SVD Approach to Data Unfolding.

Reference: Nucl. Instrum. Meth. A372, 469 (1996) [hep-ph/9509307]

TSVDUnfold implements the singular value decomposition based unfolding method (see reference). Currently, the unfolding of one-dimensional histograms is supported, with the same number of bins for the measured and the unfolded spectrum.

The unfolding procedure is based on singular value decomposition of the response matrix. The regularisation of the unfolding is implemented via a discrete minimum-curvature condition.

Monte Carlo inputs:

xini: true underlying spectrum (TH1D, n bins)
bini: reconstructed spectrum (TH1D, n bins)
Adet: response matrix (TH2D, nxn bins)

Consider the unfolding of a measured spectrum bdat with covariance matrix Bcov (if not passed explicitly, a diagonal covariance will be built given the errors of bdat). The corresponding spectrum in the Monte Carlo is given by bini, with the true underlying spectrum given by xini. The detector response is described by Adet, with Adet filled with events (not probabilities) with the true observable on the y-axis and the reconstructed observable on the x-axis.

The measured distribution can be unfolded for any combination of resolution, efficiency and acceptance effects, provided an appropriate definition of xini and Adet.

The unfolding can be performed by

TSVDUnfold *tsvdunf = new TSVDUnfold( bdat, Bcov, bini, xini, Adet );
TH1D* unfresult = tsvdunf->Unfold( kreg );

where kreg determines the regularisation of the unfolding. In general, overregularisation (too small kreg) will bias the unfolded spectrum towards the Monte Carlo input, while underregularisation (too large kreg) will lead to large fluctuations in the unfolded spectrum. The optimal regularisation can be determined following guidelines in Nucl. Instrum. Meth. A372, 469 (1996) [hep-ph/9509307] using the distribution of the |d_i|<> that can be obtained by tsvdunf->GetD() and/or using pseudo-experiments.

Covariance matrices on the measured spectrum (for either the total uncertainties or individual sources of uncertainties) can be propagated to covariance matrices using the GetUnfoldCovMatrix method, which uses pseudo experiments for the propagation. In addition, GetAdetCovMatrix allows for the propagation of the statistical uncertainties on the response matrix using pseudo experiments. The covariance matrix corresponding to Bcov is also computed as described in Nucl. Instrum. Meth. A372, 469 (1996) [hep-ph/9509307] and can be obtained from tsvdunf->GetXtau() and its (regularisation independent) inverse from tsvdunf->GetXinv(). The distribution of singular values can be retrieved using tsvdunf->GetSV().

See also the tutorial for a toy example.

Definition at line 54 of file TSVDUnfold.h.

Public Member Functions
	TSVDUnfold (const TH1D bdat, const TH1D bini, const TH1D xini, const TH2D Adet)
	Alternative constructor User provides data and MC test spectra, as well as detector response matrix, diagonal covariance matrix of measured spectrum built from the uncertainties on measured spectrum. More...

	TSVDUnfold (const TH1D bdat, TH2D Bcov, const TH1D bini, const TH1D xini, const TH2D *Adet)
	Default constructor Initialisation of TSVDUnfold User provides data and MC test spectra, as well as detector response matrix and the covariance matrix of the measured distribution. More...

	TSVDUnfold (const TSVDUnfold &other)
	Copy constructor. More...

virtual	~TSVDUnfold ()
	Destructor. More...

void	SetNormalize (Bool_t normalize)

TH1D *	Unfold (Int_t kreg)
	Perform the unfolding with regularisation parameter kreg. More...

TH2D *	GetUnfoldCovMatrix (const TH2D *cov, Int_t ntoys, Int_t seed=1)
	Determine for given input error matrix covariance matrix of unfolded spectrum from toy simulation given the passed covariance matrix on measured spectrum "cov" - covariance matrix on the measured spectrum, to be propagated "ntoys" - number of pseudo experiments used for the propagation "seed" - seed for pseudo experiments Note that this covariance matrix will contain effects of forced normalisation if spectrum is normalised to unit area. More...

TH2D *	GetAdetCovMatrix (Int_t ntoys, Int_t seed=1)
	Determine covariance matrix of unfolded spectrum from finite statistics in response matrix using pseudo experiments "ntoys" - number of pseudo experiments used for the propagation "seed" - seed for pseudo experiments. More...

