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rf610_visualerror.py File Reference

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 rf610_visualerror
 

Detailed Description

View in nbviewer Open in SWAN Likelihood and minimization: visualization of errors from a covariance matrix

import ROOT
# Setup example fit
# ---------------------------------------
# Create sum of two Gaussians pdf with factory
x = ROOT.RooRealVar("x", "x", -10, 10)
m = ROOT.RooRealVar("m", "m", 0, -10, 10)
s = ROOT.RooRealVar("s", "s", 2, 1, 50)
sig = ROOT.RooGaussian("sig", "sig", x, m, s)
m2 = ROOT.RooRealVar("m2", "m2", -1, -10, 10)
s2 = ROOT.RooRealVar("s2", "s2", 6, 1, 50)
bkg = ROOT.RooGaussian("bkg", "bkg", x, m2, s2)
fsig = ROOT.RooRealVar("fsig", "fsig", 0.33, 0, 1)
model = ROOT.RooAddPdf("model", "model", ROOT.RooArgList(
sig, bkg), ROOT.RooArgList(fsig))
# Create binned dataset
x.setBins(25)
d = model.generateBinned(ROOT.RooArgSet(x), 1000)
# Perform fit and save fit result
r = model.fitTo(d, ROOT.RooFit.Save())
# Visualize fit error
# -------------------------------------
# Make plot frame
frame = x.frame(ROOT.RooFit.Bins(40), ROOT.RooFit.Title(
"P.d.f with visualized 1-sigma error band"))
d.plotOn(frame)
# Visualize 1-sigma error encoded in fit result 'r' as orange band using linear error propagation
# ROOT.This results in an error band that is by construction symmetric
#
# The linear error is calculated as
# error(x) = Z* F_a(x) * Corr(a,a') F_a'(x)
#
# where F_a(x) = [ f(x,a+da) - f(x,a-da) ] / 2,
#
# with f(x) = the plotted curve
# 'da' = error taken from the fit result
# Corr(a,a') = the correlation matrix from the fit result
# Z = requested significance 'Z sigma band'
#
# The linear method is fast (required 2*N evaluations of the curve, N is the number of parameters),
# but may not be accurate in the presence of strong correlations (~>0.9) and at Z>2 due to linear and
# Gaussian approximations made
#
model.plotOn(frame, ROOT.RooFit.VisualizeError(
r, 1), ROOT.RooFit.FillColor(ROOT.kOrange))
# Calculate error using sampling method and visualize as dashed red line.
#
# In self method a number of curves is calculated with variations of the parameter values, sampled
# from a multi-variate Gaussian pdf that is constructed from the fit results covariance matrix.
# The error(x) is determined by calculating a central interval that capture N% of the variations
# for each valye of x, N% is controlled by Z (i.e. Z=1 gives N=68%). The number of sampling curves
# is chosen to be such that at least 100 curves are expected to be outside the N% interval, is minimally
# 100 (e.g. Z=1.Ncurve=356, Z=2.Ncurve=2156)) Intervals from the sampling method can be asymmetric,
# and may perform better in the presence of strong correlations, may take
# (much) longer to calculate
model.plotOn(
frame,
ROOT.RooFit.VisualizeError(
r,
1,
ROOT.kFALSE),
ROOT.RooFit.DrawOption("L"),
ROOT.RooFit.LineWidth(2),
ROOT.RooFit.LineColor(
ROOT.kRed))
# Perform the same type of error visualization on the background component only.
# The VisualizeError() option can generally applied to _any_ kind of
# plot (components, asymmetries, etc..)
model.plotOn(
frame, ROOT.RooFit.VisualizeError(
r, 1), ROOT.RooFit.FillColor(
ROOT.kOrange), ROOT.RooFit.Components("bkg"))
model.plotOn(
frame,
ROOT.RooFit.VisualizeError(
