Rf 7 0 9_ Barlow Beeston¶

Implementing the Barlow-Beeston method for taking into account the statistical uncertainty of a Monte-Carlo fit template.

Based on a demo by Wouter Verkerke

Author: Harshal Shende, Stephan Hageboeck (C++ version)
This notebook tutorial was automatically generated with ROOTBOOK-izer from the macro found in the ROOT repository on Monday, January 17, 2022 at 10:18 AM.

In [ ]:
import ROOT

First, construct a likelihood model with a Gaussian signal on top of a uniform background

In [ ]:
x = ROOT.RooRealVar("x", "x", -20, 20)
x.setBins(25)

meanG = ROOT.RooRealVar("meanG", "meanG", 1, -10, 10)
sigG = ROOT.RooRealVar("sigG", "sigG", 1.5, -10, 10)
g = ROOT.RooGaussian("g", "Gauss", x, meanG, sigG)
u = ROOT.RooUniform("u", "Uniform", x)

Generate the data to be fitted

In [ ]:
sigData = g.generate(x, 50)
bkgData = u.generate(x, 1000)

sumData = ROOT.RooDataSet("sumData", "Gauss + Uniform", x)
sumData.append(sigData)
sumData.append(bkgData)

Make histogram templates for signal and background. Let's take a signal distribution with low statistics and a more accurate background distribution. Normally, these come from Monte Carlo simulations, but we will just generate them.

In [ ]:
dh_sig = g.generateBinned(x, 50)
dh_bkg = u.generateBinned(x, 10000)

Case 0 - 'Rigid templates'

Construct histogram shapes for signal and background

In [ ]:
p_h_sig = ROOT.RooHistFunc("p_h_sig", "p_h_sig", x, dh_sig)
p_h_bkg = ROOT.RooHistFunc("p_h_bkg", "p_h_bkg", x, dh_bkg)

Construct scale factors for adding the two distributions

In [ ]:
Asig0 = ROOT.RooRealVar("Asig", "Asig", 1, 0.01, 5000)
Abkg0 = ROOT.RooRealVar("Abkg", "Abkg", 1, 0.01, 5000)

Construct the sum model

In [ ]:
model0 = ROOT.RooRealSumPdf("model0", "model0", [p_h_sig, p_h_bkg], [Asig0, Abkg0], True)

Case 1 - 'Barlow Beeston'

Construct parameterized histogram shapes for signal and background

In [ ]:
p_ph_sig1 = ROOT.RooParamHistFunc("p_ph_sig", "p_ph_sig", dh_sig)
p_ph_bkg1 = ROOT.RooParamHistFunc("p_ph_bkg", "p_ph_bkg", dh_bkg)

Asig1 = ROOT.RooRealVar("Asig", "Asig", 1, 0.01, 5000)
Abkg1 = ROOT.RooRealVar("Abkg", "Abkg", 1, 0.01, 5000)

Construct the sum of these

In [ ]:
model_tmp = ROOT.RooRealSumPdf("sp_ph", "sp_ph", [p_ph_sig1, p_ph_bkg1], [Asig1, Abkg1], True)

Construct the subsidiary poisson measurements constraining the histogram parameters These ensure that the bin contents of the histograms are only allowed to vary within the statistical uncertainty of the Monte Carlo.

In [ ]:
hc_sig = ROOT.RooHistConstraint("hc_sig", "hc_sig", p_ph_sig1)
hc_bkg = ROOT.RooHistConstraint("hc_bkg", "hc_bkg", p_ph_bkg1)

Construct the joint model with template PDFs and constraints

In [ ]:
model1 = ROOT.RooProdPdf("model1", "model1", {hc_sig, hc_bkg}, Conditional=(model_tmp, x))

Case 2 - 'Barlow Beeston' light (one parameter per bin for all samples)

Construct the histogram shapes, using the same parameters for signal and background This requires passing the first histogram to the second, so that their common parameters can be re-used. The first ParamHistFunc will create one parameter per bin, such as p_ph_sig2_gamma_bin_0. This allows bin 0 to fluctuate up and down. Then, the SAME parameters are connected to the background histogram, so the bins flucutate synchronously. This reduces the number of parameters.

