Multidimensional models: normalization and integration of pdfs, construction of cumulative distribution functions from pdfs in two dimensions

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 11:22 AM.

In [1]:
from __future__ import print_function
import ROOT
Welcome to JupyROOT 6.27/01

Set up model

Create observables x,y

In [2]:
x = ROOT.RooRealVar("x", "x", -10, 10)
y = ROOT.RooRealVar("y", "y", -10, 10)

Create pdf gaussx(x,-2,3), gaussy(y,2,2)

In [3]:
gx = ROOT.RooGaussian("gx", "gx", x, ROOT.RooFit.RooConst(-2), ROOT.RooFit.RooConst(3))
gy = ROOT.RooGaussian("gy", "gy", y, ROOT.RooFit.RooConst(+2), ROOT.RooFit.RooConst(2))

gxy = gx(x)*gy(y)

In [4]:
gxy = ROOT.RooProdPdf("gxy", "gxy", [gx, gy])

Retrieve raw & normalized values of RooFit pdfs

Return 'raw' unnormalized value of gx

In [5]:
print("gxy = ", gxy.getVal())
gxy =  0.4856717852477124

Return value of gxy normalized over x and y in range [-10,10]

In [6]:
nset_xy = {x, y}
print("gx_Norm[x,y] = ", gxy.getVal(nset_xy))
gx_Norm[x,y] =  0.012933200957206766

Create object representing integral over gx which is used to calculate gx_Norm[x,y] == gx / gx_Int[x,y]

In [7]:
x_and_y = {x, y}
igxy = gxy.createIntegral(x_and_y)
print("gx_Int[x,y] = ", igxy.getVal())
gx_Int[x,y] =  37.552326516436096

NB: it is also possible to do the following

Return value of gxy normalized over x in range [-10,10] (i.e. treating y as parameter)

In [8]:
nset_x = {x}
print("gx_Norm[x] = ", gxy.getVal(nset_x))
gx_Norm[x] =  0.1068955044839622

Return value of gxy normalized over y in range [-10,10] (i.e. treating x as parameter)

In [9]:
nset_y = {y}
print("gx_Norm[y] = ", gxy.getVal(nset_y))
gx_Norm[y] =  0.12098919425696865

Integarte normalizes pdf over subrange

Define a range named "signal" in x from -5,5

In [10]:
x.setRange("signal", -5, 5)
y.setRange("signal", -3, 3)
[#1] INFO:Eval -- RooRealVar::setRange(x) new range named 'signal' created with bounds [-5,5]
[#1] INFO:Eval -- RooRealVar::setRange(y) new range named 'signal' created with bounds [-3,3]

Create an integral of gxy_Norm[x,y] over x and y in range "signal" ROOT.This is the fraction of of pdf gxy_Norm[x,y] which is in the range named "signal"

In [11]:
igxy_sig = gxy.createIntegral(x_and_y, NormSet=x_and_y, Range="signal")
print("gx_Int[x,y|signal]_Norm[x,y] = ", igxy_sig.getVal())
gx_Int[x,y|signal]_Norm[x,y] =  0.5720351351990985

Construct cumulative distribution function from pdf

Create the cumulative distribution function of gx i.e. calculate Int[-10,x] gx(x') dx'

In [12]:
gxy_cdf = gxy.createCdf({x, y})

Plot cdf of gx versus x

In [13]:
hh_cdf = gxy_cdf.createHistogram("hh_cdf", x, Binning=40, YVar=dict(var=y, Binning=40))

c = ROOT.TCanvas("rf308_normintegration2d", "rf308_normintegration2d", 600, 600)

[#0] WARNING:InputArguments -- RooAbsReal::createHistogram(gxy_cdf_Int[x_prime,y_prime|CDF]_Norm[x_prime,y_prime]) WARNING extended mode requested for a non-pdf object, ignored
Info in <TCanvas::Print>: png file rf308_normintegration2d.png has been created

Draw all canvases

In [14]:
from ROOT import gROOT