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An algorithm to unfold distributions from detector to truth level.

TUnfold is used to decompose a measurement y into several sources x, given the measurement uncertainties and a matrix of migrations A. The method can be applied to a large number of problems, where the measured distribution y is a linear superposition of several Monte Carlo shapes. Beyond such a simple template fit, TUnfold has an adjustable regularisation term and also supports an optional constraint on the total number of events.

For most applications, it is better to use the derived class TUnfoldDensity instead of TUnfold. TUnfoldDensity adds various features to TUnfold, such as: background subtraction, propagation of systematic uncertainties, complex multidimensional arrangements of the bins. For innocent users, the most notable improvement of TUnfoldDensity over TUnfold are the getter functions. For TUnfold, histograms have to be booked by the user and the getter functions fill the histogram bins. TUnfoldDensity simply returns a new, already filled histogram.

If you use this software, please consider the following citation
S.Schmitt, JINST 7 (2012) T10003 [arXiv:1205.6201]
Detailed documentation and updates are available on http://www.desy.de/~sschmitt

Brief recipy to use TUnfold:

• a matrix (truth,reconstructed) is given as a two-dimensional histogram as argument to the constructor of TUnfold
• a vector of measurements is given as one-dimensional histogram using the SetInput() method
• The unfolding is performed
• either once with a fixed parameter tau, method DoUnfold(tau)
• or multiple times in a scan to determine the best chouce of tau, method ScanLCurve()
• Unfolding results are retrieved using various GetXXX() methods

Basic formulae:
χ2A=(Ax-y)TVyy-1(Ax-y)
χ2L=(x-f*x0)TLTL(x-f*x0)
χ2unf2A2χ2L+λΣi(Ax-y)i
x:result, A:probabilities, y:data, Vyy:data covariance, f:bias scale, x0:bias, L:regularisation conditions, τ:regularisation strength, λ:Lagrangian multiplier
Without area constraint, λ is set to zero, and χ2unf is minimized to determine x. With area constraint, both x and λ are determined.

Definition at line 106 of file TUnfold.h.

## Public Types

enum  EConstraint { kEConstraintNone =0 , kEConstraintArea =1 }
type of extra constraint More...

enum  EHistMap { kHistMapOutputHoriz = 0 , kHistMapOutputVert = 1 }
arrangement of axes for the response matrix (TH2 histogram) More...

enum  ERegMode {
kRegModeNone = 0 , kRegModeSize = 1 , kRegModeDerivative = 2 , kRegModeCurvature = 3 ,
kRegModeMixed = 4
}
choice of regularisation scheme More...

Public Types inherited from TObject
enum  {
kIsOnHeap = 0x01000000 , kNotDeleted = 0x02000000 , kZombie = 0x04000000 , kInconsistent = 0x08000000 ,
}

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 ))
}

## Public Member Functions

TUnfold (const TH2 *hist_A, EHistMap histmap, ERegMode regmode=kRegModeSize, EConstraint constraint=kEConstraintArea)
Set up response matrix and regularisation scheme.

TUnfold (void)
only for use by root streamer or derived classes

~TUnfold (void) override

virtual Double_t DoUnfold (Double_t tau)
perform the unfolding for a given regularisation parameter tau

Double_t DoUnfold (Double_t tau, const TH1 *hist_y, Double_t scaleBias=0.0)
perform the unfolding for a given input and regularisation

void GetBias (TH1 *bias, const Int_t *binMap=nullptr) const
get bias vector including bias scale

Double_t GetChi2A (void) const
get χ2A contribution determined in recent unfolding

Double_t GetChi2L (void) const
get χ2L contribution determined in recent unfolding

double GetDF (void) const
return the effecive number of degrees of freedom See e.g.

void GetDXDY (TH2 *dxdy) const
get matrix connecting input and output changes

void GetEmatrix (TH2 *ematrix, const Int_t *binMap=nullptr) const
get output covariance matrix, possibly cumulated over several bins

Double_t GetEpsMatrix (void) const
get numerical accuracy for Eigenvalue analysis when inverting matrices with rank problems

void GetFoldedOutput (TH1 *folded, const Int_t *binMap=nullptr) const
get unfolding result on detector level

void GetInput (TH1 *inputData, const Int_t *binMap=nullptr) const
Input vector of measurements.

void GetInputInverseEmatrix (TH2 *ematrix)
get inverse of the measurement's covariance matrix

void GetL (TH2 *l) const
get matrix of regularisation conditions

virtual Double_t GetLcurveX (void) const
get value on x-axis of L-curve determined in recent unfolding

virtual Double_t GetLcurveY (void) const
get value on y-axis of L-curve determined in recent unfolding

void GetLsquared (TH2 *lsquared) const
get matrix of regularisation conditions squared

Int_t GetNdf (void) const
get number of degrees of freedom determined in recent unfolding

void GetNormalisationVector (TH1 *s, const Int_t *binMap=nullptr) const
histogram of truth bins, determined from suming over the response matrix

Int_t GetNpar (void) const
get number of truth parameters determined in recent unfolding

Int_t GetNr (void) const
get number of regularisation conditions

void GetOutput (TH1 *output, const Int_t *binMap=nullptr) const
get output distribution, possibly cumulated over several bins

void GetProbabilityMatrix (TH2 *A, EHistMap histmap) const
get matrix of probabilities

Double_t GetRhoAvg (void) const
get average global correlation determined in recent unfolding

Double_t GetRhoI (TH1 *rhoi, const Int_t *binMap=nullptr, TH2 *invEmat=nullptr) const
get global correlation coefficiencts, possibly cumulated over several bins

void GetRhoIJ (TH2 *rhoij, const Int_t *binMap=nullptr) const
get correlation coefficiencts, possibly cumulated over several bins

Double_t GetRhoMax (void) const
get maximum global correlation determined in recent unfolding

TVectorD GetSqrtEvEmatrix (void) const

double GetSURE (void) const
return Stein's unbiased risk estimator See e.g.

Double_t GetTau (void) const
return regularisation parameter

TClassIsA () const override

Int_t RegularizeBins (int start, int step, int nbin, ERegMode regmode)
add regularisation conditions for a group of bins

Int_t RegularizeBins2D (int start_bin, int step1, int nbin1, int step2, int nbin2, ERegMode regmode)
add regularisation conditions for 2d unfolding

Int_t RegularizeCurvature (int left_bin, int center_bin, int right_bin, Double_t scale_left=1.0, Double_t scale_right=1.0)
add a regularisation condition on the curvature of three truth bin

Int_t RegularizeDerivative (int left_bin, int right_bin, Double_t scale=1.0)
add a regularisation condition on the difference of two truth bin

Int_t RegularizeSize (int bin, Double_t scale=1.0)
add a regularisation condition on the magnitude of a truth bin

virtual Int_t ScanLcurve (Int_t nPoint, Double_t tauMin, Double_t tauMax, TGraph **lCurve, TSpline **logTauX=nullptr, TSpline **logTauY=nullptr, TSpline **logTauCurvature=nullptr)
scan the L curve, determine tau and unfold at the final value of tau

virtual Int_t ScanSURE (Int_t nPoint, Double_t tauMin, Double_t tauMax, TGraph **logTauSURE=nullptr, TGraph **df_chi2A=nullptr, TGraph **lCurve=nullptr)
minimize Stein's unbiased risk estimator "SURE" using successive calls to DoUnfold at various tau.

void SetBias (const TH1 *bias)
set bias vector

void SetConstraint (EConstraint constraint)
set type of area constraint

void SetEpsMatrix (Double_t eps)
set numerical accuracy for Eigenvalue analysis when inverting matrices with rank problems

virtual Int_t SetInput (const TH1 *hist_y, Double_t scaleBias=0.0, Double_t oneOverZeroError=0.0, const TH2 *hist_vyy=nullptr, const TH2 *hist_vyy_inv=nullptr)
Define input data for subsequent calls to DoUnfold(tau)

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.

