A Likelihood Fitting Technique for Nuclear Physics Data Analysis

Transcription

1 A Likelihood Fitting Technique for Nuclear Physics Data Analysis Priyashree Roy Nuclear Physics Seminar 0/24/204 Priyashree Roy, Florida State University 24 Oct 204, FSU / 3

2 Outline Fitting Techniques Fitting Techniques 2 ML Fit Example 3 Priyashree Roy, Florida State University 24 Oct 204, FSU / 3

3 Motivation Fitting Techniques Good fitting techniques are essential to extract maximum and most accurate information from experiments. "I can prove it or disprove it! What do you want me to do?" Priyashree Roy, Florida State University 24 Oct 204, FSU 2 / 3

4 Motivation Fitting Techniques Good fitting techniques are essential to extract maximum and most accurate information from experiments. Method of least squares (χ 2 fits)- Most common fitting technique for data with huge statistics. Event-based maximum likelihood fitting (ML fits)- very useful for extracting maximum information from low statistics datasets. Priyashree Roy, Florida State University 24 Oct 204, FSU 2 / 3

5 The Method of Least Squares Bin the data and plot histograms. Based on the assumption that each datapoint is Gaussian distributed, minimize χ 2 = ( yexperiment i y theory i (α j) σ i ) 2 and get the best estimate of the fit parameters α j. P.R. Bevington, Data Reduction and Analysis for the Physical Sciences (McGraw-Hill, 992) Priyashree Roy, Florida State University 24 Oct 204, FSU 3 / 3

6 The Method of Least Squares Bin the data and plot histograms. Based on the assumption that each datapoint is Gaussian distributed, minimize χ 2 = ( yexperiment i y theory i (α j) σ i ) 2 and get the best estimate of the fit parameters α j. High statistics - easy to fit Bad fit - datapoints not Gaussian distributed. Priyashree Roy, Florida State University 24 Oct 204, FSU 3 / 3

7 The Method of Least Squares Bin the data and plot histograms. Based on the assumption that each datapoint is Gaussian distributed, minimize χ 2 = ( yexperiment i y theory i (α j) σ i ) 2 and get the best estimate of the fit parameters α j. For example: x=cosθ fitting an angular distributiony theory (a,b) = a+bcos 2 (θ) Calculate χ 2 and minimize it to get the best values of a and b. L. Lyons, Statistics for Nuclear and Particle Physicists (CUP, 986) Priyashree Roy, Florida State University 24 Oct 204, FSU 3 / 3

8 The Method of Least Squares Disadvantage - Each bin should have enough statistics so that it is Gaussian distributed. Need coarse binning for data with low statistics. This can lead to loss of information. Priyashree Roy, Florida State University 24 Oct 204, FSU 4 / 3

9 The Event-based Maximum Likelihood Method We calculate the probability density for observing each event as a function of the fit parameter, P(y i (α)), and construct a likelihood L - L = P(y i (α)) over all events. i Maximize L as a function of the fit parameter α to find the best value forα. Priyashree Roy, Florida State University 24 Oct 204, FSU 5 / 3

10 The Event-based Maximum Likelihood Method We calculate the probability density for observing each event as a function of the fit parameter, P(y i (α)), and construct a likelihood L - L = P(y i (α)) over all events. i Maximize L as a function of the fit parameter α to find the best value forα. Example : Fitting the angular distribution (slide 2 data) - Probability density, y i ( b a ) = 2[+( b 3a )][+(b a )cos2 θ i ] for the i th event. Likelihood, L( b a ) = 2[+( b i 3a )][+(b a )cos2 θ i ] Event-based - no loss of information due to binning! L. Lyons, Statistics for Nuclear and Particle Physicists (CUP, 986) Priyashree Roy, Florida State University 24 Oct 204, FSU 5 / 3

11 ML orχ 2 - which one to use? Cons - χ 2 fitting Prosfastest and easiest. Gives goodness-of-fit indication. Makes (incorrect) Gaussian error assumption on low statistics bins. Binning problem - misses information for feature size<bin size. Pros- Cons - Most robust. ML fitting No Gaussian error assumption. No loss of information. No goodness-of-fit indication. Needs Monte-Carlo studies for verification. Computationally expensive for large N. Priyashree Roy, Florida State University 24 Oct 204, FSU 6 / 3

