continuous simulation of beyond-standard-model processes with multiple parameters
DESCRIPTION
Continuous simulation of Beyond-Standard-Model processes with multiple parameters. Jiahang Zhong (University of Oxford * ) Shih-Chang Lee (Academia Sinica) ACAT 2011, 5-9 September, London. * Was in Academia Sinica and Nanjing University. Motivation. - PowerPoint PPT PresentationTRANSCRIPT
Continuous simulation of
Beyond-Standard-Model processes with multiple
parameters
Jiahang Zhong (University of Oxford*)
Shih-Chang Lee (Academia Sinica)
ACAT 2011, 5-9 September, London
* Was in Academia Sinica and Nanjing University
2
Motivation
Many Beyond Standard Model (BSM) processes are defined by more than one free parameters
Masses of hypothetical particlesCoupling constants…
Grid ScanScan the parameter spacewith grid pointsSimulate a sample of events on each point
ACAT 2011, 5-9 September, London
Var1
Var2
Jiahang ZHONG
ACAT 2011, 5-9 September, London 3
Motivation
The difficulties of the grid-scan approach:Curse of dimensionality
Npoints~Nd
Hard to go beyond 2DCostly for finer granularity
Var1
Var2
Jiahang ZHONG
ACAT 2011, 5-9 September, London 4
Motivation
The difficulties of the grid-scan approach:Curse of dimensionality
Npoints~Nd
Hard to go beyond 2DCostly for finer granularity
Large statistics requiredSamples at different points are treated independentlyConsiderable statistics neededwithin each sample
Var1
Var2
Pass
Fail~10k evts
Jiahang ZHONG
ACAT 2011, 5-9 September, London 5
Motivation
The difficulties of the grid-scan approach:Curse of dimensionality
Npoints~Nd
Hard to go beyond 2DCostly for finer granularity
Large statistics requiredSamples at different points are treated independentlyConsiderable statistics neededwithin each sample
DiscretenessConsiderable space between pointsSmoothing/interpolation neededConsequent systematic uncertainties
Var1
Var2
~TeV
~100GeV
Jiahang ZHONG
6
Motivation
Grid-scan:Curse of dimensionality
Large statistics needed
Discreteness
The aim of Continuous MCCompetent for multivariate parameter space
Less events to be simulated
Continuous estimation of signal yield over the parameter space
ACAT 2011, 5-9 September, LondonJiahang ZHONG
The usage of multivariate BSM simulation is to estimate signal yields over the parameter space.
Yields: N(x)=L* σ(x) * ε(x)
L: Luminosity.Irrelevant to x (the free parameters)
σ: Cross section, branching ratio. Easy to calculate with event generators
ε: Detector acceptance, offline efficiencyNeed large amount and expensive detector simulation
Therefore, our method is focused on easing the estimation of ε
Motivation
ACAT 2011, 5-9 September, London 7Jiahang ZHONG
The procedure
Event generation
ACAT 2011, 5-9 September, London 8
Var1
Var2
Var1
Var2
Grid Scan Continuous MC
O(10d) space points O(100k) space points
O(10k) events/point O(1) events/point
Jiahang ZHONG
The procedure
Bayesian Neural Network (BNN) is used to fit the efficiency ε
Desirable features of NN fitting
Non-parametric modeling
Smooth over the parameter space
Unbinned fitting
Suffer less from dimensionality
Correlation between the variables
Jiahang ZHONG ACAT 2011, 5-9 September, London 9
Unbinned fitting vs. Binned Histogram
The procedure
Bayesian implementations of NN further provide
Automatic complexity control of NN topology during training
Probabilistic output
Uncertainty estimation of the output
Uncertainty of the output estimated based on the p.d.f. of the NN parameters.