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 TObjectClone (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 TObjectDrawClone (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 TObjectFindObject (const char *name) const
Must be redefined in derived classes.

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

virtual Option_tGetDrawOption () 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_tGetOption () 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, void *vp)
Only called by placement new when throwing an exception.

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

void operator delete[] (void *ptr, void *vp)
Only called by placement new[] when throwing an exception.

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)

TObjectoperator= (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 TClassClass ()

static const char * Class_Name ()

static constexpr Version_t Class_Version ()

static const char * DeclFileName ()

static const char * GetTUnfoldVersion (void)
return a string describing the TUnfold version

Static Public Member Functions inherited from TObject
static TClassClass ()

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 Member Functions

void AddMSparse (TMatrixDSparse *dest, Double_t f, const TMatrixDSparse *src) const
add a sparse matrix, scaled by a factor, to another scaled matrix

Bool_t AddRegularisationCondition (Int_t i0, Double_t f0, Int_t i1=-1, Double_t f1=0., Int_t i2=-1, Double_t f2=0.)
add a row of regularisation conditions to the matrix L

Bool_t AddRegularisationCondition (Int_t nEle, const Int_t *indices, const Double_t *rowData)
add a row of regularisation conditions to the matrix L

void ClearHistogram (TH1 *h, Double_t x=0.) const
Initialize bin contents and bin errors for a given histogram.

virtual void ClearResults (void)
reset all results

TMatrixDSparseCreateSparseMatrix (Int_t nrow, Int_t ncol, Int_t nele, Int_t *row, Int_t *col, Double_t *data) const
create a sparse matrix, given the nonzero elements

virtual Double_t DoUnfold (void)
core unfolding algorithm

void ErrorMatrixToHist (TH2 *ematrix, const TMatrixDSparse *emat, const Int_t *binMap, Bool_t doClear) const
add up an error matrix, also respecting the bin mapping

const TMatrixDSparseGetAx (void) const
vector of folded-back result

Int_t GetBinFromRow (int ix) const
converts matrix row to truth histogram bin number

const TMatrixDSparseGetDXDAM (int i) const
matrix contributions of the derivative dx/dA

const TMatrixDSparseGetDXDAZ (int i) const
vector contributions of the derivative dx/dA

const TMatrixDSparseGetDXDtauSquared (void) const
vector of derivative dx/dtauSquared, using internal bin counting

const TMatrixDSparseGetDXDY (void) const
matrix of derivatives dx/dy

const TMatrixDSparseGetE (void) const
matrix E, using internal bin counting

const TMatrixDSparseGetEinv (void) const
matrix E-1, using internal bin counting

Int_t GetNx (void) const
returns internal number of output (truth) matrix rows

Int_t GetNy (void) const
returns the number of measurement bins

virtual TString GetOutputBinName (Int_t iBinX) const
Get bin name of an outpt bin.

Double_t GetRhoIFromMatrix (TH1 *rhoi, const TMatrixDSparse *eOrig, const Int_t *binMap, TH2 *invEmat) const

Int_t GetRowFromBin (int ix) const
converts truth histogram bin number to matrix row

const TMatrixDSparseGetVxx (void) const
covariance matrix of the result

const TMatrixDSparseGetVxxInv (void) const
inverse of covariance matrix of the result

const TMatrixDSparseGetVyyInv (void) const
inverse of covariance matrix of the data y

const TMatrixDGetX (void) const
vector of the unfolding result

TMatrixDSparseInvertMSparseSymmPos (const TMatrixDSparse *A, Int_t *rank) const
get the inverse or pseudo-inverse of a positive, sparse matrix

TMatrixDSparseMultiplyMSparseM (const TMatrixDSparse *a, const TMatrixD *b) const
multiply sparse matrix and a non-sparse matrix

TMatrixDSparseMultiplyMSparseMSparse (const TMatrixDSparse *a, const TMatrixDSparse *b) const
multiply two sparse matrices

TMatrixDSparseMultiplyMSparseMSparseTranspVector (const TMatrixDSparse *m1, const TMatrixDSparse *m2, const TMatrixTBase< Double_t > *v) const
calculate a sparse matrix product M1*V*M2T where the diagonal matrix V is given by a vector

TMatrixDSparseMultiplyMSparseTranspMSparse (const TMatrixDSparse *a, const TMatrixDSparse *b) const
multiply a transposed Sparse matrix with another Sparse matrix

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 ()

## Static Protected Member Functions

static void DeleteMatrix (TMatrixD **m)
delete matrix and invalidate pointer

static void DeleteMatrix (TMatrixDSparse **m)
delete sparse matrix and invalidate pointer

## Protected Attributes

TMatrixDSparsefA
response matrix A

Double_t fBiasScale
scale factor for the bias

EConstraint fConstraint
type of constraint to use for the unfolding

TArrayI fHistToX
mapping of histogram bins to matrix indices

TMatrixDSparsefL
regularisation conditions L

ERegMode fRegMode
type of regularisation

TArrayD fSumOverY
truth vector calculated from the non-normalized response matrix

Double_t fTauSquared
regularisation parameter tau squared

TMatrixDSparsefVyy
covariance matrix Vyy corresponding to y

TMatrixDfX0
bias vector x0

TArrayI fXToHist
mapping of matrix indices to histogram bins

TMatrixDfY
input (measured) data y

## Private Member Functions

void InitTUnfold (void)
initialize data menbers, for use in constructors

## Private Attributes

TMatrixDSparsefAx
result x folded back A*x

Double_t fChi2A
chi**2 contribution from (y-Ax)Vyy-1(y-Ax)

TMatrixDSparsefDXDAM [2]
matrix contribution to the of derivative dx_k/dA_ij

TMatrixDSparsefDXDAZ [2]
vector contribution to the of derivative dx_k/dA_ij

TMatrixDSparsefDXDtauSquared
derivative of the result wrt tau squared

TMatrixDSparsefDXDY
derivative of the result wrt dx/dy

TMatrixDSparsefE
matrix E

TMatrixDSparsefEinv
matrix E^(-1)

Double_t fEpsMatrix
machine accuracy used to determine matrix rank after eigenvalue analysis

Int_t fIgnoredBins
number of input bins which are dropped because they have error=nullptr

Double_t fLXsquared
chi**2 contribution from (x-s*x0)TLTL(x-s*x0)

Int_t fNdf
number of degrees of freedom

Double_t fRhoAvg
average global correlation coefficient

Double_t fRhoMax
maximum global correlation coefficient

TMatrixDSparsefVxx
covariance matrix Vxx

TMatrixDSparsefVxxInv
inverse of covariance matrix Vxx-1

TMatrixDSparsefVyyInv
inverse of the input covariance matrix Vyy-1

TMatrixDfX
unfolding result x

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

#include <TUnfold.h>

Inheritance diagram for TUnfold:
[legend]

## ◆ EConstraint

 enum TUnfold::EConstraint

type of extra constraint

Enumerator
kEConstraintNone

use no extra constraint

kEConstraintArea

enforce preservation of the area

Definition at line 112 of file TUnfold.h.

