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Statistical methods in LHC data analysis part I.2 Luca Lista INFN Napoli Contents • Hypothesis testing • Neyman-Pearson lemma and likelihood ratio • Multivariate analysis (elements) • Chi-square fits and goodness-of-fit • Confidence intervals • Feldman-Cousins ordering Luca Lista Statistical methods in LHC data analysis 2 Multivariate discrimination • The problem: – A signal and a background are characterized by n variables with different distributions for the two cases: signal, and background • Generalization to more than two cases is an easy extension – Given a measurement (event) of n variables with some discriminating power, (x1, …, xn), identify (discriminate) the event as signal or background • Property of discriminator: – Selection efficiency: probability of right answer – Misidentification probability (for background) – Purity: fraction of signal in a positively identified sample • Depends on the signal and background composition! It is not a property of the discriminator only – Fake rate: fraction of background in a positively identified sample, = 1 - Purity Luca Lista Statistical methods in LHC data analysis 3 Test of Hypothesis terminology • Naming by statisticians is usually less natural for physics applications than previous slide • H0 = null hypothesis – E.g.: a sample contains only background; a particle is a pion; etc. • H1 = alternative hypothesis – E.g.: a sample contains background + signal; or a particle is a muon; etc. • = significance level: probability to reject H1 if true (error of first kind), i.e. assuming H1 – = 1 – efficiency for signal • = probability to reject H0 if true (error of second kind), i.e. assuming H0 – = efficiency for background Luca Lista Statistical methods in LHC data analysis 4 Cut analysis • Cut on one (or more) variables: – If x xcut – Else, if x xcut signal background Efficiency (1-) Mis-id probability() xcut Luca Lista x Statistical methods in LHC data analysis 5 Efficiency vs mis-id • Varying the cut both the efficiency and mis-id change Efficiency 1 0 Luca Lista 0 Mis-id Statistical methods in LHC data analysis 1 6 Variations on cut analyses • Cut on multiple variables – AND/OR of single cuts • Multi-dimensional cuts: – Linear cuts – Piece-wise linear cuts – Non-linear combinations • At some point, hard to find optimal cut values, or too many cuts required – How to determine the cuts, looking at control samples? – Control samples could be MC, or selected data decays – Note: cut selection must be done a-priori, before looking at data, to avoid biases! Luca Lista Statistical methods in LHC data analysis 7 Straight cuts or something else? • Straight cuts may not be optimal in all cases Luca Lista Statistical methods in LHC data analysis 8 Neyman-Pearson lemma • Fixing the signal efficiency (1- ), a selection based on the likelihood ratio gives the lowest possible mis-id probability (): (x) = L(x|H1) / L(x|H0) > k • If we can’t use the likelihood ratio, we can choose other discriminators, or “test statistics”: • A test statistic is any function of x (like (x)) that allows to discriminate the two hypotheses Luca Lista Statistical methods in LHC data analysis 9 Likelihood ratio discriminator • We make the ratio of likelihoods defined in the two hypotheses: • Q may also depend on a number of unknown parameters (1,…,N) • Best discriminator, if the multi-dimensional likelihood is perfectly known (Neyman-Pearson lemma) • Great effort in getting the correct ratio – E.g.: Matrix Element Tecnhniques for top mass and singletop at Tevatron Luca Lista Statistical methods in LHC data analysis 10 Likelihood factorization • We make the ratio of likelihoods defined in the two hypotheses assuming PDF factorized as product of 1-D PDF: If PDF can be factorized into independent components • Approximate in case of non perfectly factorized PDF – E.g.: correlation • A rotation or other judicious transformations in the variables’ space may be used to remove the correlation – Sometimes even different for s and b hypotheses Luca Lista Statistical methods in LHC data analysis 11 Building projective PDF’s • PDF’s for likelihood discriminator – If not uncorrelated, need to find uncorrelated variables first, otherwise plain PDF product is suboptimal Luca Lista Statistical methods in LHC data analysis 12 Likelihood ratio output • Good separation achieved in this case TMVA L > 0.5 Luca Lista Statistical methods in LHC data analysis 13 Fisher discriminator • Combine a number of variables into a single discriminator • Equivalent to project the distribution along a line • Use the linear combination of inputs that maximizes the distance of the means of the