FourBinInstructional.py

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## \file
## \ingroup tutorial_roostats
## \notebook
## This example is a generalization of the on/off problem.
##
##  This example is a generalization of the on/off problem.
## It's a common setup for SUSY searches.  Imagine that one has two
## variables "x" and "y" (eg. missing ET and SumET), see figure.
## The signal region has high values of both of these variables (top right).
## One can see low values of "x" or "y" acting as side-bands.  If we
## just used "y" as a sideband, we would have the on/off problem.
##  - In the signal region we observe non events and expect s+b events.
##  - In the region with low values of "y" (bottom right)
##    we observe noff events and expect tau*b events.
## Note the significance of tau.  In the background only case:
##
## ~~~{.cpp}
##    tau ~ <expectation off> / <expectation on>
## ~~~
##
## If tau is known, this model is sufficient, but often tau is not known exactly.
## So one can use low values of "x" as an additional constraint for tau.
## Note that this technique critically depends on the notion that the
## joint distribution for "x" and "y" can be factorized.
## Generally, these regions have many events, so it the ratio can be
## measured very precisely there.  So we extend the model to describe the
## left two boxes... denoted with "bar".
##   - In the upper left we observe nonbar events and expect bbar events
##   - In the bottom left we observe noffbar events and expect tau bbar events
## Note again we have:
##
## ~~~{.cpp}
##    tau ~ <expectation off bar> / <expectation on bar>
## ~~~
##
## One can further expand the model to account for the systematic associated
## to assuming the distribution of "x" and "y" factorizes (eg. that
## tau is the same for off/on and offbar/onbar). This can be done in several
## ways, but here we introduce an additional parameter rho, which so that
## one set of models will use tau and the other tau*rho. The choice is arbitrary,
## but it has consequences on the numerical stability of the algorithms.
## The "bar" measurements typically have more events (& smaller relative errors).
## If we choose
##
## ~~~{.cpp}
## <expectation noffbar> = tau * rho * <expectation noonbar>
## ~~~
##
## the product tau*rho will be known very precisely (~1/sqrt(bbar)) and the contour
## in those parameters will be narrow and have a non-trivial tau~1/rho shape.
## However, if we choose to put rho on the non/noff measurements (where the
## product will have an error `~1/sqrt(b))`, the contours will be more amenable
## to numerical techniques.  Thus, here we choose to define:
##
## ~~~{.cpp}
##    tau := <expectation off bar> / (<expectation on bar>)
##    rho := <expectation off> / (<expectation on> * tau)
##
## ^ y
## |
## |---------------------------+
## |               |           |
## |     nonbar    |    non    |
## |      bbar     |    s+b    |
## |               |           |
## |---------------+-----------|
## |               |           |
## |    noffbar    |    noff   |
## |    tau bbar   | tau b rho |
## |               |           |
## +-----------------------------> x
## ~~~
##
## Left in this way, the problem is under-constrained.  However, one may
## have some auxiliary measurement (usually based on Monte Carlo) to
## constrain rho.  Let us call this auxiliary measurement that gives
## the nominal value of rho "rhonom".  Thus, there is a 'constraint' term in
## the full model: P(rhonom | rho).  In this case, we consider a Gaussian
## constraint with standard deviation sigma.
##
## In the example, the initial values of the parameters are:
##
## ~~~{.cpp}
##   - s    = 40
##   - b    = 100
##   - tau  = 5
##   - bbar = 1000
##   - rho  = 1
##   (sigma for rho = 20%)
## ~~~
##
## and in the toy dataset:
##
## ~~~{.cpp}
##    - non = 139
##    - noff = 528
##    - nonbar = 993
##    - noffbar = 4906
##    - rhonom = 1.27824
## ~~~
##
## Note, the covariance matrix of the parameters has large off-diagonal terms.
## Clearly s,b are anti-correlated.  Similarly, since noffbar >> nonbar, one would
## expect bbar,tau to be anti-correlated.
##
## This can be seen below.
##
## ~~~{.cpp}
##             GLOBAL      b    bbar   rho      s     tau
##         b  0.96820   1.000  0.191 -0.942 -0.762 -0.209
##      bbar  0.91191   0.191  1.000  0.000 -0.146 -0.912
##       rho  0.96348  -0.942  0.000  1.000  0.718 -0.000
##         s  0.76250  -0.762 -0.146  0.718  1.000  0.160
##       tau  0.92084  -0.209 -0.912 -0.000  0.160  1.000
## ~~~
##
## Similarly, since tau*rho appears as a product, we expect rho,tau
## to be anti-correlated. When the error on rho is significantly
## larger than 1/sqrt(bbar), tau is essentially known and the
## correlation is minimal (tau mainly cares about bbar, and rho about b,s).
## In the alternate parametrization (bbar* tau * rho) the correlation coefficient
## for rho,tau is large (and negative).
##
## The code below uses best-practices for RooFit & RooStats as of June 2010.
##
## It proceeds as follows:
##  - create a workspace to hold the model
##  - use workspace factory to quickly create the terms of the model
##  - use workspace factory to define total model (a prod pdf)
##  - create a RooStats ModelConfig to specify observables, parameters of interest
##  - add to the ModelConfig a prior on the parameters for Bayesian techniques
##    note, the pdf it is factorized for parameters of interest & nuisance params
##  - visualize the model
##  - write the workspace to a file
##  - use several of RooStats IntervalCalculators & compare results
##
## \macro_image
## \macro_output
## \macro_code
##
## \date July 2022
## \authors Artem Busorgin, Kyle Cranmer and Tanja Rommerskirchen (C++ version)
 