## ◆ EHistMap

 enum TUnfold::EHistMap

arrangement of axes for the response matrix (TH2 histogram)

Enumerator
kHistMapOutputHoriz

truth level on x-axis of the response matrix

kHistMapOutputVert

truth level on y-axis of the response matrix

Definition at line 142 of file TUnfold.h.

## ◆ ERegMode

 enum TUnfold::ERegMode

choice of regularisation scheme

Enumerator
kRegModeNone

no regularisation, or defined later by RegularizeXXX() methods

kRegModeSize

regularise the amplitude of the output distribution

kRegModeDerivative

regularize the 1st derivative of the output distribution

kRegModeCurvature

regularize the 2nd derivative of the output distribution

kRegModeMixed

mixed regularisation pattern

Definition at line 122 of file TUnfold.h.

## ◆ TUnfold() [1/2]

 TUnfold::TUnfold ( const TH2 * hist_A, EHistMap histmap, ERegMode regmode = kRegModeSize, EConstraint constraint = kEConstraintArea )

Set up response matrix and regularisation scheme.

Parameters
 [in] hist_A matrix of MC events that describes the migrations [in] histmap mapping of the histogram axes [in] regmode (default=kRegModeSize) global regularisation mode [in] constraint (default=kEConstraintArea) type of constraint

Treatment of overflow bins in the matrix hist_A

• Events reconstructed in underflow or overflow bins are counted as inefficiency. They have to be filled properly.
• Events where the truth level is in underflow or overflow bins are treated as a part of the generator level distribution. The full truth level distribution (including underflow and overflow) is unfolded.

If unsure, do the following:

• store evens where the truth is in underflow or overflow (sometimes called "fakes") in a separate TH1. Ensure that the truth-level underflow and overflow bins of hist_A are all zero.
• the fakes are background to the measurement. Use the classes TUnfoldSys and TUnfoldDensity instead of the plain TUnfold for subtracting background

Definition at line 1699 of file TUnfold.cxx.

## ◆ TUnfold() [2/2]

 TUnfold::TUnfold ( void )

only for use by root streamer or derived classes

Definition at line 238 of file TUnfold.cxx.

## ◆ ~TUnfold()

 TUnfold::~TUnfold ( void )
override

## Member Function Documentation

 void TUnfold::AddMSparse ( TMatrixDSparse * dest, Double_t f, const TMatrixDSparse * src ) const
protected

add a sparse matrix, scaled by a factor, to another scaled matrix

Parameters
 [in,out] dest destination matrix [in] f scaling factor [in] src matrix to be added to dest

a replacement for (*dest) += f * (*src) which suffered from a bug in old root versions

Definition at line 915 of file TUnfold.cxx.

 Bool_t TUnfold::AddRegularisationCondition ( Int_t i0, Double_t f0, Int_t i1 = -1, Double_t f1 = 0., Int_t i2 = -1, Double_t f2 = 0. )
protected

add a row of regularisation conditions to the matrix L

Parameters
 [in] i0 truth histogram bin number [in] f0 entry in the matrix L, column i0 [in] i1 truth histogram bin number [in] f1 entry in the matrix L, column i1 [in] i2 truth histogram bin number [in] f2 entry in the matrix L, column i2

the arguments are used to form one row (k) of the matrix L, where
Lk,i0=f0 and Lk,i1=f1 and Lk,i2=f2
negative indexes i0,i1,i2 are ignored

Definition at line 1917 of file TUnfold.cxx.

 Bool_t TUnfold::AddRegularisationCondition ( Int_t nEle, const Int_t * indices, const Double_t * rowData )
protected

add a row of regularisation conditions to the matrix L

Parameters
 [in] nEle number of valid entries in indeces and rowData [in] indices column numbers of L to fill [in] rowData data to fill into the new row of L

returns true if a row was added, false otherwise
A new row k is added to the matrix L, its dimension is expanded. The new elements Lki are filled from the array rowData[] where the indices i which are taken from the array indices[].

Definition at line 1954 of file TUnfold.cxx.

## ◆ Class()

 static TClass * TUnfold::Class ( )
static
Returns
TClass describing this class

## ◆ Class_Name()

 static const char * TUnfold::Class_Name ( )
static
Returns
Name of this class

## ◆ Class_Version()

 static constexpr Version_t TUnfold::Class_Version ( )
inlinestaticconstexpr
Returns
Version of this class

Definition at line 356 of file TUnfold.h.

## ◆ ClearHistogram()

 void TUnfold::ClearHistogram ( TH1 * h, Double_t x = 0. ) const
protected

Initialize bin contents and bin errors for a given histogram.

Parameters
 [out] h histogram [in] x new histogram content

all histgram errors are set to zero, all contents are set to x

Definition at line 3680 of file TUnfold.cxx.

## ◆ ClearResults()

 void TUnfold::ClearResults ( void )
protectedvirtual

reset all results

Reimplemented in TUnfoldSys.

Definition at line 208 of file TUnfold.cxx.

## ◆ CreateSparseMatrix()

 TMatrixDSparse * TUnfold::CreateSparseMatrix ( Int_t nrow, Int_t ncol, Int_t nel, Int_t * row, Int_t * col, Double_t * data ) const
protected

create a sparse matrix, given the nonzero elements

Parameters
 [in] nrow number of rows [in] ncol number of columns [in] nel number of non-zero elements [in] row row indexes of non-zero elements [in] col column indexes of non-zero elements [in] data non-zero elements data

return pointer to a new sparse matrix

shortcut to new TMatrixDSparse() followed by SetMatrixArray()

Definition at line 578 of file TUnfold.cxx.

## ◆ DeclFileName()

 static const char * TUnfold::DeclFileName ( )
inlinestatic
Returns
Name of the file containing the class declaration

Definition at line 356 of file TUnfold.h.

## ◆ DeleteMatrix() [1/2]

 void TUnfold::DeleteMatrix ( TMatrixD ** m )
staticprotected

delete matrix and invalidate pointer

Parameters
 [in,out] m pointer to a matrix-pointer

if the matrix pointer os non-zero, thematrix id deleted. The matrox pointer is set to zero.

