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8. Hypotheses 8.4 Two more things. Inclusion of systematic errors LHR methods needs a prediction (from MC simulation) for the expected numbers of s and b in each bin („channel“) Statistical p.d.f.´s for these numbers are poissonian (or gaussian, if large)
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8. Hypotheses 8.4 Two more things • Inclusion of systematic errors • LHR methods needs a prediction (from MC simulation) for the expected • numbers of s and b in each bin („channel“) • Statistical p.d.f.´s for these numbers are poissonian (or gaussian, if large) • Prediction of s and b also have systematic uncertainties • finite MC statistics • theoretical uncertainties in production cross section • uncertainties from detector efficiencies and acceptances • uncertainty in integrated luminosity • … • some of these uncertainties can be correlated between channels • tough job: determine these systematic uncertainties • statistical procedure: convolute the (estimated) p.d.f.´s for systematics • (usually assumed gaussian) with the poissonians statistical p.d.f.´s K. Desch – Statistical methods of data analysis SS10
8. Hypotheses 8.4 Two more things (rather) easy-to-userootclass Tlimit() public: TLimit() TLimit(constTLimit&) virtual ~TLimit() staticTClass* Class() staticTConfidenceLevel* ComputeLimit(TLimitDataSource* data, Int_tnmc = 50000, boolstat = false, TRandom* generator = 0) staticTConfidenceLevel* ComputeLimit(Double_t s, Double_t b, Int_t d, Int_tnmc = 50000, boolstat = false, TRandom* generator = 0) staticTConfidenceLevel* ComputeLimit(TH1* s, TH1* b, TH1* d, Int_tnmc = 50000, boolstat = false, TRandom* generator = 0) staticTConfidenceLevel* ComputeLimit(Double_t s, Double_t b, Int_t d, TVectorD* se, TVectorD* be, TObjArray*, Int_tnmc = 50000, boolstat = false, TRandom* generator = 0) staticTConfidenceLevel* ComputeLimit(TH1* s, TH1* b, TH1* d, TVectorD* se, TVectorD* be, TObjArray*, Int_tnmc = 50000, boolstat = false, TRandom* generator = 0) onlyneedsvectorsofsignal, background, observeddata (and theirerrors) and computes (e.g.) CLb, CLs+b,CLs, exptectedCLb, CLs+b,CLs and muchmore… K. Desch – Statistical methods of data analysis SS10
8. Hypotheses 8.4 Two more things The „look elsewhere“ effect The LHR test is a „either-or“ test of two hypotheses (e.g. „Higgs at 114 GeV“ or „no Higgs at 114 GeV“) When the question of a discovery of a new particle is asked, often many „signal“ hypotheses are tested against the background hypothesis simultaneously (e.g. m=105, m=108, m=111, m=114, …) The probability that any of these hypotheses yields a „false-positve“ result is larger than the probability for a single hypothesis to be false-positive This is the „look elsewhere“ effect If the probabilites are small, the 1-CLb can simply be multiplied by the number of different hypotheses that are tested simultaneously In case there is continous „test mass“ in principle infinitely many hypotheses are tested – but they are correlated (excess for mtest = 114 will also cause excess for mtest = 114.5) K. Desch – Statistical methods of data analysis SS10
8. Hypotheses 8.4 Two more things The „look elsewhere“ effect (ctd.) need an „effective“ number of tested hypotheses hard to quantify exactly Ansatz: two hypotheses are uncorrelated if their reconstructed mass distributions do not overlap Estimate: effective number of hypotheses = range of test masses / average mass resolution K. Desch – Statistical methods of data analysis SS10
9. Classification Task: how to find needles in haystacks? how to enrich a sample of events with „signal-like“ events? Why is it important? Only one out of 1011 LHC events contains a Higgs (decay) if it exists. Existence („discovery“) can only be shown if a statistically significant excess can be extracted from the data Most particle physics analyses require separation of a signal from background(s) based on a set of discriminating variables K. Desch – Statistical methods of data analysis SS10
9. Classification • Different types of input information (discriminating variables) • multivariate analysis • Combine these variables to extract the maximum of discriminating power • between signal and background • Kinematic variables (masses, momenta, decay angles, …) • Event properties (jet/lepton multiplicity, sum of charges, …) • Event shape (sphericity, …) • Detector response (silicon hits, dE/dx, Cherenkov angle, shower profiles, …) • etc. K. Desch – Statistical methods of data analysis SS10
x2 x2 x2 H1 H1 H1 H0 H0 H0 x1 x1 x1 9. Classification Suppose data sample with two types of events:H0, H1 We have found discriminating input variables x1, x2, … What decision boundary should we use to select events of type H1? Rectangular cuts Linear boundary Nonlinear boundary(ies) K. Desch – Statistical methods of data analysis SS10
9. Classification [H.Voss] K. Desch – Statistical methods of data analysis SS10
9. Classification [H.Voss] K. Desch – Statistical methods of data analysis SS10
9. Classification [H.Voss] K. Desch – Statistical methods of data analysis SS10
9. Classification [H.Voss] K. Desch – Statistical methods of data analysis SS10
9. Classification [H.Voss] K. Desch – Statistical methods of data analysis SS10
9. Classification [H.Voss] K. Desch – Statistical methods of data analysis SS10
9. Classification [H.Voss] K. Desch – Statistical methods of data analysis SS10
9. Classification MVA algorithms Finding the optimal Multivariate Analysis (MVA) algorithm is not trivial Large variety of different algorithms exist K. Desch – Statistical methods of data analysis SS10
9. Classification (Projective) Likelihood-Selection [H.Voss] K. Desch – Statistical methods of data analysis SS10
9. Classification (Projective) Likelihood-Selection [H.Voss] K. Desch – Statistical methods of data analysis SS10
