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 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. 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...

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...

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...

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

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

Int_t	GetKReg () const

TH1D *	GetSV () const
	Returns singular values vector. 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 *	GetXinv () const
	Returns the computed inverse of the covariance matrix. More...

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

void	SetNormalize (Bool_t normalize)

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

Public Member Functions inherited from TObject
	TObject ()
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	TObject copy ctor. More...

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	Returns kTRUE if object inherits from class "classname". More...

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void	InvertBit (UInt_t f)

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

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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. More...

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). More...

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void	Obsolete (const char method, const char asOfVers, const char *removedFromVers) const
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void	operator delete (void *ptr)
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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. More...

virtual void	Paint (Option_t *option="")
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virtual void	Pop ()
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virtual void	Print (Option_t *option="") const
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virtual Int_t	Read (const char *name)
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virtual void	RecursiveRemove (TObject *obj)
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void	ResetBit (UInt_t f)

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...

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)

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	SysError (const char method, const char msgfmt,...) const
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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. More...

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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. More...

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

static void	H2M (const TH2D *histo, TMatrixD &mat)
	Fill 2D histogram into matrix. 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	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 void	RegularisedSymMatInvert (TMatrixDSym &mat, Double_t eps=1e-3)
	naive regularised inversion cuts off small elements More...

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

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

Private Attributes
const TH2D *	fAdet

TH2D *	fBcov

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

const TH1D *	fBini

Int_t	fDdim
	Truth and reconstructed dimensions. More...

TH1D *	fDHist
	Regularisation parameter. More...

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

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

Int_t	fNdim

Bool_t	fNormalize
	Derivative for curvature matrix. More...

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

TH1D *	fToyhisto

TH2D *	fToymat
	Toy MC histogram. More...

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

const TH1D *	fXini

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

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

Additional Inherited Members
Public Types inherited from TObject
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Protected Member Functions inherited from TObject
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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 79 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 132 of file TSVDUnfold.cxx.

◆ TSVDUnfold() [3/3]

TSVDUnfold::TSVDUnfold ( const TSVDUnfold & other )

Copy constructor.

Definition at line 177 of file TSVDUnfold.cxx.

◆ ~TSVDUnfold()

TSVDUnfold::~TSVDUnfold ( )

virtual

Destructor.

Definition at line 202 of file TSVDUnfold.cxx.

Member Function Documentation

◆ CompProd()

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

staticprivate

Multiply entries of two vectors.

Definition at line 708 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 886 of file TSVDUnfold.cxx.

◆ FillCurvatureMatrix()

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

private

Definition at line 725 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 517 of file TSVDUnfold.cxx.

◆ GetBCov()

TH2D * TSVDUnfold::GetBCov ( ) const

Returns the covariance matrix.

Definition at line 618 of file TSVDUnfold.cxx.

◆ GetCurvature()

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

staticprivate

Compute curvature of vector.

Definition at line 718 of file TSVDUnfold.cxx.

◆ GetD()

TH1D * TSVDUnfold::GetD ( ) const

Returns d vector (for choosing appropriate regularisation)

Definition at line 582 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 593 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 411 of file TSVDUnfold.cxx.

◆ GetXinv()

TH2D * TSVDUnfold::GetXinv ( ) const

Returns the computed inverse of the covariance matrix.

Definition at line 610 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 602 of file TSVDUnfold.cxx.

◆ H2M()

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

staticprivate

Fill 2D histogram into matrix.

Definition at line 650 of file TSVDUnfold.cxx.

◆ H2V()

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

staticprivate

Fill 1D histogram into vector.

Definition at line 626 of file TSVDUnfold.cxx.

◆ H2Verr()

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

staticprivate

Fill 1D histogram errors into vector.

Definition at line 634 of file TSVDUnfold.cxx.

◆ InitHistos()

void TSVDUnfold::InitHistos ( )

private

Definition at line 813 of file TSVDUnfold.cxx.

◆ M2H()

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

staticprivate

Fill 2D histogram into matrix.

Definition at line 662 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 690 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 833 of file TSVDUnfold.cxx.

