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// @(#)root/hist:$Name: $:$Id: TLimit.cc,v 1.3 2010/11/26 13:30:05 auterman Exp $
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// Author: Christophe.Delaere@cern.ch 21/08/2002
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///////////////////////////////////////////////////////////////////////////
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//
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// TLimit
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//
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// Class to compute 95% CL limits
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//
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// adapted from the mclimit code from Tom Junk (CLs method)
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// see http://root.cern.ch/root/doc/TomJunk.pdf
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// see http://cern.ch/thomasj/searchlimits/ecl.html
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// see: Tom Junk,NIM A434, p. 435-443, 1999
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//
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// see also the following interesting references:
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// Alex Read, "Presentation of search results: the CLs technique"
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// Journal of Physics G: Nucl. Part. Phys. 28 2693-2704 (2002).
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// http://www.iop.org/EJ/abstract/0954-3899/28/10/313/
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//
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// A nice article is also available in the CERN yellow report with the proceeding
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// of the 2000 CERN workshop on confidence intervals.
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//
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// Alex Read, "Modified Frequentist Analysis of Search Results (The CLs Method)"
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// CERN 2000-005 (30 May 2000)
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//
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//
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///////////////////////////////////////////////////////////////////////////
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#include "TLimit.h"
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#include "TArrayD.h"
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#include "TOrdCollection.h"
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#include "TConfidenceLevel.h"
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#include "TLimitDataSource.h"
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#include "TRandom3.h"
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#include "TH1.h"
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#include "TObjArray.h"
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#include "TMath.h"
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#include "TIterator.h"
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#include "TObjString.h"
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#include "TClassTable.h"
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#include "Riostream.h"
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ClassImp(TLimit)
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TArrayD *TLimit::fgTable = new TArrayD(0);
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TOrdCollection *TLimit::fgSystNames = new TOrdCollection();
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TConfidenceLevel *TLimit::ComputeLimit(TLimitDataSource * data,
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Int_t nmc, bool stat,
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TRandom * generator,
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Double_t(*statistic) (Double_t,
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Double_t,
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Double_t))
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{
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// class TLimit
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// ------------
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// Algorithm to compute 95% C.L. limits using the Likelihood ratio
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// semi-bayesian method.
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// It takes signal, background and data histograms wrapped in a
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// TLimitDataSource as input and runs a set of Monte Carlo experiments in
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// order to compute the limits. If needed, inputs are fluctuated according
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// to systematics. The output is a TConfidenceLevel.
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//
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// class TLimitDataSource
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// ----------------------
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//
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// Takes the signal, background and data histograms as well as different
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// systematics sources to form the TLimit input.
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//
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// class TConfidenceLevel
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// ----------------------
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//
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// Final result of the TLimit algorithm. It is created just after the
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// time-consuming part and can be stored in a TFile for further processing.
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// It contains light methods to return CLs, CLb and other interesting
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// quantities.
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//
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// The actual algorithm...
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// From an input (TLimitDataSource) it produces an output TConfidenceLevel.
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// For this, nmc Monte Carlo experiments are performed.
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// As usual, the larger this number, the longer the compute time,
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// but the better the result.
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//Begin_Html
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/*
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<FONT SIZE=+0>
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<p>Supposing that there is a plotfile.root file containing 3 histograms
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(signal, background and data), you can imagine doing things like:</p>
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<p>
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<BLOCKQUOTE><PRE>
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TFile* infile=new TFile("plotfile.root","READ");
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infile->cd();
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TH1D* sh=(TH1D*)infile->Get("signal");
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TH1D* bh=(TH1D*)infile->Get("background");
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TH1D* dh=(TH1D*)infile->Get("data");
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TLimitDataSource* mydatasource = new TLimitDataSource(sh,bh,dh);
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TConfidenceLevel *myconfidence = TLimit::ComputeLimit(mydatasource,50000);
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cout << " CLs : " << myconfidence->CLs() << endl;
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cout << " CLsb : " << myconfidence->CLsb() << endl;
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cout << " CLb : " << myconfidence->CLb() << endl;
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cout << "< CLs > : " << myconfidence->GetExpectedCLs_b() << endl;
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cout << "< CLsb > : " << myconfidence->GetExpectedCLsb_b() << endl;
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cout << "< CLb > : " << myconfidence->GetExpectedCLb_b() << endl;
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delete myconfidence;
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delete mydatasource;
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infile->Close();
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</PRE></BLOCKQUOTE></p>
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<p></p>
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<p>More informations can still be found on
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<a HREF="http://cern.ch/aleph-proj-alphapp/doc/tlimit.html">this</a> page.</p>
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</FONT>
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*/
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//End_Html
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// The final object returned...
