1. RooFit A tool kit for data modeling in ROOT
Wouter Verkerke (NIKHEF)
David Kirkby (UC Irvine)
Wouter Verkerke, NIKHEF

2. Focus: coding a probability density function
Focus on one practical aspect of many data analysis in HEP: How do you formulate your p.d.f. in ROOT
For ‘simple’ problems (gauss, polynomial), ROOT built-in models well sufficient
But if you want to do unbinned ML fits, use non-trivial functions, or work with multidimensional functions you are quickly running into trouble
Wouter Verkerke, NIKHEF

3. A recent example
BaBar experiment at SLAC: Extract sin(2b) from time dependent CP violation of B decay: e+e-  Y(4s)  BB
Reconstruct both Bs, measure decay time difference
Physics of interest is in decay time dependent oscillation
Many issues arise
Standard ROOT function framework clearly insufficient to handle such complicated functions  must develop new framework
Normalization of p.d.f. not always trivial to calculate  may need numeric integration techniques
Unbinned fit, >2 dimensions, many events  computation performance important  must try optimize code for acceptable performance
Simultaneous fit to control samples to account for detector performance
Wouter Verkerke, NIKHEF

4. A recent example
Initial approach BaBar: write it from scratch (in FORTRAN!)
Does its designated job quite well, but took a long time to develop
Possible because sin(2b) effort supported by O(50) people.
Optimization of ML calculations hand-coded  error prone and not easy to maintain
Difficult to transfer knowledge/code from one analysis to another.
A better solution: A modeling language in C++ that integrates seamlessly into ROOT
Recycle code and knowledge
Development of RooFit package
Started 5 years ago.
Very successful  virtually everybody in BaBar uses it  now in standard ROOT distribution
Wouter Verkerke, NIKHEF

5. RooFit – a data modeling language for ROOT
Extension to ROOT – (Almost) no overlap with existing functionality
C++ command line interface & macros
Data management & histogramming
Graphics interface
I/O support
MINUIT
ToyMC data Generation
Data/Model
Fitting
Data Modeling
Model Visualization
Wouter Verkerke, NIKHEF

6. Data modeling - Desired functionality
Building/Adjusting Models
Easy to write basic PDFs ( normalization)
Easy to compose complex models (modular design)
Reuse of existing functions
Flexibility – No arbitrary implementation-related restrictions
A n a l y s i s w o r k c y c l e
Using Models
Fitting : Binned/Unbinned (extended) MLL fits, Chi2 fits
Toy MC generation: Generate MC datasets from any model
Visualization: Slice/project model & data in any possible way
Speed – Should be as fast or faster than hand-coded model
Wouter Verkerke, NIKHEF

7. Data modeling – OO representation
Idea: represent math symbols as C++ objects
Result: 1 line of code per symbol in a function (the C++ constructor) rather than 1 line of code per function
Mathematical concept
RooFit class
variable
RooRealVar
function
RooAbsReal
PDF
RooAbsPdf
space point
RooArgSet
integral
RooRealIntegral
list of space points
RooAbsData
Wouter Verkerke, NIKHEF

8. Data modeling – Constructing composite objects
Straightforward correlation between mathematical representation of formula and RooFit code
Math 
RooGaussian g
RooFit
diagram 
RooRealVar x
RooRealVar m
RooFormulaVar sqrts   
RooRealVar s
RooFit
code
 RooRealVar x(“x”,”x”,-10,10) ;
 RooRealVar m(“m”,”mean”,0) ;
 RooRealVar s(“s”,”sigma”,2,0,10) ;
 RooFormulaVar sqrts(“sqrts”,”sqrt(s)”,s) ;
 RooGaussian g(“g”,”gauss”,x,m,sqrts) ;
Wouter Verkerke, NIKHEF

9. Model building – (Re)using standard components
RooFit provides a collection of compiled standard PDF classes
RooBMixDecay
Physics inspired
ARGUS,Crystal Ball, Breit-Wigner, Voigtian, B/D-Decay,….
RooPolynomial
RooHistPdf
Non-parametric
Histogram, KEYS
RooArgusBG
RooGaussian
Basic
Gaussian, Exponential, Polynomial,… Chebychev polynomial
Easy to extend the library: each p.d.f. is a separate C++ class
Wouter Verkerke, NIKHEF

10. Importance of a good and extendable model library
If ‘new’ p.d.f.s are made available in a easy-to-use form, more people will use them
Example 1: non-parametric kernel estimation technique
See article by Kyle Cranmer
People like it, but are put off by the prospect of thoroughly reading the entire article, coding the concept from scratch, and then using it.
In RooFit switching from a histogram-based shape to a KEYS shape is a one-line code change (RooHistPdf  RooKeysPdf)
Example 2: Chebychev polynomials
Its easy to convince people to try them if all they need to do is a one-line code change (RooPolynomial  RooChebychev)
Wouter Verkerke, NIKHEF