◆ SetNormalize()

void TSVDUnfold::SetNormalize ( Bool_t normalize )

inline

Definition at line 66 of file TSVDUnfold.h.

◆ Unfold()

TH1D * TSVDUnfold::Unfold ( Int_t kreg )

Perform the unfolding with regularisation parameter kreg.

Definition at line 243 of file TSVDUnfold.cxx.

◆ V2H()

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

staticprivate

Fill vector into 1D histogram.

Definition at line 642 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 674 of file TSVDUnfold.cxx.

Member Data Documentation

◆ fAdet

const TH2D* TSVDUnfold::fAdet

private

Definition at line 141 of file TSVDUnfold.h.

◆ fBcov

TH2D* TSVDUnfold::fBcov

private

Definition at line 138 of file TSVDUnfold.h.

◆ fBdat

const TH1D* TSVDUnfold::fBdat

private

Computed inverse of covariance matrix.

Definition at line 137 of file TSVDUnfold.h.

◆ fBini

const TH1D* TSVDUnfold::fBini

private

Definition at line 139 of file TSVDUnfold.h.

◆ fDdim

Int_t TSVDUnfold::fDdim

private

Truth and reconstructed dimensions.

Definition at line 128 of file TSVDUnfold.h.

◆ fDHist

TH1D* TSVDUnfold::fDHist

private

Regularisation parameter.

Definition at line 131 of file TSVDUnfold.h.

◆ fKReg

Int_t TSVDUnfold::fKReg

private

Normalize unfolded spectrum to 1.

Definition at line 130 of file TSVDUnfold.h.

◆ fMatToyMode

Bool_t TSVDUnfold::fMatToyMode

private

Internal switch for covariance matrix propagation.

Definition at line 147 of file TSVDUnfold.h.

◆ fNdim

Int_t TSVDUnfold::fNdim

private

Definition at line 127 of file TSVDUnfold.h.

◆ fNormalize

Bool_t TSVDUnfold::fNormalize

private

Derivative for curvature matrix.

Definition at line 129 of file TSVDUnfold.h.

◆ fSVHist

TH1D* TSVDUnfold::fSVHist

private

Distribution of d (for checking regularization)

Definition at line 132 of file TSVDUnfold.h.

◆ fToyhisto

TH1D* TSVDUnfold::fToyhisto

private

Definition at line 144 of file TSVDUnfold.h.

◆ fToymat

TH2D* TSVDUnfold::fToymat

private

Toy MC histogram.

Definition at line 145 of file TSVDUnfold.h.

◆ fToyMode

Bool_t TSVDUnfold::fToyMode

private

Toy MC detector response matrix.

Definition at line 146 of file TSVDUnfold.h.

◆ fXini

const TH1D* TSVDUnfold::fXini

private

Definition at line 140 of file TSVDUnfold.h.

◆ fXinv

TH2D* TSVDUnfold::fXinv

private

Computed regularized covariance matrix.

Definition at line 134 of file TSVDUnfold.h.

◆ fXtau

TH2D* TSVDUnfold::fXtau

private

Distribution of singular values.

Definition at line 133 of file TSVDUnfold.h.

Libraries for TSVDUnfold:

[legend]

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

◆ CompProd()

◆ ComputeChiSquared()

◆ FillCurvatureMatrix()

◆ GetAdetCovMatrix()

◆ GetBCov()

◆ GetCurvature()

◆ GetD()

◆ GetKReg()

◆ GetSV()

◆ GetUnfoldCovMatrix()

◆ GetXinv()

◆ GetXtau()

◆ H2M()

◆ H2V()

◆ H2Verr()

◆ InitHistos()

◆ M2H()

◆ MatDivVec()

◆ RegularisedSymMatInvert()

◆ SetNormalize()

◆ Unfold()

◆ V2H()

◆ VecDiv()

Member Data Documentation

◆ fAdet

◆ fBcov

◆ fBdat

◆ fBini

◆ fDdim

◆ fDHist

◆ fKReg

◆ fMatToyMode

◆ fNdim

◆ fNormalize

◆ fSVHist

◆ fToyhisto

◆ fToymat

◆ fToyMode

◆ fXini

◆ fXinv

◆ fXtau