Compound fit models

1D-case

Ostap offers a very easy way to build the compound fit models from the individual components. E.g.the case of the trivial fit model that consists of one signal and one background components:

signal      = ... 
background  = ...
model    = Fit1D ( signal = signal , background = backround ) ## <-- HERE! 
dataset  = ...
result , frame = model.fitTo ( dataset , draw = True ) ## fit and vizualize

The fit model can contains several signal and backround components, and also other components :

model    = Fit1D ( signal = signal ,
                   background       = backround , 
                   othersignals     = [ ... ]   , 
                   otherbackgrounds = [ ... ]   , 
                   others           = [ ... ]   )

In this case several signal, backgrounds and/or others components can be combined into single signal, backround and/or others components:

model    = Fit1D ( combine_signals     = True , 
                   combine_backgrounds = True , 
                   combine_others      = True , ... )

In practice it is very convinients approach is several signal/background/other componens are specified.

On default extended `RooAddPdf' fit model is created, however , one can force non-extended model:

model    = Fit1D ( extended = False , ... )

In this case one can also instruct the class Fit1D to create recursive (default) or non-recursive fit fractions:

model    = Fit1D ( extended = False , recursive = False , ... )

All components (signal/background/others) can be specified as Ostap-based models. Also one can provide them in a form of bare RooAbsPdf, but for this case one needs to provide also xvar-variable

mass  = ROOT.RooRealVar('mass','mass',2,3)
gauss = ROOT.RooGaussian( 'Gauss', 'Gauss', mass , ... )
model = Fit1D( signal = gauss , xvar = mass , ... )

For background components there is also an alternative way to specify it:

• None : RooPolynomial of zero degree (uniform distribution) will be created and used as background component
  • Attention: background=None does not imply the absence of background component
• negative integer n : Ostap model PolyPos_pdf will be created and used as background component. This model corresponds to the positive polynomial of degree -n. The polinomial is constrained to be non-negative for the whole considered interval of xvar. This constraint allows rather robust and stable fits, especialy for the low-statistics case.
• non-negative integer n: Ostap model Bkg_pdf, that is a product of the exponential function and the positive polynomial of degree n will be created and used as background component. Note:
  • The background=0 case corresponds to simple exponential backtround
  • Since the polynomial is constrained to be non-negative this PDF is very stable and robust, especually for the low-statistic case,
• as RooAbsReal object, in this case it is interpreted as the exponental slope

Actually, the separation into signal, background and other components is a bit arbitrary. However it is helpful for

• to define the meaningful names for the fit parameters
• to separate different components for visualisation, since different styles (lines, colors, etc) are used for differentr categories

Access to the model components

The individual components can be accessed using python properties

gaudd = Gauss_pdf ( ... 
model = Fit1D ( signal = gauss , ... ) 
print model.signal.sigma  ##   get sigma of Gauss 
print model.signal.mean   ##   get mean  of Gauss

Fit parameters

The parametters of the created RooAddPdf can be accessed via python properties, e.g. for extended fits:

gaudd = Gauss_pdf ( ... 
model = Fit1D ( signal = gauss , ... ) 
print 'signal yield(s):' , model.S   
print 'background(s):'   , model.B
print 'others: '         , model.C
model.S = 100   ## set value of signal component to be 100 events 
model.B.fix(50) ## fix the yield of the background component at 50  events 
model.draw()

Depending on the number of corresponsing componens and flags combine_signals, combine_backgrounds, combine_others these properties can be scalar values or arrays/tuples.

For combine_signal=True, combine_backgrounds =True, combine_others=True cases oen also gets properrties fS , fB and fC that corresponds to the fractions of individual signal/backgroud/others components for the compound signal/ signal/backgroud/others.

For non-extended fits, the main parameters are fractions:

gaudd = Gauss_pdf ( ... 
model = Fit1D ( signal = gauss , ... , extended = False ) 
...
print 'fractions:' , model.F

Extended multi-component fit model

model_ext1 = Models.Fit1D (
    name             = 'EXT1'     , 
    signal           =   signal_1 ,
    othersignals     = [ signal_2 , signal_3 ] ,
    background       =   wide_1   ,
    otherbackgrounds = [ wide_2 ] ,
    others           = [ narrow_1 , narrow_2 ] ,
    )

One can define some initial setting for fit-parameters:

model_ext1.S[0].value = 5000
model_ext1.S[1].value = 5000 
model_ext1.S[2].value = 5000 

model_ext1.B[0].value = 1700
model_ext1.B[1].value = 2300 

model_ext1.C[0].value =  500
model_ext1.C[1].value =  400

The fit itself is trivial

r, f = model_ext1.fitTo ( dataset , draw = False , silent = True )
r, f = model_ext1.fitTo ( dataset , draw = False , silent = True )

And accessing fit results is also simple:

print 'Signals               [S]:' , model_ext1.S
print 'Backgrounds           [B]:' , model_ext1.B
print 'Components            [C]:' , model_ext1.C
print 'Fractions             [F]:' , model_ext1.F
print 'Signal fractions     [fS]:' , model_ext1.fS
print 'Background fractions [fB]:' , model_ext1.fB
print 'Component fractions  [fC]:' , model_ext1.fC
print 'Yields           [yields]:' , model_ext1.yields
print 'Fractions     [fractions]:' , model_ext1.fractions

