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Statistics in ROOT

Statistics in ROOT. René Brun, Anna Kreshuk, Lorenzo Moneta PH/SFT group, CERN http://root.cern.ch ftp://root.cern.ch/root/phystat05.ppt. Contents. User interface Data storage and access Analysis Visualization New Math libraries Future plans. ROOT’s user interface . C++ in batch mode

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Statistics in ROOT

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  1. Statistics in ROOT René Brun, Anna Kreshuk, Lorenzo Moneta PH/SFT group, CERN http://root.cern.ch ftp://root.cern.ch/root/phystat05.ppt PHYSTAT 05, Oxford

  2. Contents • User interface • Data storage and access • Analysis • Visualization • New Math libraries • Future plans PHYSTAT 05, Oxford

  3. ROOT’s user interface • C++ in batch mode • C++ interpreted code with CINT – the C++ interpreter • in the command line: • loading a macro: • C++ compiled code via CINT • Python: • Access to ROOT from Python • Access to Python from ROOT root -b -q myMacro.C > myMacro.log root[0] for (int i=0; i<10; i++) cout<<“hello ”<<i<<endl; root[1].L mySmallMacro.C; root[2] myFunction(1, 2, 3); root[].L myScript.C+ Creating shared library /home/…/MyScript_C.so >>>from ROOT import TLorentzVector >>> l = TLorentzVector root [0] TPython::LoadMacro(“MyPyClass.py”); root [1] MyPyClass mpc; PHYSTAT 05, Oxford

  4. ROOT and external libraries • Using external libraries from ROOT: • rootcint – utility to link compiled C/C++ objects with CINT C/C++ interpreter • Example: • In the Makefile of MyLibrary, rootcint generates the dictionary for MyClass • Load and use MyLibrary in a ROOT session: root[] .L MyLibrary.so root[] MyClass *mc = new MyClass(); PHYSTAT 05, Oxford

  5. Data storage and access • Allows to analyze Terabytes of data • Can select entries from different physical locations and collect them into the analysis dataset TTree1 TTree2 Dataset to analyze TTreeN Branches of a TTree are read independently, so the variables not needed for the analysis are not loaded into memory V1 V2 …………V23 ………….....V99 PHYSTAT 05, Oxford

  6. Histograms • 1-2-3 dimensional histograms • Errors for each bin can be computed: • Default: as sqrt(bin content) • As sqrt(sum of squares of weights of the bin) • 1-2 dimensional profile histograms • Mean value of Y and its standard deviation for each bin in X PHYSTAT 05, Oxford

  7. Analysis of TTrees • TTree::Draw method and TTreeViewer - an easy way to examine the tree: • Producing histograms of user-defined expressions in up to 4 dimensions • Expressions – C++ formulas • Selections – expressions, user-defined macros or graphical cuts Tree.Draw(“sqrt(x):y”, “x>0 && y<1”); Tree.Draw(“2*TMath::Log(x)”, cut1 || cut2); Examples: PHYSTAT 05, Oxford

  8. Fitting - interface • Minimization packages: Minuit and Fumili • Fitting can be done: • Directly in those packages with a user-defined function to minimize • Through the general interface of • TH1::Fit (binned data) – Chisquare and Loglikelihood methods • TGraph::Fit (unbinned data) • TGraphErrors::Fit (data with errors) • TGraphAsymmErrors::Fit (taking into account asymmetry of errors) • TTree::Fit and TTree::UnbinnedFit • RooFitpackage for object-oriented data modeling. Distributed with ROOT starting from version 5.02-00 PHYSTAT 05, Oxford

  9. Linear Fitting (1) • New class TLinearFitter • Used to fit functions linear in the parameters • 10-15 times faster than Minuit, depending on the fitting function • Simple to use in a multidimensional case • Example: • Expressions with such syntax can be used in all the Fit interface functions lfitter.SetFormula(“1 ++ x0 ++ sqrt(x1) ++ exp(x2) ++ x3 ++ x4”); PHYSTAT 05, Oxford

  10. Linear Fitting (2) Robust least trimmed squares fitting • Based on the subset of h cases (out of n) whose least squares fit possesses the smallest sum of squared residuals • High breakdown point – • smallest proportion of outliers that can cause the estimator • to produce values arbitrarily far from the true parameters Graph.Fit(“pol3”, “rob=0.75”, -2, 2); 2nd parameter – fraction h of the good points PHYSTAT 05, Oxford

  11. Smoothing and peak finding • TSpectrum class: • 1 and 2-dim background estimation • smoothing • deconvolution • peak search and fitting • Graph smoothers: • Kernel smoother • Lowess • “Super smoother” • Splines – cubic and quintic PHYSTAT 05, Oxford

