Statistical Analysis Tools for Particle Physics
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1 Statistical Analysis Tools for Particle Physics IDPASC School of Flavour Physics Valencia, 2-7 May, 2013 Glen Cowan Physics Department Royal Holloway, University of London G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools 1
2 Outline Multivariate methods for HEP Event selection as a statistical test Neyman-Pearson lemma and likelihood ratio test Some multivariate classifiers: Linear Neural networks Boosted Decision Trees Support Vector Machines G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools 2
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5 A simulated SUSY event in ATLAS high p T muons high p T jets of hadrons p p missing transverse energy G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools page 5
6 Background events This event from Standard Model ttbar production also has high p T jets and muons, and some missing transverse energy. can easily mimic a SUSY event. G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools page 6
7 Event selection as a statistical test For each event we measure a set of numbers: x = ( x, ) x 1 = jet p T x 2 = missing energy x 3 = particle i.d. measure,... x follows some n-dimensional joint probability density, which depends on the type of event produced, i.e., was it pp tt, pp ~ g ~ g, x p( x H 0 ) j E.g. hypotheses H 0, H 1,... Often simply signal, background 1,x n ( x ) p H 1 x i G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools page 7
8 Finding an optimal decision boundary In particle physics usually start by making simple cuts : H 0 x i < c i x j < c j H 1 Maybe later try some other type of decision boundary: H 0 H 0 H 1 H 1 G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools page 8
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36 Overtraining If decision boundary is too flexible it will conform too closely to the training points overtraining. Monitor by applying classifier to independent test sample. training sample independent test sample G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools page 36
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40 Neural network example from LEP II Signal: e + e W + W (often 4 well separated hadron jets) Background: e + e qqgg (4 less well separated hadron jets) input variables based on jet structure, event shape,... none by itself gives much separation. Neural network output: (Garrido, Juste and Martinez, ALEPH ) G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools page 40
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46 Kernel-based PDE (KDE, Parzen window) Consider d dimensions, N training events, x 1,..., x N, estimate f (x) with kernel bandwidth (smoothing parameter) Use e.g. Gaussian kernel: Need to sum N terms to evaluate function (slow); faster algorithms only count events in vicinity of x (k-nearest neighbor, range search). G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools 46
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50 Particle i.d. in MiniBooNE Detector is a 12-m diameter tank of mineral oil exposed to a beam of neutrinos and viewed by 1520 photomultiplier tubes: Search for ν µ to ν e oscillations required particle i.d. using information from the PMTs. H.J. Yang, MiniBooNE PID, DNP06 G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools page 50
51 Decision trees Out of all the input variables, find the one for which with a single cut gives best improvement in signal purity: where w i. is the weight of the ith event. Resulting nodes classified as either signal/background. Iterate until stop criterion reached based on e.g. purity or minimum number of events in a node. The set of cuts defines the decision boundary. Example by MiniBooNE experiment, B. Roe et al., NIM 543 (2005) 577 G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools page 51
52 Finding the best single cut The level of separation within a node can, e.g., be quantified by the Gini coefficient, calculated from the (s or b) purity as: For a cut that splits a set of events a into subsets b and c, one can quantify the improvement in separation by the change in weighted Gini coefficients: where, e.g., Choose e.g. the cut to the maximize Δ; a variant of this scheme can use instead of Gini e.g. the misclassification rate: G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools page 52
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62 Monitoring overtraining From MiniBooNE example: Performance stable after a few hundred trees. G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools page 62
63 Comparison of boosting algorithms A number of boosting algorithms on the market; differ in the update rule for the weights. G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools page 63
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65 Support Vector Machines Map input variables into high dimensional feature space: x φ Maximize distance between separating hyperplanes (margin) subject to constraints allowing for some misclassification. Final classifier only depends on scalar products of φ(x): So only need kernel Bishop ch 7 G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools page 65
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79 Summary on multivariate methods Particle physics has used several multivariate methods for many years: linear (Fisher) discriminant neural networks naive Bayes and has in the last several years started to use a few more k-nearest neighbour boosted decision trees support vector machines The emphasis is often on controlling systematic uncertainties between the modeled training data and Nature to avoid false discovery. Although many classifier outputs are "black boxes", a discovery at 5σ significance with a sophisticated (opaque) method will win the competition if backed up by, say, 4σ evidence from a cut-based method. G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools page 79
80 Quotes I like Keep it simple. As simple as possible. Not any simpler. A. Einstein If you believe in something you don't understand, you suffer,... Stevie Wonder G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools page 80
81 Extra slides G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools page 81
82 Two distinct event selection problems In some cases, the event types in question are both known to exist. Example: separation of different particle types (electron vs muon) Use the selected sample for further study. In other cases, the null hypothesis H 0 means "Standard Model" events, and the alternative H 1 means "events of a type whose existence is not yet established" (to do so is the goal of the analysis). Many subtle issues here, mainly related to the heavy burden of proof required to establish presence of a new phenomenon. Typically require p-value of background-only hypothesis below ~ 10 7 (a 5 sigma effect) to claim discovery of "New Physics". G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools page 82
83 Using classifier output for discovery f(y) background signal N(y) background search region excess? Normalized to unity y Normalized to expected number of events Discovery = number of events found in search region incompatible with background-only hypothesis. p-value of background-only hypothesis can depend crucially distribution f(y b) in the "search region". y cut y G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools page 83
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85 Single top quark production (CDF/D0) Top quark discovered in pairs, but SM predicts single top production. Use many inputs based on jet properties, particle i.d.,... Pair-produced tops are now a background process. signal (blue + green) G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools page 85
86 Different classifiers for single top Also Naive Bayes and various approximations to likelihood ratio,... Final combined result is statistically significant (>5σ level) but not easy to understand classifier outputs. G. Cowan IDPASC School of Flavour Physics, Valencia, 2-7 May 2013 / Statistical Analysis Tools page 86
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