Multidimensional models: using the likelihood ratio technique to construct a signal enhanced one-dimensional projection of a multi-dimensional pdf
Author: Clemens Lange, Wouter Verkerke (C++ version)
This notebook tutorial was automatically generated with ROOTBOOK-izer from the macro found in the ROOT repository on Wednesday, April 17, 2024 at 11:18 AM.
import ROOT
Create observables
x = ROOT.RooRealVar("x", "x", -5, 5)
y = ROOT.RooRealVar("y", "y", -5, 5)
z = ROOT.RooRealVar("z", "z", -5, 5)
Create signal pdf gauss(x)gauss(y)gauss(z)
gx = ROOT.RooGaussian("gx", "gx", x, 0.0, 1.0)
gy = ROOT.RooGaussian("gy", "gy", y, 0.0, 1.0)
gz = ROOT.RooGaussian("gz", "gz", z, 0.0, 1.0)
sig = ROOT.RooProdPdf("sig", "sig", [gx, gy, gz])
Create background pdf poly(x)poly(y)poly(z)
px = ROOT.RooPolynomial("px", "px", x, [-0.1, 0.004])
py = ROOT.RooPolynomial("py", "py", y, [0.1, -0.004])
pz = ROOT.RooPolynomial("pz", "pz", z)
bkg = ROOT.RooProdPdf("bkg", "bkg", [px, py, pz])
Create composite pdf sig+bkg
fsig = ROOT.RooRealVar("fsig", "signal fraction", 0.1, 0.0, 1.0)
model = ROOT.RooAddPdf("model", "model", [sig, bkg], [fsig])
data = model.generate({x, y, z}, 20000)
Make plain projection of data and pdf on x observable
frame = x.frame(Title="Projection of 3D data and pdf on X", Bins=40)
data.plotOn(frame)
model.plotOn(frame)
<cppyy.gbl.RooPlot object at 0xa18a360>
[#1] INFO:Plotting -- RooAbsReal::plotOn(model) plot on x integrates over variables (y,z)
Calculate projection of signal and total likelihood on (y,z) observables i.e. integrate signal and composite model over x
sigyz = sig.createProjection({x})
totyz = model.createProjection({x})
Construct the log of the signal / signal+background probability
llratio_func = ROOT.RooFormulaVar("llratio", "log10(@0)-log10(@1)", [sigyz, totyz])
Calculate the llratio value for each event in the dataset
data.addColumn(llratio_func)
<cppyy.gbl.RooRealVar object at 0xa294e10>
Extract the subset of data with large signal likelihood
dataSel = data.reduce(Cut="llratio>0.7")
[#1] INFO:InputArguments -- The formula llratio>0.7 claims to use the variables (x,y,z,llratio) but only (llratio) seem to be in use. inputs: llratio>0.7
Make plot frame
frame2 = x.frame(Title="Same projection on X with LLratio(y,z)>0.7", Bins=40)
Plot select data on frame
dataSel.plotOn(frame2)
<cppyy.gbl.RooPlot object at 0xa624520>
Generate large number of events for MC integration of pdf projection
mcprojData = model.generate({x, y, z}, 10000)
Calculate LL ratio for each generated event and select MC events with llratio)0.7
mcprojData.addColumn(llratio_func)
mcprojDataSel = mcprojData.reduce(Cut="llratio>0.7")
[#1] INFO:InputArguments -- The formula llratio>0.7 claims to use the variables (x,y,z,llratio) but only (llratio) seem to be in use. inputs: llratio>0.7
Project model on x, projected observables (y,z) with Monte Carlo technique on set of events with the same llratio cut as was applied to data
model.plotOn(frame2, ProjWData=mcprojDataSel)
c = ROOT.TCanvas("rf316_llratioplot", "rf316_llratioplot", 800, 400)
c.Divide(2)
c.cd(1)
ROOT.gPad.SetLeftMargin(0.15)
frame.GetYaxis().SetTitleOffset(1.4)
frame.Draw()
c.cd(2)
ROOT.gPad.SetLeftMargin(0.15)
frame2.GetYaxis().SetTitleOffset(1.4)
frame2.Draw()
c.SaveAs("rf316_llratioplot.png")
[#1] INFO:Plotting -- RooAbsReal::plotOn(model) plot on x averages using data variables (y,z) [#1] INFO:Plotting -- RooAbsReal::plotOn(model) only the following components of the projection data will be used: (y,z) [#1] INFO:Fitting -- using CPU computation library compiled with -mavx2
Info in <TCanvas::Print>: png file rf316_llratioplot.png has been created
Draw all canvases
from ROOT import gROOT
gROOT.GetListOfCanvases().Draw()