import ROOT
 
doBayesian = False
doFeldmanCousins = False
doMCMC = False
 
# let's time this challenging example
t = ROOT.TStopwatch()
t.Start()
 
# set RooFit random seed for reproducible results
ROOT.RooRandom.randomGenerator().SetSeed(4357)
 
# make model
wspace = ROOT.RooWorkspace("wspace")
wspace.factory("Poisson::on(non[0,1000], sum::splusb(s[40,0,100],b[100,0,300]))")
wspace.factory("Poisson::off(noff[0,5000], prod::taub(b,tau[5,3,7],rho[1,0,2]))")
wspace.factory("Poisson::onbar(nonbar[0,10000], bbar[1000,500,2000])")
wspace.factory("Poisson::offbar(noffbar[0,1000000], prod::lambdaoffbar(bbar, tau))")
wspace.factory("Gaussian::mcCons(rhonom[1.,0,2], rho, sigma[.2])")
wspace.factory("PROD::model(on,off,onbar,offbar,mcCons)")
wspace.defineSet("obs", "non,noff,nonbar,noffbar,rhonom")
 
wspace.factory("Uniform::prior_poi({s})")
wspace.factory("Uniform::prior_nuis({b,bbar,tau, rho})")
wspace.factory("PROD::prior(prior_poi,prior_nuis)")
 
# ----------------------------------
# Control some interesting variations
# define parameers of interest
# for 1-d plots
wspace.defineSet("poi", "s")
wspace.defineSet("nuis", "b,tau,rho,bbar")
 
# for 2-d plots to inspect correlations:
# wspace.defineSet("poi","s,rho")
 
# test simpler cases where parameters are known.
# wspace["tau"].setConstant()
# wspace["rho"].setConstant()
# wspace["b"].setConstant()
# wspace["bbar"].setConstant()
 
# inspect workspace
# wspace.Print()
 
# ----------------------------------------------------------
# Generate toy data
# generate toy data assuming current value of the parameters
# import into workspace.
# add Verbose() to see how it's being generated
data = wspace["model"].generate(wspace.set("obs"), 1)
# data.Print("v")
wspace.Import(data)
 
# ----------------------------------
# Now the statistical tests
# model config
modelConfig = ROOT.RooStats.ModelConfig("FourBins")
modelConfig.SetWorkspace(wspace)
modelConfig.SetPdf(wspace["model"])
modelConfig.SetPriorPdf(wspace["prior"])
modelConfig.SetParametersOfInterest(wspace.set("poi"))
modelConfig.SetNuisanceParameters(wspace.set("nuis"))
wspace.Import(modelConfig)
wspace.writeToFile("FourBin.root")
 
# -------------------------------------------------
# If you want to see the covariance matrix uncomment
# wspace["model"].fitTo(data)
 
# use ProfileLikelihood
plc = ROOT.RooStats.ProfileLikelihoodCalculator(data, modelConfig)
plc.SetConfidenceLevel(0.95)
plInt = plc.GetInterval()
msglevel = ROOT.RooMsgService.instance().globalKillBelow()
ROOT.RooMsgService.instance().setGlobalKillBelow(ROOT.RooFit.FATAL)
plInt.LowerLimit(wspace["s"])  # get ugly print out of the way. Fix.
ROOT.RooMsgService.instance().setGlobalKillBelow(msglevel)
 