Extended fit model with compound components

model_ext2 = Models.Fit1D (
    name                = 'EXT2'     , 
    signal              =   signal_1 ,
    othersignals        = [ signal_2 , signal_3 ] ,
    background          =   wide_1   ,
    otherbackgrounds    = [ wide_2 ] ,
    others              = [ narrow_1 , narrow_2 ] ,
    #
    combine_signals     = True   , ## <-- HERE 
    combine_backgrounds = True   , ## <-- HERE 
    combine_others      = True   , ## <-- HERE

Setting the initial values of fit parameters is trivial:

model_ext2.S  = 5000
model_ext2.B  = 4200
model_ext2.C  =  700

model_ext2.fS[0].value = 0.33 
model_ext2.fS[1].value = 0.50

model_ext2.fB[0].value = 0.40 
model_ext2.fC[0].value = 0.60

The fit itself and access to fit parameters is the same as above.

Non-extended multi-component fit model with non-recursive fit-fractions

model_ne1 = Models.Fit1D (
    name                = 'NE1'                   , 
    signal              =   signal_1              , 
    othersignals        = [ signal_2 , signal_3 ] ,
    background          =   wide_1   ,
    otherbackgrounds    = [ wide_2 ] ,
    others              = [ narrow_1 , narrow_2 ] ,
    ##
    extended            = False , ## <--- HERE
    recursive           = False   ## <--- HERE 
    )

Setting the initial values of fit-fractions:

model_ne1.F[0].value = 0.25
model_ne1.F[1].value = 0.25
model_ne1.F[2].value = 0.25
model_ne1.F[3].value = 0.08
model_ne1.F[4].value = 0.12
model_ne1.F[5].value = 0.05

The fit itself and access to fit parameters is the same as above.

Non-extended multi-component fit model with recursive fit-fractions

model_ne2 = Models.Fit1D (
    name                = 'NE2'                   , 
    signal              =   signal_1              , 
    othersignals        = [ signal_2 , signal_3 ] ,
    background          =   wide_1   ,
    otherbackgrounds    = [ wide_2 ] ,
    others              = [ narrow_1 , narrow_2 ] , 
    ##
    extended            = False , ## <-- HERE 
    recursive           = True  , ## <-- HERE 
    )

Setting initial fit parameters:

model_ne2.F[0].value = 0.25
model_ne2.F[1].value = 0.33
model_ne2.F[2].value = 0.50
model_ne2.F[3].value = 0.37
model_ne2.F[4].value = 0.74
model_ne2.F[5].value = 0.50

The fit itself and access to fit parameters is the same as above.

Non-extended fit model with compound components and non-recursive fit-fractions

model_ne3 = Models.Fit1D (
    name                = 'NE2'                   , 
    signal              =   signal_1              , 
    othersignals        = [ signal_2 , signal_3 ] ,
    background          =   wide_1   ,
    otherbackgrounds    = [ wide_2 ] ,
    others              = [ narrow_1 , narrow_2 ] , 
    ##
    combine_signals     = True   , ## <--- HERE 
    combine_backgrounds = True   , ## <--- HERE 
    combine_others      = True   , ## <--- HERE 
    ##
    extended            = False  , ## <--- HERE 
    recursive           = False    ## <--- HERE 
    )

Setting the initial values:

model_ne3. F[0].value = 0.75
model_ne3. F[1].value = 0.30
model_ne3.fS[0].value = 0.33
model_ne3.fS[1].value = 0.50
model_ne3.fB[0].value = 0.41
model_ne3.fC[0].value = 0.58

The fit itself and access to fit parameters is the same as above.

Non-extended fit model with compound components and recursive fit-fractions

model_ne4 = Models.Fit1D (
    name                = 'NE4'                   , 
    signal              =   signal_1              , 
    othersignals        = [ signal_2 , signal_3 ] ,
    background          =   wide_1   ,
    otherbackgrounds    = [ wide_2 ] ,
    others              = [ narrow_1 , narrow_2 ] , 
    ##
    combine_signals     = True   , ## <--- HERE 
    combine_backgrounds = True   , ## <--- HERE
    combine_others      = True   , ## <--- HERE 
    ##
    extended            = False  , ## <--- HERE 
    recursive           = True     ## <--- HERE

Setting the initial values:

model_ne4. F[0].value = 0.75
model_ne4. F[1].value = 0.80
model_ne4.fS[0].value = 0.33
model_ne4.fS[1].value = 0.50
model_ne4.fB[0].value = 0.41
model_ne4.fC[0].value = 0.50

The fit itself and access to fit parameters is the same as above.

All the ways to deal with Fit1D objects are illustrated here:

The corresponding output can be inspected here

2D-case

Generic 2D-case

Symmetric 2D-case

3D-case

Generic 3D-case

Symmetric 3D-case

Mixed-symmetry3D-case