Int_t	GetKReg () const

TH1D *	GetD () const
	Returns d vector (for choosing appropriate regularisation) More...

TH1D *	GetSV () const
	Returns singular values vector. More...

TH2D *	GetXtau () const
	Returns the computed regularized covariance matrix corresponding to total uncertainties on measured spectrum as passed in the constructor. More...

TH2D *	GetXinv () const
	Returns the computed inverse of the covariance matrix. More...

TH2D *	GetBCov () const
	Returns the covariance matrix. More...

Double_t	ComputeChiSquared (const TH1D &truspec, const TH1D &unfspec)
	Helper routine to compute chi-squared between distributions using the computed inverse of the covariance matrix for the unfolded spectrum as given in paper. More...

Public Member Functions inherited from TObject
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virtual void	AppendPad (Option_t *option="")
	Append graphics object to current pad. More...

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	Browse object. May be overridden for another default action. More...

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virtual void	Clear (Option_t *="")

virtual TObject *	Clone (const char *newname="") const
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	Compare abstract method. More...

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	Computes distance from point (px,py) to the object. More...

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

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

virtual TObject *	DrawClone (Option_t *option="") const
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virtual void	Execute (const char method, const char params, Int_t *error=0)
	Execute method on this object with the given parameter string, e.g. More...

virtual void	Execute (TMethod method, TObjArray params, Int_t *error=0)
	Execute method on this object with parameters stored in the TObjArray. More...

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

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

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

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

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

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

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virtual Option_t *	GetOption () const

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

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

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

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

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

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

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

virtual Bool_t	IsSortable () const

Bool_t	IsOnHeap () const

Bool_t	IsZombie () const

virtual Bool_t	Notify ()
	This method must be overridden to handle object notification. More...

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

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

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

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

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

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

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

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

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

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

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

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

virtual Int_t	Write (const char *name=0, Int_t option=0, Int_t bufsize=0) const
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void *	operator new (size_t sz)

void *	operator new[] (size_t sz)

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

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

void	operator delete (void *ptr)
	Operator delete. More...

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

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

void	SetBit (UInt_t f)

void	ResetBit (UInt_t f)

Bool_t	TestBit (UInt_t f) const

Int_t	TestBits (UInt_t f) const

void	InvertBit (UInt_t f)

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

virtual void	Warning (const char method, const char msgfmt,...) const
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virtual void	Error (const char method, const char msgfmt,...) const
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	Use this method to declare a method obsolete. More...

Private Member Functions
void	FillCurvatureMatrix (TMatrixD &tCurv, TMatrixD &tC) const

void	InitHistos ()

Static Private Member Functions
static Double_t	GetCurvature (const TVectorD &vec, const TMatrixD &curv)
	Compute curvature of vector. More...

static void	H2V (const TH1D *histo, TVectorD &vec)
	Fill 1D histogram into vector. More...

static void	H2Verr (const TH1D *histo, TVectorD &vec)
	Fill 1D histogram errors into vector. More...

static void	V2H (const TVectorD &vec, TH1D &histo)
	Fill vector into 1D histogram. More...

static void	H2M (const TH2D *histo, TMatrixD &mat)
	Fill 2D histogram into matrix. More...

static void	M2H (const TMatrixD &mat, TH2D &histo)
	Fill 2D histogram into matrix. More...

static TMatrixD	MatDivVec (const TMatrixD &mat, const TVectorD &vec, Int_t zero=0)
	Divide matrix entries by vector. More...

static TVectorD	CompProd (const TVectorD &vec1, const TVectorD &vec2)
	Multiply entries of two vectors. More...

static TVectorD	VecDiv (const TVectorD &vec1, const TVectorD &vec2, Int_t zero=0)
	Divide entries of two vectors. More...

static void	RegularisedSymMatInvert (TMatrixDSym &mat, Double_t eps=1e-3)
	naive regularised inversion cuts off small elements More...

Private Attributes
Int_t	fNdim

Int_t	fDdim
	Truth and reconstructed dimensions. More...

Bool_t	fNormalize
	Derivative for curvature matrix. More...