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TConfidenceLevel *result = new TConfidenceLevel(nmc);
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// The random generator used...
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TRandom *myrandom = generator ? generator : new TRandom3;
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// Compute some total quantities on all the channels
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Int_t nbins = 0;
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Int_t maxbins = 0;
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Double_t nsig = 0;
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Double_t nbg = 0;
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Int_t ncand = 0;
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Int_t i;
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for (i = 0; i <= data->GetSignal()->GetLast(); i++) {
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nbins += ((TH1D *) (data->GetSignal()->At(i)))->GetNbinsX();
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maxbins = ((TH1D *) (data->GetSignal()->At(i)))->GetNbinsX() > maxbins ?
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((TH1D *) (data->GetSignal()->At(i)))->GetNbinsX() + 1 : maxbins;
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nsig += ((TH1D *) (data->GetSignal()->At(i)))->Integral();
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nbg += ((TH1D *) (data->GetBackground()->At(i)))->Integral();
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ncand += (Int_t) ((TH1D *) (data->GetCandidates()->At(i)))->Integral();
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}
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result->SetBtot(nbg);
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result->SetStot(nsig);
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result->SetDtot(ncand);
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Double_t buffer = 0;
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fgTable->Set(maxbins * (data->GetSignal()->GetLast() + 1));
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for (Int_t channel = 0; channel <= data->GetSignal()->GetLast(); channel++)
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for (Int_t bin = 0;
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bin <= ((TH1D *) (data->GetSignal()->At(channel)))->GetNbinsX();
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bin++) {
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Double_t s = (Double_t) ((TH1D *) (data->GetSignal()->At(channel)))->GetBinContent(bin);
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Double_t b = (Double_t) ((TH1D *) (data->GetBackground()->At(channel)))->GetBinContent(bin);
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Double_t d = (Double_t) ((TH1D *) (data->GetCandidates()->At(channel)))->GetBinContent(bin);
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// Compute the value of the "-2lnQ" for the actual data
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if ((b == 0) && (s > 0)) {
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//cout << "WARNING: Ignoring bin " << bin << " of channel "
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// << channel << " which has s=" << s << " but b=" << b << endl;
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//cout << " Maybe the MC statistic has to be improved..." << endl;
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}
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if ((s > 0) && (b > 0))
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buffer += statistic(s, b, d);
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// precompute the log(1+s/b)'s in an array to speed up computation
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// background-free bins are set to have a maximum t.s. value
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// for protection (corresponding to s/b of about 5E8)
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if ((s > 0) && (b > 0))
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fgTable->AddAt(statistic(s, b, 1), (channel * maxbins) + bin);
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else if ((s > 0) && (b == 0))
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fgTable->AddAt(20, (channel * maxbins) + bin);
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}
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result->SetTSD(buffer);
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// accumulate MC experiments. Hold the test statistic function fixed, but
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// fluctuate s and b within syst. errors for computing probabilities of
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// having that outcome. (Alex Read's prescription -- errors are on the ensemble,
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// not on the observed test statistic. This technique does not split outcomes.)
