Overview
A common method used in High Energy Physics to perform measurements is the maximum Likelihood method, exploiting discriminating variables to disentangle signal from background. The crucial point for such an analysis to be reliable is to use an exhaustive list of sources of events combined with an accurate description of all the Probability Density Functions (PDF).
To assess the validity of the fit, a convincing quality check is to explore further the data sample by examining the distributions of control variables. A control variable can be obtained for instance by removing one of the discriminating variables before performing again the maximum Likelihood fit: this removed variable is a control variable. The expected distribution of this control variable, for signal, is to be compared to the one extracted, for signal, from the data sample. In order to be able to do so, one must be able to unfold from the distribution of the whole data sample.
The TSPlot method allows to reconstruct the distributions for the control variable, independently for each of the various sources of events, without making use of any a priori knowledge on this variable. The aim is thus to use the knowledge available for the discriminating variables to infer the behaviour of the individual sources of events with respect to the control variable.
TSPlot is optimal if the control variable is uncorrelated with the discriminating variables.
A detail description of the formalism itself, called , is given in [1].
The method
The technique is developped in the above context of a maximum Likelihood method making use of discriminating variables.
One considers a data sample in which are merged several species of events. These species represent various signal components and background components which all together account for the data sample. The different terms of the log-Likelihood are:
The class TSPlot allows to reconstruct the true distribution of a control variable for each of the species from the sole knowledge of the PDFs of the discriminating variables . The plots obtained thanks to the TSPlot class are called .
Some properties and checks
Beside reproducing the true distribution, bear remarkable properties:
Different steps followed by TSPlot
Illustrations
To illustrate the technique, one considers an example derived from the analysis where have been first used (charmless B decays). One is dealing with a data sample in which two species are present: the first is termed signal and the second background. A maximum Likelihood fit is performed to obtain the two yields and . The fit relies on two discriminating variables collectively denoted which are chosen within three possible variables denoted , and . The variable which is not incorporated in is used as the control variable . The six distributions of the three variables are assumed to be the ones depicted in Fig. 1.
A data sample being built through a Monte Carlo simulation based on the distributions shown in Fig. 1, one obtains the three distributions of Fig. 2. Whereas the distribution of clearly indicates the presence of the signal, the distribution of and are less obviously populated by signal.
Chosing and as discriminating variables to determine and through a maximum Likelihood fit, one builds, for the control variable which is unknown to the fit, the two for signal and background shown in Fig. 3. One observes that the for signal reproduces correctly the PDF even where the latter vanishes, although the error bars remain sizeable. This results from the almost complete cancellation between positive and negative weights: the sum of weights is close to zero while the sum of weights squared is not. The occurence of negative weights occurs through the appearance of the covariance matrix, and its negative components, in the definition of Eq. (2).
A word of caution is in order with respect to the error bars. Whereas their sum in quadrature is identical to the statistical uncertainties of the yields determined by the fit, and if, in addition, they are asymptotically correct, the error bars should be handled with care for low statistics and/or for too fine binning. This is because the error bars do not incorporate two known properties of the PDFs: PDFs are positive definite and can be non-zero in a given x-bin, even if in the particular data sample at hand, no event is observed in this bin. The latter limitation is not specific to , rather it is always present when one is willing to infer the PDF at the origin of an histogram, when, for some bins, the number of entries does not guaranty the applicability of the Gaussian regime. In such situations, a satisfactory practice is to attach allowed ranges to the histogram to indicate the upper and lower limits of the PDF value which are consistent with the actual observation, at a given confidence level.
Chosing and as discriminating variables to determine and through a maximum Likelihood fit, one builds, for the control variable which is unknown to the fit, the two for signal and background shown in Fig. 4. In the for signal one observes that error bars are the largest in the regions where the background is the largest.