Definition at line 188 of file TUnfold.cxx.

## ◆ DeleteMatrix() [2/2]

 void TUnfold::DeleteMatrix ( TMatrixDSparse ** m )
staticprotected

delete sparse matrix and invalidate pointer

Parameters
 [in,out] m pointer to a matrix-pointer

if the matrix pointer os non-zero, thematrix id deleted. The matrox pointer is set to zero.

Definition at line 200 of file TUnfold.cxx.

## ◆ DoUnfold() [1/3]

 Double_t TUnfold::DoUnfold ( Double_t tau )
virtual

perform the unfolding for a given regularisation parameter tau

Parameters
 [in] tau regularisation parameter

this method sets tau and then calls the core unfolding algorithm

Definition at line 2491 of file TUnfold.cxx.

## ◆ DoUnfold() [2/3]

 Double_t TUnfold::DoUnfold ( Double_t tau_reg, const TH1 * input, Double_t scaleBias = 0.0 )

perform the unfolding for a given input and regularisation

Parameters
 [in] tau_reg regularisation parameter [in] input input distribution with uncertainties [in] scaleBias (default=0.0) scale factor applied to the bias

This is a shortcut for { SetInput(input,scaleBias); DoUnfold(tau); }

Definition at line 2235 of file TUnfold.cxx.

## ◆ DoUnfold() [3/3]

 Double_t TUnfold::DoUnfold ( void )
protectedvirtual

core unfolding algorithm

Definition at line 246 of file TUnfold.cxx.

## ◆ ErrorMatrixToHist()

 void TUnfold::ErrorMatrixToHist ( TH2 * ematrix, const TMatrixDSparse * emat, const Int_t * binMap, Bool_t doClear ) const
protected

add up an error matrix, also respecting the bin mapping

Parameters
 [in,out] ematrix error matrix histogram [in] emat error matrix stored with internal mapping (member fXToHist) [in] binMap mapping of histogram bins [in] doClear if true, ematrix is cleared prior to adding elements of emat to it.

the array binMap is explained with the method GetOutput(). The matrix emat must have dimension NxN where N=fXToHist.size() The flag doClear may be used to add covariance matrices from several uncertainty sources.

Definition at line 3379 of file TUnfold.cxx.

## ◆ GetAx()

 const TMatrixDSparse * TUnfold::GetAx ( void ) const
inlineprotected

vector of folded-back result

Definition at line 247 of file TUnfold.h.

## ◆ GetBias()

 void TUnfold::GetBias ( TH1 * out, const Int_t * binMap = nullptr ) const

get bias vector including bias scale

Parameters
 [out] out histogram to store the scaled bias vector. The bin contents are overwritten [in] binMap (default=nullptr) array for mapping truth bins to histogram bins

This method returns the bias vector times scaling factor, f*x0

The use of binMap is explained with the documentation of the GetOutput() method

Definition at line 2935 of file TUnfold.cxx.

## ◆ GetBinFromRow()

 Int_t TUnfold::GetBinFromRow ( int ix ) const
inlineprotected

converts matrix row to truth histogram bin number

Definition at line 235 of file TUnfold.h.

## ◆ GetChi2A()

 Double_t TUnfold::GetChi2A ( void ) const
inline

get χ2A contribution determined in recent unfolding

Definition at line 328 of file TUnfold.h.

## ◆ GetChi2L()

 Double_t TUnfold::GetChi2L ( void ) const

get χ2L contribution determined in recent unfolding

Definition at line 3231 of file TUnfold.cxx.

## ◆ GetDF()

 double TUnfold::GetDF ( void ) const

return the effecive number of degrees of freedom See e.g.

arXiv:1612.09415 and the references therein

Here, DF is calculated using the dependence of the unfolding result x on the data y

This calculation is done assuming a CONSTANT data variance. I.e. the uncertainties reported to TUnfold in "SetInput()" ought to be independent of the measurements. This is NOT true for standard Poisson-distributed data. In practice the impact is expected to be small

Definition at line 3749 of file TUnfold.cxx.

## ◆ GetDXDAM()

 const TMatrixDSparse * TUnfold::GetDXDAM ( int i ) const
inlineprotected

matrix contributions of the derivative dx/dA

Definition at line 251 of file TUnfold.h.

## ◆ GetDXDAZ()

 const TMatrixDSparse * TUnfold::GetDXDAZ ( int i ) const
inlineprotected

vector contributions of the derivative dx/dA

Definition at line 253 of file TUnfold.h.

## ◆ GetDXDtauSquared()

 const TMatrixDSparse * TUnfold::GetDXDtauSquared ( void ) const
inlineprotected

vector of derivative dx/dtauSquared, using internal bin counting

Definition at line 264 of file TUnfold.h.

## ◆ GetDXDY() [1/2]

 void TUnfold::GetDXDY ( TH2 * dxdy ) const

get matrix connecting input and output changes

get matrix describing gow the result changes with the input data

Parameters
 [out] dxdy two-dimensional histogram to store the matrix connecting the output and input data. The bin contents are overwritten for those bins where dxdy is non-zero.

Definition at line 3038 of file TUnfold.cxx.

## ◆ GetDXDY() [2/2]

 const TMatrixDSparse * TUnfold::GetDXDY ( void ) const
inlineprotected

matrix of derivatives dx/dy

Definition at line 249 of file TUnfold.h.

## ◆ GetE()

 const TMatrixDSparse * TUnfold::GetE ( void ) const
inlineprotected

matrix E, using internal bin counting

Definition at line 257 of file TUnfold.h.

## ◆ GetEinv()

 const TMatrixDSparse * TUnfold::GetEinv ( void ) const
inlineprotected

matrix E-1, using internal bin counting

Definition at line 255 of file TUnfold.h.

## ◆ GetEmatrix()

 void TUnfold::GetEmatrix ( TH2 * ematrix, const Int_t * binMap = nullptr ) const

get output covariance matrix, possibly cumulated over several bins

Parameters
 [out] ematrix histogram to store the covariance. The bin contents are overwritten. [in] binMap (default=nullptr) array for mapping truth bins to histogram bins

The use of binMap is explained with the documentation of the GetOutput() method

Definition at line 3446 of file TUnfold.cxx.

## ◆ GetEpsMatrix()

 Double_t TUnfold::GetEpsMatrix ( void ) const
inline

get numerical accuracy for Eigenvalue analysis when inverting matrices with rank problems

Definition at line 351 of file TUnfold.h.

## ◆ GetFoldedOutput()

 void TUnfold::GetFoldedOutput ( TH1 * out, const Int_t * binMap = nullptr ) const

get unfolding result on detector level

Parameters
 [out] out histogram to store the correlation coefficiencts. The bin contents and errors are overwritten. [in] binMap (default=nullptr) array for mapping truth bins to histogram bins

This method returns the unfolding output folded by the response matrix, i.e. the vector Ax.

The use of binMap is explained with the documentation of the GetOutput() method

Definition at line 2962 of file TUnfold.cxx.

