Binomial fitter for the division of two histograms.

Use when you need to calculate a selection's efficiency from two histograms, one containing all entries, and one containing the subset of these entries that pass the selection, and when you have a parametrization available for the efficiency as a function of the variable(s) under consideration.

A very common problem when estimating efficiencies is that of error estimation: when no other information is available than the total number of events N and the selected number n, the best estimate for the selection efficiency p is n/N. Standard binomial statistics dictates that the uncertainty (this presupposes sufficiently high statistics that an approximation by a normal distribution is reasonable) on p, given N, is

\[ \sqrt{\frac{p(1-p)}{N}} \]

However, when p is estimated as n/N, fluctuations from the true p to its estimate become important, especially for low numbers of events, and giving rise to biased results.

When fitting a parametrized efficiency, these problems can largely be overcome, as a hypothesized true efficiency is available by construction. Even so, simply using the corresponding uncertainty still presupposes that Gaussian errors yields a reasonable approximation. When using, instead of binned efficiency histograms, the original numerator and denominator histograms, a binned maximum likelihood can be constructed as the product of bin-by-bin binomial probabilities to select n out of N events. Assuming that a correct parametrization of the efficiency is provided, this construction in general yields less biased results (and is much less sensitive to binning details).

A generic use of this method is given below (note that the method works for 2D and 3D histograms as well):

{
    TH1* denominator;              // denominator histogram
    TH1* numerator;                // corresponding numerator histogram
    TF1* eff;                      // efficiency parametrization
    ....                           // set step sizes and initial parameter
    ....                           //   values for the fit function
    ....                           // possibly also set ranges, see TF1::SetRange()
    TBinomialEfficiencyFitter* f = new TBinomialEfficiencyFitter(
                                 numerator, denominator);
    Int_t status = f->Fit(eff, "I");
    if (status == 0) {
      // if the fit was successful, display bin-by-bin efficiencies
      // as well as the result of the fit
      numerator->Sumw2();
      TH1* hEff = dynamic_cast<TH1*>(numerator->Clone("heff"));
      hEff->Divide(hEff, denominator, 1.0, 1.0, "B");
      hEff->Draw("E");
      eff->Draw("same");
    }
}

Note that this method cannot be expected to yield reliable results when using weighted histograms (because the likelihood computation will be incorrect).

Definition at line 42 of file TBinomialEfficiencyFitter.h.

Public Member Functions
	TBinomialEfficiencyFitter ()
	default constructor

	TBinomialEfficiencyFitter (const TH1 numerator, const TH1 denominator)
	Constructor.

	~TBinomialEfficiencyFitter () override
	destructor

Double_t	EvaluateFCN (const Double_t *par)

TFitResultPtr	Fit (TF1 f1, Option_t option="")
	Carry out the fit of the given function to the given histograms.

ROOT::Fit::Fitter *	GetFitter ()
	Provide access to the underlying fitter object.

TClass *	IsA () const override

void	Set (const TH1 numerator, const TH1 denominator)
	Initialize with a new set of inputs.

void	SetPrecision (Double_t epsilon)
	Set the required integration precision, see TF1::Integral()

void	Streamer (TBuffer &) override
	Stream an object of class TObject.

void	StreamerNVirtual (TBuffer &ClassDef_StreamerNVirtual_b)

Public Member Functions inherited from TObject
	TObject ()
	TObject constructor.

	TObject (const TObject &object)
	TObject copy ctor.

virtual	~TObject ()
	TObject destructor.

void	AbstractMethod (const char *method) const
	Use this method to implement an "abstract" method that you don't want to leave purely abstract.

virtual void	AppendPad (Option_t *option="")
	Append graphics object to current pad.

virtual void	Browse (TBrowser *b)
	Browse object. May be overridden for another default action.