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// keep the tstats as sum log(1+s/b). convert to -2lnQ when preparing the results
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// (reason -- like to keep the < signs right)
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Double_t *tss = new Double_t[nmc];
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Double_t *tsb = new Double_t[nmc];
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Double_t *lrs = new Double_t[nmc];
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Double_t *lrb = new Double_t[nmc];
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Double_t *sig = new Double_t[nmc];
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Double_t *bgd = new Double_t[nmc];
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for (i = 0; i < nmc; i++) {
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tss[i] = 0;
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tsb[i] = 0;
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lrs[i] = 0;
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lrb[i] = 0;
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sig[i] = 0;
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bgd[i] = 0;
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// fluctuate signal and background
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TLimitDataSource *fluctuated = Fluctuate(data, !i, myrandom, stat);
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for (Int_t channel = 0;
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channel <= fluctuated->GetSignal()->GetLast(); channel++) {
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for (Int_t bin = 0;
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bin <=((TH1D *) (fluctuated->GetSignal()->At(channel)))->GetNbinsX();
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bin++) {
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if ((Double_t) ((TH1D *) (fluctuated->GetSignal()->At(channel)))->GetBinContent(bin) != 0) {
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// s+b hypothesis
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Double_t rate = (Double_t) ((TH1D *) (fluctuated->GetSignal()->At(channel)))->GetBinContent(bin) +
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(Double_t) ((TH1D *) (fluctuated->GetBackground()->At(channel)))->GetBinContent(bin);
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Double_t rand = myrandom->Poisson(rate);
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tss[i] += rand * fgTable->At((channel * maxbins) + bin);
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Double_t s = (Double_t) ((TH1D *) (fluctuated->GetSignal()->At(channel)))->GetBinContent(bin);
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Double_t b = (Double_t) ((TH1D *) (fluctuated->GetBackground()->At(channel)))->GetBinContent(bin);
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//std::cout<<"ch="<<channel<<", bin="<<bin<<", s = "<<s<<", b="<<b<<std::endl;
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sig[i]+=s;
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bgd[i]+=b;
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if ((s > 0) && (b > 0))
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lrs[i] += statistic(s, b, rand) - s;
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else if ((s > 0) && (b == 0))
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lrs[i] += 20 * rand - s;
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// b hypothesis
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rate = (Double_t) ((TH1D *) (fluctuated->GetBackground()->At(channel)))->GetBinContent(bin);
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rand = myrandom->Poisson(rate);
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tsb[i] += rand * fgTable->At((channel * maxbins) + bin);
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if ((s > 0) && (b > 0))
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lrb[i] += statistic(s, b, rand) - s;
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else if ((s > 0) && (b == 0))
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lrb[i] += 20 * rand - s;
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}
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}
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}
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lrs[i] = TMath::Exp(lrs[i]);
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lrb[i] = TMath::Exp(lrb[i]);
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if (data != fluctuated)
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delete fluctuated;
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}
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// lrs and lrb are the LR's (no logs) = prob(s+b)/prob(b) for
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// that choice of s and b within syst. errors in the ensemble. These are
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// the MC experiment weights for relating the s+b and b PDF's of the unsmeared
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// test statistic (in which cas one can use another test statistic if one likes).
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result->SetSig(sig);
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result->SetBgd(bgd);
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// Now produce the output object.
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// The final quantities are computed on-demand form the arrays tss, tsb, lrs and lrb.
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result->SetTSS(tss);
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result->SetTSB(tsb);
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result->SetLRS(lrs);
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result->SetLRB(lrb);
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if (!generator)
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delete myrandom;
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return result;
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}
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TLimitDataSource *TLimit::Fluctuate(TLimitDataSource * input, bool init,
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TRandom * generator, bool stat)
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{
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// initialisation: create a sorted list of all the names of systematics
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if (init) {
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// create a "map" with the systematics names
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TIterator *errornames = input->GetErrorNames()->MakeIterator();
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TObjArray *listofnames = 0;
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while ((listofnames = ((TObjArray *) errornames->Next()))) {
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TObjString *name = NULL;
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TIterator *loniter = listofnames->MakeIterator();
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while ((name = (TObjString *) (loniter->Next())))
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if ((fgSystNames->IndexOf(name)) < 0)
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fgSystNames->AddLast(name);
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delete loniter;
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}
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delete errornames;
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fgSystNames->Sort();
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}
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// if there are no systematics, just returns the input as "fluctuated" output
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if ((fgSystNames->GetSize() <= 0)&&(!stat))
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return input;
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// if there are only stat, just fluctuate stats.
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if (fgSystNames->GetSize() <= 0) {
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TLimitDataSource *result = new TLimitDataSource();
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result->SetOwner();
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for (Int_t channel = 0; channel <= input->GetSignal()->GetLast(); channel++) {
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TH1D *newsignal = (TH1D *) (input->GetSignal()->At(channel))->Clone();
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if(stat)
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for(int i=1; i<=newsignal->GetNbinsX(); i++) {
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newsignal->SetBinContent(i,newsignal->GetBinContent(i)+generator->Gaus(0,newsignal->GetBinError(i)));
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}
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newsignal->SetDirectory(0);
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TH1D *newbackground = (TH1D *) (input->GetBackground()->At(channel))->Clone();
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if(stat)
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for(int i=1; i<=newbackground->GetNbinsX(); i++)
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newbackground->SetBinContent(i,newbackground->GetBinContent(i)+generator->Gaus(0,newbackground->GetBinError(i)));
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newbackground->SetDirectory(0);
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TH1D *newcandidates = new TH1D(*(TH1D *) (input->GetCandidates()));
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newcandidates->SetDirectory(0);
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result->AddChannel(newsignal, newbackground, newcandidates);
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}
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return result;
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}
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// Find a choice for the random variation and
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// re-toss all random numbers if any background or signal
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// goes negative. (background = 0 is bad too, so put a little protection
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// around it -- must have at least 10% of the bg estimate).
