# Rf 6 1 0_Visualerror¶

Likelihood and minimization: visualization of errors from a covariance matrix

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 Friday, May 13, 2022 at 09:41 AM.

In [1]:
import ROOT

Welcome to JupyROOT 6.27/01


## Setup example fit¶

Create sum of two Gaussians pdf with factory

In [2]:
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", [sig, bkg], [fsig])


Create binned dataset

In [3]:
x.setBins(25)
d = model.generateBinned({x}, 1000)


Perform fit and save fit result

In [4]:
r = model.fitTo(d, Save=True)

[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: activating const optimization
[#1] INFO:Minimization --  The following expressions will be evaluated in cache-and-track mode: (sig,bkg)
**********
**    1 **SET PRINT           1
**********
**********
**********
PARAMETER DEFINITIONS:
NO.   NAME         VALUE      STEP SIZE      LIMITS
1 fsig         3.30000e-01  1.00000e-01    0.00000e+00  1.00000e+00
2 m            0.00000e+00  2.00000e+00   -1.00000e+01  1.00000e+01
3 m2          -1.00000e+00  2.00000e+00   -1.00000e+01  1.00000e+01
4 s            2.00000e+00  5.00000e-01    1.00000e+00  5.00000e+01
5 s2           6.00000e+00  2.50000e+00    1.00000e+00  5.00000e+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=2770.05 FROM MIGRAD    STATUS=INITIATE       20 CALLS          21 TOTAL
EDM= unknown      STRATEGY= 1      NO ERROR MATRIX
EXT PARAMETER               CURRENT GUESS       STEP         FIRST
NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE
1  fsig         3.30000e-01   1.00000e-01   2.14988e-01  -9.80006e+00
2  m            0.00000e+00   2.00000e+00   2.01358e-01  -3.95919e+01
3  m2          -1.00000e+00   2.00000e+00   2.02430e-01   3.98515e+01
4  s            2.00000e+00   5.00000e-01   7.46809e-02   2.95415e+01
5  s2           6.00000e+00   2.50000e+00   1.74125e-01   4.56667e+01
ERR DEF= 0.5
MIGRAD WILL VERIFY CONVERGENCE AND ERROR MATRIX.
COVARIANCE MATRIX CALCULATED SUCCESSFULLY
FCN=2767.65 FROM MIGRAD    STATUS=CONVERGED     125 CALLS         126 TOTAL
EDM=1.56678e-05    STRATEGY= 1      ERROR MATRIX ACCURATE
EXT PARAMETER                                   STEP         FIRST
NO.   NAME      VALUE            ERROR          SIZE      DERIVATIVE
1  fsig         2.98883e-01   6.74303e-02   2.39863e-03   7.00359e-03
2  m            3.08816e-01   2.09098e-01   6.83546e-04  -2.93512e-02
3  m2          -1.31219e+00   3.64277e-01   9.46152e-04   8.97852e-02
4  s            1.78229e+00   2.51997e-01   9.25951e-04  -2.49363e-02
5  s2           5.51197e+00   4.85498e-01   6.94615e-04   1.27501e-01
ERR DEF= 0.5
EXTERNAL ERROR MATRIX.    NDIM=  25    NPAR=  5    ERR DEF=0.5
4.580e-03 -3.800e-03 -1.557e-02  1.331e-02  2.642e-02
-3.800e-03  4.373e-02 -3.141e-03 -1.195e-02 -2.959e-02
-1.557e-02 -3.141e-03  1.328e-01 -4.509e-02 -1.100e-01
1.331e-02 -1.195e-02 -4.509e-02  6.354e-02  7.253e-02
2.642e-02 -2.959e-02 -1.100e-01  7.253e-02  2.358e-01
PARAMETER  CORRELATION COEFFICIENTS
NO.  GLOBAL      1      2      3      4      5
1  0.89430   1.000 -0.269 -0.632  0.780  0.804
2  0.43384  -0.269  1.000 -0.041 -0.227 -0.291
3  0.70478  -0.632 -0.041  1.000 -0.491 -0.621
4  0.78303   0.780 -0.227 -0.491  1.000  0.593
5  0.82883   0.804 -0.291 -0.621  0.593  1.000
**********
**    7 **SET ERR         0.5
**********
**********
**    8 **SET PRINT           1
**********
**********
**    9 **HESSE        2500
**********
COVARIANCE MATRIX CALCULATED SUCCESSFULLY
FCN=2767.65 FROM HESSE     STATUS=OK             31 CALLS         157 TOTAL
EDM=1.56685e-05    STRATEGY= 1      ERROR MATRIX ACCURATE
EXT PARAMETER                                INTERNAL      INTERNAL
NO.   NAME      VALUE            ERROR       STEP SIZE       VALUE
1  fsig         2.98883e-01   6.78952e-02   4.79727e-04  -4.13956e-01
2  m            3.08816e-01   2.09026e-01   1.36709e-04   3.08865e-02
3  m2          -1.31219e+00   3.67042e-01   1.89230e-04  -1.31599e-01
4  s            1.78229e+00   2.53102e-01   1.85190e-04  -1.31741e+00
5  s2           5.51197e+00   4.89300e-01   1.38923e-04  -9.54177e-01
ERR DEF= 0.5
EXTERNAL ERROR MATRIX.    NDIM=  25    NPAR=  5    ERR DEF=0.5
4.644e-03 -3.811e-03 -1.596e-02  1.350e-02  2.692e-02
-3.811e-03  4.370e-02 -2.970e-03 -1.206e-02 -2.967e-02
-1.596e-02 -2.970e-03  1.348e-01 -4.619e-02 -1.129e-01
1.350e-02 -1.206e-02 -4.619e-02  6.410e-02  7.402e-02
2.692e-02 -2.967e-02 -1.129e-01  7.402e-02  2.395e-01
PARAMETER  CORRELATION COEFFICIENTS
NO.  GLOBAL      1      2      3      4      5
1  0.89584   1.000 -0.268 -0.638  0.782  0.807
2  0.43319  -0.268  1.000 -0.039 -0.228 -0.290
3  0.71012  -0.638 -0.039  1.000 -0.497 -0.628
4  0.78518   0.782 -0.228 -0.497  1.000  0.597
5  0.83175   0.807 -0.290 -0.628  0.597  1.000
[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: deactivating const optimization