The results above can be obtained by running the tutorial TestSPlot.C
TSPlot() | |
TSPlot(Int_t nx, Int_t ny, Int_t ne, Int_t ns, TTree* tree) | |
virtual | ~TSPlot() |
void | TObject::AbstractMethod(const char* method) const |
virtual void | TObject::AppendPad(Option_t* option = "") |
virtual void | Browse(TBrowser* b) |
static TClass* | Class() |
virtual const char* | TObject::ClassName() const |
virtual void | TObject::Clear(Option_t* = "") |
virtual TObject* | TObject::Clone(const char* newname = "") const |
virtual Int_t | TObject::Compare(const TObject* obj) const |
virtual void | TObject::Copy(TObject& object) const |
virtual void | TObject::Delete(Option_t* option = "")MENU |
virtual Int_t | TObject::DistancetoPrimitive(Int_t px, Int_t py) |
virtual void | TObject::Draw(Option_t* option = "") |
virtual void | TObject::DrawClass() constMENU |
virtual TObject* | TObject::DrawClone(Option_t* option = "") constMENU |
virtual void | TObject::Dump() constMENU |
virtual void | TObject::Error(const char* method, const char* msgfmt) const |
virtual void | TObject::Execute(const char* method, const char* params, Int_t* error = 0) |
virtual void | TObject::Execute(TMethod* method, TObjArray* params, Int_t* error = 0) |
virtual void | TObject::ExecuteEvent(Int_t event, Int_t px, Int_t py) |
virtual void | TObject::Fatal(const char* method, const char* msgfmt) const |
void | FillSWeightsHists(Int_t nbins = 50) |
void | FillXvarHists(Int_t nbins = 100) |
void | FillYpdfHists(Int_t nbins = 100) |
void | FillYvarHists(Int_t nbins = 100) |
virtual TObject* | TObject::FindObject(const char* name) const |
virtual TObject* | TObject::FindObject(const TObject* obj) const |
virtual Option_t* | TObject::GetDrawOption() const |
static Long_t | TObject::GetDtorOnly() |
virtual const char* | TObject::GetIconName() const |
virtual const char* | TObject::GetName() const |
Int_t | GetNevents() |
Int_t | GetNspecies() |
virtual char* | TObject::GetObjectInfo(Int_t px, Int_t py) const |
static Bool_t | TObject::GetObjectStat() |
virtual Option_t* | TObject::GetOption() const |
void | GetSWeights(TMatrixD& weights) |
void | GetSWeights(Double_t* weights) |
TH1D* | GetSWeightsHist(Int_t ixvar, Int_t ispecies, Int_t iyexcl = -1) |
TObjArray* | GetSWeightsHists() |
virtual const char* | TObject::GetTitle() const |
TString* | GetTreeExpression() |
TString* | GetTreeName() |
TString* | GetTreeSelection() |
virtual UInt_t | TObject::GetUniqueID() const |
TH1D* | GetXvarHist(Int_t ixvar) |
TObjArray* | GetXvarHists() |
TH1D* | GetYpdfHist(Int_t iyvar, Int_t ispecies) |
TObjArray* | GetYpdfHists() |
TH1D* | GetYvarHist(Int_t iyvar) |
TObjArray* | GetYvarHists() |
virtual Bool_t | TObject::HandleTimer(TTimer* timer) |
virtual ULong_t | TObject::Hash() const |
virtual void | TObject::Info(const char* method, const char* msgfmt) const |
virtual Bool_t | TObject::InheritsFrom(const char* classname) const |
virtual Bool_t | TObject::InheritsFrom(const TClass* cl) const |
virtual void | TObject::Inspect() constMENU |
void | TObject::InvertBit(UInt_t f) |
virtual TClass* | IsA() const |
virtual Bool_t | TObject::IsEqual(const TObject* obj) const |
virtual Bool_t | IsFolder() const |
Bool_t | TObject::IsOnHeap() const |
virtual Bool_t | TObject::IsSortable() const |
Bool_t | TObject::IsZombie() const |
virtual void | TObject::ls(Option_t* option = "") const |
void | MakeSPlot(Option_t* option = "v") |
void | TObject::MayNotUse(const char* method) const |
virtual Bool_t | TObject::Notify() |
void | TObject::Obsolete(const char* method, const char* asOfVers, const char* removedFromVers) const |
static void | TObject::operator delete(void* ptr) |
static void | TObject::operator delete(void* ptr, void* vp) |
static void | TObject::operator delete[](void* ptr) |
static void | TObject::operator delete[](void* ptr, void* vp) |
void* | TObject::operator new(size_t sz) |
void* | TObject::operator new(size_t sz, void* vp) |
void* | TObject::operator new[](size_t sz) |
void* | TObject::operator new[](size_t sz, void* vp) |
TObject& | TObject::operator=(const TObject& rhs) |
virtual void | TObject::Paint(Option_t* option = "") |
virtual void | TObject::Pop() |
virtual void | TObject::Print(Option_t* option = "") const |
virtual Int_t | TObject::Read(const char* name) |
virtual void | TObject::RecursiveRemove(TObject* obj) |
void | RefillHist(Int_t type, Int_t var, Int_t nbins, Double_t min, Double_t max, Int_t nspecies = -1) |
void | TObject::ResetBit(UInt_t f) |
virtual void | TObject::SaveAs(const char* filename = "", Option_t* option = "") constMENU |
virtual void | TObject::SavePrimitive(ostream& out, Option_t* option = "") |
void | TObject::SetBit(UInt_t f) |
void | TObject::SetBit(UInt_t f, Bool_t set) |
virtual void | TObject::SetDrawOption(Option_t* option = "")MENU |
static void | TObject::SetDtorOnly(void* obj) |
void | SetInitialNumbersOfSpecies(Int_t* numbers) |
void | SetNEvents(Int_t ne) |
void | SetNSpecies(Int_t ns) |
void | SetNX(Int_t nx) |
void | SetNY(Int_t ny) |
static void | TObject::SetObjectStat(Bool_t stat) |
void | SetTree(TTree* tree) |
void | SetTreeSelection(const char* varexp = "", const char* selection = "", Long64_t firstentry = 0) |
virtual void | TObject::SetUniqueID(UInt_t uid) |
virtual void | ShowMembers(TMemberInspector& insp) |
virtual void | Streamer(TBuffer& b) |
void | StreamerNVirtual(TBuffer& b) |