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bool retoss = kTRUE;
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Double_t *serrf = NULL;
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Double_t *berrf = NULL;
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do {
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Double_t *toss = new Double_t[fgSystNames->GetSize()];
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for (Int_t i = 0; i < fgSystNames->GetSize(); i++)
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toss[i] = generator->Gaus(0, 1);
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retoss = kFALSE;
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serrf = new Double_t[(input->GetSignal()->GetLast()) + 1];
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berrf = new Double_t[(input->GetSignal()->GetLast()) + 1];
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for (Int_t channel = 0;
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channel <= input->GetSignal()->GetLast();
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channel++) {
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serrf[channel] = 0;
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berrf[channel] = 0;
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bool AsymmetricErrors = ( ((TH1D *) (input->GetErrorOnSignalNeg()->At(channel)))->GetNbinsX() > 0);
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for (Int_t bin = 0;
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bin <=((TH1D *) (input->GetErrorOnSignal()->At(channel)))->GetNbinsX();
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bin++) {
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Double_t s_toss=toss[fgSystNames->BinarySearch((TObjString*) (((TObjArray *) (input->GetErrorNames()->At(channel)))->At(bin)))];
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Double_t b_toss=toss[fgSystNames->BinarySearch((TObjString*) (((TObjArray *) (input->GetErrorNames()->At(channel)))->At(bin)))];
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Double_t s_scale=((TH1D *) (input->GetErrorOnSignal()->At(channel)))->GetBinContent(bin);
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Double_t b_scale=((TH1D *) (input->GetErrorOnBackground()->At(channel)))->GetBinContent(bin);
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if (AsymmetricErrors && s_toss<0) {
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s_scale=((TH1D *) (input->GetErrorOnSignalNeg()->At(channel)))->GetBinContent(bin);
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}
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if (AsymmetricErrors && b_toss<0) {
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b_scale=((TH1D *) (input->GetErrorOnBackgroundNeg()->At(channel)))->GetBinContent(bin);
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}
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serrf[channel] += s_scale * s_toss;
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berrf[channel] += b_scale * b_toss;
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}
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if ((serrf[channel] < -1.0) || (berrf[channel] < -0.9)) {
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retoss = kTRUE;
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continue;
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}
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}
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delete[]toss;
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} while (retoss);
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// adjust the fluctuated signal and background counts with a legal set
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// of random fluctuations above.
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TLimitDataSource *result = new TLimitDataSource();
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result->SetOwner();
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for (Int_t channel = 0; channel <= input->GetSignal()->GetLast();
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channel++) {
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TH1D *newsignal = new TH1D(*(TH1D *) (input->GetSignal()->At(channel)));
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if(stat)
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for(int i=1; i<=newsignal->GetNbinsX(); i++) {
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newsignal->SetBinContent(i,newsignal->GetBinContent(i)+generator->Gaus(0,newsignal->GetBinError(i)));
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}
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newsignal->Scale(1 + serrf[channel]);
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newsignal->SetDirectory(0);
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TH1D *newbackground = new TH1D(*(TH1D *) (input->GetBackground()->At(channel)));
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if(stat)
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for(int i=1; i<=newbackground->GetNbinsX(); i++)
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newbackground->SetBinContent(i,newbackground->GetBinContent(i)+generator->Gaus(0,newbackground->GetBinError(i)));
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newbackground->Scale(1 + berrf[channel]);
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newbackground->SetDirectory(0);
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TH1D *newcandidates = new TH1D(*(TH1D *) (input->GetCandidates()));
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newcandidates->SetDirectory(0);
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result->AddChannel(newsignal, newbackground, newcandidates);
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}
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delete[] serrf;
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delete[] berrf;
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return result;
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}
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|