11. Model building – (Re)using standard components
Library p.d.f.s can be adjusted on the fly.
Just plug in any function expression you like as input variable
Works universally, even for classes you write yourself
Maximum flexibility of library shapes keeps library small
g(x;m,s)
m(y;a0,a1)
g(x,y;a0,a1,s)
RooPolyVar m(“m”,y,RooArgList(a0,a1)) ;
RooGaussian g(“g”,”gauss”,x,m,s) ;
Wouter Verkerke, NIKHEF

12. Handling of p.d.f normalization
Normalization of (component) p.d.f.s to unity is often a good part of the work of writing a p.d.f.
RooFit handles most normalization issues transparently to the user
P.d.f can advertise (multiple) analytical expressions for integrals
When no analytical expression is provided, RooFit will automatically perform numeric integration to obtain normalization
More complicated that it seems: even if gauss(x,m,s) can be integrated analytically over x, gauss(f(x),m,s) cannot. Such use cases are automatically recognized.
Multi-dimensional integrals can be combination of numeric and p.d.f-provided analytical partial integrals
Variety of numeric integration techniques is interfaced
Adaptive trapezoid, Gauss-Kronrod, VEGAS MC…
User can override parameters globally or per p.d.f. as necessary
Wouter Verkerke, NIKHEF

13. Model building – (Re)using standard components
Most physics models can be composed from ‘basic’ shapes
RooBMixDecay
RooPolynomial
RooHistPdf
RooArgusBG
RooGaussian +
RooAddPdf
Wouter Verkerke, NIKHEF

14. Model building – (Re)using standard components
Most physics models can be composed from ‘basic’ shapes
RooBMixDecay
RooProdPdf h(“h”,”h”, RooArgSet(f,g))
RooPolynomial
RooHistPdf
RooArgusBG
RooProdPdf k(“k”,”k”,g, Conditional(f,x))
RooGaussian *
RooProdPdf
Wouter Verkerke, NIKHEF

15. Using models - Overview
All RooFit models provide universal and complete fitting and Toy Monte Carlo generating functionality
Model complexity only limited by available memory and CPU power
Fitting/plotting a 5-D model as easy as using a 1-D model
Most operations are one-liners
Fitting
Generating
data = gauss.generate(x,1000)
RooAbsPdf
gauss.fitTo(data)
RooDataSet
RooAbsData
Wouter Verkerke, NIKHEF

16. Fitting functionality
pdf.fitTo(data)
Binned or unbinned ML fit?
For RooFit this is the essentially the same!
Nice example of C++ abstraction through inheritance
Fitting interface takes abstract RooAbsData object
Supply unbinned data object (created from TTree)  unbinned fit
Supply histogram object (created from THx)  binned fit
Binned
Unbinned x y z 1 3 5 2 4 6
RooDataSet
RooDataHist
RooAbsData
Wouter Verkerke, NIKHEF

17. Automated vs. hand-coded optimization of p.d.f.
Automatic pre-fit PDF optimization
Prior to each fit, the PDF is analyzed for possible optimizations
Optimization algorithms:
Detection and precalculation of constant terms in any PDF expression
Function caching and lazy evaluation
Factorization of multi-dimensional problems where ever possible
Optimizations are always tailored to the specific use in each fit.
Possible because OO structure of p.d.f. allows automated analysis of structure
No need for users to hard-code optimizations
Keeps your code understandable, maintainable and flexible without sacrificing performance
Optimization concepts implemented by RooFit are applied consistently and completely to all PDFs
Speedup of factor 3-10 reported in realistic complex fits
Fit parallelization on multi-CPU hosts
Option for automatic parallelization of fit function on multi-CPU hosts (no explicit or implicit support from user PDFs needed)
Wouter Verkerke, NIKHEF

18. MC Event generation pdf.generate(vars,nevt)
Sample “toy Monte Carlo” samples from any PDF
Accept/reject sampling method used by default
MC generation method automatically optimized
PDF can advertise internal generator if more efficient methods exists (e.g. Gaussian)
Each generator request will use the most efficient accept-reject / internal generator combination available
Operator PDFs (sum,product,…) distribute generation over components whenever possible
Generation order for products with conditional PDFs is sorted out automatically
Possible because OO structure of p.d.f. allows automated analysis of structure
Can also specify template data as input to generator
Look more carefully at accept/reject side bar
pdf.generate(vars,data)
Wouter Verkerke, NIKHEF

19. Using models – Plotting
Model visualization geared towards ‘publication plots’ not interactive browsing  emphasis on 1-dimensional plots
Simplest case: plotting a 1-D model over data
Modular structure of composite p.d.f.s allows easy access to components for plotting
Can show Poisson confidence intervals instead of sqrt(N) errors
RooPlot* frame = mes.frame() ;
data->plotOn(frame) ;
pdf->plotOn(frame) ;
pdf->plotOn(frame,Components(“bkg”))
frame->Draw() ;
Adaptive spacing of curve points to achieve 1‰ precision
Can store plot with data and all curves as single object
Wouter Verkerke, NIKHEF