  12. Multivariate methods (1) • Minimum Covariance Determinant Estimator – a highly robust estimator of multivariate location and scatter • Class TRobustEstimator • High breakdown point • Algorithm similar to Least Trimmed Squares regression PHYSTAT 05, Oxford

  13. Multivariate methods (2) • TPrincipal - principal components analysis • TMultiDimFit – approximates a multidimensional function with monomials, Chebyshev or Legendre polynomials • TMultiLayerPerceptron – a neural networks class • All multivariate methods can take input data from a TTree PHYSTAT 05, Oxford

  14. Confidence intervals • TLimit – computes 95% C.L. limits using the Likelihood ratio semi-Bayesian method • TRolke – computes confidence intervals for the rate of the Poisson in the presence of background and efficiency with a fully frequentist treatment of uncertainties. • TFeldmanCousins – calculate the C.L. upper limit using the Feldman-Cousins method PHYSTAT 05, Oxford

  15. Small useful algorithms • In the namespace TMath: • Most probability distribution functions, their densities and inverses • Special functions • Mean and Median – also for weighted datasets, Variance and K-th order statistic • Kolmogorov-Smirnov test PHYSTAT 05, Oxford

  16. Linear algebra and quadratic programming • Linear algebra package: • General, symmetric and sparse matrices • Matrix decompositions • Eigenvalue analysis • Quadratic programming library: • Dense and sparse data • Gondzio and Mehrotra solving methods PHYSTAT 05, Oxford

  17. Graphs • 1-d: • TGraph • TGraphErrors • TGraphAsymmErrors • TMultiGraph – a collection of graphs • 2-d: • TGraph2D • TGraph2DErrors PHYSTAT 05, Oxford

  18. ROOT Math Packages PHYSTAT 05, Oxford

  19. Library with the basic Math functionality build-able as a standalone library no dependency on others ROOT packages no external dependency Main content of MathCore: Basic and commonly used mathematical functions Special and statistics (pdf, cdf) functions Interfaces to function and algorithm classes Basic implementation of some numerical algorithms 3D and LorentzVectors Random numbers MathCore PHYSTAT 05, Oxford

  20. MathMore • Library with extra mathematical functionalities • Current content: • C++ interface to functions and algorithms from the Gnu Scientific Library (GSL) • Mathematical functions implemented using GSL • Algorithms currently present: • adaptive numerical integration, derivation, root finders, interpolation,1D minimization • repository for needed and useful extra Math functionality • could include other useful math libraries PHYSTAT 05, Oxford

  21. Summary and Future plans • First versions of MathCore and MathMore libraries are being released • Transition phase, over in 2-3 months • Next addition will be new random number package • Improvement of the fitting interface • Statistical algorithms to add: • sPlot • Loess - locally weighted polynomial regression • Cluster analysis • Boxplot and spiderplot • Interface with R? PHYSTAT 05, Oxford

  22. Mathematical Functions • Special functions • use proposed C++ standard interface: • double cyl_bessel_i (double nu, double x); • Statistical functions • Probability density functions (pdf) • Cumulative dist. (lower tail and upper tail) • Inverse of cumulative distributions • Coherent naming scheme (also proposed to C++ standard) • chisquared_pdf, chisquared_prob, chisquared_quant, • Chisquared_prob_inv, chisquare_quant_inv PHYSTAT 05, Oxford

  23. Mathematical Functions (cont) • New functions with better precision than old one in ROOT • Extensive tests of numerical accuracy • Comparison with other libraries (Nag, Mathematica) PHYSTAT 05, Oxford

  24. Numerical Algorithm • New C++ classes and interfaces for describing algorithms and functions • Integrator classes • Implementation based on GSL (QGS) for definite and indefinite integration • Move of functionality currently in ROOT TF1 inside new classes in MathCore • Easier to use for all clients PHYSTAT 05, Oxford

  25. Physics and Geometry Vectors • Classes for 3D Vectors and LorentzVectors with their operations and transformations • Merge old ROOT and CLHEP • New classes with cleaner interfaces, generic on the scalar type and the based coordinates • (cartesian, polar, cylindrical, etc..) • Classes for 3D rotations and Lorentz transformations • Have also rotations based on quaternion • Work done in collaboration with Fermilab group PHYSTAT 05, Oxford

  26. Minimization • New C++ version of Minuit being introduced in ROOT • Same algorithms translated in C++ plus some added functionality • Fumili minimizer, single side bounds • Going under extensive validation tests before after PHYSTAT 05, Oxford

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