virtual void | TObject::SysError(const char* method, const char* msgfmt) const |
Bool_t | TObject::TestBit(UInt_t f) const |
Int_t | TObject::TestBits(UInt_t f) const |
virtual void | TObject::UseCurrentStyle() |
virtual void | TObject::Warning(const char* method, const char* msgfmt) const |
virtual Int_t | TObject::Write(const char* name = 0, Int_t option = 0, Int_t bufsize = 0) |
virtual Int_t | TObject::Write(const char* name = 0, Int_t option = 0, Int_t bufsize = 0) const |
virtual void | TObject::DoError(int level, const char* location, const char* fmt, va_list va) const |
void | TObject::MakeZombie() |
void | SPlots(Double_t* covmat, Int_t i_excl) |
enum TObject::EStatusBits { | kCanDelete | |
kMustCleanup | ||
kObjInCanvas | ||
kIsReferenced | ||
kHasUUID | ||
kCannotPick | ||
kNoContextMenu | ||
kInvalidObject | ||
}; | ||
enum TObject::[unnamed] { | kIsOnHeap | |
kNotDeleted | ||
kZombie | ||
kBitMask | ||
kSingleKey | ||
kOverwrite | ||
kWriteDelete | ||
}; |
TMatrixD | fMinmax | mins and maxs of variables for histogramming |
Int_t | fNSpecies | Number of species |
Int_t | fNevents | Total number of events |
Double_t* | fNumbersOfEvents | [fNSpecies] estimates of numbers of events in each species |
Int_t | fNx | Number of control variables |
Int_t | fNy | Number of discriminating variables |
TMatrixD | fPdfTot | ! |
TMatrixD | fSWeights | computed sWeights |
TObjArray | fSWeightsHists | histograms of weighted variables |
TString* | fSelection | Selection on the tree |
TTree* | fTree | ! |
TString* | fTreename | The name of the data tree |
TString* | fVarexp | Variables used for splot |
TMatrixD | fXvar | ! |
TObjArray | fXvarHists | histograms of control variables |
TMatrixD | fYpdf | ! |
TObjArray | fYpdfHists | histograms of pdfs |
TMatrixD | fYvar | ! |
TObjArray | fYvarHists | histograms of discriminating variables |
normal TSPlot constructor nx : number of control variables ny : number of discriminating variables ne : total number of events ns : number of species tree: input data
Set the initial number of events of each species - used as initial estimates in minuit
Calculates the sWeights The option controls the print level "Q" - no print out "V" - prints the estimated #of events in species - default "VV" - as "V" + the minuit printing + sums of weights for control
Returns the matrix of sweights. It is assumed that the array passed in the argurment is big enough
Returns the array of histograms of x variables (not weighted) If histograms have not already been filled, they are filled with default binning 100.
Returns the histogram of variable #ixvar If histograms have not already been filled, they are filled with default binning 100.
Returns the array of histograms of y variables. If histograms have not already been filled, they are filled with default binning 100.
Returns the histogram of variable iyvar.If histograms have not already been filled, they are filled with default binning 100.
Fills the histograms of pdf-s of y variables with binning nbins
Returns the array of histograms of pdf's of y variables with binning nbins If histograms have not already been filled, they are filled with default binning 100.
Returns the histogram of the pdf of variable #iyvar for species #ispecies, binning nbins If histograms have not already been filled, they are filled with default binning 100.
The order of histograms in the array: x0_species0, x0_species1,..., x1_species0, x1_species1,..., y0_species0, y0_species1,... If the histograms have already been filled with a different binning, they are refilled and all histograms are deleted
Returns an array of all histograms of variables, weighted with sWeights If histograms have not been already filled, they are filled with default binning 50 The order of histograms in the array: x0_species0, x0_species1,..., x1_species0, x1_species1,..., y0_species0, y0_species1,...
The Fill...Hist() methods fill the histograms with the real limits on the variables This method allows to refill the specified histogram with user-set boundaries min and max Parameters: type = 1 - histogram of x variable #nvar = 2 - histogram of y variable #nvar = 3 - histogram of y_pdf for y #nvar and species #nspecies = 4 - histogram of x variable #nvar, species #nspecies, WITH sWeights = 5 - histogram of y variable #nvar, species #nspecies, WITH sWeights
Returns the histogram of a variable, weithed with sWeights If histograms have not been already filled, they are filled with default binning 50 If parameter ixvar!=-1, the histogram of x-variable #ixvar is returned for species ispecies If parameter ixvar==-1, the histogram of y-variable #iyexcl is returned for species ispecies If the histogram has already been filled and the binning is different from the parameter nbins all histograms with old binning will be deleted and refilled.
Specifies the variables from the tree to be used for splot Variables fNx, fNy, fNSpecies and fNEvents should already be set! In the 1st parameter it is assumed that first fNx variables are x(control variables), then fNy y variables (discriminating variables), then fNy*fNSpecies ypdf variables (probability distribution functions of dicriminating variables for different species). The order of pdfs should be: species0_y0, species0_y1,... species1_y0, species1_y1,...species[fNSpecies-1]_y0... The 2nd parameter allows to make a cut TTree::Draw method description contains more details on specifying expression and selection