A Likelihood Fitting Technique for Nuclear Physics Data...

Fitting TechniquesApplications in Hadron Spectroscopy

Outlook

A Likelihood Fitting Technique for Nuclear PhysicsData Analysis

Priyashree Roy

Nuclear Physics Seminar

10/24/2014

Priyashree Roy, Florida State University 24 Oct 2014, FSU 1 / 13


Outlook

Outline

1 Fitting Techniques

2 Applications in Hadron SpectroscopyResearch MotivationML Fit Example 1ML Fit Example 2

3 Outlook



Outlook

Motivation

"I can prove it or disprove it !What do you want me to do?"

Good fitting techniques are essential to extract maximum andmost accurate information from experiments.



Outlook

Motivation

Good fitting techniques are essential to extract maximum andmost accurate information from experiments.

Method of least squares (χ2 fits)-Most common fitting techniquefor data with huge statistics.

Event-based maximum likelihoodfitting (ML fits)- very useful forextracting maximum informationfrom low statistics datasets.



Outlook

The Method of Least Squares

Bin the data and plot histograms. Based on the as-sumption that each datapoint is Gaussian distributed,minimize

χ2 =∑

(yexperimenti

−ytheoryi

(αj)

σi)2

and get the best estimate of the fit parametersαj . P.R. Bevington, ’Data Reduction and Analysis for the Physical

Sciences’ (McGraw-Hill, 1992)



Outlook



χ2 =∑

(yexperimenti

−ytheoryi

(αj)

σi)2

and get the best estimate of the fit parametersαj . High statistics - easy to fit

Bad fit - datapoints not Gaussiandistributed.



Outlook



χ2 =∑

(yexperimenti

−ytheoryi

(αj)

σi)2

and get the best estimate of the fit parametersαj .

For example: fitting an angular distribution-

ytheory(a, b) = a+ bcos2(θ)Calculateχ2 and minimize it to get the best values ofa and b.

L. Lyons, ’Statistics for Nuclear and Particle Physicists’(CUP, 1986)

x = cosθ



Outlook


Disadvantage -Each bin should have enough statistics so that it is Gaussiandistributed. Need coarse binning for data with low statistics. This can lead toloss of information.



Outlook

The Event-based Maximum Likelihood Method

We calculate the probability density for observingeach eventas a function ofthe fit parameter,P (yi(α)), and construct a likelihood L -L =

∏

i

P (yi(α)) over all events.

Maximize L as a function of the fit parameterα to find the best value forα.



Outlook

The Event-based Maximum Likelihood Method

We calculate the probability density for observingeach eventas a function ofthe fit parameter,P (yi(α)), and construct a likelihood L -L =

∏

i

P (yi(α)) over all events.

Maximize L as a function of the fit parameterα to find the best value forα.

Example : Fitting the angular distribution (slide 2 data) -

Probability density,yi( ba ) =1

2[1+( b3a

)][1 + ( b

a)cos2θi] for theith event.

Likelihood,L( ba) =

∏

i

12[1+( b

3a)][1 + ( b

a)cos2θi]

Event-based - no loss of information due to binning !

L. Lyons, ’Statistics for Nuclear and Particle Physicists’(CUP, 1986)



Outlook

ML or χ2 - which one to use ?

χ2 fitting

Pros-

fastest and easiest.

Gives goodness-of-fit indication.

Cons -

Makes (incorrect) Gaussian errorassumption on low statistics bins.

Binning problem - missesinformation for feature size< binsize.

ML fitting

Pros-

Most robust.

No Gaussian error assumption.

No loss of information.

Cons -

No goodness-of-fit indication.Needs Monte-Carlo studies forverification.

Computationally expensive forlarge N.



Outlook

Research MotivationML Fit Example 1ML Fit Example 2

Outline



3 Outlook



Outlook


Why are Polarization Observables Important?

Atomic Spectrum of Hydrogen

Eγ [MeV]

γp → pπ0 CLAS@JLab

ELSAMAMIGRAALSPring-8...