ULong_t	CheckedHash ()
	Check and record whether this class has a consistent Hash/RecursiveRemove setup (*) and then return the regular Hash value for this object.

virtual const char *	ClassName () const
	Returns name of class to which the object belongs.

virtual void	Clear (Option_t *="")

virtual TObject *	Clone (const char *newname="") const
	Make a clone of an object using the Streamer facility.

virtual Int_t	Compare (const TObject *obj) const
	Compare abstract method.

virtual void	Copy (TObject &object) const
	Copy this to obj.

virtual void	Delete (Option_t *option="")
	Delete this object.

virtual Int_t	DistancetoPrimitive (Int_t px, Int_t py)
	Computes distance from point (px,py) to the object.

virtual void	Draw (Option_t *option="")
	Default Draw method for all objects.

virtual void	DrawClass () const
	Draw class inheritance tree of the class to which this object belongs.

virtual TObject *	DrawClone (Option_t *option="") const
	Draw a clone of this object in the current selected pad with: `gROOT->SetSelectedPad(c1)`.

virtual void	Dump () const
	Dump contents of object on stdout.

virtual void	Error (const char method, const char msgfmt,...) const
	Issue error message.

virtual void	Execute (const char method, const char params, Int_t *error=nullptr)
	Execute method on this object with the given parameter string, e.g.

virtual void	Execute (TMethod method, TObjArray params, Int_t *error=nullptr)
	Execute method on this object with parameters stored in the TObjArray.

virtual void	ExecuteEvent (Int_t event, Int_t px, Int_t py)
	Execute action corresponding to an event at (px,py).

virtual void	Fatal (const char method, const char msgfmt,...) const
	Issue fatal error message.

virtual TObject *	FindObject (const char *name) const
	Must be redefined in derived classes.

virtual TObject *	FindObject (const TObject *obj) const
	Must be redefined in derived classes.

virtual Option_t *	GetDrawOption () const
	Get option used by the graphics system to draw this object.

virtual const char *	GetIconName () const
	Returns mime type name of object.

virtual const char *	GetName () const
	Returns name of object.

virtual char *	GetObjectInfo (Int_t px, Int_t py) const
	Returns string containing info about the object at position (px,py).

virtual Option_t *	GetOption () const

virtual const char *	GetTitle () const
	Returns title of object.

virtual UInt_t	GetUniqueID () const
	Return the unique object id.

virtual Bool_t	HandleTimer (TTimer *timer)
	Execute action in response of a timer timing out.

virtual ULong_t	Hash () const
	Return hash value for this object.

Bool_t	HasInconsistentHash () const
	Return true is the type of this object is known to have an inconsistent setup for Hash and RecursiveRemove (i.e.

virtual void	Info (const char method, const char msgfmt,...) const
	Issue info message.

virtual Bool_t	InheritsFrom (const char *classname) const
	Returns kTRUE if object inherits from class "classname".

virtual Bool_t	InheritsFrom (const TClass *cl) const
	Returns kTRUE if object inherits from TClass cl.

virtual void	Inspect () const
	Dump contents of this object in a graphics canvas.

void	InvertBit (UInt_t f)

Bool_t	IsDestructed () const
	IsDestructed.

virtual Bool_t	IsEqual (const TObject *obj) const
	Default equal comparison (objects are equal if they have the same address in memory).

virtual Bool_t	IsFolder () const
	Returns kTRUE in case object contains browsable objects (like containers or lists of other objects).

R__ALWAYS_INLINE Bool_t	IsOnHeap () const

virtual Bool_t	IsSortable () const

R__ALWAYS_INLINE Bool_t	IsZombie () const

virtual void	ls (Option_t *option="") const
	The ls function lists the contents of a class on stdout.

void	MayNotUse (const char *method) const
	Use this method to signal that a method (defined in a base class) may not be called in a derived class (in principle against good design since a child class should not provide less functionality than its parent, however, sometimes it is necessary).

virtual Bool_t	Notify ()
	This method must be overridden to handle object notification (the base implementation is no-op).

void	Obsolete (const char method, const char asOfVers, const char *removedFromVers) const
	Use this method to declare a method obsolete.

void	operator delete (void *ptr)
	Operator delete.

void	operator delete[] (void *ptr)
	Operator delete [].

void *	operator new (size_t sz)

void *	operator new (size_t sz, void *vp)

void *	operator new[] (size_t sz)

void *	operator new[] (size_t sz, void *vp)