Int_t	fKReg
	Normalize unfolded spectrum to 1. More...

TH1D *	fDHist
	Regularisation parameter. More...

TH1D *	fSVHist
	Distribution of d (for checking regularization) More...

TH2D *	fXtau
	Distribution of singular values. More...

TH2D *	fXinv
	Computed regularized covariance matrix. More...

const TH1D *	fBdat
	Computed inverse of covariance matrix. More...

TH2D *	fBcov

const TH1D *	fBini

const TH1D *	fXini

const TH2D *	fAdet

TH1D *	fToyhisto

TH2D *	fToymat
	Toy MC histogram. More...

Bool_t	fToyMode
	Toy MC detector response matrix. More...

Bool_t	fMatToyMode
	Internal switch for covariance matrix propagation. More...

Additional Inherited Members
Public Types inherited from TObject
enum	EStatusBits { kCanDelete = BIT(0), kMustCleanup = BIT(3), kObjInCanvas = BIT(3), kIsReferenced = BIT(4), kHasUUID = BIT(5), kCannotPick = BIT(6), kNoContextMenu = BIT(8), kInvalidObject = BIT(13) }

enum	{ kIsOnHeap = 0x01000000, kNotDeleted = 0x02000000, kZombie = 0x04000000, kBitMask = 0x00ffffff }

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

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Protected Member Functions inherited from TObject
void	MakeZombie ()

virtual void	DoError (int level, const char location, const char fmt, va_list va) const
	Interface to ErrorHandler (protected). More...

#include <TSVDUnfold.h>

Inheritance diagram for TSVDUnfold:

[legend]

Constructor & Destructor Documentation

TSVDUnfold::TSVDUnfold	(	const TH1D *	bdat,
		const TH1D *	bini,
		const TH1D *	xini,
		const TH2D *	Adet
	)

Alternative constructor User provides data and MC test spectra, as well as detector response matrix, diagonal covariance matrix of measured spectrum built from the uncertainties on measured spectrum.

Definition at line 79 of file TSVDUnfold.cxx.

TSVDUnfold::TSVDUnfold	(	const TH1D *	bdat,
		TH2D *	Bcov,
		const TH1D *	bini,
		const TH1D *	xini,
		const TH2D *	Adet
	)

Default constructor Initialisation of TSVDUnfold User provides data and MC test spectra, as well as detector response matrix and the covariance matrix of the measured distribution.

Definition at line 132 of file TSVDUnfold.cxx.

TSVDUnfold::TSVDUnfold ( const TSVDUnfold & other )

Copy constructor.

Definition at line 177 of file TSVDUnfold.cxx.

TSVDUnfold::~TSVDUnfold ( )

virtual

Destructor.

Definition at line 202 of file TSVDUnfold.cxx.

Member Function Documentation

TVectorD TSVDUnfold::CompProd	(	const TVectorD &	vec1,
		const TVectorD &	vec2
	)

staticprivate

Multiply entries of two vectors.

Definition at line 708 of file TSVDUnfold.cxx.

Referenced by Unfold().

Double_t TSVDUnfold::ComputeChiSquared	(	const TH1D &	truspec,
		const TH1D &	unfspec
	)

Helper routine to compute chi-squared between distributions using the computed inverse of the covariance matrix for the unfolded spectrum as given in paper.

Definition at line 886 of file TSVDUnfold.cxx.

void TSVDUnfold::FillCurvatureMatrix	(	TMatrixD &	tCurv,
		TMatrixD &	tC
	)		const

private

Definition at line 725 of file TSVDUnfold.cxx.

Referenced by Unfold().

TH2D * TSVDUnfold::GetAdetCovMatrix	(	Int_t	ntoys,
		Int_t	seed = `1`
	)

Determine covariance matrix of unfolded spectrum from finite statistics in response matrix using pseudo experiments "ntoys" - number of pseudo experiments used for the propagation "seed" - seed for pseudo experiments.

Definition at line 517 of file TSVDUnfold.cxx.

TH2D * TSVDUnfold::GetBCov ( ) const

Returns the covariance matrix.

Definition at line 618 of file TSVDUnfold.cxx.

