Rf 2 0 2_Extendedmlfit¶

Addition and convolution: setting up an extended maximum likelihood fit

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, January 19, 2022 at 10:12 AM.

In [ ]:
import ROOT

Set up 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])

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])

Method 1 - Construct extended composite model¶

Sum the composite signal and background into an extended pdf nsigsig+nbkgbkg

In [ ]:
nsig = ROOT.RooRealVar("nsig", "number of signal events", 500, 0.0, 10000)
nbkg = ROOT.RooRealVar("nbkg", "number of background events", 500, 0, 10000)
model = ROOT.RooAddPdf("model", "(g1+g2)+a", [bkg, sig], [nbkg, nsig])

Sample, fit and plot extended model¶

Generate a data sample of expected number events in x from model = model.expectedEvents() = nsig+nbkg

In [ ]:
data = model.generate({x})

Fit model to data, ML term automatically included

In [ ]:
model.fitTo(data)

Plot data and PDF overlaid, expected number of events for pdf projection normalization rather than observed number of events (==data.numEntries())

In [ ]:
xframe = x.frame(Title="extended ML fit example")
data.plotOn(xframe)
model.plotOn(xframe, Normalization=dict(scaleFactor=1.0, scaleType=ROOT.RooAbsReal.RelativeExpected))

Overlay the background component of model with a dashed line

In [ ]:
model.plotOn(
xframe,
Components={bkg},
LineStyle=":",
Normalization=dict(scaleFactor=1.0, scaleType=ROOT.RooAbsReal.RelativeExpected),
)

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

In [ ]:
ras_bkg_sig2 = {bkg, sig2}
model.plotOn(
xframe,
Components=ras_bkg_sig2,
LineStyle=":",
Normalization=dict(scaleFactor=1.0, scaleType=ROOT.RooAbsReal.RelativeExpected),
)

Print structure of composite pdf

In [ ]:
model.Print("t")

Method 2 - Construct extended components first¶

Associated nsig/nbkg as expected number of events with sig/bkg

In [ ]:
esig = ROOT.RooExtendPdf("esig", "extended signal pdf", sig, nsig)
ebkg = ROOT.RooExtendPdf("ebkg", "extended background pdf", bkg, nbkg)

Sum extended components without coefs¶

Construct sum of two extended pdf (no coefficients required)

In [ ]:
model2 = ROOT.RooAddPdf("model2", "(g1+g2)+a", [ebkg, esig])

Draw the frame on the canvas

In [ ]:
c = ROOT.TCanvas("rf202_extendedmlfit", "rf202_extendedmlfit", 600, 600)