two classes while minimizing the variance within each class: Sir Ronald Aylmer Fisher (1890-1962) • The maximization problem can be solved with linear algebra Luca Lista Statistical methods in LHC data analysis 14 Rewriting Fisher discriminant • • • m1, m2 are the two samples’ average vectors 1, 2 are the two samples’ covariance matrices Transform with linear vector of coefficients w – w is normal to the discriminator hyperplane “between classes scatter matrix” “within classes scatter matrix” Luca Lista Statistical methods in LHC data analysis 15 Maximizing the Fisher discriminant • Either compute derivatives w.r.t. wi • Equivalent to solve the eigenvalues problem: Luca Lista Statistical methods in LHC data analysis 16 Fisher in the previous example • Not always optimal: it’s linear cut, after all…! F>0 Luca Lista Statistical methods in LHC data analysis 17 Other discriminator methods • Artificial Neural Networks • Boosted Decision Trees • Those topics are beyond the scope of this tutorial – A brief sketch will be given just for completeness • More details in TMVA package – http://tmva.sourceforge.net/ Luca Lista Statistical methods in LHC data analysis 18 Artificial Neural Networks • Artificial simplified model of how neurons work Input layer x0 Hidden layers Output layer w01 w02 x1 w11 w12 w21 w22 x2 … y w2n xp Luca Lista w0n w1n () Activation function Statistical methods in LHC data analysis 19 Network vs other discriminators • Artificial neural network with a single hidden layer may approximate any analytical function within a given approximation if the number of neurons is sufficiently high • Adding more hidden layers can make the approximation more efficient – i.e.: smaller total number of neurons • Demonstration in: – H. N. Mhaskar, Neural Computation, Vol. 8, No. 1, Pages 164-177 (1996), Neural Networks for Optimal Approximation of Smooth and Analytic Functions: “We prove that neural networks with a single hidden layer are capable of providing an optimal order of approximation for functions assumed to possess a given number of derivatives, if the activation function evaluated by each principal element satisfies certain technical conditions” Luca Lista Statistical methods in LHC data analysis 20 (Boosted) Decision Trees • • • • • Select as usual a set of discriminating variables Progressively split the sample according to subsequent cuts o single discriminating variables Optimize the splitting cuts in order to obtain the best signal/background separation Repeat splitting until the sample contains mostly signal or background, and the statistics on the split samples is too low to continue Many different trees are need to be combined for a robust and effective discrimination (“forest”) Branch Branch Leaf Branch Leaf Leaf Leaf Decision tree Luca Lista Statistical methods in LHC data analysis 21 A strongly non linear case y x Luca Lista Statistical methods in LHC data analysis 22 Classifiers separation Projective Likelihood ratio Fisher BDT Neural Network Luca Lista Statistical methods in LHC data analysis 23 Cutting on classifiers output (I) Fisher > 0 Luca Lista Statistical methods in LHC data analysis L > 0.5 24 Cutting on classifiers output (II) NN > 0 Luca Lista BDT > 0 Statistical methods in LHC data analysis 25 Jerzy Neyman’s confidence intervals • • • • • • Scan an unknown parameter Given , compute the interval [x1, x2] that contain x with a probability C.L. = 1- Ordering rule needed! Invert the confidence belt, and find the interval [1, 2] for a given experimental outcome of x A fraction 1- of the experiments will produce x such that the corresponding interval [1, 2] contains the true value of (coverage probability) Note that the random variables are [1, 2], not Luca Lista From PDG statistics review RooStats::NeymanConstruction Statistical methods in LHC data analysis 26 Ordering rule • Different possible choices of the interval giving the same are 1- are possible • For fixed = 0 we can have different choices f(x|0) f(x|0) /2 1- 1- Upper limit choice Luca Lista x Central interval Statistical methods in LHC data analysis /2 x 27 Feldman-Cousins ordering • Find the contour of the likelihood ratio that gives an area • R = {x : L(x|θ) / L(x|θbest) > k} f(x|0) RooStats::FeldmanCousins f(x|0)/f(x| best(x)) 1- x Luca Lista Statistical methods in LHC data analysis 28 “Flip-flopping” • When to quote a central value or upper limit? • E.g.: – “Quote a 90% C.L. upper limit of the measurement is below 3; quote a central value otherwise” • Upper limit central interval decided according to observed data • This produces incorrect coverage! • Feldman-Cousins interval ordering guarantees the correct coverage Luca Lista Statistical methods in LHC data analysis 29 “Flip-flopping” with Gaussian PDF • Assume Gaussian with a fixed width: =1 90% 5% 90% 10% 10% 5% 5% x = x 1.64485 < x + 1.28155 x Coverage is 85% for low ! 