Rf 2 0 1_Composite

Addition and convolution: composite pdf with signal and background component

pdf = f_bkg * bkg(x,a0,a1) + (1-fbkg) * (f_sig1 * sig1(x,m,s1 + (1-f_sig1) * sig2(x,m,s2)))

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 Monday, January 17, 2022 at 09:53 AM.

In [ ]:
import ROOT

Setup component pdfs

Declare observable x

In [ ]:
x = ROOT.RooRealVar("x", "x", 0, 10)

Create two Gaussian PDFs g1(x,mean1,sigma) anf g2(x,mean2,sigma) and their parameters

In [ ]:
mean = ROOT.RooRealVar("mean", "mean of gaussians", 5)
sigma1 = ROOT.RooRealVar("sigma1", "width of gaussians", 0.5)
sigma2 = ROOT.RooRealVar("sigma2", "width of gaussians", 1)

sig1 = ROOT.RooGaussian("sig1", "Signal component 1", x, mean, sigma1)
sig2 = ROOT.RooGaussian("sig2", "Signal component 2", x, mean, sigma2)

Build Chebychev polynomial pdf

In [ ]:
a0 = ROOT.RooRealVar("a0", "a0", 0.5, 0.0, 1.0)
a1 = ROOT.RooRealVar("a1", "a1", -0.2, 0.0, 1.0)
bkg = ROOT.RooChebychev("bkg", "Background", x, [a0, a1])

Method 1 - Two RooAddPdfs

Add signal components

Sum the signal components into a composite signal pdf

In [ ]:
sig1frac = ROOT.RooRealVar("sig1frac", "fraction of component 1 in signal", 0.8, 0.0, 1.0)
sig = ROOT.RooAddPdf("sig", "Signal", [sig1, sig2], [sig1frac])

Add signal and background

Sum the composite signal and background

In [ ]:
bkgfrac = ROOT.RooRealVar("bkgfrac", "fraction of background", 0.5, 0.0, 1.0)
model = ROOT.RooAddPdf("model", "g1+g2+a", [bkg, sig], [bkgfrac])

Sample, fit and plot model

Generate a data sample of 1000 events in x from model

In [ ]:
data = model.generate({x}, 1000)

Fit model to data

In [ ]:
model.fitTo(data)

Plot data and PDF overlaid

In [ ]:
xframe = x.frame(Title="Example of composite pdf=(sig1+sig2)+bkg")
data.plotOn(xframe)
model.plotOn(xframe)

Overlay the background component of model with a dashed line

In [ ]:
model.plotOn(xframe, Components={bkg}, LineStyle="--")

Overlay the background+sig2 components of model with a dotted line

In [ ]:
model.plotOn(xframe, Components={bkg, sig2}, LineStyle=":")

Print structure of composite pdf

In [ ]:
model.Print("t")

Method 2 - One RooAddPdf with recursive fractions

Construct sum of models on one go using recursive fraction interpretations

model2 = bkg + (sig1 + sig2)

In [ ]:
model2 = ROOT.RooAddPdf("model", "g1+g2+a", [bkg, sig1, sig2], [bkgfrac, sig1frac], True)

NB: Each coefficient is interpreted as the fraction of the left-hand component of the i-th recursive sum, i.e.

sum4 = A + ( B + ( C + D) with fraction fA, and fC expands to

sum4 = fAA + (1-fA)(fBB + (1-fB)(fCC + (1-fC)D))

Plot recursive addition model

In [ ]:
model2.plotOn(xframe, LineColor="r", LineStyle="--")
model2.plotOn(xframe, Components={bkg, sig2}, LineColor="r", LineStyle="--")
model2.Print("t")

Draw the frame on the canvas

In [ ]:
c = ROOT.TCanvas("rf201_composite", "rf201_composite", 600, 600)
ROOT.gPad.SetLeftMargin(0.15)
xframe.GetYaxis().SetTitleOffset(1.4)
xframe.Draw()

c.SaveAs("rf201_composite.png")

Draw all canvases

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