PWA Model Selection using a genetic algorithm Stephan Schmeing , Sebastian Neubert, Karl Bicker September 19th 2013 School on Concepts of Modern Amplitude Analysis Techniques Flecken-Zechlin, Germany
Introduction: Partial-Wave Analysis at COMPASS Genetic Algorithm for Model Selection First Results Conclusion Outlook Outline Introduction: Partial-Wave Analysis at COMPASS Genetic Algorithm for Model Selection First Results Conclusion Outlook Stephan Schmeing — PWA Model Selection using a genetic algorithm 2/15
Introduction: Partial-Wave Analysis at COMPASS Genetic Algorithm for Model Selection First Results Conclusion Outlook Partial-Wave Analysis at COMPASS Motivation Diffractive Dissociation in 5 π No ”bump hunting” by eye possible ⇒ Partial-Wave Analysis (PWA): J P -Decomposition of mass spectrum using angular distribution Stephan Schmeing — PWA Model Selection using a genetic algorithm 3/15
Introduction: Partial-Wave Analysis at COMPASS Genetic Algorithm for Model Selection First Results Conclusion Outlook Partial-Wave Analysis at COMPASS Partial-Wave Analysis Partial-Wave Analysis Parametrisation of cross section (simplified): Waves ρ ij ( m X ) φ i ( τ ) φ j ( τ ) ∗ � σ ( τ ) = σ 0 i , j 11-dimensional maximum likelihood fit to experimental kinematic distributions For the calculation the sum over all waves has to be truncated Truncation introduces systematic errors Optimal model for truncation has to be found Stephan Schmeing — PWA Model Selection using a genetic algorithm 4/15
Introduction: Partial-Wave Analysis at COMPASS Genetic Algorithm for Model Selection First Results Conclusion Outlook Partial-Wave Analysis at COMPASS Partial-Wave Analysis Spin Density Matrix Stephan Schmeing — PWA Model Selection using a genetic algorithm 5/15
Introduction: Partial-Wave Analysis at COMPASS Genetic Algorithm for Model Selection First Results Conclusion Outlook Partial-Wave Analysis at COMPASS Partial-Wave Analysis Model Requirements The model should describe the data well 1 The number of parameters should be as small as possible 2 Correlations between parameters should be minimal 3 Stephan Schmeing — PWA Model Selection using a genetic algorithm 6/15
Introduction: Partial-Wave Analysis at COMPASS Genetic Algorithm for Model Selection First Results Conclusion Outlook Partial-Wave Analysis at COMPASS Partial-Wave Analysis Model Selection Traditional way: Compare log(likelihood) for different truncations Use physical arguments and preexisting knowledge Trial and error Introduces bias Has no methodical handle on systematic errors Too many possibilities for 5 π case ⇒ Use an algorithm for model selection Stephan Schmeing — PWA Model Selection using a genetic algorithm 7/15
Introduction: Partial-Wave Analysis at COMPASS Genetic Algorithm for Model Selection First Results Conclusion Outlook Genetic Algorithm for Model Selection Principle Working Principle Stephan Schmeing — PWA Model Selection using a genetic algorithm 8/15
Introduction: Partial-Wave Analysis at COMPASS Genetic Algorithm for Model Selection First Results Conclusion Outlook Genetic Algorithm for Model Selection Ranking Criterion Goodness-of-Fit Criterion Log(likelihood) alone cannot be used to quantify model quality, since more parameters tend to give better log(likelihood) Use Bayes’ theorem to judge model quality Evidence Best fit likelihood · Occam factor � P ( Data | A k P ( A k P ( Data | M k ) ML , M k ) · ML | M k ) σ A k | Data � Additional factor to supress small waves with large errors is introduced A number of approximations are needed to calculate this Stephan Schmeing — PWA Model Selection using a genetic algorithm 9/15