12 Outline Fitting Techniques ML Fit Example Fitting Techniques 2 ML Fit Example 3 Priyashree Roy, Florida State University 24 Oct 204, FSU 6 / 3

13 ML Fit Example Why are Polarization Observables Important? Volker Crede, Nuclear Physics Seminar Talk, Sept 26, 204 Atomic Spectrum of Hydrogen γp pπ 0 E γ [MeV] CLAS@JLab ELSA MAMI GRAAL SPring-8... Priyashree Roy, Florida State University 24 Oct 204, FSU 7 / 3

14 ML Fit Example Spin Observables for γ p pπ + π at CLAS, JLAB 5 independent kinematic variables needed - E γ,φ π, cos(θ + π ), cos(θ c.m. + p ),m π + π For linearly polarized photon beam and transversely polarized protons, reaction rateσ = σ 0 {(+Λ x P x +Λ y P y ) W. Roberts et al., Phys. Rev. C 7, (2005) +δ l [sin2β(i s +Λ x P s x +Λ y P s y) +cos2β(i c +Λ x P c x +Λ y P c y)]} δ l : deg. of beam polarization, Λ : deg. of target polarization Priyashree Roy, Florida State University 24 Oct 204, FSU 8 / 3

15 ML Fit Example Event-Based Qfactor Method for Signal-Bkg Separation Pion mass distribution from butanol Counts Signal + Background Polarized Target: Butanol has free polarized p + unpolarized bound p and n. Reactions from bound nucleons contribute to the background. C and CH 2 to study background MM (MeV) Signal-background separation - Assign an event-based dilution factor or "Qfactor" to each event which shows the chance that it came from the signal distribution. Priyashree Roy, Florida State University 24 Oct 204, FSU 9 / 3

16 ML Fit Example Event-Based Qfactor Method with Likelihood Fits Pictorial depiction of NN search A multivariate analysis - For each event ("seed event"), find N nearest neighbors in 4-D kinematic phase space (E γ,θ,φ,cos(θ p ) c.m. ). Plot mass distribution of then + events and fit. Priyashree Roy, Florida State University 24 Oct 204, FSU 0 / 3

17 ML Fit Example Event-Based Qfactor Method with Likelihood Fits Pictorial depiction of NN search A multivariate analysis - For each event ("seed event"), find N nearest neighbors in 4-D kinematic phase space (E γ,θ,φ,cos(θ p ) c.m. ). Plot mass distribution of then + events and fit. Since N is small (300), use ML method to fit the mass distribution. L = [f Signal (m i,α)+f Bkg (m i,β)] i f Q seed event = Signal (m 0,α best ) [f Signal (m 0,α best )+f Bkg (m 0,β best )], m 0- seed event s mass. Priyashree Roy, Florida State University 24 Oct 204, FSU 0 / 3

18 ML Fit Example Event-Based Qfactor Method with Likelihood Fits Pictorial depiction of NN search MM_butanol Signal (QF) Background(QF) MM_car MM_car_scaled Signal-background separation using Qfactors MM(MeV) A multivariate analysis - For each event ("seed event"), find N nearest neighbors in 4-D kinematic phase space (E γ,θ,φ,cos(θ p ) c.m. ). Plot mass distribution of then + events and fit. Since N is small (300), use ML method to fit the mass distribution. L = [f Signal (m i,α)+f Bkg (m i,β)] i f Q seed event = Signal (m 0,α best ) [f Signal (m 0,α best )+f Bkg (m 0,β best )], m 0- seed event s mass. Computation time reasonably minimized- fits 0,000 events in 30 min. Priyashree Roy, Florida State University 24 Oct 204, FSU 0 / 3

19 I Fitting Techniques ML Fit Example Polarization Observables: χ 2 vs. Unbinned ML Fit For linearly pol. beam and unpolarized target, σ σ σ +σ = 2δ av l [I c cos(2φ lab )+I s sin(2φ lab )] 2+(δ l δ l )[[Ic cos(2φ lab )+I s sin(2φ lab )] Unbinned ML fit to the angular distribution allowed extraction of I s,c in 4D (E γ, φ π, cos(θ + π ), cos(θ c.m. + p )). Not possible withχ 2 fits. c 0 0.0< cos(θ π +) <-0.8 Preliminary 0.0< cos(θ π +) < < cos(θ π +) < < cos(θ π +) < < cos(θ π +) < < cos(θ π +) <-0.0 g8b g9b Solid curve -Fourier t to g8b 0.2< cos(θ π +) < < cos(θ π +) < < cos(θ π +) < < cos(θ π +) < φ π + Priyashree Roy, Florida State University 24 Oct 204, FSU / 3

