1. 15th September 2005PHYSTAT 05, Oxford 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

2. PHYSTAT 05, Oxford2 15th September 2005 Contents User interface Data storage and access Analysis Visualization New Math libraries Future plans

3. PHYSTAT 05, Oxford3 15th September 2005 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 >>> from ROOT import TLorentzVector >>> l = TLorentzVector root [0] TPython::LoadMacro(“MyPyClass.py”); root [1] MyPyClass mpc; 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

4. PHYSTAT 05, Oxford4 15th September 2005 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();

5. PHYSTAT 05, Oxford5 15th September 2005 Data storage and access Allows to analyze Terabytes of data Can select entries from different physical locations and collect them into the analysis dataset Dataset to analyze V1 V2 …………V23 ………….....V99 Branches of a TTree are read independently, so the variables not needed for the analysis are not loaded into memory TTree1 TTree2 TTreeN

6. PHYSTAT 05, Oxford6 15th September 2005 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

7. PHYSTAT 05, Oxford7 15th September 2005 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 Examples: Tree.Draw(“sqrt(x):y”, “x>0 && y<1”); Tree.Draw(“2*TMath::Log(x)”, cut1 || cut2);

8. PHYSTAT 05, Oxford8 15th September 2005 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  RooFit package for object-oriented data modeling. Distributed with ROOT starting from version 5.02-00

9. PHYSTAT 05, Oxford9 15th September 2005 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”);

10. PHYSTAT 05, Oxford10 15th September 2005 Linear Fitting (2) Based on the subset of h cases (out of n) whose least squares fit possesses the smallest sum of squared residuals Robust least trimmed squares fitting  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); 2 nd parameter – fraction h of the good points

11. PHYSTAT 05, Oxford11 15th September 2005 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

12. PHYSTAT 05, Oxford12 15th September 2005 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

13. PHYSTAT 05, Oxford13 15th September 2005 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

14. PHYSTAT 05, Oxford14 15th September 2005 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

15. PHYSTAT 05, Oxford15 15th September 2005 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

16. PHYSTAT 05, Oxford16 15th September 2005 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

17. PHYSTAT 05, Oxford17 15th September 2005 Graphs 1-d:  TGraph  TGraphErrors  TGraphAsymmErrors  TMultiGraph – a collection of graphs 2-d:  TGraph2D  TGraph2DErrors

18. PHYSTAT 05, Oxford18 15th September 2005 ROOT Math Packages

19. PHYSTAT 05, Oxford19 15th September 2005 MathCore 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

20. PHYSTAT 05, Oxford20 15th September 2005 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

21. PHYSTAT 05, Oxford21 15th September 2005 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?

22. PHYSTAT 05, Oxford22 15th September 2005 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

23. PHYSTAT 05, Oxford23 15th September 2005 Mathematical Functions (cont) New functions with better precision than old one in ROOT  Extensive tests of numerical accuracy  Comparison with other libraries (Nag, Mathematica)

24. PHYSTAT 05, Oxford24 15th September 2005 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

25. PHYSTAT 05, Oxford25 15th September 2005 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

26. PHYSTAT 05, Oxford26 15th September 2005 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 beforeafter