## ◆ GetInput()

 void TUnfold::GetInput ( TH1 * out, const Int_t * binMap = nullptr ) const

Input vector of measurements.

Parameters
 [out] out histogram to store the measurements. Bin content and bin errors are overwritte. [in] binMap (default=nullptr) array for mapping truth bins to histogram bins

Bins which had an uncertainty of zero in the call to SetInput() maye acquire bin contents or bin errors different from the original settings in SetInput().

The use of binMap is explained with the documentation of the GetOutput() method

Definition at line 3069 of file TUnfold.cxx.

## ◆ GetInputInverseEmatrix()

 void TUnfold::GetInputInverseEmatrix ( TH2 * out )

get inverse of the measurement's covariance matrix

Parameters
 [out] out histogram to store the inverted covariance

Definition at line 3098 of file TUnfold.cxx.

## ◆ GetL()

 void TUnfold::GetL ( TH2 * out ) const

get matrix of regularisation conditions

Parameters
 [out] out histogram to store the regularisation conditions. the bincontents are overwritten

The histogram should have dimension nr (y-axis) times nx (x-axis). nr corresponds to the number of regularisation conditions, it can be obtained using the method GetNr(). nx corresponds to the number of histogram bins in the response matrix along the truth axis.

Definition at line 3191 of file TUnfold.cxx.

## ◆ GetLcurveX()

 Double_t TUnfold::GetLcurveX ( void ) const
virtual

get value on x-axis of L-curve determined in recent unfolding

x=log10(GetChi2A())

Definition at line 3251 of file TUnfold.cxx.

## ◆ GetLcurveY()

 Double_t TUnfold::GetLcurveY ( void ) const
virtual

get value on y-axis of L-curve determined in recent unfolding

y=log10(GetChi2L())

Definition at line 3260 of file TUnfold.cxx.

## ◆ GetLsquared()

 void TUnfold::GetLsquared ( TH2 * out ) const

get matrix of regularisation conditions squared

Parameters
 [out] out histogram to store the squared matrix of regularisation conditions. the bin contents are overwritten

This returns the square matrix LTL as a histogram

The histogram should have dimension nx times nx, where nx corresponds to the number of histogram bins in the response matrix along the truth axis.

Definition at line 3151 of file TUnfold.cxx.

## ◆ GetNdf()

 Int_t TUnfold::GetNdf ( void ) const
inline

get number of degrees of freedom determined in recent unfolding

This returns the number of valid measurements minus the number of unfolded truth bins. If the area constraint is active, one further degree of freedom is subtracted

Definition at line 338 of file TUnfold.h.

## ◆ GetNormalisationVector()

 void TUnfold::GetNormalisationVector ( TH1 * out, const Int_t * binMap = nullptr ) const

histogram of truth bins, determined from suming over the response matrix

Parameters
 [out] out histogram to store the truth bins. The bin contents are overwritten [in] binMap (default=nullptr) array for mapping truth bins to histogram bins

This vector is also used to initialize the bias x0. However, the bias vector may be changed using the SetBias() method.

The use of binMap is explained with the documentation of the GetOutput() method

Definition at line 2910 of file TUnfold.cxx.

## ◆ GetNpar()

 Int_t TUnfold::GetNpar ( void ) const

get number of truth parameters determined in recent unfolding

empty bins of the response matrix or bins which can not be unfolded due to rank deficits are not counted

Definition at line 3242 of file TUnfold.cxx.

## ◆ GetNr()

 Int_t TUnfold::GetNr ( void ) const

get number of regularisation conditions

Ths returns the number of regularisation conditions, useful for booking a histogram for a subsequent call of GetL().

Definition at line 3176 of file TUnfold.cxx.

## ◆ GetNx()

 Int_t TUnfold::GetNx ( void ) const
inlineprotected

returns internal number of output (truth) matrix rows

Definition at line 229 of file TUnfold.h.

## ◆ GetNy()

 Int_t TUnfold::GetNy ( void ) const
inlineprotected

returns the number of measurement bins

Definition at line 237 of file TUnfold.h.

## ◆ GetOutput()

 void TUnfold::GetOutput ( TH1 * output, const Int_t * binMap = nullptr ) const

get output distribution, possibly cumulated over several bins

Parameters
 [out] output existing output histogram. content and errors will be updated. [in] binMap (default=nullptr) array for mapping truth bins to histogram bins

If nonzero, the array binMap must have dimension n+2, where n corresponds to the number of bins on the truth axis of the response matrix (the histogram specified with the TUnfold constructor). The indexes of binMap correspond to the truth bins (including underflow and overflow) of the response matrix. The element binMap[i] specifies the histogram number in output where the corresponding truth bin will be stored. It is possible to specify the same output bin number for multiple indexes, in which case these bins are added. Set binMap[i]=-1 to ignore an unfolded truth bin. The uncertainties are calculated from the corresponding parts of the covariance matrix, properly taking care of added truth bins.
If the pointer binMap is zero, the bins are mapped one-to-one. Truth bin zero (underflow) is stored in the output underflow, truth bin 1 is stored in bin number 1, etc.

Definition at line 3289 of file TUnfold.cxx.

## ◆ GetOutputBinName()

 TString TUnfold::GetOutputBinName ( Int_t iBinX ) const
protectedvirtual

Get bin name of an outpt bin.

Parameters
 [in] iBinX bin number

Return value: name of the bin
For TUnfold and TUnfoldSys, this function simply returns the bin number as a string. This function really only makes sense in the context of TUnfoldDensity, where binnig schemes are implemented using the class TUnfoldBinning, and non-trivial bin names are returned.

Reimplemented in TUnfoldDensity.

Definition at line 1667 of file TUnfold.cxx.

## ◆ GetProbabilityMatrix()

 void TUnfold::GetProbabilityMatrix ( TH2 * A, EHistMap histmap ) const

get matrix of probabilities

Parameters
 [out] A two-dimensional histogram to store the probabilities (normalized response matrix). The bin contents are overwritten for those bins where A is nonzero [in] histmap specify axis along which the truth bins are oriented

Definition at line 3010 of file TUnfold.cxx.

## ◆ GetRhoAvg()

 Double_t TUnfold::GetRhoAvg ( void ) const
inline

get average global correlation determined in recent unfolding

Definition at line 326 of file TUnfold.h.

## ◆ GetRhoI()

 Double_t TUnfold::GetRhoI ( TH1 * rhoi, const Int_t * binMap = nullptr, TH2 * invEmat = nullptr ) const

get global correlation coefficiencts, possibly cumulated over several bins

Parameters
 [out] rhoi histogram to store the global correlation coefficients. The bin contents are overwritten. [in] binMap (default=nullptr) array for mapping truth bins to histogram bins [out] invEmat (default=nullptr) histogram to store the inverted covariance matrix

for a given bin, the global correlation coefficient is defined as
ρi=sqrt(1-1/(Vii*V-1ii))
such that the calculation of global correlation coefficients possibly involves the inversion of a covariance matrix.

return value: maximum global correlation coefficient

The use of binMap is explained with the documentation of the GetOutput() method

Definition at line 3504 of file TUnfold.cxx.