Introduction: Partial-Wave Analysis at COMPASS Genetic Algorithm for Model Selection First Results Conclusion Outlook Genetic Algorithm for Model Selection Optimization of Algorithm Optimization Criteria Full search space has to be explored: After a short starting phase the 1 average evidence should fluctuate around constant well below maximum evidence As few as possible created models should be invalid (for example due 2 to not converging fits) Final result should be (close to) optimal solution: Manually 3 improvement should not be possible Stephan Schmeing — PWA Model Selection using a genetic algorithm 10/15
Introduction: Partial-Wave Analysis at COMPASS Genetic Algorithm for Model Selection First Results Conclusion Outlook First Results Conditions Data from COMPASS 2004 hadron pilot run Use a pool of 284 Waves Run 100 generations with 50 models each Stephan Schmeing — PWA Model Selection using a genetic algorithm 11/15
Introduction: Partial-Wave Analysis at COMPASS Genetic Algorithm for Model Selection First Results Conclusion Outlook First Results First Results Number of waves 40 Waveset size optimizes 38 around 34 waves 36 Finally chosen waveset 34 contains 31 waves 32 30 0 10 20 30 40 50 60 70 80 90 Generation Stephan Schmeing — PWA Model Selection using a genetic algorithm 12/15
Introduction: Partial-Wave Analysis at COMPASS Genetic Algorithm for Model Selection First Results Conclusion Outlook First Results First Results Average evidence varies 10 3 × slightly around constant Evidence 1832 not far from 1830 maximum(1834 · 10 3 ) 1828 Currently only between 1826 16 and 32% of the created models are valid 1824 Simple manual breeding 1822 step can still increase 1820 final result 0 10 20 30 40 50 60 70 80 90 Generation Stephan Schmeing — PWA Model Selection using a genetic algorithm 13/15
Introduction: Partial-Wave Analysis at COMPASS Genetic Algorithm for Model Selection First Results Conclusion Outlook Conclusion Conclusion A genetic algorithm for model selection has been implemented in the framework of the ROOTPWA toolkit: ( http://sourceforge.net/projects/rootpwa/ ) A first partial-wave analysis using the algorithm has been performed The algorithm converged to a finite number of waves in the model High congruence between TOP 20 models ⇒ Goodness-of-Fit Criterion works Stephan Schmeing — PWA Model Selection using a genetic algorithm 14/15
Introduction: Partial-Wave Analysis at COMPASS Genetic Algorithm for Model Selection First Results Conclusion Outlook Outlook Outlook Tuning of algorithm parameters and selection/mutation methods Tests of results with simulated dataset Transfer to other decay channels Stephan Schmeing — PWA Model Selection using a genetic algorithm 15/15
Backup Backup PWA formula 1 Waves j ( τ ) ∗ σ ( τ ) = σ 0 � � ij ( m X ) φ ǫ i ( τ ) φ ǫ ρ ǫ ǫ = − 1 i , j Ranks ir T ǫ ∗ Spin density matrix: ρ ǫ ij ( m X ) = � T ǫ jr r = 1 Stephan Schmeing — PWA Model Selection using a genetic algorithm 1/3
Backup Backup Evidence Waves � ln P ( Data | M k ) � ln P ( Data | A k ML , M k ) − ln V k ( 2 π ) d | C A k | Data | + � A + ln ln S a a = Dimension: d number of real parameters = 2 · number of complex parameters ∞ � − x −| T a | 2 � 1 � Significance: S a = 2 π exp dx √ 2 σ 2 a 5 σ a Probability of the intensity of wave to be more than 5 σ larger than zero d V k 2 + 1 ) r d − 1 2 Parameter volume: A = d π Γ( d ( d − 1 ) -dimensional hypersphere with radius r = √ N events (neglecting interference between waves) Stephan Schmeing — PWA Model Selection using a genetic algorithm 2/3
Backup Backup Interpretation of Evidence Evidence not normalised ⇒ No absolut interpretation Relative intepretation ⇒ Bayes-Faktor: B 10 = P ( Data | M 1 ) P ( Data | M 0 ) ln B 10 B 10 Evidence 0 to 1 1 to 3 Not worth mentioning 1 to 3 3 to 20 Positive 3 to 5 20 to 150 Strong ≥ 5 ≥ 150 Very strong Stephan Schmeing — PWA Model Selection using a genetic algorithm 3/3
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