Gram-Charlierand Edgeworthexpansions for nongaussiancorrelations in femtoscopy

Transcription

1 Gram-Charlierand Edgeworthepansions or nongaussianorrelations in emtosopy Zimányi9 Winter Shool on eavy Ion Physis Mihiel de Kok University o Stellenbosh South Aria

2 Eperimental Femtosopy Fireball Position p Momentum Detetor r Sr Ψ, r Wave untion Fourier Transorm p p p C r Idential,non-interating Relative distane distribution partiles Correlation untion C d r [ ] S r Ψ, r

3 First Approimation: Gaussian Assume Gaussian shape or orrelator: C λ ep Out, long and side out, long, side i R i i Measuring Gaussian Radii through itting R i out out long side out long long

4 igh-statistis Eperimental Correlation untions: Not Gaussian! C STAR AuAu GeV out GeV/ Data: Measured D Correlation untion are not Gaussian. The traditional approah: itting o non-gaussian untions. Systemati desriptions beyond Gaussian: armonis Pratt & Danielewiz, Edgeworthand Gram-Charlierseries Reerene: T. Csörgőand S. egyi, Phys. Lett. B 89, 5.

5 Derivation o Gram-Charlierseries Assume one dimension, C g with Moments: Cumulants: g d r Ε[ r r ] g d Ε[ ] Ε[ ] E[ ] r K i We want to use umulantsto go beyond the Gaussian. R i out out long side out long long

6 Mean First our Cumulants Variane Skewness Kurtosis

7 Why Cumulants? 6 σ σ σ σ σ Cumulantsare invariant under translation Cumulants are simpler than moments One-dimensional Gaussian: Moments o a Gaussian Cumulants σ σ π ep g

8 r r G i d d log r r G i d d 6 Generating untion d e g i i i G i!! K Moment generating untion Fourier Transorm. Cumulant generating untion Log o Fourier Transorm. Moments: Cumulants:!!! ] log[ i i i i G K Moments to Cumulants: G

9 Reerene untion F G K!!! Measured orrelation untion Want to approimate g in terms o a reerene untion Generating untions o g and : Start with a Taylor epansion in the Fourier Spae g * *! ep! i i i d e F! ep! i i i d e g G

10 Gram-Charlier Series K! ' '! '! g Coeiients are determined by the moments/umulants Useul property o Fourier transorms Epansion in the derivatives o a reerene untion ' ' ' F i F i F i F K!!! F G

11 Determining the Coeiients G F!!! K Taking logs on both sides and epanding log F log G Coeiients in terms o CumulantDierenes: *! Cumulant dierenes to Coeiients 6

12 Ininite Formal Series Trunate series to orm a partial sum, rom ininity to k ow good is this approimation in pratie?! d d g Partial Sums! d d g k k Trunate to k terms

13 Kurtosis We will now use analytial untions or the orrelatorto test the Gram-Charlier epansion. Negative kurtosis Zero kurtosis Positive kurtosis log Gaussian [ ] vs. Negative Kurtosis Zero Kurtosis Positive Kurtosis Beta Distribution Gaussian yperseant Student s t NormalInverse Gaussian

14 Gram-CharlierType A Series: Gaussian reerene untion Gaussian gives Orthogonal Polynomials; Rodrigues ormula or ermite polynomials. Gram-Charlier Series is not neessarily orthogonal! ] [ r d d r ]!!! [ K g σ σ π ep

15 Gaussian Negative-Kurtosis g Beta g k k! 6 g Gram-Charlier Beta 6 th order g 6 Negative probabilities

16 Positive-kurtosis g Gaussian k k yperseant g! th Gram-Charlier yperseant g 6 th Gram-Charlieris worse 8 th Gram-Charlier 6 8 yperseant g yperseant g

17 6 6 6! 5 5 5!!! EdgeworthEpansion Same series; dierent trunation Assume that unknown orrelator g is the sum o n variables.! ep n n i n n G n G σ σ Trunate aording to order in n instead o a number o terms Reordering o terms ! 6 6! 6 9 9! 7 7! 5 5! 5 6 6!!! Gram-Charlier Edgeworth

18 Edgeworthdoes better Gaussian yperseant g th orderare the same yperseant g Gram-Charlier6 terms Edgeworth6 th order in n 6 yperseant 6 yperseant g g

19 Asymptoti Series Interim Summary Edgeworth and Gram-Charlier have the same onvergene Gaussian reerene will not onverge or positive kurtosis. Negative kurtosis will onverge, but will have negative tails. Dierent reerene untion or dierent measured kurtosis Negative kurtosis g: use Beta Distribution or. Solves negative probabilities.. Great onvergene. Small positive kurtosis g: use Edgeworth Epansion or Large positive kurtosis g: use Student s t Distribution or and ildebrandt polynomials, investigate urther...

20 ildebrandt Polynomials m, ab m a Student s t distribtion: Orthogonal polynomials: S S S 6 m m m m m Γ m m m m m 6 5Γ m6 m Γ m m m 6 Γ m Student s t distribution has limited number o moments m-. ildebrandt polynomials don t eist or higher orders.

21 Orthogonalityvs. Gram-Charlier Pearson amily: Orthogonal and Gram-Charlier Choose: Either Gram-Charlier derivatives o reerene or Orthogonal Polynomials Gram-Charlier αδ e αδ K π a δ δ Normal Inverse Gaussian Finite moments and simple umulants Construt polynomials or take derivatives Pearson Family Orthogonal Polynomials

22 Strategies or Positive kurtosis: Comparison Gauss-Edgeworth 6 yperseant g ildebrandt 6 yperseant g NIG Gram-Charlier 6 yperseant g NIG Polynomials 6 yperseant g

23 Strategies or Positive kurtosis: Dierene Gauss-Edgeworth. ildebrandt. 6 g Partial Sum-yperseant NIG Gram-Charlier.5 NIG Polynomials.

24 Conlusions The epansions are not based on itting; this might be an advantage in higher dimensions. For measured distributions g lose to Gaussian, the Edgeworth epansion perorms better than Gram-Charlier. For highly nongaussiandistributions g, both series epansions ail. Choosing nongaussianreerene untions an signiiantly improve desription. Negative kurtosis g: use Beta distribution or Positive kurtosis g: hoose reerene to losely resemble g Cumulants and Moments are only a good idea i the shape is nearly Gaussian.

25 Smoothness property All derivatives should be zero at the endpoints o the reerene untion No surae terms in partial integration. Ensures oeiient are only dependent on the moments/umulants ' ''

26 n Orthogonality? Rodrigues ormula: Orthogonal Polynomials [ ] d n w n P d Corretion untion to ensure smooth ontat d d w dy λ y d n Sturm-Liouville Euation g d

27 Pearson s Dierential Euation ' w a a b b b a a [ ] # # # P Pw D w wd w D P I the degree o the orretion untion w is greater than, the last euation would be impossible.

28 Gaussian Pearson Family Impossible Student s t F-Ratio Beta Inverse Gamma Gamma Skewness