of background tagged events as a function of signal events

In this section we are showing the signal weight RMS for the events in the toys that are categorized as signal-like events. Since, we are tagging the events based on the SSSV and Combinatorial contributions there are two set of plots for each ct bin. In the table below the RMS vs NSignal we can see the plots of the weight distribution of the tagged events for a given number of signal events.

CT Bins Bin 0 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7

signal weight distributions for tagged signal, tagged combinatorial and tagged sssv events. The stacks are created as a function of the number of signal events.

CT bins Bin 0 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7
Stacked hists
sPlotOnToys BDT0.365 cheb1ExpOneGaus SignalWeightDistForSigSSSVCombTaggedEvents stacks ctBin0.png
Stacked histograms
sPlotOnToys BDT0.365 cheb1ExpOneGaus SignalWeightDistForSigSSSVCombTaggedEvents stacks ctBin1.png
Stacked histograms
sPlotOnToys BDT0.365 cheb1ExpOneGaus SignalWeightDistForSigSSSVCombTaggedEvents stacks ctBin2.png
Stacked histograms
sPlotOnToys BDT0.365 cheb1ExpOneGaus SignalWeightDistForSigSSSVCombTaggedEvents stacks ctBin3.png
Stacked histograms
sPlotOnToys BDT0.365 cheb1ExpOneGaus SignalWeightDistForSigSSSVCombTaggedEvents stacks ctBin4.png
Stacked histograms
sPlotOnToys BDT0.365 cheb1ExpOneGaus SignalWeightDistForSigSSSVCombTaggedEvents stacks ctBin5.png
Stacked histograms
sPlotOnToys BDT0.365 cheb1ExpOneGaus SignalWeightDistForSigSSSVCombTaggedEvents stacks ctBin6.png
Stacked histograms
sPlotOnToys BDT0.365 cheb1ExpOneGaus SignalWeightDistForSigSSSVCombTaggedEvents stacks ctBin7.png
Stacked histograms
No stacked hists
sPlotOnToys BDT0.365 cheb1ExpOneGaus SignalWeightDistForSigSSSVCombTaggedEvents nostacks ctBin0.png
No stacked histograms
sPlotOnToys BDT0.365 cheb1ExpOneGaus SignalWeightDistForSigSSSVCombTaggedEvents nostacks ctBin1.png
No stacked histograms
sPlotOnToys BDT0.365 cheb1ExpOneGaus SignalWeightDistForSigSSSVCombTaggedEvents nostacks ctBin2.png
No stacked histograms
sPlotOnToys BDT0.365 cheb1ExpOneGaus SignalWeightDistForSigSSSVCombTaggedEvents nostacks ctBin3.png
No stacked histograms
sPlotOnToys BDT0.365 cheb1ExpOneGaus SignalWeightDistForSigSSSVCombTaggedEvents nostacks ctBin4.png
No stacked histograms
sPlotOnToys BDT0.365 cheb1ExpOneGaus SignalWeightDistForSigSSSVCombTaggedEvents nostacks ctBin5.png
No stacked histograms
sPlotOnToys BDT0.365 cheb1ExpOneGaus SignalWeightDistForSigSSSVCombTaggedEvents nostacks ctBin6.png
No stacked histograms
sPlotOnToys BDT0.365 cheb1ExpOneGaus SignalWeightDistForSigSSSVCombTaggedEvents nostacks ctBin7.png
No stacked histograms

Signal weight RMS of signal tagged events as a function of signal events

We are showing in this section the corresponding plots of the signal weight RMS of the signal tagged events, as shown for the background tagged events.

CT Bins Bin 0 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7

Deploying analytical toys

Running the 3D RMS fit in all the different ct bins indicated that for most of the bins the fit of the suggested model is failing. Further investigation yielded that the reason behind the failed fits is the limitation in toys due to the method used till now for generation. Till this moment bootstrap toys have been used in order to have desired toys for testing the sPlot procedure. However, the background sample used for the analysis bdt cut is very limited in statistics. For this reason the usage of analytical model generated toys is being used. The procedure and validation of the analytical toys before deployed for the RMS fit can be found here AnalyticalModel. The first test that was run on the newly created toys are to check how many points to expect in each ct bin to be fitted with the 3D rms model.

Bin 0 Bin 1 Bin 2
Bin 3 Bin 4 Bin 5
Bin 6 Bin 7  

From those plots in the table it can be seen that only the ct bin 2< have the necessary amount of events in order to allow the fit to be performed, and therefore the fit was successfully converged in those bins only. The pValue from the fit is not that good. However in the following table the fit parameters as a function of the number of bins is being shown.

sPlotOnToys BDT0.365 cheb1ExpOneGaus RMS3DFitParameterEvolutionForAllCtBin v2.png
Fit parameter evolution as a function of ct bins from the 3D RMS fit model

the corresponding projections for the bin that are managed to be fit with the 3D model are the following:

sPlotOnToys BDT0.365 cheb1ExpOneGaus RMS3DFitProjection ctBin3 v1.png
RMS fit with 3D model ct bin 3
sPlotOnToys BDT0.365 cheb1ExpOneGaus RMS3DFitProjection ctBin4 v1.png
RMS fit with 3D model ct bin 4
sPlotOnToys BDT0.365 cheb1ExpOneGaus RMS3DFitProjection ctBin5 v1.png
RMS fit with 3D mode ct bin 5
sPlotOnToys BDT0.365 cheb1ExpOneGaus RMS3DFitProjection ctBin6 v1.png
RMS fit with 3D model ct bin 6
sPlotOnToys BDT0.365 cheb1ExpOneGaus RMS3DFitProjection ctBin7 v1.png
RMS fit with 3D model ct bin 7