# use FeldmaCousins (takes ~20 min)
fc = ROOT.RooStats.FeldmanCousins(data, modelConfig)
fc.SetConfidenceLevel(0.95)
# number counting: dataset always has 1 entry with N events observed
fc.FluctuateNumDataEntries(False)
fc.UseAdaptiveSampling(True)
fc.SetNBins(40)
fcInt = ROOT.RooStats.PointSetInterval()
if doFeldmanCousins:  # takes 7 minutes
    fcInt = fc.GetInterval()
 
# use BayesianCalculator (only 1-d parameter of interest, slow for this problem)
bc = ROOT.RooStats.BayesianCalculator(data, modelConfig)
bc.SetConfidenceLevel(0.95)
bInt = ROOT.RooStats.SimpleInterval()
if doBayesian and len(wspace.set("poi")) == 1:
    bInt = bc.GetInterval()
else:
    print("Bayesian Calc. only supports on parameter of interest")
 
# use MCMCCalculator  (takes about 1 min)
# Want an efficient proposal function, so derive it from covariance
# matrix of fit
fit = wspace["model"].fitTo(data, Save=True)
ph = ROOT.RooStats.ProposalHelper()
ph.SetVariables(fit.floatParsFinal())
ph.SetCovMatrix(fit.covarianceMatrix())
ph.SetUpdateProposalParameters(ROOT.kTRUE)  # auto-create mean vars and add mappings
ph.SetCacheSize(100)
pf = ph.GetProposalFunction()
 
mc = ROOT.RooStats.MCMCCalculator(data, modelConfig)
mc.SetConfidenceLevel(0.95)
mc.SetProposalFunction(pf)
mc.SetNumBurnInSteps(500)  # first N steps to be ignored as burn-in
mc.SetNumIters(50000)
mc.SetLeftSideTailFraction(0.5)  # make a central interval
mcInt = ROOT.RooStats.MCMCInterval()
if doMCMC:
    mcInt = mc.GetInterval()
 
# ----------------------------------
# Make some plots
c1 = ROOT.gROOT.Get("c1")
if not c1:
    c1 = ROOT.TCanvas("c1")
 
if doBayesian and doMCMC:
    c1.Divide(3)
    c1.cd(1)
elif doBayesian or doMCMC:
    c1.Divide(2)
    c1.cd(1)
 
lrplot = ROOT.RooStats.LikelihoodIntervalPlot(plInt)
lrplot.Draw()
 
if doBayesian and len(wspace.set("poi")) == 1:
    c1.cd(2)
    # the plot takes a long time and print lots of error
    # using a scan it is better
    bc.SetScanOfPosterior(20)
    bplot = bc.GetPosteriorPlot()
    bplot.Draw()
 
if doMCMC:
    if doBayesian and len(wspace.set("poi")) == 1:
        c1.cd(3)
    else:
        c1.cd(2)
    mcPlot = ROOT.RooStats.MCMCIntervalPlot(mcInt)
    mcPlot.Draw()
 
# ----------------------------------
# querry intervals
print(
    "Profile Likelihood interval on s = [{}, {}]".format(plInt.LowerLimit(wspace["s"]), plInt.UpperLimit(wspace["s"]))
)
# Profile Likelihood interval on s = [12.1902, 88.6871]
 
if doBayesian and len(wspace.set("poi")) == 1:
    print("Bayesian interval on s = [{}, {}]".format(bInt.LowerLimit(), bInt.UpperLimit()))
 
if doFeldmanCousins:
    print(
        "Feldman Cousins interval on s = [{}, {}]".format(fcInt.LowerLimit(wspace["s"]), fcInt.UpperLimit(wspace["s"]))
    )
    # Feldman Cousins interval on s = [18.75 +/- 2.45, 83.75 +/- 2.45]
 
if doMCMC:
    print("MCMC interval on s = [{}, {}]".format(mcInt.LowerLimit(wspace["s"]), mcInt.UpperLimit(wspace["s"])))
    # MCMC interval on s = [15.7628, 84.7266]
 
t.Stop()
t.Print()
 
c1.SaveAs("FourBinInstructional.png")
 
# TODO: The calculators have to be destructed first. Otherwise, we can get
# segmentation faults depending on the destruction order, which is random in
# Python. Probably the issue is that some object has a non-owning pointer to
# another object, which it uses in its destructor. This should be fixed either
# in the design of RooStats in C++, or with phythonizations.
del plc
del bc
del mc