TObject &	operator= (const TObject &rhs)
	TObject assignment operator.

virtual void	Paint (Option_t *option="")
	This method must be overridden if a class wants to paint itself.

virtual void	Pop ()
	Pop on object drawn in a pad to the top of the display list.

virtual void	Print (Option_t *option="") const
	This method must be overridden when a class wants to print itself.

virtual Int_t	Read (const char *name)
	Read contents of object with specified name from the current directory.

virtual void	RecursiveRemove (TObject *obj)
	Recursively remove this object from a list.

void	ResetBit (UInt_t f)

virtual void	SaveAs (const char filename="", Option_t option="") const
	Save this object in the file specified by filename.

virtual void	SavePrimitive (std::ostream &out, Option_t *option="")
	Save a primitive as a C++ statement(s) on output stream "out".

void	SetBit (UInt_t f)

void	SetBit (UInt_t f, Bool_t set)
	Set or unset the user status bits as specified in f.

virtual void	SetDrawOption (Option_t *option="")
	Set drawing option for object.

virtual void	SetUniqueID (UInt_t uid)
	Set the unique object id.

void	StreamerNVirtual (TBuffer &ClassDef_StreamerNVirtual_b)

virtual void	SysError (const char method, const char msgfmt,...) const
	Issue system error message.

R__ALWAYS_INLINE Bool_t	TestBit (UInt_t f) const

Int_t	TestBits (UInt_t f) const

virtual void	UseCurrentStyle ()
	Set current style settings in this object This function is called when either TCanvas::UseCurrentStyle or TROOT::ForceStyle have been invoked.

virtual void	Warning (const char method, const char msgfmt,...) const
	Issue warning message.

virtual Int_t	Write (const char *name=nullptr, Int_t option=0, Int_t bufsize=0)
	Write this object to the current directory.

virtual Int_t	Write (const char *name=nullptr, Int_t option=0, Int_t bufsize=0) const
	Write this object to the current directory.

Static Public Member Functions
static TClass *	Class ()

static const char *	Class_Name ()

static constexpr Version_t	Class_Version ()

static const char *	DeclFileName ()

Static Public Member Functions inherited from TObject
static TClass *	Class ()

static const char *	Class_Name ()

static constexpr Version_t	Class_Version ()

static const char *	DeclFileName ()

static Longptr_t	GetDtorOnly ()
	Return destructor only flag.

static Bool_t	GetObjectStat ()
	Get status of object stat flag.

static void	SetDtorOnly (void *obj)
	Set destructor only flag.

static void	SetObjectStat (Bool_t stat)
	Turn on/off tracking of objects in the TObjectTable.

Protected Attributes
Bool_t	fAverage
	True if the fit function must be averaged over the bin.

TH1 *	fDenominator
	Denominator histogram.

Double_t	fEpsilon
	Precision required for function integration (option "I")

Bool_t	fFitDone
	Set to kTRUE when the fit has been done.

ROOT::Fit::Fitter *	fFitter
	pointer to the real fitter

TF1 *	fFunction
	Function to fit.

TH1 *	fNumerator
	Numerator histogram.

Bool_t	fRange
	True if the fit range must be taken from the function range.

Private Member Functions
void	ComputeFCN (Double_t &f, const Double_t *par)
	Compute the likelihood.

Additional Inherited Members
Public Types inherited from TObject
enum	{ kIsOnHeap = 0x01000000 , kNotDeleted = 0x02000000 , kZombie = 0x04000000 , kInconsistent = 0x08000000 , kBitMask = 0x00ffffff }

enum	{ kSingleKey = (1ULL << ( 0 )) , kOverwrite = (1ULL << ( 1 )) , kWriteDelete = (1ULL << ( 2 )) }

enum	EDeprecatedStatusBits { kObjInCanvas = (1ULL << ( 3 )) }

enum	EStatusBits { kCanDelete = (1ULL << ( 0 )) , kMustCleanup = (1ULL << ( 3 )) , kIsReferenced = (1ULL << ( 4 )) , kHasUUID = (1ULL << ( 5 )) , kCannotPick = (1ULL << ( 6 )) , kNoContextMenu = (1ULL << ( 8 )) , kInvalidObject = (1ULL << ( 13 )) }