Volker Crede, Nuclear Physics Seminar Talk, Sept 26, 2014



Outlook


Spin Observables for~γ~p → pπ+π− at CLAS, JLAB

5 independent kinematic variablesneeded -Eγ , φ∗π+ , cos(θ

∗

π+), cos(θc.m.p ), mπ+π−

For linearly polarized photon beam and transversely polarized protons,reaction rate-σ = σ0{(1 + ΛxPx + ΛyPy)

+δl[sin2β(Is + ΛxP

s

x+ ΛyP

s

y)

+cos2β(Ic + ΛxPc

x+ ΛyP

c

y)]}

δl : deg. of beam polarization,Λ : deg. of target polarization

W. Robertset al., Phys. Rev. C71, 055201 (2005)



Outlook


Event-Based Qfactor Method for Signal-Bkg Separation

MM (MeV)0 50 100 150 200 250 300 350 400 450 500

Counts

0

1000

2000

3000

4000

5000

6000

7000

Signal +

Background

Pion mass distribution

from butanol

Polarized Target: Butanolhas free polarized p + unpolarizedbound p and n.Reactions from bound nucleonscontribute to the background.

C andCH2 to study background.

Signal-background separation -Assign anevent-based dilution fac-tor or "Qfactor" to each event whichshows the chance that it came fromthe signal distribution.



Outlook


Event-Based Qfactor Method with Likelihood FitsPictorial depiction

of NN search

A multivariate analysis - For each event ("seed event"), find N nearestneighbors in 4-D kinematic phase space (Eγ , θ∗, φ∗, cos(θp)c.m.). Plot massdistribution of theN + 1 events and fit.



Outlook



of NN search


Since N is small (300), use ML method to fit the mass distribution.L =

∏

i

[fSignal(mi, α) + fBkg(mi, β)]

Qseed−event =fSignal(m0,α

best)[fSignal(m0,αbest)+fBkg(m0,βbest)]

, m0- seed event’s mass.



Outlook



of NN search

MM (MeV)0 50 100 150 200 250 300 350 400 450 5000

1000

2000

3000

4000

5000

6000

7000MM_butanolSignal (QF)Background (QF)MM_carMM_car_scaled

Signal-background

separation using Qfactors


Since N is small (300), use ML method to fit the mass distribution.L =

∏

i

[fSignal(mi, α) + fBkg(mi, β)]

Qseed−event =fSignal(m0,α

best)[fSignal(m0,αbest)+fBkg(m0,βbest)]

, m0- seed event’s mass.

Computation time reasonably minimized-fits 10,000 events in 30 min.



Outlook


Polarization Observables:χ2 vs. Unbinned ML Fit

Prelim

inary

)


Outlook


ObservableIc in ~γ~p → pπ+π−, Eγ : 1.1− 1.2 GeV

Prelim

inary

g8b

g9b

Solid curve -Fourier fit to g8b

)


Outlook


ObservableIs in ~γ~p → pπ+π−, Eγ : 1.1− 1.2 GeV

Prelim

inary

-1

-1

1

g8b

g9b

Solid curve - Fourier fit to g8b

-1

-1

)


Outlook

Outline



3 Outlook



Outlook

Outlook

Unbinned ML technique - very useful to optimize fit parameters of anygeneral distribution with low statistics.Assumption - the model or fit function has the right form. Works verywell for the extraction of polarization observables since the fit functionis well-known.

Bayesian analysis goes one step further - compare modelsquantitatively !Example: New particle in the mass spectrum ?Model 1 - No new particle, fit function :fBackground

Model 2 - A new particle, fit function :fNarrowGaussian + fBackground

Construct "evidence ratios" using Bayes’ theorem to compare themodels.

Learn more about Bayesian analysis in the next talk by Raditya !



Outlook

Outlook

Unbinned ML technique - very useful to optimize fit parameters of anygeneral distribution with low statistics.Assumption - the model or fit function has the right form. Works verywell for the extraction of polarization observables since the fit functionis well-known.

Bayesian analysis goes one step further - compare modelsquantitatively.Example: New particle in the mass spectrum ?Model 1 - No new particle, fit function :fBackground

Model 2 - A new particle, fit function :fNarrowGaussian + fBackground

Construct "evidence ratios" using Bayes’ theorem to compare themodels.

Learn more about Bayesian analysis in the next talk by Raditya !

D.G. Irelandet al., arXiv:0709.3154v2 [hep-ph] 12 Dec 2007



Outlook

Thank You


Fitting TechniquesApplications in Hadron SpectroscopyResearch MotivationML Fit Example 1ML Fit Example 2

Outlook

A Likelihood Fitting Technique for Nuclear Physics Data...

Documents

Transcript of A Likelihood Fitting Technique for Nuclear Physics Data...