Double_t TSVDUnfold::GetCurvature	(	const TVectorD &	vec,
		const TMatrixD &	curv
	)

staticprivate

Compute curvature of vector.

Definition at line 718 of file TSVDUnfold.cxx.

Referenced by Unfold().

TH1D * TSVDUnfold::GetD ( ) const

Returns d vector (for choosing appropriate regularisation)

Definition at line 582 of file TSVDUnfold.cxx.

Int_t TSVDUnfold::GetKReg ( ) const

inline

Definition at line 94 of file TSVDUnfold.h.

Referenced by GetAdetCovMatrix(), GetUnfoldCovMatrix(), and Unfold().

TH1D * TSVDUnfold::GetSV ( ) const

Returns singular values vector.

Definition at line 593 of file TSVDUnfold.cxx.

TH2D * TSVDUnfold::GetUnfoldCovMatrix	(	const TH2D *	cov,
		Int_t	ntoys,
		Int_t	seed = `1`
	)

Determine for given input error matrix covariance matrix of unfolded spectrum from toy simulation given the passed covariance matrix on measured spectrum "cov" - covariance matrix on the measured spectrum, to be propagated "ntoys" - number of pseudo experiments used for the propagation "seed" - seed for pseudo experiments Note that this covariance matrix will contain effects of forced normalisation if spectrum is normalised to unit area.

Definition at line 411 of file TSVDUnfold.cxx.

TH2D * TSVDUnfold::GetXinv ( ) const

Returns the computed inverse of the covariance matrix.

Definition at line 610 of file TSVDUnfold.cxx.

TH2D * TSVDUnfold::GetXtau ( ) const

Returns the computed regularized covariance matrix corresponding to total uncertainties on measured spectrum as passed in the constructor.

Note that this covariance matrix will not contain the effects of forced normalization if spectrum is normalized to unit area.

Definition at line 602 of file TSVDUnfold.cxx.

void TSVDUnfold::H2M	(	const TH2D *	histo,
		TMatrixD &	mat
	)

staticprivate

Fill 2D histogram into matrix.

Definition at line 650 of file TSVDUnfold.cxx.

Referenced by Unfold().

void TSVDUnfold::H2V	(	const TH1D *	histo,
		TVectorD &	vec
	)

staticprivate

Fill 1D histogram into vector.

Definition at line 626 of file TSVDUnfold.cxx.

Referenced by Unfold().

void TSVDUnfold::H2Verr	(	const TH1D *	histo,
		TVectorD &	vec
	)

staticprivate

Fill 1D histogram errors into vector.

Definition at line 634 of file TSVDUnfold.cxx.

Referenced by Unfold().

void TSVDUnfold::InitHistos ( )

private

Definition at line 813 of file TSVDUnfold.cxx.

Referenced by Unfold().

void TSVDUnfold::M2H	(	const TMatrixD &	mat,
		TH2D &	histo
	)

staticprivate

Fill 2D histogram into matrix.

Definition at line 662 of file TSVDUnfold.cxx.

Referenced by Unfold().

TMatrixD TSVDUnfold::MatDivVec	(	const TMatrixD &	mat,
		const TVectorD &	vec,
		Int_t	zero = `0`
	)

staticprivate

Divide matrix entries by vector.

Definition at line 690 of file TSVDUnfold.cxx.

void TSVDUnfold::RegularisedSymMatInvert	(	TMatrixDSym &	mat,
		Double_t	eps = `1e-3`
	)

staticprivate

naive regularised inversion cuts off small elements

Definition at line 833 of file TSVDUnfold.cxx.

void TSVDUnfold::SetNormalize ( Bool_t normalize )

inline

Definition at line 74 of file TSVDUnfold.h.

TH1D * TSVDUnfold::Unfold ( Int_t kreg )

Perform the unfolding with regularisation parameter kreg.

Definition at line 243 of file TSVDUnfold.cxx.

Referenced by GetAdetCovMatrix(), and GetUnfoldCovMatrix().

void TSVDUnfold::V2H	(	const TVectorD &	vec,
		TH1D &	histo
	)

staticprivate

Fill vector into 1D histogram.

Definition at line 642 of file TSVDUnfold.cxx.

Referenced by Unfold().