## ◆ GetRhoIFromMatrix()

 Double_t TUnfold::GetRhoIFromMatrix ( TH1 * rhoi, const TMatrixDSparse * eOrig, const Int_t * binMap, TH2 * invEmat ) const
protected

Definition at line 3553 of file TUnfold.cxx.

## ◆ GetRhoIJ()

 void TUnfold::GetRhoIJ ( TH2 * rhoij, const Int_t * binMap = nullptr ) const

get correlation coefficiencts, possibly cumulated over several bins

Parameters
 [out] rhoij histogram to store the correlation coefficiencts. The bin contents are overwritten. [in] binMap (default=nullptr) array for mapping truth bins to histogram bins

The use of binMap is explained with the documentation of the GetOutput() method

Definition at line 3461 of file TUnfold.cxx.

## ◆ GetRhoMax()

 Double_t TUnfold::GetRhoMax ( void ) const
inline

get maximum global correlation determined in recent unfolding

Definition at line 324 of file TUnfold.h.

## ◆ GetRowFromBin()

 Int_t TUnfold::GetRowFromBin ( int ix ) const
inlineprotected

converts truth histogram bin number to matrix row

Definition at line 233 of file TUnfold.h.

## ◆ GetSqrtEvEmatrix()

 TVectorD TUnfold::GetSqrtEvEmatrix ( void ) const

Definition at line 2509 of file TUnfold.cxx.

## ◆ GetSURE()

 double TUnfold::GetSURE ( void ) const

return Stein's unbiased risk estimator See e.g.

arXiv:1612.09415

A minimum in the SURE variable is a good choice of regularisation strength

NOTE: the calculation of SURE depends on the calculation of DF. See the method GetDF() for caveats with Poisson-distributed data.

Definition at line 3732 of file TUnfold.cxx.

## ◆ GetTau()

 Double_t TUnfold::GetTau ( void ) const

return regularisation parameter

Definition at line 3223 of file TUnfold.cxx.

## ◆ GetTUnfoldVersion()

 const char * TUnfold::GetTUnfoldVersion ( void )
static

return a string describing the TUnfold version

The version is reported in the form Vmajor.minor Changes of the minor version number typically correspond to bug-fixes. Changes of the major version may result in adding or removing data attributes, such that the streamer methods are not compatible between different major versions.

Definition at line 3717 of file TUnfold.cxx.

## ◆ GetVxx()

 const TMatrixDSparse * TUnfold::GetVxx ( void ) const
inlineprotected

covariance matrix of the result

Definition at line 243 of file TUnfold.h.

## ◆ GetVxxInv()

 const TMatrixDSparse * TUnfold::GetVxxInv ( void ) const
inlineprotected

inverse of covariance matrix of the result

Definition at line 245 of file TUnfold.h.

## ◆ GetVyyInv()

 const TMatrixDSparse * TUnfold::GetVyyInv ( void ) const
inlineprotected

inverse of covariance matrix of the data y

Definition at line 259 of file TUnfold.h.

## ◆ GetX()

 const TMatrixD * TUnfold::GetX ( void ) const
inlineprotected

vector of the unfolding result

Definition at line 241 of file TUnfold.h.

## ◆ InitTUnfold()

 void TUnfold::InitTUnfold ( void )
private

initialize data menbers, for use in constructors

Definition at line 144 of file TUnfold.cxx.

## ◆ InvertMSparseSymmPos()

 TMatrixDSparse * TUnfold::InvertMSparseSymmPos ( const TMatrixDSparse * A, Int_t * rankPtr ) const
protected

get the inverse or pseudo-inverse of a positive, sparse matrix

Parameters
 [in] A the sparse matrix to be inverted, has to be positive [in,out] rankPtr if zero, suppress calculation of pseudo-inverse otherwise the rank of the matrix is returned in *rankPtr

return value: 0 or a new sparse matrix

• if(rankPtr==nullptr) return the inverse if it exists, or return 0
• else return a (pseudo-)inverse and store the rank of the matrix in *rankPtr

the matrix inversion is optimized in performance for the case where a large submatrix of A is diagonal

Definition at line 992 of file TUnfold.cxx.

## ◆ IsA()

 TClass * TUnfold::IsA ( ) const
inlineoverridevirtual
Returns
TClass describing current object

Reimplemented from TObject.

Reimplemented in TUnfoldDensity, and TUnfoldSys.

Definition at line 356 of file TUnfold.h.

## ◆ MultiplyMSparseM()

 TMatrixDSparse * TUnfold::MultiplyMSparseM ( const TMatrixDSparse * a, const TMatrixD * b ) const
protected

multiply sparse matrix and a non-sparse matrix

Parameters
 [in] a sparse matrix [in] b matrix

returns a new sparse matrix a*b.
A replacement for: new TMatrixDSparse(a,TMatrixDSparse::kMult,b) the root implementation had problems in older versions of root

Definition at line 760 of file TUnfold.cxx.

## ◆ MultiplyMSparseMSparse()

 TMatrixDSparse * TUnfold::MultiplyMSparseMSparse ( const TMatrixDSparse * a, const TMatrixDSparse * b ) const
protected

multiply two sparse matrices

Parameters
 [in] a sparse matrix [in] b sparse matrix

returns a new sparse matrix a*b.
A replacement for: new TMatrixDSparse(a,TMatrixDSparse::kMult,b) the root implementation had problems in older versions of root

Definition at line 603 of file TUnfold.cxx.

## ◆ MultiplyMSparseMSparseTranspVector()

 TMatrixDSparse * TUnfold::MultiplyMSparseMSparseTranspVector ( const TMatrixDSparse * m1, const TMatrixDSparse * m2, const TMatrixTBase< Double_t > * v ) const
protected

calculate a sparse matrix product M1*V*M2T where the diagonal matrix V is given by a vector

Parameters
 [in] m1 pointer to sparse matrix with dimension I*K [in] m2 pointer to sparse matrix with dimension J*K [in] v pointer to vector (matrix) with dimension K*1

returns a sparse matrix R with elements rijkM1ikVkM2jk

Definition at line 819 of file TUnfold.cxx.

## ◆ MultiplyMSparseTranspMSparse()

 TMatrixDSparse * TUnfold::MultiplyMSparseTranspMSparse ( const TMatrixDSparse * a, const TMatrixDSparse * b ) const
protected

multiply a transposed Sparse matrix with another Sparse matrix

Parameters
 [in] a sparse matrix (to be transposed) [in] b sparse matrix

returns a new sparse matrix aT*b
this is a replacement for the root constructors new TMatrixDSparse(TMatrixDSparse(TMatrixDSparse::kTransposed,*a),TMatrixDSparse::kMult,*b)

Definition at line 677 of file TUnfold.cxx.