Analytical chi2 fit with the use of sPlot templates

In this section what is shown is arising from the fact that the difference between the sPlot bin content (expectation value of bin) is very different from the bin content of the MC template (expectation value). So, even if fixing the RMS with the analytical model a significant bias due to the difference is going to be present. For this reason, the use of sPlot templates has been attempted. The procedure is the following, after having validated that the analytical and bootstrap toys are equivalent, for the limited in statistics background sample we used the analytical models and for the signal we used the full analysis MC. The signal MC was sampled for a set of different lifetime values (accept/reject) and with the combination of the analytical background samples (hybrid toys) a set of toys has been generated. Those toys have been fitted with the same mass model for the sPlot subtraction. The resulting sPlots for each value of the lifetime were used to sample the bin content distribution for the ct histogram. The mean value of the bin content along with either the mean value of the error distribution or the RMS value of the bin content, were used to create an sPlot template for each lifetime value. Those templates subsequently were used to be fitted on toys generated from the analytical models to test the fitter performance. By creating the templates from sPlots an assumption is being folded in for the background distribution. This yields to a source of systematics that need to be studied if this approach is successful. The following chi2 scans have been obtained.

Chi2 scan with mean of error distribution Chi2 scan with rms of bin content distribution
sPlotOnToys che1ExpOneGaus chi2ScansWithsPlotTemplates meanError 8ctBins.png
Chi2 scan for 4 toys with template error to be the mean error from the sPlot error distribution
sPlotOnToys che1ExpOneGaus chi2ScansWithsPlotTemplates rnsError 8ctBins.png
Chi2 scan for 4 toys (same as left) with template error to be the rms of the sPlot bin content distribution

it can be seen that both options show similar shapes in the fits. Which is also the expected. For the performance now on 1000 toys only the mean error option has been attempted with the following result.

Chi2 pulls/residuals with sPlot templates with mean error
Pull/Residual plots for chi2 analytical fit with sPlot templates with mean error

sPlot template comparison

The next step is to compare the sPlot templates generated from toys with the signal only MC templates that were used till today. Since, we generated more than 1000 sPlot templates to compare all of them would have been very difficult. For this reason we chose 3 lifetime values one below the reference, one around the reference and the last one beyond the reference. (tau = 0.75,1.5,2.5). In the following comparison we see the templates with the RMS of the bin content distribution used as error, and the templates that use the mean error returned from sPlot.

sPlot templates (Mean error) vs MC signal template sPlot template (RMS error) vs MC signal template
chi2TemplateComparison meanErrorAsErrorsOnsPlotTemplateVsMCTemplates.png
sPlot template vs signal MC template
chi2TemplateComparison rmsAsErrorsOnsPlotTemplateVsMCTemplates.png
sPlot template vs signal MC template

Error distribution from sPlot templates

From the previous section it's clear that the most affecting factor in the chi2 is not the bin content of the template but rather the error estimation from sPlot. For this reason the distributions of the same three lifetime above have been generated. Along with the error distribution 5 lines are added to represent the rms from the sPlot bin content distribution with and without the 0 entries toys, the MC Poisson uncertainty, and finally the mean of the error distribution.

Tau = 0.75 Tau = 1.5 Tau = 2.5
meanErrorsPlotTemplate originalErrorDist lifetime0.750000.png
Error distribution for tau = 0.75 ps
meanErrorsPlotTemplate originalErrorDist lifetime1.500000.png
Error distribution for tau = 1.5 ps
meanErrorsPlotTemplate originalErrorDist lifetime2.500000.png
Error distribution for tau = 2.5 ps

sPlot templates pulls residuals

With the sPlot templates at hand the next step is now to freeze the error used from the templates and test them on toys for the performance. From here on the error used is the RMS from the bin content distribution. The toys used for the sPlot templates are the 10k toys that were generated from the analytical models. Finally, the scan range was set from [0.5-5.0]ps with a step of 0.01ps. Additionally, 4 different configuration have been applied to be evaluated. (1) Using sPlot template for expectation value and as error the RMS from the template, (2) the MC template expectation value and the per toy error from the fitted toys, (3) sPlot expectation value from template and error from the per toy returned error, (4) the MC template expectation value with the sPlot template error. Finally, only the fits that managed to determine the error within the scan range were kept to avoid large variations on the pull distribution.

sPlot expectation value and error MC expectation value and per toy error sPlot expectation value and per toy error MC expectation value and sPlot template error
residualsPullsForsPlotTemplates withRMSError.png
sPlot bin content and error in chi2
residualsPullsForMCTemplates withRMSError.png
MC bin content and toy error in chi2
residualsPullsForsPlotBinContentAndMCError withRMSError.png
sPlot bin content and toy error in chi2
residualsPullsForMCBinContentAndsPlotError withRMSError.png
MC bin content with sPlot template error

Debugging event excess on sPlot template pulls/residuals

In the sPlot template pulls/residuals it can be seen that in the 0.6-0.8 bin there is an excess of events that is unphysical. For this reason further investigation has been made in order to check what is the source of this excess. The check was to extract the toys that are falling into this bin and check what the fitted value returned by the sPlot templates is:

Fitted value in excess bin toys

the fact that all templates are fitted with the same value is a bit suspicious. Therefore the scans for those toys have been produced to see if the fact that the 2.12ps template is fitting all of the is a systematic effect or not.

-- IoannisXiotidis - 2021-02-03

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PNGpng chi2TemplateComparison_meanErrorAsErrorsOnsPlotTemplateVsMCTemplates.png r1 manage 19.6 K 2021-03-08 - 16:50 IoannisXiotidis  
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