TVectorD TSVDUnfold::VecDiv	(	const TVectorD &	vec1,
		const TVectorD &	vec2,
		Int_t	zero = `0`
	)

staticprivate

Divide entries of two vectors.

Definition at line 674 of file TSVDUnfold.cxx.

Member Data Documentation

const TH2D* TSVDUnfold::fAdet

private

Definition at line 149 of file TSVDUnfold.h.

Referenced by GetAdetCovMatrix(), GetUnfoldCovMatrix(), InitHistos(), and Unfold().

TH2D* TSVDUnfold::fBcov

private

Definition at line 146 of file TSVDUnfold.h.

Referenced by GetBCov(), Unfold(), and ~TSVDUnfold().

const TH1D* TSVDUnfold::fBdat

private

Computed inverse of covariance matrix.

Definition at line 145 of file TSVDUnfold.h.

Referenced by GetUnfoldCovMatrix(), and Unfold().

const TH1D* TSVDUnfold::fBini

private

Definition at line 147 of file TSVDUnfold.h.

Referenced by Unfold().

Int_t TSVDUnfold::fDdim

private

Truth and reconstructed dimensions.

Definition at line 136 of file TSVDUnfold.h.

Referenced by FillCurvatureMatrix(), and TSVDUnfold().

TH1D* TSVDUnfold::fDHist

private

Regularisation parameter.

Definition at line 139 of file TSVDUnfold.h.

Referenced by GetD(), InitHistos(), Unfold(), and ~TSVDUnfold().

Int_t TSVDUnfold::fKReg

private

Normalize unfolded spectrum to 1.

Definition at line 138 of file TSVDUnfold.h.

Referenced by GetKReg(), and Unfold().

Bool_t TSVDUnfold::fMatToyMode

private

Internal switch for covariance matrix propagation.

Definition at line 155 of file TSVDUnfold.h.

Referenced by GetAdetCovMatrix(), and Unfold().

Int_t TSVDUnfold::fNdim

private

Definition at line 135 of file TSVDUnfold.h.

Referenced by GetAdetCovMatrix(), GetUnfoldCovMatrix(), InitHistos(), TSVDUnfold(), and Unfold().

Bool_t TSVDUnfold::fNormalize

private

Derivative for curvature matrix.

Definition at line 137 of file TSVDUnfold.h.

Referenced by SetNormalize(), and Unfold().

TH1D* TSVDUnfold::fSVHist

private

Distribution of d (for checking regularization)

Definition at line 140 of file TSVDUnfold.h.

Referenced by GetSV(), InitHistos(), Unfold(), and ~TSVDUnfold().

TH1D* TSVDUnfold::fToyhisto

private

Definition at line 152 of file TSVDUnfold.h.

Referenced by GetUnfoldCovMatrix(), Unfold(), and ~TSVDUnfold().

TH2D* TSVDUnfold::fToymat

private

Toy MC histogram.

Definition at line 153 of file TSVDUnfold.h.

Referenced by GetAdetCovMatrix(), Unfold(), and ~TSVDUnfold().

Bool_t TSVDUnfold::fToyMode

private

Toy MC detector response matrix.

Definition at line 154 of file TSVDUnfold.h.

Referenced by GetUnfoldCovMatrix(), and Unfold().

const TH1D* TSVDUnfold::fXini

private

Definition at line 148 of file TSVDUnfold.h.

Referenced by GetAdetCovMatrix(), and Unfold().

TH2D* TSVDUnfold::fXinv

private

Computed regularized covariance matrix.

Definition at line 142 of file TSVDUnfold.h.

Referenced by ComputeChiSquared(), GetXinv(), InitHistos(), Unfold(), and ~TSVDUnfold().

TH2D* TSVDUnfold::fXtau

private

Distribution of singular values.

Definition at line 141 of file TSVDUnfold.h.

Referenced by GetXtau(), InitHistos(), Unfold(), and ~TSVDUnfold().

Collaboration diagram for TSVDUnfold:

[legend]

The documentation for this class was generated from the following files:

hist/hist/inc/TSVDUnfold.h
hist/hist/src/TSVDUnfold.cxx

Public Member Functions

Private Member Functions

Static Private Member Functions

Private Attributes

Additional Inherited Members

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation