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 );

TH1D

1-D histogram with a double per channel (see TH1 documentation)}

Definition TH1.h:620

TSVDUnfold

SVD Approach to Data Unfolding.

Definition TSVDUnfold.h:46

TSVDUnfold::Unfold

TH1D * Unfold(Int_t kreg)

Perform the unfolding with regularisation parameter kreg.

Definition TSVDUnfold.cxx:241

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 46 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.

	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.

	TSVDUnfold (const TSVDUnfold &other)
	Copy constructor.

	~TSVDUnfold () override
	Destructor.

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.

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.

TH2D *	GetBCov () const
	Returns the covariance matrix.

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

Int_t	GetKReg () const

TH1D *	GetSV () const
	Returns singular values vector.

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.

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

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

TClass *	IsA () const override

void	SetNormalize (Bool_t normalize)

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

void	StreamerNVirtual (TBuffer &ClassDef_StreamerNVirtual_b)

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

Public Member Functions inherited from TObject
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	TObject copy ctor.

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R__ALWAYS_INLINE Bool_t	IsZombie () const

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void	ResetBit (UInt_t f)

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Int_t	TestBits (UInt_t f) const

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

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

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

void	InitHistos ()

Static Private Member Functions
static TVectorD	CompProd (const TVectorD &vec1, const TVectorD &vec2)
	Multiply entries of two vectors.

static Double_t	GetCurvature (const TVectorD &vec, const TMatrixD &curv)
	Compute curvature of vector.

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

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

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

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

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

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

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

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

Private Attributes
Class members
Int_t	fNdim
	! Truth and reconstructed dimensions

Int_t	fDdim
	! Derivative for curvature matrix

Bool_t	fNormalize
	! Normalize unfolded spectrum to 1

Int_t	fKReg
	! Regularisation parameter

TH1D *	fDHist
	! Distribution of d (for checking regularization)

TH1D *	fSVHist
	! Distribution of singular values

TH2D *	fXtau
	! Computed regularized covariance matrix

TH2D *	fXinv
	! Computed inverse of covariance matrix

Input histos
const TH1D *	fBdat
	Measured distribution (data)

TH2D *	fBcov
	Covariance matrix of measured distribution (data)

const TH1D *	fBini
	Reconstructed distribution (MC)

const TH1D *	fXini
	Truth distribution (MC)

const TH2D *	fAdet
	Detector response matrix.

Evaluation of covariance matrices
TH1D *	fToyhisto
	! Toy MC histogram

TH2D *	fToymat
	! Toy MC detector response matrix

Bool_t	fToyMode
	! Internal switch for covariance matrix propagation

Bool_t	fMatToyMode
	! Internal switch for evaluation of statistical uncertainties from response matrix

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 <TSVDUnfold.h>

Inheritance diagram for TSVDUnfold:

[legend]

Constructor & Destructor Documentation

◆ TSVDUnfold() [1/3]

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 77 of file TSVDUnfold.cxx.

◆ TSVDUnfold() [2/3]

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 130 of file TSVDUnfold.cxx.

◆ TSVDUnfold() [3/3]

TSVDUnfold::TSVDUnfold ( const TSVDUnfold & other )

Copy constructor.

Definition at line 175 of file TSVDUnfold.cxx.

◆ ~TSVDUnfold()

TSVDUnfold::~TSVDUnfold ( )

override

Destructor.

Definition at line 200 of file TSVDUnfold.cxx.

Member Function Documentation

◆ Class()

static TClass * TSVDUnfold::Class ( )

static

Returns: TClass describing this class

◆ Class_Name()

static const char * TSVDUnfold::Class_Name ( )

static

Returns: Name of this class

◆ Class_Version()

static constexpr Version_t TSVDUnfold::Class_Version ( )

inlinestaticconstexpr

Returns: Version of this class

Definition at line 156 of file TSVDUnfold.h.

◆ CompProd()

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

staticprivate

Multiply entries of two vectors.

Definition at line 706 of file TSVDUnfold.cxx.

◆ ComputeChiSquared()

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 884 of file TSVDUnfold.cxx.

◆ DeclFileName()

static const char * TSVDUnfold::DeclFileName ( )

inlinestatic

Returns: Name of the file containing the class declaration

Definition at line 156 of file TSVDUnfold.h.

◆ FillCurvatureMatrix()

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

private

Definition at line 723 of file TSVDUnfold.cxx.

◆ GetAdetCovMatrix()

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 515 of file TSVDUnfold.cxx.

◆ GetBCov()

TH2D * TSVDUnfold::GetBCov ( ) const

Returns the covariance matrix.

Definition at line 616 of file TSVDUnfold.cxx.

◆ GetCurvature()

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

staticprivate

Compute curvature of vector.

Definition at line 716 of file TSVDUnfold.cxx.

◆ GetD()

TH1D * TSVDUnfold::GetD ( ) const

Returns d vector (for choosing appropriate regularisation)

Definition at line 580 of file TSVDUnfold.cxx.

◆ GetKReg()

Int_t TSVDUnfold::GetKReg ( ) const

inline

Definition at line 86 of file TSVDUnfold.h.

◆ GetSV()

TH1D * TSVDUnfold::GetSV ( ) const

Returns singular values vector.

Definition at line 591 of file TSVDUnfold.cxx.

◆ GetUnfoldCovMatrix()

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 409 of file TSVDUnfold.cxx.

◆ GetXinv()

TH2D * TSVDUnfold::GetXinv ( ) const

Returns the computed inverse of the covariance matrix.

Definition at line 608 of file TSVDUnfold.cxx.

◆ GetXtau()

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 600 of file TSVDUnfold.cxx.

◆ H2M()

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

staticprivate

Fill 2D histogram into matrix.

Definition at line 648 of file TSVDUnfold.cxx.

◆ H2V()

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

staticprivate

Fill 1D histogram into vector.

Definition at line 624 of file TSVDUnfold.cxx.

◆ H2Verr()

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

staticprivate

Fill 1D histogram errors into vector.

Definition at line 632 of file TSVDUnfold.cxx.

◆ InitHistos()

void TSVDUnfold::InitHistos ( )

private

Definition at line 811 of file TSVDUnfold.cxx.

◆ IsA()

TClass * TSVDUnfold::IsA ( ) const

inlineoverridevirtual

Returns: TClass describing current object

Reimplemented from TObject.

Definition at line 156 of file TSVDUnfold.h.

◆ M2H()

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

staticprivate

Fill 2D histogram into matrix.

Definition at line 660 of file TSVDUnfold.cxx.

◆ MatDivVec()

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

staticprivate

Divide matrix entries by vector.

Definition at line 688 of file TSVDUnfold.cxx.

◆ RegularisedSymMatInvert()

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

staticprivate

naive regularised inversion cuts off small elements

Definition at line 831 of file TSVDUnfold.cxx.

◆ SetNormalize()

void TSVDUnfold::SetNormalize ( Bool_t normalize )

inline

Definition at line 66 of file TSVDUnfold.h.

◆ Streamer()

void TSVDUnfold::Streamer ( TBuffer & R__b )

overridevirtual

Stream an object of class TObject.

Reimplemented from TObject.

◆ StreamerNVirtual()

void TSVDUnfold::StreamerNVirtual ( TBuffer & ClassDef_StreamerNVirtual_b )

inline

Definition at line 156 of file TSVDUnfold.h.

◆ Unfold()

TH1D * TSVDUnfold::Unfold ( Int_t kreg )

Perform the unfolding with regularisation parameter kreg.

Definition at line 241 of file TSVDUnfold.cxx.

◆ V2H()

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

staticprivate

Fill vector into 1D histogram.

Definition at line 640 of file TSVDUnfold.cxx.

◆ VecDiv()

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

staticprivate

Divide entries of two vectors.

Definition at line 672 of file TSVDUnfold.cxx.

Member Data Documentation

◆ fAdet

const TH2D* TSVDUnfold::fAdet

private

Detector response matrix.

Definition at line 144 of file TSVDUnfold.h.

◆ fBcov

TH2D* TSVDUnfold::fBcov

private

Covariance matrix of measured distribution (data)

Definition at line 141 of file TSVDUnfold.h.

◆ fBdat

const TH1D* TSVDUnfold::fBdat

private

Measured distribution (data)

Definition at line 140 of file TSVDUnfold.h.

◆ fBini

const TH1D* TSVDUnfold::fBini

private

Reconstructed distribution (MC)

Definition at line 142 of file TSVDUnfold.h.

◆ fDdim

Int_t TSVDUnfold::fDdim

private

! Derivative for curvature matrix

Definition at line 129 of file TSVDUnfold.h.

◆ fDHist

TH1D* TSVDUnfold::fDHist

private

! Distribution of d (for checking regularization)

Definition at line 132 of file TSVDUnfold.h.

◆ fKReg

Int_t TSVDUnfold::fKReg

private

! Regularisation parameter

Definition at line 131 of file TSVDUnfold.h.

◆ fMatToyMode

Bool_t TSVDUnfold::fMatToyMode

private

! Internal switch for evaluation of statistical uncertainties from response matrix

Definition at line 152 of file TSVDUnfold.h.

◆ fNdim

Int_t TSVDUnfold::fNdim

private

! Truth and reconstructed dimensions

Definition at line 128 of file TSVDUnfold.h.

◆ fNormalize

Bool_t TSVDUnfold::fNormalize

private

! Normalize unfolded spectrum to 1

Definition at line 130 of file TSVDUnfold.h.

◆ fSVHist

TH1D* TSVDUnfold::fSVHist

private

! Distribution of singular values

Definition at line 133 of file TSVDUnfold.h.

◆ fToyhisto

TH1D* TSVDUnfold::fToyhisto

private

! Toy MC histogram

Definition at line 149 of file TSVDUnfold.h.

◆ fToymat

TH2D* TSVDUnfold::fToymat

private

! Toy MC detector response matrix

Definition at line 150 of file TSVDUnfold.h.

◆ fToyMode

Bool_t TSVDUnfold::fToyMode

private

! Internal switch for covariance matrix propagation

Definition at line 151 of file TSVDUnfold.h.

◆ fXini

const TH1D* TSVDUnfold::fXini

private

Truth distribution (MC)

Definition at line 143 of file TSVDUnfold.h.

◆ fXinv

TH2D* TSVDUnfold::fXinv

private

! Computed inverse of covariance matrix

Definition at line 135 of file TSVDUnfold.h.

◆ fXtau

TH2D* TSVDUnfold::fXtau

private

! Computed regularized covariance matrix

Definition at line 134 of file TSVDUnfold.h.

Libraries 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

Static Public Member Functions

Private Member Functions

Static Private Member Functions

Private Attributes

Additional Inherited Members

Constructor & Destructor Documentation

◆ TSVDUnfold() [1/3]

◆ TSVDUnfold() [2/3]

◆ TSVDUnfold() [3/3]

◆ ~TSVDUnfold()

Member Function Documentation

◆ Class()

◆ Class_Name()

◆ Class_Version()

◆ CompProd()

◆ ComputeChiSquared()

◆ DeclFileName()

◆ FillCurvatureMatrix()

◆ GetAdetCovMatrix()

◆ GetBCov()

◆ GetCurvature()

◆ GetD()

◆ GetKReg()

◆ GetSV()

◆ GetUnfoldCovMatrix()

◆ GetXinv()

◆ GetXtau()

◆ H2M()

◆ H2V()

◆ H2Verr()

◆ InitHistos()

◆ IsA()

◆ M2H()

◆ MatDivVec()

◆ RegularisedSymMatInvert()

◆ SetNormalize()

◆ Streamer()

◆ StreamerNVirtual()

◆ Unfold()

◆ V2H()

◆ VecDiv()

Member Data Documentation

◆ fAdet

◆ fBcov

◆ fBdat

◆ fBini

◆ fDdim

◆ fDHist

◆ fKReg

◆ fMatToyMode

◆ fNdim

◆ fNormalize

◆ fSVHist

◆ fToyhisto

◆ fToymat

◆ fToyMode

◆ fXini

◆ fXinv

◆ fXtau