Protected Types inherited from TObject
enum	{ kOnlyPrepStep = (1ULL << ( 3 )) }

Protected Member Functions inherited from TObject
virtual void	DoError (int level, const char location, const char fmt, va_list va) const
	Interface to ErrorHandler (protected).

void	MakeZombie ()

#include <TBinomialEfficiencyFitter.h>

Inheritance diagram for TBinomialEfficiencyFitter:

[legend]

Constructor & Destructor Documentation

◆ TBinomialEfficiencyFitter() [1/2]

TBinomialEfficiencyFitter::TBinomialEfficiencyFitter ( )

default constructor

Definition at line 101 of file TBinomialEfficiencyFitter.cxx.

◆ TBinomialEfficiencyFitter() [2/2]

TBinomialEfficiencyFitter::TBinomialEfficiencyFitter	(	const TH1 *	numerator,
		const TH1 *	denominator
	)

Constructor.

Note that no objects are copied, so it is up to the user to ensure that the histogram pointers remain valid.

Both histograms need to be "consistent". This is not checked here, but in TBinomialEfficiencyFitter::Fit().

Definition at line 121 of file TBinomialEfficiencyFitter.cxx.

◆ ~TBinomialEfficiencyFitter()

TBinomialEfficiencyFitter::~TBinomialEfficiencyFitter ( )

override

destructor

Definition at line 131 of file TBinomialEfficiencyFitter.cxx.

Member Function Documentation

◆ Class()

static TClass * TBinomialEfficiencyFitter::Class ( )

static

Returns: TClass describing this class

◆ Class_Name()

static const char * TBinomialEfficiencyFitter::Class_Name ( )

static

Returns: Name of this class

◆ Class_Version()

static constexpr Version_t TBinomialEfficiencyFitter::Class_Version ( )

inlinestaticconstexpr

Returns: Version of this class

Definition at line 73 of file TBinomialEfficiencyFitter.h.

◆ ComputeFCN()

void TBinomialEfficiencyFitter::ComputeFCN	(	Double_t &	f,
		const Double_t *	par
	)

private

Compute the likelihood.

Definition at line 329 of file TBinomialEfficiencyFitter.cxx.

◆ DeclFileName()

static const char * TBinomialEfficiencyFitter::DeclFileName ( )

inlinestatic

Returns: Name of the file containing the class declaration

Definition at line 73 of file TBinomialEfficiencyFitter.h.

◆ EvaluateFCN()

Double_t TBinomialEfficiencyFitter::EvaluateFCN ( const Double_t * par )

inline

Definition at line 67 of file TBinomialEfficiencyFitter.h.

◆ Fit()

TFitResultPtr TBinomialEfficiencyFitter::Fit	(	TF1 *	f1,
		Option_t *	option = `""`
	)

Carry out the fit of the given function to the given histograms.

If option "I" is used, the fit function will be averaged over the bin (the default is to evaluate it simply at the bin center).

If option "R" is used, the fit range will be taken from the fit function (the default is to use the entire histogram).

If option "S" a TFitResult object is returned and it can be used to obtain additional fit information, like covariance or correlation matrix.

Note that all parameter values, limits, and step sizes are copied from the input fit function f1 (so they should be set before calling this method. This is particularly relevant for the step sizes, taken to be the "error" set on input, as a null step size usually fixes the corresponding parameter. That is protected against, but in such cases an arbitrary starting step size will be used, and the reliability of the fit should be questioned). If parameters are to be fixed, this should be done by specifying non-null parameter limits, with lower limits larger than upper limits.

On output, f1 contains the fitted parameters and errors, as well as the number of degrees of freedom, and the goodness-of-fit estimator as given by S. Baker and R. Cousins, Nucl. Instr. Meth. A221 (1984) 437.

Definition at line 195 of file TBinomialEfficiencyFitter.cxx.