3 Luca Lista Statistical methods in LHC data analysis x From Feldman and Cousins’ paper 30 Feldman-Cousins approach • Define range such that: – P(x|) / P(x|best(x)) > k best = max(x, 0) best = x for x 0 Usual errors Asymmetric errors Upper limits Solution can be found numerically x Luca Lista Will see more when talking about upper limits… Statistical methods in LHC data analysis 31 Binomial Confidence Interval • Using the proper Neyman belt inversion, e.g. Feldman Cousins method, avoids odd problems, like null errors when estimating efficiencies equal to 0 or 1, that would occur using the central limit formula: Luca Lista Statistical methods in LHC data analysis 32 Binned fits: minimum2 • • Bin entries can be approximated by Gaussian for sufficiently large number of entries with r.m.s. equal to ni (Neyman): The expected number of entries i is often approximated as the value of a continuous function f at the center xi of the bin: • • • • • Denominator ni could be replaced by i=f(xi; 1, …, n) (Pearson) Usually simpler to implement than un-binned ML fits Analytic solution exists for linear and other simple problems Un-binned ML fits unpractical for large sample size Binned fits can give poor results for small number of entries Luca Lista Statistical methods in LHC data analysis 33 Fit quality • The value of the Maximum Likelihood obtained in a fit w.r.t its expected distributions don’t give any information about the goodness of the fit • Chi-square test – The2 of a fit with a Gaussian underlying model should be distributed according to a known PDF n is the number of degrees of freedom – Sometimes this is not the case, if the model can’t be sufficiently approximated with a Gaussian – The integral of the right-most tail (P(2>X)=) is one example of socalled ‘p-value’ • Beware! p-values are not the “probability of the fit hypothesis” – This would be a Bayesian probability, with a different meaning, and should be computed in a different way ( next lecture)! Luca Lista Statistical methods in LHC data analysis 34 Binned likelihood • • • Assume our sample is a binned histogram from an event counting experiment (obeying Poissonian statistics), with no need of a Gaussian approximation We can build a likelihood function multiplying Poisson distributions for the number of entries in each bin, {ni} having expected number of entries depending on some unknown parameters: i(1, …k) We can minimize the following quantity: Luca Lista Statistical methods in LHC data analysis 35 Binned likelihood ratio • A better alternative to the (Gaussian-inspired, Neyman and Pearson’s) 2 has been proposed by Baker and Cousins using the likelihood ratio: • Same minimum value as previous slide, since a constant term has been added to the log-likelihood • It also provides a goodness-of-fit information, and asymptotically obeys chi-squared distribution with k-n degrees of freedom (Wilks’ theorem) S. Baker and R. Cousins, Clarification of the Use of Chi-square and Likelihood Functions in Fits to Histograms, NIM 221:437 (1984) Luca Lista Statistical methods in LHC data analysis 36 Combining measurements with2 • Two measurements with different uncorrelated (Gaussian) errors: • Build 2: • Minimize 2: • Estimate m as: • Error estimate: Luca Lista Statistical methods in LHC data analysis 37 Covariance and cov. matrix • Definitions: – Covariance: – Correlation: • Correlated n-dimensional Gaussian: • where: Luca Lista Statistical methods in LHC data analysis 38 Two-dimensional Gaussian • Product of two independent Gaussians with different • Rotation in the (x, y) plane Luca Lista Statistical methods in LHC data analysis 39 Two-dimensional Gaussian (cont.) • Rotation preserves the metrics: • Covariance in rotated coordinates: Luca Lista Statistical methods in LHC data analysis 40 Two-dimensional Gaussian (cont.) • A pictorial view of an iso-probability contour y y x y Luca Lista x Statistical methods in LHC data analysis x 41 1D projections • PDF projections are (1D) Gaussians: • Areas of 1 and 2 contours differ in 1D and 2D! y P1D P2D 1 0.6827 0.3934 2 0.9545 0.8647 3 0.9973 0.9889 1.515 0.8702 0.6827 2.486 0.9871 0.9545 3.439 0.9994 0.9973 Luca Lista 2 1 x 1 2 Statistical methods in LHC data analysis 42 Generalization of 2 to n dimensions • If we have n measurements, (m1, …, mn) with a nn covariance matrix (Cij) , the chi-squared can be generalized as follows: • More details on the PDG statistics review Luca Lista Statistical methods in LHC data analysis 43 Combining correlated