## Visualize fit error¶

Make plot frame

In [5]:
frame = x.frame(Bins=40, Title="P.d.f with visualized 1-sigma error band")
d.plotOn(frame)

Out[5]:
<cppyy.gbl.RooPlot object at 0x9957890>

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

In [6]:
model.plotOn(frame, VisualizeError=(r, 1), FillColor="kOrange")

Out[6]:
<cppyy.gbl.RooPlot object at 0x9957890>
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.


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

In [7]:
model.plotOn(frame, VisualizeError=(r, 1, False), DrawOption="L", LineWidth=2, LineColor="r")

Out[7]:
<cppyy.gbl.RooPlot object at 0x9957890>
[#1] INFO:Plotting -- RooAbsReal::plotOn(model) INFO: visualizing 1-sigma uncertainties in parameters (m,s,fsig,m2,s2) from fit result fitresult_model_genData using 315 samplings.


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..)

In [8]:
model.plotOn(frame, VisualizeError=(r, 1), FillColor="kOrange", Components="bkg")
model.plotOn(
frame,
VisualizeError=(r, 1, False),
DrawOption="L",
LineWidth=2,
LineColor="r",
Components="bkg",
LineStyle="--",
)

Out[8]:
<cppyy.gbl.RooPlot object at 0x9957890>
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) directly selected PDF components: (bkg)
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) indirectly selected PDF components: ()
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) directly selected PDF components: (bkg)
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) indirectly selected PDF components: ()
[#1] INFO:Plotting -- RooAbsReal::plotOn(model) INFO: visualizing 1-sigma uncertainties in parameters (m,s,fsig,m2,s2) from fit result fitresult_model_genData using 315 samplings.


Overlay central value

In [9]:
model.plotOn(frame)
model.plotOn(frame, Components="bkg", LineStyle="--")
d.plotOn(frame)
frame.SetMinimum(0)

[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) directly selected PDF components: (bkg)
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) indirectly selected PDF components: ()


## Visualize partial fit error¶

Make plot frame

In [10]:
frame2 = x.frame(Bins=40, 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

In [11]:
model.plotOn(frame2, VisualizeError=(r, {m, m2}, 2), FillColor="c")
model.plotOn(frame2, Components="bkg", VisualizeError=(r, {m, m2}, 2), FillColor="c")

model.plotOn(frame2)
model.plotOn(frame2, Components="bkg", LineStyle="--")
frame2.SetMinimum(0)

[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) directly selected PDF components: (bkg)
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) indirectly selected PDF components: ()
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) directly selected PDF components: (bkg)
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) indirectly selected PDF components: ()


Make plot frame

In [12]:
frame3 = x.frame(Bins=40, Title="Visualization of 2-sigma partial error from (s,s2)")


Propagate partial error due to yield parameter using linear and sampling method

In [13]:
model.plotOn(frame3, VisualizeError=(r, {s, s2}, 2), FillColor="g")
model.plotOn(frame3, Components="bkg", VisualizeError=(r, {fsig}, 2), FillColor="g")

model.plotOn(frame3)
model.plotOn(frame3, Components="bkg", LineStyle="--")
frame3.SetMinimum(0)

[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) directly selected PDF components: (bkg)
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) indirectly selected PDF components: ()
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) directly selected PDF components: (bkg)
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) indirectly selected PDF components: ()


Make plot frame

In [14]:
frame4 = x.frame(Bins=40, Title="Visualization of 2-sigma partial error from fsig")


Propagate partial error due to yield parameter using linear and sampling method

In [15]:
model.plotOn(frame4, VisualizeError=(r, {fsig}, 2), FillColor="m")
model.plotOn(frame4, Components="bkg", VisualizeError=(r, {fsig}, 2), FillColor="m")

model.plotOn(frame4)
model.plotOn(frame4, Components="bkg", LineStyle="--")
frame4.SetMinimum(0)

c = ROOT.TCanvas("rf610_visualerror", "rf610_visualerror", 800, 800)
c.Divide(2, 2)
c.cd(1)
frame.GetYaxis().SetTitleOffset(1.4)
frame.Draw()
c.cd(2)
frame2.GetYaxis().SetTitleOffset(1.6)
frame2.Draw()
c.cd(3)
frame3.GetYaxis().SetTitleOffset(1.6)
frame3.Draw()
c.cd(4)
frame4.GetYaxis().SetTitleOffset(1.6)
frame4.Draw()

c.SaveAs("rf610_visualerror.png")

[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) directly selected PDF components: (bkg)
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) indirectly selected PDF components: ()
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#0] WARNING:Plotting -- Fill color or style was set for plotting "model", but these only have an effect when 'DrawOption("F")' for fill is used at the same time.
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) directly selected PDF components: (bkg)
[#1] INFO:Plotting -- RooAbsPdf::plotOn(model) indirectly selected PDF components: ()

Info in <TCanvas::Print>: png file rf610_visualerror.png has been created


Draw all canvases

In [16]:
from ROOT import gROOT
gROOT.GetListOfCanvases().Draw()