Statistical fluctuation of the training sample
Choice of NN topology
Impact of fitting goodness at certain space point x
Jiahang ZHONG ACAT 2011, 5-9 September, London 10
Demo
Production of right-handed W boson and Majorana neutrino
Di-lepton final state2 leptons (e,μ)pT>20GeV, |eta|<2.5
cone20/pT<0.1
Two free parametersWR mass [500GeV,1500GeV]
NR mass [0, M(WR)]
Affect both the cross-section and efficiency
11
Demo
Continuous SimulationGenerated 100k events, each with random { M(WR), M(NR) }
Put each event through the selection criteria, and assign target value 1/0 if it pass/failFeed all events to a BNN, with { M(WR), M(NR) } as the input variablesUse the trained BNN as a function to provide ε±σε
Reference grid-scanA grid with 100GeV step in M(WR) and 50GeV step in M(NR) (171 samples in total)Sufficient statistics in each sample to achieve precise reference values
Jiahang ZHONG ACAT 2011, 5-9 September, London 12
Demo
The BNN fitted efficiency Reference from grid-scan
Jiahang ZHONG ACAT 2011, 5-9 September, London 13
Demo
The difference between fitted values and reference values
Jiahang ZHONG ACAT 2011, 5-9 September, London 14
Demo
Uncertainty estimated by the BNN.
Jiahang ZHONG ACAT 2011, 5-9 September, London 15
Demo
The real deviations vs. estimated uncertainties (Nσ)
Jiahang ZHONG ACAT 2011, 5-9 September, London 16
Summary
New approach to simulate multivariate BSM processesMore space points, less eventsUse BNN fitting to obtain smooth yield estimation
Performance tested byThe deviation between BNN and reference valuesThis deviation vs. BNN uncertainty
Limitation: the assumption of smooth distributionNot sensitive to local abrupt changesLess performance across physics boundary.
17ACAT 2011, 5-9 September, LondonJiahang ZHONG
完Thank you!
18ACAT 2011, 5-9 September, LondonJiahang ZHONG
Backup
More detailed documentation of this methodhttp://arxiv.org/abs/1107.0166
The Bayesian Neural Network in TMVA/ROOThttp://www.sciencedirect.com/science/article/pii/S0010465511002682
19
Links
ACAT 2011, 5-9 September, LondonJiahang ZHONG
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A black-box of discriminator A white-box of non-parametric fitting tool
A multivariate function y(x)Generic function approximator (analog to polynomial in 1D)
Training unbinned MLE fitting
y: NN output, a probability, [0,1]t: Target value, 1=pass, 0=fail
BackupHow does BNN fitting work
i
tt yyL 1)1( l
ljlj xwa )1()1(
)1()1(jj afh
)tanh()()1( xxf
j
jiji xwa )2()2(
)2()2(ii afy
xxf )()2(
ACAT 2011, 5-9 September, LondonJiahang ZHONG
Backup: Bayesian implementation of NN(I)
21
Probability fitting
Unbinned fitting
Full usage of every event
Extrapolation/Interpolation
Fit y as probability function
Bernoulli likelihood
1,0y
i
iiii xytxytL ))(1log(*)1()(log*)log(
HistogramBNN
ACAT 2011, 5-9 September, LondonJiahang ZHONG
Backup: Bayesian implementation of NN (II)
22
Uncertainty estimation
Training:
Most probable value wMP
P(w|D)Probability of other w
Prediction
Probability
Uncertainty of y
Avoid excessive extrapolation (non-trivial for multivariate analysis)
wDwwx'x' dPPP )|(*),|y()|y(
HistogramBNN
ACAT 2011, 5-9 September, LondonJiahang ZHONG
Backup: Bayesian implementation of NN (III)
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RegulatorOvertraining is possible due to excessive complexity of NN
Early stopUse half input sample as monitor
Manual decision of when to stop excessive fitting
RegulatorPrior knowledge that “simpler” model is preferred
Adaptive during training
Save the monitor sample!!!
Early stop
Regulator
ACAT 2011, 5-9 September, LondonJiahang ZHONG