measurements • Correlation coefficient 0: • Build 2 including correlation terms: • Minimization gives: Luca Lista Statistical methods in LHC data analysis 44 Correlated errors • The “common error” C is defined as: H. Greenlee, Combining CDF and D0 Physics Results, Fermilab Workshop on Confidence Limits, March 28, 2000 • Using error propagation, this also implies that: • The previous formulas now become: Luca Lista Statistical methods in LHC data analysis 45 Toy Monte Carlo • Generate a large number of experiments according to the fit model, with fixed parameters () • Fit all the toy samples as if they where the real data samples • Study the distributions of the fit quantities • Parameter pulls: p = (est - )/ – Verify the absence of bias: p = 0 – Verify the correct error estimate : (p) = 1 • Statistical uncertainty will depend on number of the Toy Monte Carlo experiments • Distribution of maximum likelihood (or -2lnL) gives no information about the quality of the fit • Goodness of fit for ML in more than one dimension is still an open and debated issue • Often preferred likelihood ratio w.r.t. a null hypothesis – Asymptotically distributed as a chi-square – Determine the C.L. of the fit to real data as fraction of toy cases with worse value of maximum log-likelihood-ratio Luca Lista Statistical methods in LHC data analysis 46 Kolmogorov - Smirnov test • Assume you have a sample {x1, …, xn}, you want to test if the set is compatible with being produced by random variables obeying a PDF f(x) • The test consists in building the cumulative distribution for the set and the PDF: • The distance between the two cumulative distribution is evaluated as: Luca Lista Statistical methods in LHC data analysis 47 Kolmogorov-Smirnov test in a picture 1 D n F(x) Fn(x) 0 x x x 1 Luca Lista 2 … Statistical methods in LHC data analysis x n 48 Kolmogorov distribution • For large n: – Dn converges to zero (small Dn = good agreement) – K=n Dn has a distribution that is independent on f(x) known as Kolmogorov distribution (related to Brownian motion) • Kolmogorov distribution is: • Caveat with KS test: – Very common in HEP, but not always appropriate – If the shape or parameters of the PDF f(x) are determined from the sample (i.e.: with a fit) the distribution of nDn may deviate from the Kolmogorov distribution. – A toy Monte Carlo method could be used in those case to evaluate the distribution of n Dn Luca Lista Statistical methods in LHC data analysis 49 Two sample KS test • We can test whether two samples {x1, …, xn}, {y1, …, ym}, follow the same distribution using the distance: • The variable that follows asymptotically the Kolmogorov distribution is, in this case: Luca Lista Statistical methods in LHC data analysis 50 A concrete 2 example Electro-Weak precision tests Electro-Weak precision tests • SM inputs from LEP (Aleph, Delphi, L3, Opal), SLC (SLD), Tevatron (CDF, D0). Luca Lista Statistical methods in LHC data analysis 52 Higgs mass prediction • • • Global 2 analysis, using ZFitter for detailed SM calculations Correlation terms not negligible, even cross-experiment (LEP energy…) Higgs mass prediction from indirect effect on radiative corrections Luca Lista Statistical methods in LHC data analysis 53 References • • • • • Gary J. Feldman, Robert D. Cousins, “Unified approach to the classical statistical analysis of small signals”, Phys. Rev. D 57, 3873 - 3889 (1998) J. Friedman, T. Hastie and R. Tibshirani, “The Elements of Statistical Learning”, Springer Series in Statistics, 2001. A. Webb, “Statistical Pattern Recognition”, 2nd Edition, J. Wiley & Sons Ltd, 2002. L.I. Kuncheva, “Combining Pattern Classifiers”, J. Wiley & Sons, 2004. Artificial Neural Networks – – • Bing Cheng and D. M. Titterington, Statist. Sci. Volume 9, Number 1 (1994), 2-30, Neural Networks: A Review from a Statistical Perspective Robert P.W. Duin, Learned from Neural Networks http://ict.ewi.tudelft.nl/~duin/papers/asci_00_NNReview.pdf Boosted decision trees – – – – – R.E. Schapire, The boosting approach to machine learning: an overview, MSRI Workshop on Nonlinear Estimation and Classification, 2002. Y. Freund, R.E. Schapire, A short introduction to boosting, J. Jpn. Soc. Artif. Intell. 14 (5) (1999) 771 Byron P. Roe et al, Boosted Decision Trees as an Alternative to Artificial Neural Networks for Particle Identification B.P Roe et al., Nucl.Instrum.Meth. A543 (2005) 577-584 Boosted decision trees as an alternative to artificial neural networks for particle identification http://arxiv.org/abs/physics/0408124 Bauer and Kohavi, Machine Learning 36 (1999), “An empirical comparison of voting classification algorithms” Luca Lista Statistical methods in LHC data analysis 54