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Statistics in ROOT
René Brun, Anna Kreshuk, Lorenzo Moneta
PH/SFT group, CERN
http://root.cern.ch
ftp://root.cern.ch/root/phystat05.ppt
15th September 2005
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Contents
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User interface
Data storage and access
Analysis
Visualization
New Math libraries
Future plans
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ROOT’s user interface

C++ in batch mode
root -b -q myMacro.C > myMacro.log
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C++ interpreted code with CINT – the C++ interpreter

in the command line:
root[0] for (int i=0; i<10; i++) cout<<“hello ”<<i<<endl;

loading a macro:
root[1] .L mySmallMacro.C;
root[2] myFunction(1, 2, 3);
C++ compiled code via CINT
root[] .L myScript.C+
Creating shared library /home/…/MyScript_C.so
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Python:

Access to ROOT from Python

Access to Python from ROOT
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>>> from ROOT import TLorentzVector
>>> l = TLorentzVector
root [0] TPython::LoadMacro(“MyPyClass.py”);
root [1] MyPyClass mpc;
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ROOT and external libraries

Using external libraries from ROOT:
– utility to link compiled C/C++ objects with
CINT C/C++ interpreter
 Example:
 rootcint
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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();
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Data storage and access
• Allows
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
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to analyze Terabytes of
data
• Can select entries from
different physical locations and
collect them into the analysis
dataset
V1 V2 …………V23 ………….....V99
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Histograms
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1-2-3 dimensional histograms

Errors for each bin can be computed:
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Default: as sqrt(bin content)
As sqrt(sum of squares of weights of the bin)
1-2 dimensional profile histograms
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Mean value of Y and its standard deviation for each bin in X
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Analysis of TTrees
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TTree::Draw method and TTreeViewer - an easy way to examine the tree:
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Producing histograms of user-defined expressions in up to 4 dimensions
Expressions – C++ formulas
Selections – expressions, user-defined macros or graphical cuts
Examples:
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Tree.Draw(“sqrt(x):y”, “x>0 && y<1”);
Tree.Draw(“2*TMath::Log(x)”, cut1 || cut2);
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Fitting - interface
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Minimization packages: Minuit and Fumili
Fitting can be done:
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Directly in those packages with a user-defined function to minimize
 Through the general interface of
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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
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Linear Fitting (1)
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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
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Example:
lfitter.SetFormula(“1 ++ x0 ++ sqrt(x1) ++ exp(x2) ++ x3 ++ x4”);
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Expressions with such syntax can be used in all the
Fit interface functions
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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);
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2nd parameter –
fraction h of the
good points
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Smoothing and peak finding
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TSpectrum class:
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Graph smoothers:
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1 and 2-dim background
estimation
smoothing
deconvolution
peak search and fitting
Kernel smoother
Lowess
“Super smoother”
Splines – cubic and quintic
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Multivariate methods (1)
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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
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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

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Confidence intervals
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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
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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
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Linear algebra and quadratic
programming
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Linear algebra package:
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General, symmetric and
sparse matrices
Matrix decompositions
Eigenvalue analysis
Quadratic programming
library:
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Dense and sparse data
Gondzio and Mehrotra
solving methods
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Graphs
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1-d:
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TGraph
TGraphErrors
TGraphAsymmErrors
TMultiGraph – a collection
of graphs
2-d:

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TGraph2D
TGraph2DErrors
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ROOT Math Packages
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MathCore
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Library with the basic Math functionality
build-able as a standalone library
 no
dependency on others ROOT packages
 no external dependency
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Main content of MathCore:
 Basic

and commonly used mathematical functions
Special and statistics (pdf, cdf) functions
 Interfaces
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to function and algorithm classes
Basic implementation of some numerical algorithms
 3D
and LorentzVectors
 Random numbers
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MathMore
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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
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Summary and Future plans

First versions of MathCore and MathMore libraries are
being released
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Next addition will be new random number package
Improvement of the fitting interface
Statistical algorithms to add:
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Transition phase, over in 2-3 months
sPlot
Loess - locally weighted polynomial regression
Cluster analysis
Boxplot and spiderplot
Interface with R?
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Mathematical Functions
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Special functions
 use
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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)
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chisquared_pdf, chisquared_prob, chisquared_quant,
Chisquared_prob_inv, chisquare_quant_inv
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Mathematical Functions (cont)
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New functions with better precision than old one
in ROOT
 Extensive
tests of numerical accuracy
 Comparison with other libraries (Nag, Mathematica)
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Numerical Algorithm
New C++ classes and interfaces for
describing algorithms and functions
 Integrator classes
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 Implementation
based on GSL (QGS) for
definite and indefinite integration
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Move of functionality currently in ROOT
TF1 inside new classes in MathCore
 Easier
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to use for all clients
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Physics and Geometry Vectors
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Classes for 3D Vectors and LorentzVectors with their
operations and transformations
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New classes with cleaner interfaces, generic on the
scalar type and the based coordinates
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(cartesian, polar, cylindrical, etc..)
Classes for 3D rotations and Lorentz transformations
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Merge old ROOT and CLHEP
Have also rotations based on quaternion
Work done in collaboration with Fermilab group
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Minimization

New C++ version of Minuit being introduced in ROOT
 Same algorithms translated in C++ plus some added
functionality
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Fumili minimizer, single side bounds
Going under extensive validation tests
before
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after
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