20 I I Fitting Techniques ML Fit Example ObservableI c in γ p pπ + π,e γ :..2 GeV c 0 0.0< cos(θ π +) < < cos(θ π +) < < cos(θ π +) < < cos(θp c.m. ) < < cos(θp c.m. ) <.0 Preliminary 0.2< cos(θ π +) < < cos(θ π +) < < cos(θ π +) < < cos(θ π +) <-0.0 c 0 g8b g9b Solid curve - Fourier fit to g8b 0.4< cos(θ π +) < < cos(θ π +) < < cos(θ π +) < φ π + 0.0< cos(θ π +) <-0.8 Preliminary 0.0< cos(θ π +) < < cos(θ π +) < < cos(θ π +) < < cos(θ π +) < < cos(θ π +) < < cos(θ π +) <-0.0 g8b g9b Solid curve -Fourier fit to g8b 0.4< cos(θ π +) < < cos(θ π +) < < cos(θ π +) < φ π + 4-dim. phase space: (E γ,φ π +, cos(θ π + ), cos(θ c.m. p )) I =I 0 {δ l [I s sin(2β)+i c cos(2β)]} β : angle between beam pol. and reaction plane. Good agreement of experimental data. Priyashree Roy, Florida State University 24 Oct 204, FSU 2 / 3

21 I I Fitting Techniques ML Fit Example ObservableI s in γ p pπ + π,e γ :..2 GeV s 0 0.0< cos(θ π +) <-0.8 Preliminary 0.0< cos(θ π +) < < cos(θp c.m. ) < < cos(θp c.m. ) <.0-0.8< cos(θ π +) < < cos(θ π +) < < cos(θ π +) < < cos(θ π +) < < cos(θ π +) <-0.0 s g8b g9b Solid curve - Fourier fit to g8b 0.4< cos(θ π +) < < cos(θ π +) < < cos(θ π +) < φ π + 0.0< cos(θ π +) < Preliminary 0.0< cos(θ π +) < < cos(θ π +) < < cos(θ π +) < < cos(θ π +) < < cos(θ π +) < < cos(θ π +) <-0.0 g8b g9b Solid curve - Fourier fit to g8b 0.4< cos(θ π +) < < cos(θ π +) < < cos(θ π +) < φ π + 4-dim. phase space: (E γ,φ π +, cos(θ π + ), cos(θ c.m. p )) I =I 0 {δ l [I s sin(2β)+i c cos(2β)]} β : angle between beam pol. and reaction plane. Good agreement of experimental data. Priyashree Roy, Florida State University 24 Oct 204, FSU 2 / 3

22 Outline Fitting Techniques Fitting Techniques 2 ML Fit Example 3 Priyashree Roy, Florida State University 24 Oct 204, FSU 2 / 3

23 Fitting Techniques Unbinned ML technique - very useful to optimize fit parameters of any general distribution with low statistics. Assumption - the model or fit function has the right form. Works very well for the extraction of polarization observables since the fit function is well-known. Bayesian analysis goes one step further - compare models quantitatively! Example: New particle in the mass spectrum? Model - No new particle, fit function : f Background Model 2 - A new particle, fit function : f NarrowGaussian +f Background Construct "evidence ratios" using Bayes theorem to compare the models. Learn more about Bayesian analysis in the next talk by Raditya! Priyashree Roy, Florida State University 24 Oct 204, FSU 3 / 3

24 Fitting Techniques Unbinned ML technique - very useful to optimize fit parameters of any general distribution with low statistics. Assumption - the model or fit function has the right form. Works very well for the extraction of polarization observables since the fit function is well-known. Bayesian analysis goes one step further - compare models quantitatively. Example: New particle in the mass spectrum? Model - No new particle, fit function : f Background Model 2 - A new particle, fit function : f NarrowGaussian +f Background Construct "evidence ratios" using Bayes theorem to compare the models. D.G. Ireland et al., arxiv: v2 [hep-ph] 2 Dec 2007 Learn more about Bayesian analysis in the next talk by Raditya! Priyashree Roy, Florida State University 24 Oct 204, FSU 3 / 3

25 Thank You Priyashree Roy, Florida State University 24 Oct 204, FSU 3 / 3