20. Visualizing multi-dimensional models
IMHO: Visualization of multi-dim. models much more complicated than visualization of multi-dim. data
Simplest scenario: projection plots
Simple example M(m,t) = f[S(m)*T(t)]+(1-f)[B(m)*U(t)]
Provide intuitive behavior: projection of p.d.f. automatically follows projection of data: 1-D plot frame remembers ‘hidden’ variables in dataset. Subsequent p.d.f.s in same frame are automatically projected
Works identically for non-factorizable p.d.f.s! Necessary integrals are calculated automatically either analytical or numerically
mB distribution
Dt distribution
RooPlot* frame = dt.frame() ;
data.plotOn(frame) ;
model.plotOn(frame) ;
model.plotOn(frame,Components(“bkg”)) ; 
Wouter Verkerke, NIKHEF

21. Using models – plotting multi-dimensional models
Projection plot usually doesn’t capture full information content of your p.d.f.  equally easy to plot a ‘slice’
mB distribution
Dt distribution
RooPlot* frame = dt.frame() ;
data.plotOn(frame) ;
model.plotOn(frame) ;
model.plotOn(frame,Components(“bkg”)) ; 
RooPlot* frame = dt.frame() ;
dt.setRange(“sel”,5.27,5.30) ;
data.plotOn(frame,CutRange(“sel”)) ;
model.plotOn(frame,ProjectionRange(“sel”));
model.plotOn(frame,ProjectionRange(“sel”), Components(“bkg”)) ;
More complicated ‘slice’ views such as a likelihood ratio plot only take O(3) lines of extra code
Wouter Verkerke, NIKHEF

22. Using models – plotting
Can also use template data to show ‘average’ curve
Convenient for visualization of conditional p.d.f.s
Example plot t distribution of following model over data, averaging over dt values of data
model.plotOn(frame,ProjWData(dtData)) ;
model.plotOn(frame,Components("bkg"), ProjWData(dtData),LineStyle(kDashed)) ;
Wouter Verkerke, NIKHEF

23. Advanced features – Task automation
Support for routine task automation, e.g. useful in finding confidence intervals
Accumulate
fit statistics
Input model
Generate toy MC
Fit model
Distribution of
- parameter values
- parameter errors
- parameter pulls
Repeat N times
// Instantiate MC study manager
RooMCStudy mgr(inputModel) ;
// Generate and fit 100 samples of 1000 events
mgr.generateAndFit(100,1000) ;
// Plot distribution of sigma parameter
mgr.plotParam(sigma)->Draw()
Wouter Verkerke, NIKHEF

24. RooFit at SourceForge - roofit.sourceforge.net
RooFit moved to SourceForge to facilitate access and communication with non-BaBar users
Code access
CVS repository via pserver
File distribution sets for production versions
Wouter Verkerke, NIKHEF

25. RooFit at SourceForge - Documentation
Five separate tutorials
More than 250 slides and 20 macros in total
Documentation
Comprehensive set of tutorials (PPT slide show + example macros)
Also available by end of 2005:
Pedagogical User Guide + Reference Guide (~100 pages document)
Class reference in THtml style
Wouter Verkerke, NIKHEF

26. Selection of BaBar publications in 2004 using RooFit
Improved Measurement of the CKM angle alpha using B0r+r-
Measurement of branching fractions and charge asymmetries in B+ decays to hp+, hK+, hr+ and h'p+, and search for B0 decays to hK0 and hw
Branching Fraction and CP Asymmetries in B0KSKSKS
Measurement of CP Asymmetries in B0fK0 and B0K+K-K0S
Measurement of Branching Fractions and Time-Dependent CP-Violating Asymmetries in Bh' K Decays
Improved Measurement of Time-Dependent CP Violation in B0 to (ccbar)K0 Decays (‘sin2b’)
Measurements of the Branching Fraction and CP-Violating Asymmetries in B0f0(980)KS Decays
Measurement of Time-dependent CP-Violating Asymmetries in B0K*g, K*KSp0 Decays
Study of the decay B0r+r- and constraints on the CKM angle alpha.
Measurement of CP-violating Asymmetries in B0K0Sp0 Decays
Measurement of Time-Dependent CP Asymmetries in B0fK0
Search for B+/-  [K-/+ p+/-]D K+/- and upper limit on the bu amplitude in B+/-  DK+/-
Limits on the Decay-Rate Difference of Neutral B Mesons and on CP, T, and CPT Violation in B0B0bar Oscillations
Wouter Verkerke, NIKHEF

27. Summary RooFit adds a powerful data modeling language to ROOT
Mathematical objects (variables, functions…) represented as C++ objects
Complex models are easily composed from library of standard components
New fundamental components can be written easily
Powerful tools for fitting, Toy MC generation and visualization are easy to use
Compact syntax
Universal functionality (almost no arbitrary / implementation related restrictions)
Automated function optimization analysis and implementation delivers industrial strength performance
RooFit makes the road from a ‘simple’ to a really elaborate/complex p.d.f. less steep:
RooFit is distributed with ROOT starting with ROOT version v
Wouter Verkerke, NIKHEF