r,
1,
ROOT.kFALSE),
ROOT.RooFit.DrawOption("L"),
ROOT.RooFit.LineWidth(2),
ROOT.RooFit.LineColor(
ROOT.kRed),
ROOT.RooFit.Components("bkg"),
ROOT.RooFit.LineStyle(
ROOT.kDashed))
# Overlay central value
model.plotOn(frame)
model.plotOn(frame, ROOT.RooFit.Components("bkg"),
ROOT.RooFit.LineStyle(ROOT.kDashed))
d.plotOn(frame)
frame.SetMinimum(0)
# Visualize partial fit error
# ------------------------------------------------------
# Make plot frame
frame2 = x.frame(ROOT.RooFit.Bins(40), ROOT.RooFit.Title(
"Visualization of 2-sigma partial error from (m,m2)"))
# Visualize partial error. For partial error visualization the covariance matrix is first reduced as follows
# ___ -1
# Vred = V22 = V11 - V12 * V22 * V21
#
# Where V11,V12,V21, represent a block decomposition of the covariance matrix into observables that
# are propagated (labeled by index '1') and that are not propagated (labeled by index '2'), V22bar
# is the Shur complement of V22, as shown above
#
# (Note that Vred is _not_ a simple sub-matrix of V)
# Propagate partial error due to shape parameters (m,m2) using linear and
# sampling method
model.plotOn(frame2, ROOT.RooFit.VisualizeError(
r, ROOT.RooArgSet(m, m2), 2), ROOT.RooFit.FillColor(ROOT.kCyan))
model.plotOn(frame2, ROOT.RooFit.Components("bkg"), ROOT.RooFit.VisualizeError(
r, ROOT.RooArgSet(m, m2), 2), ROOT.RooFit.FillColor(ROOT.kCyan))
model.plotOn(frame2)
model.plotOn(frame2, ROOT.RooFit.Components("bkg"),
ROOT.RooFit.LineStyle(ROOT.kDashed))
frame2.SetMinimum(0)
# Make plot frame
frame3 = x.frame(ROOT.RooFit.Bins(40), ROOT.RooFit.Title(
"Visualization of 2-sigma partial error from (s,s2)"))
# Propagate partial error due to yield parameter using linear and sampling
# method
model.plotOn(frame3, ROOT.RooFit.VisualizeError(
r, ROOT.RooArgSet(s, s2), 2), ROOT.RooFit.FillColor(ROOT.kGreen))
model.plotOn(frame3, ROOT.RooFit.Components("bkg"), ROOT.RooFit.VisualizeError(
r, ROOT.RooArgSet(s, s2), 2), ROOT.RooFit.FillColor(ROOT.kGreen))
model.plotOn(frame3)
model.plotOn(frame3, ROOT.RooFit.Components("bkg"),
ROOT.RooFit.LineStyle(ROOT.kDashed))
frame3.SetMinimum(0)
# Make plot frame
frame4 = x.frame(ROOT.RooFit.Bins(40), ROOT.RooFit.Title(
"Visualization of 2-sigma partial error from fsig"))
# Propagate partial error due to yield parameter using linear and sampling
# method
model.plotOn(frame4, ROOT.RooFit.VisualizeError(
r, ROOT.RooArgSet(fsig), 2), ROOT.RooFit.FillColor(ROOT.kMagenta))
model.plotOn(frame4, ROOT.RooFit.Components("bkg"), ROOT.RooFit.VisualizeError(
r, ROOT.RooArgSet(fsig), 2), ROOT.RooFit.FillColor(ROOT.kMagenta))
model.plotOn(frame4)
model.plotOn(frame4, ROOT.RooFit.Components("bkg"),
ROOT.RooFit.LineStyle(ROOT.kDashed))
frame4.SetMinimum(0)
c = ROOT.TCanvas("rf610_visualerror", "rf610_visualerror", 800, 800)
c.Divide(2, 2)
c.cd(1)
ROOT.gPad.SetLeftMargin(0.15)
frame.GetYaxis().SetTitleOffset(1.4)
frame.Draw()
c.cd(2)
ROOT.gPad.SetLeftMargin(0.15)
frame2.GetYaxis().SetTitleOffset(1.6)
frame2.Draw()
c.cd(3)
ROOT.gPad.SetLeftMargin(0.15)
frame3.GetYaxis().SetTitleOffset(1.6)
frame3.Draw()
c.cd(4)
ROOT.gPad.SetLeftMargin(0.15)
frame4.GetYaxis().SetTitleOffset(1.6)
frame4.Draw()
c.SaveAs("rf610_visualerror.png")
Date
February 2018
Authors
Clemens Lange, Wouter Verkerke (C++ version)

Definition in file rf610_visualerror.py.