# rf703_effpdfprod¶

Special pdf's: using a product of an (acceptance) efficiency and a pdf as 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, November 30, 2022 at 11:24 AM.

In [1]:
import ROOT

Welcome to JupyROOT 6.27/01


## Define observables and decay pdf¶

Declare observables

In [2]:
t = ROOT.RooRealVar("t", "t", 0, 5)


Make pdf

In [3]:
tau = ROOT.RooRealVar("tau", "tau", -1.54, -4, -0.1)
model = ROOT.RooExponential("model", "model", t, tau)


## Define efficiency function¶

Use error function to simulate turn-on slope

In [4]:
eff = ROOT.RooFormulaVar("eff", "0.5*(TMath::Erf((t-1)/0.5)+1)", [t])


## Define decay pdf with efficiency¶

Multiply pdf(t) with efficiency in t

In [5]:
modelEff = ROOT.RooEffProd("modelEff", "model with efficiency", model, eff)


## Plot efficiency, pdf¶

In [6]:
frame1 = t.frame(Title="Efficiency")
eff.plotOn(frame1, LineColor="r")

frame2 = t.frame(Title="Pdf with and without efficiency")

model.plotOn(frame2, LineStyle="--")
modelEff.plotOn(frame2)

Out[6]:
<cppyy.gbl.RooPlot object at 0x8d44190>
[#1] INFO:NumericIntegration -- RooRealIntegral::init(modelEff_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)


## Generate toy data, fit model eff to data¶

Generate events. If the input pdf has an internal generator, internal generator is used and an accept/reject sampling on the efficiency is applied.

In [7]:
data = modelEff.generate({t}, 10000)


Fit pdf. The normalization integral is calculated numerically.

In [8]:
modelEff.fitTo(data)

Out[8]:
<cppyy.gbl.RooFitResult object at 0x(nil)>
[#1] INFO:NumericIntegration -- RooRealIntegral::init(modelEff_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)
[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: activating const optimization
[#1] INFO:Minimization --  The following expressions have been identified as constant and will be precalculated and cached: (eff)
**********
**    1 **SET PRINT           1
**********
**********
**********
PARAMETER DEFINITIONS:
NO.   NAME         VALUE      STEP SIZE      LIMITS
1 tau         -1.54000e+00  3.90000e-01   -4.00000e+00 -1.00000e-01
**********
**    3 **SET ERR         0.5
**********
**********
**    4 **SET PRINT           1
**********
**********
**    5 **SET STR           1
**********
NOW USING STRATEGY  1: TRY TO BALANCE SPEED AGAINST RELIABILITY
**********
**********
FIRST CALL TO USER FUNCTION AT NEW START POINT, WITH IFLAG=4.
START MIGRAD MINIMIZATION.  STRATEGY  1.  CONVERGENCE WHEN EDM .LT. 1.00e-03
FCN=9696.74 FROM MIGRAD    STATUS=INITIATE        4 CALLS           5 TOTAL
EDM= unknown      STRATEGY= 1      NO ERROR MATRIX
EXT PARAMETER               CURRENT GUESS       STEP         FIRST
NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE
1  tau         -1.54000e+00   3.90000e-01   2.09076e-01   1.80888e+02
ERR DEF= 0.5
MIGRAD WILL VERIFY CONVERGENCE AND ERROR MATRIX.
COVARIANCE MATRIX CALCULATED SUCCESSFULLY
FCN=9695.84 FROM MIGRAD    STATUS=CONVERGED      12 CALLS          13 TOTAL
EDM=2.41592e-05    STRATEGY= 1      ERROR MATRIX ACCURATE
EXT PARAMETER                                   STEP         FIRST
NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE
1  tau         -1.55887e+00   1.40931e-02   5.05936e-04   6.58163e-01
ERR DEF= 0.5
EXTERNAL ERROR MATRIX.    NDIM=  25    NPAR=  1    ERR DEF=0.5
1.986e-04
**********
**    7 **SET ERR         0.5
**********
**********
**    8 **SET PRINT           1
**********
**********
**    9 **HESSE         500
**********
COVARIANCE MATRIX CALCULATED SUCCESSFULLY
FCN=9695.84 FROM HESSE     STATUS=OK              5 CALLS          18 TOTAL
EDM=2.41576e-05    STRATEGY= 1      ERROR MATRIX ACCURATE
EXT PARAMETER                                INTERNAL      INTERNAL
NO.   NAME      VALUE            ERROR       STEP SIZE       VALUE
1  tau         -1.55887e+00   1.40931e-02   1.01187e-04   2.54601e-01
ERR DEF= 0.5
EXTERNAL ERROR MATRIX.    NDIM=  25    NPAR=  1    ERR DEF=0.5
1.986e-04
[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: deactivating const optimization


Plot generated data and overlay fitted pdf

In [9]:
frame3 = t.frame(Title="Fitted pdf with efficiency")
data.plotOn(frame3)
modelEff.plotOn(frame3)

c = ROOT.TCanvas("rf703_effpdfprod", "rf703_effpdfprod", 1200, 400)
c.Divide(3)
c.cd(1)
frame1.GetYaxis().SetTitleOffset(1.4)
frame1.Draw()
c.cd(2)
frame2.GetYaxis().SetTitleOffset(1.6)
frame2.Draw()
c.cd(3)
frame3.GetYaxis().SetTitleOffset(1.6)
frame3.Draw()

c.SaveAs("rf703_effpdfprod.png")

[#1] INFO:NumericIntegration -- RooRealIntegral::init(modelEff_Int[t]) using numeric integrator RooIntegrator1D to calculate Int(t)

Info in <TCanvas::Print>: png file rf703_effpdfprod.png has been created


Draw all canvases

In [10]:
from ROOT import gROOT
gROOT.GetListOfCanvases().Draw()