## ◆ RegularizeBins()

 Int_t TUnfold::RegularizeBins ( int start, int step, int nbin, ERegMode regmode )

add regularisation conditions for a group of bins

Parameters
 [in] start first bin number [in] step step size [in] nbin number of bins [in] regmode regularisation mode (one of: kRegModeSize, kRegModeDerivative, kRegModeCurvature)

add regularisation conditions for a group of equidistant bins. There are nbin bins, starting with bin start and with a distance of step between bins.

Return value: number of regularisation conditions which could not be added.
Conditions which are not added typically correspond to bin numbers where the truth can not be unfolded (either response matrix is empty or the data do not constrain).

Definition at line 2143 of file TUnfold.cxx.

## ◆ RegularizeBins2D()

 Int_t TUnfold::RegularizeBins2D ( int start_bin, int step1, int nbin1, int step2, int nbin2, ERegMode regmode )

add regularisation conditions for 2d unfolding

Parameters
 [in] start_bin first bin number [in] step1 step size, 1st dimension [in] nbin1 number of bins, 1st dimension [in] step2 step size, 2nd dimension [in] nbin2 number of bins, 2nd dimension [in] regmode regularisation mode (one of: kRegModeSize, kRegModeDerivative, kRegModeCurvature)

add regularisation conditions for a grid of bins. The start bin is start_bin. Along the first (second) dimension, there are nbin1 (nbin2) bins and adjacent bins are spaced by step1 (step2) units.

Return value: number of regularisation conditions which could not be added. Conditions which are not added typically correspond to bin numbers where the truth can not be unfolded (either response matrix is empty or the data do not constrain).

Definition at line 2204 of file TUnfold.cxx.

## ◆ RegularizeCurvature()

 Int_t TUnfold::RegularizeCurvature ( int left_bin, int center_bin, int right_bin, Double_t scale_left = 1.0, Double_t scale_right = 1.0 )

add a regularisation condition on the curvature of three truth bin

Parameters
 [in] left_bin bin number [in] center_bin bin number [in] right_bin bin number [in] scale_left (default=1) scale factor [in] scale_right (default=1) scale factor

this adds one row to L, where the element left_bin takes the value -scale_left, the element right_bin takes the value -scale_right and the element center_bin takes the value scale_left+scale_right

return value: 0 if ok, 1 if the condition has not been added. Conditions which are not added typically correspond to bin numbers where the truth can not be unfolded (either response matrix is empty or the data do not constrain).

The RegularizeXXX() methods can be used to set up a custom matrix of regularisation conditions. In this case, start with an empty matrix L (argument regmode=kRegModeNone in the constructor)

Definition at line 2098 of file TUnfold.cxx.

## ◆ RegularizeDerivative()

 Int_t TUnfold::RegularizeDerivative ( int left_bin, int right_bin, Double_t scale = 1.0 )

add a regularisation condition on the difference of two truth bin

Parameters
 [in] left_bin bin number [in] right_bin bin number [in] scale (default=1) scale factor

this adds one row to L, where the element left_bin takes the value -scale and the element right_bin takes the value +scale

return value: 0 if ok, 1 if the condition has not been added. Conditions which are not added typically correspond to bin numbers where the truth can not be unfolded (either response matrix is empty or the data do not constrain).

The RegularizeXXX() methods can be used to set up a custom matrix of regularisation conditions. In this case, start with an empty matrix L (argument regmode=kRegModeNone in the constructor)

Definition at line 2059 of file TUnfold.cxx.

## ◆ RegularizeSize()

 Int_t TUnfold::RegularizeSize ( int bin, Double_t scale = 1.0 )

add a regularisation condition on the magnitude of a truth bin

Parameters
 [in] bin bin number [in] scale (default=1) scale factor

this adds one row to L, where the element bin takes the value scale

return value: 0 if ok, 1 if the condition has not been added. Conditions which are not added typically correspond to bin numbers where the truth can not be unfolded (either response matrix is empty or the data do not constrain).

The RegularizeXXX() methods can be used to set up a custom matrix of regularisation conditions. In this case, start with an empty matrix L (argument regmode=kRegModeNone in the constructor)

Definition at line 2025 of file TUnfold.cxx.

## ◆ ScanLcurve()

 Int_t TUnfold::ScanLcurve ( Int_t nPoint, Double_t tauMin, Double_t tauMax, TGraph ** lCurve, TSpline ** logTauX = nullptr, TSpline ** logTauY = nullptr, TSpline ** logTauCurvature = nullptr )
virtual

scan the L curve, determine tau and unfold at the final value of tau

Parameters
 [in] nPoint number of points used for the scan [in] tauMin smallest tau value to study [in] tauMax largest tau value to study. If tauMin=tauMax=nullptr, a scan interval is determined automatically. [out] lCurve if nonzero, a new TGraph is returned, containing the L-curve [out] logTauX if nonzero, a new TSpline is returned, to parameterize the L-curve's x-coordinates as a function of log10(tau) [out] logTauY if nonzero, a new TSpline is returned, to parameterize the L-curve's y-coordinates as a function of log10(tau) [out] logTauCurvature if nonzero, a new TSpline is returned of the L-curve curvature as a function of log10(tau)

return value: the coordinate number in the logTauX,logTauY graphs corresponding to the "final" choice of tau

Recommendation: always check logTauCurvature, it should be a peaked function (similar to a Gaussian), the maximum corresponding to the final choice of tau. Also, check the lCurve it should be approximately L-shaped. If in doubt, adjust tauMin and tauMax until the results are satisfactory.

Definition at line 2558 of file TUnfold.cxx.

## ◆ ScanSURE()

 Int_t TUnfold::ScanSURE ( Int_t nPoint, Double_t tauMin, Double_t tauMax, TGraph ** logTauSURE = nullptr, TGraph ** df_chi2A = nullptr, TGraph ** lCurve = nullptr )
virtual

minimize Stein's unbiased risk estimator "SURE" using successive calls to DoUnfold at various tau.

Optionally, also the L-curve and its curvature are calculated for comparison. See description of GetSURE() See e.g. arXiv:1612.09415 for the definition of SURE

Parameters
 [in] nPoint : number of points [in] tauMin : lower end of scan-range [in] tauMax : upper end of scan-range [out] logTauSURE : scan result, SURE as a function of log(tau) [out] df_chi2A : parametric plot of DF against chi2A [out] lCurve : parametric plot (lCurve)

return value: index of the "best" point

if tauMin is less than zero of if tauMin is not loer than tauMax, then the scan range is determined automatically if tau=nullptr is included in the scan, then the first x-coordinate

Definition at line 3785 of file TUnfold.cxx.

## ◆ SetBias()

 void TUnfold::SetBias ( const TH1 * bias )

set bias vector

Parameters
 [in] bias histogram with new bias vector

the initial bias vector is determined from the response matrix but may be changed by using this method

Definition at line 1895 of file TUnfold.cxx.

## ◆ SetConstraint()

 void TUnfold::SetConstraint ( EConstraint constraint )

set type of area constraint

results of a previous unfolding are reset

Definition at line 3211 of file TUnfold.cxx.