In [ ]:
p_ph_sig2 = ROOT.RooParamHistFunc("p_ph_sig2", "p_ph_sig2", dh_sig)
p_ph_bkg2 = ROOT.RooParamHistFunc("p_ph_bkg2", "p_ph_bkg2", dh_bkg, p_ph_sig2, True)

Asig2 = ROOT.RooRealVar("Asig", "Asig", 1, 0.01, 5000)
Abkg2 = ROOT.RooRealVar("Abkg", "Abkg", 1, 0.01, 5000)

As before, construct the sum of signal2 and background2

In [ ]:
model2_tmp = ROOT.RooRealSumPdf("sp_ph", "sp_ph", [p_ph_sig2, p_ph_bkg2], [Asig2, Abkg2], True)

Construct the subsidiary poisson measurements constraining the statistical fluctuations

In [ ]:
hc_sigbkg = ROOT.RooHistConstraint("hc_sigbkg", "hc_sigbkg", {p_ph_sig2, p_ph_bkg2})

Construct the joint model

In [ ]:
model2 = ROOT.RooProdPdf("model2", "model2", hc_sigbkg, Conditional=(model2_tmp, x))

** Fit all models to data and plot *

In [ ]:
result0 = model0.fitTo(sumData, PrintLevel=0, Save=True)
result1 = model1.fitTo(sumData, PrintLevel=0, Save=True)
result2 = model2.fitTo(sumData, PrintLevel=0, Save=True)

can = ROOT.TCanvas("can", "", 1500, 600)
can.Divide(3, 1)

pt = ROOT.TPaveText(-19.5, 1, -2, 25)
pt.SetFillStyle(0)
pt.SetBorderSize(0)

can.cd(1)
frame = x.frame(Title="No template uncertainties")

Plot data to enable automatic determination of model0 normalisation:

In [ ]:
sumData.plotOn(frame)
model0.plotOn(frame, LineColor="b", VisualizeError=result0)

Plot data again to show it on top of model0 error bands:

In [ ]:
sumData.plotOn(frame)

Plot model components

In [ ]:
model0.plotOn(frame, LineColor="b")
p_ph_sig_set = {p_h_sig}
p_ph_bkg_set = {p_h_bkg}
model0.plotOn(frame, Components=p_ph_sig_set, LineColor="kAzure")
model0.plotOn(frame, Components=p_ph_bkg_set, LineColor="r")
model0.paramOn(frame)

sigData.plotOn(frame, MarkerColor="b")
frame.Draw()

pt_text1 = [
"No template uncertainties",
"are taken into account.",
"for the parameters A, since",
"the only source of errors",
"are the data statistics.",
]
for text in pt_text1:

pt.DrawClone()

can.cd(2)
frame = x.frame(Title="Barlow Beeston for Sig & Bkg separately")
sumData.plotOn(frame)
model1.plotOn(frame, LineColor="b", VisualizeError=result1)

Plot data again to show it on top of error bands:

In [ ]:
sumData.plotOn(frame)
model1.plotOn(frame, LineColor="b")
p_ph_sig1_set = {p_ph_sig1}
p_ph_bkg1_set = {p_ph_bkg1}
model1.plotOn(frame, Components=p_ph_sig1_set, LineColor="kAzure")
model1.plotOn(frame, Components=p_ph_bkg1_set, LineColor="r")
model1.paramOn(frame, Parameters={Asig1, Abkg1})

sigData.plotOn(frame, MarkerColor="b")
frame.Draw()

pt.Clear()
pt_text2 = [
"With gamma parameters, the",
"signal & background templates",
"Note how the blue signal",
"template changes its shape.",
"of the scale parameters A.",
]

for text in pt_text2:

pt.DrawClone()

can.cd(3)
frame = x.frame(Title="Barlow Beeston light for (Sig+Bkg)")
sumData.plotOn(frame)
model2.plotOn(frame, LineColor="b", VisualizeError=result2)

Plot data again to show it on top of model0 error bands:

In [ ]:
sumData.plotOn(frame)
model2.plotOn(frame, LineColor="b")
p_ph_sig2_set = {p_ph_sig2}
p_ph_bkg2_set = {p_ph_bkg2}
model2.plotOn(frame, Components=p_ph_sig2_set, LineColor="kAzure")
model2.plotOn(frame, Components=p_ph_bkg2_set, LineColor="r")
model2.paramOn(frame, Parameters={Asig2, Abkg2})

sigData.plotOn(frame, MarkerColor="b")
frame.Draw()

pt.Clear()
pt_text3 = [
"When signal and background",
"template share one gamma para-",
"meter per bin, they adapt less.",
"The errors of the A parameters",
"also shrink slightly.",
]
for text in pt_text3:
pt.DrawClone()

print("Asig [normal ] = {} +/- {}".format(Asig0.getVal(), Asig0.getError()))
print("Asig [BB     ] = {} +/- {}".format(Asig1.getVal(), Asig1.getError()))
print("Asig [BBlight] = {} +/- {}".format(Asig2.getVal(), Asig2.getError()))

can.SaveAs("rf709_BarlowBeeston.png")

Draw all canvases

In [ ]:
%jsroot on
from ROOT import gROOT
gROOT.GetListOfCanvases().Draw()