## ◆ SetEpsMatrix()

 void TUnfold::SetEpsMatrix ( Double_t eps )

set numerical accuracy for Eigenvalue analysis when inverting matrices with rank problems

Definition at line 3703 of file TUnfold.cxx.

## ◆ SetInput()

 Int_t TUnfold::SetInput ( const TH1 * input, Double_t scaleBias = 0.0, Double_t oneOverZeroError = 0.0, const TH2 * hist_vyy = nullptr, const TH2 * hist_vyy_inv = nullptr )
virtual

Define input data for subsequent calls to DoUnfold(tau)

Parameters
 [in] input input distribution with uncertainties [in] scaleBias (default=nullptr) scale factor applied to the bias [in] oneOverZeroError (default=nullptr) for bins with zero error, this number defines 1/error. [in] hist_vyy (default=nullptr) if non-zero, this defines the data covariance matrix [in] hist_vyy_inv (default=nullptr) if non-zero and hist_vyy is set, defines the inverse of the data covariance matrix. This feature can be useful for repeated unfoldings in cases where the inversion of the input covariance matrix is lengthy

Return value: nError1+10000*nError2

• nError1: number of bins where the uncertainty is zero. these bins either are not used for the unfolding (if oneOverZeroError==nullptr) or 1/uncertainty is set to oneOverZeroError.
• nError2: return values>10000 are fatal errors, because the unfolding can not be done. The number nError2 corresponds to the number of truth bins which are not constrained by data points.

Reimplemented in TUnfoldSys.

Definition at line 2274 of file TUnfold.cxx.

## ◆ Streamer()

 void TUnfold::Streamer ( TBuffer & R__b )
overridevirtual

Stream an object of class TObject.

Reimplemented from TObject.

Reimplemented in TUnfoldDensity, and TUnfoldSys.

## ◆ StreamerNVirtual()

 void TUnfold::StreamerNVirtual ( TBuffer & ClassDef_StreamerNVirtual_b )
inline

Definition at line 356 of file TUnfold.h.

## ◆ fA

 TMatrixDSparse* TUnfold::fA
protected

response matrix A

Definition at line 153 of file TUnfold.h.

## ◆ fAx

 TMatrixDSparse* TUnfold::fAx
private

result x folded back A*x

Definition at line 190 of file TUnfold.h.

## ◆ fBiasScale

 Double_t TUnfold::fBiasScale
protected

scale factor for the bias

Definition at line 161 of file TUnfold.h.

## ◆ fChi2A

 Double_t TUnfold::fChi2A
private

chi**2 contribution from (y-Ax)Vyy-1(y-Ax)

Definition at line 192 of file TUnfold.h.

## ◆ fConstraint

 EConstraint TUnfold::fConstraint
protected

type of constraint to use for the unfolding

Definition at line 173 of file TUnfold.h.

## ◆ fDXDAM

 TMatrixDSparse* TUnfold::fDXDAM[2]
private

matrix contribution to the of derivative dx_k/dA_ij

Definition at line 202 of file TUnfold.h.

## ◆ fDXDAZ

 TMatrixDSparse* TUnfold::fDXDAZ[2]
private

vector contribution to the of derivative dx_k/dA_ij

Definition at line 204 of file TUnfold.h.

## ◆ fDXDtauSquared

 TMatrixDSparse* TUnfold::fDXDtauSquared
private

derivative of the result wrt tau squared

Definition at line 206 of file TUnfold.h.

## ◆ fDXDY

 TMatrixDSparse* TUnfold::fDXDY
private

derivative of the result wrt dx/dy

Definition at line 208 of file TUnfold.h.

## ◆ fE

 TMatrixDSparse* TUnfold::fE
private

matrix E

Definition at line 212 of file TUnfold.h.

## ◆ fEinv

 TMatrixDSparse* TUnfold::fEinv
private

matrix E^(-1)

Definition at line 210 of file TUnfold.h.

## ◆ fEpsMatrix

 Double_t TUnfold::fEpsMatrix
private

machine accuracy used to determine matrix rank after eigenvalue analysis

Definition at line 180 of file TUnfold.h.

## ◆ fHistToX

 TArrayI TUnfold::fHistToX
protected

mapping of histogram bins to matrix indices

Definition at line 169 of file TUnfold.h.

## ◆ fIgnoredBins

 Int_t TUnfold::fIgnoredBins
private

number of input bins which are dropped because they have error=nullptr

Definition at line 178 of file TUnfold.h.

## ◆ fL

 TMatrixDSparse* TUnfold::fL
protected

regularisation conditions L

Definition at line 155 of file TUnfold.h.

## ◆ fLXsquared

 Double_t TUnfold::fLXsquared
private

chi**2 contribution from (x-s*x0)TLTL(x-s*x0)

Definition at line 194 of file TUnfold.h.

## ◆ fNdf

 Int_t TUnfold::fNdf
private

number of degrees of freedom

Definition at line 200 of file TUnfold.h.

## ◆ fRegMode

 ERegMode TUnfold::fRegMode
protected

type of regularisation

Definition at line 175 of file TUnfold.h.

## ◆ fRhoAvg

 Double_t TUnfold::fRhoAvg
private

average global correlation coefficient

Definition at line 198 of file TUnfold.h.

## ◆ fRhoMax

 Double_t TUnfold::fRhoMax
private

maximum global correlation coefficient

Definition at line 196 of file TUnfold.h.

## ◆ fSumOverY

 TArrayD TUnfold::fSumOverY
protected

truth vector calculated from the non-normalized response matrix

Definition at line 171 of file TUnfold.h.

## ◆ fTauSquared

 Double_t TUnfold::fTauSquared
protected

regularisation parameter tau squared

Definition at line 165 of file TUnfold.h.

## ◆ fVxx

 TMatrixDSparse* TUnfold::fVxx
private

covariance matrix Vxx

Definition at line 184 of file TUnfold.h.

## ◆ fVxxInv

 TMatrixDSparse* TUnfold::fVxxInv
private

inverse of covariance matrix Vxx-1

Definition at line 186 of file TUnfold.h.

## ◆ fVyy

 TMatrixDSparse* TUnfold::fVyy
protected

covariance matrix Vyy corresponding to y

Definition at line 159 of file TUnfold.h.

## ◆ fVyyInv

 TMatrixDSparse* TUnfold::fVyyInv
private

inverse of the input covariance matrix Vyy-1

Definition at line 188 of file TUnfold.h.

## ◆ fX

 TMatrixD* TUnfold::fX
private

unfolding result x

Definition at line 182 of file TUnfold.h.

## ◆ fX0

 TMatrixD* TUnfold::fX0
protected

bias vector x0

Definition at line 163 of file TUnfold.h.

## ◆ fXToHist

 TArrayI TUnfold::fXToHist
protected

mapping of matrix indices to histogram bins

Definition at line 167 of file TUnfold.h.

## ◆ fY

 TMatrixD* TUnfold::fY
protected

input (measured) data y

Definition at line 157 of file TUnfold.h.

• hist/unfold/inc/TUnfold.h
• hist/unfold/src/TUnfold.cxx