Gram- Charlier and Edgeworth expansions for nongaussian correlations in femtoscopy

1. Gram-Charlier and Edgeworth expansions for nongaussian correlations in femtoscopy Zimányi 2009 Winter School on Heavy Ion Physics Michiel de Kock University of Stellenbosch South Africa

2. Experimental Femtoscopy Fireball Momentum Detector Position Wave function Fourier Transform Identical,non-interacting particles Relative distance distribution Correlation function

3. First Approximation: Gaussian • Assume Gaussian shape for correlator: • Out, long and side • Measuring Gaussian Radii through fitting

4. High-Statistics Experimental Correlation functions: Not Gaussian! • Measured 3D Correlation function are not Gaussian. • The traditional approach: fitting of non-Gaussian functions. • Systematic descriptions beyond Gaussian: Harmonics (Pratt & Danielewicz, http://arxiv.org/abs/nucl-th/0612076v1) Edgeworthand Gram-Charlierseries Reference: T. Csörgőand S. Hegyi, Phys. Lett. B 489, 15 (2000). STAR Au+Au 200 GeV Data: http://drupl.star.bnl.gov/STAR/files/starpublications/50/data.htm

5. Derivation of Gram-Charlier series • Assume one dimension, • with • Moments: • Cumulants: • We want to use cumulants to go beyond the Gaussian.

6. First four Cumulants Mean Variance Kurtosis Skewness

7. Why Cumulants? • Cumulantsare invariant under translation • Cumulants are simpler than moments • One-dimensional Gaussian: Moments of a Gaussian Cumulants

8. Generating function Moment generating function (Fourier Transform). Cumulant generating function (Log of Fourier Transform). Moments: Cumulants: Moments to Cumulants:

9. Reference function Measured correlation function • Want to approximate g in terms of a reference function Generating functions of g and f: Start with a Taylor expansion in the Fourier Space

10. Gram-Charlier Series Useful property of Fourier transforms Expansion in the derivatives of a reference function Coefficients are determined by the moments/cumulants

11. Determining the Coefficients Taking logs on both sides and expanding Coefficients in terms of Cumulant Differences: Cumulant differences to Coefficients

12. Partial Sums Infinite Formal Series Truncate series to form a partial sum, from infinity to k How good is this approximation in practice? Truncate to k terms

13. Kurtosis We will now use analytical functions for the correlator to test the Gram-Charlier expansion. Negative kurtosis Zero kurtosis Positive kurtosis Gaussian

14. Gram-Charlier Type A Series:Gaussian reference function Gaussian gives Orthogonal Polynomials; Rodriguesformula for Hermitepolynomials. Gram-Charlier Series is not necessarily orthogonal!

15. Negative-Kurtosis g(q) Gaussian Beta Gram-Charlier (6th order) Beta Negative probabilities

16. Positive-kurtosis g(q) 4th Gram-Charlier Gaussian Hypersecant Hypersecant 6th Gram-Charlier is worse 8th Gram-Charlier Hypersecant Hypersecant

17. Edgeworth Expansion • Same series; different truncation • Assume that unknown correlator g(q) is the sum of n variables. Truncate according to order in n instead of a number of terms (Reordering of terms). Gram-Charlier Edgeworth

18. Edgeworth does better 4thorderare the same Gaussian Hypersecant Hypersecant Gram-Charlier (6 terms) Edgeworth (6th order in n) Hypersecant Hypersecant

19. Interim Summary • Asymptotic Series • Edgeworth and Gram-Charlier have the same convergence • Gaussian reference will not converge for positive kurtosis. • Negative kurtosis will converge, but will have negative tails. Different reference function for different measured kurtosis • Negative kurtosis g(q): use Beta Distribution for f(q) • Solves negative probabilities. • Great convergence . • Small positive kurtosis g(q): use Edgeworth Expansion for f(q) • Large positive kurtosis g(q): use Student’s t Distribution for f(q) and Hildebrandt polynomials, investigate further...

20. Hildebrandt Polynomials Student’s t distribtion: Orthogonal polynomials: • Student’s t distribution has limited number of moments (2m-1). • Hildebrandt polynomials don’t exist for higher orders.

21. Gram-Charlier Orthogonal Polynomials Orthogonality vs. Gram-Charlier • Pearson family: Orthogonal and Gram-Charlier • Choose: Either Gram-Charlier (derivatives of reference) or Orthogonal Polynomials Pearson Family Normal Inverse Gaussian • Finite moments and simple cumulants • Construct polynomials or take derivatives

22. Strategies for Positive kurtosis: Comparison Gauss-Edgeworth Hypersecant Hildebrandt Hypersecant NIG Gram-Charlier Hypersecant NIG Polynomials Hypersecant

23. Strategies for Positive kurtosis: Difference Gauss-Edgeworth Hildebrandt Partial Sum-Hypersecant NIG Gram-Charlier NIG Polynomials

24. Conclusions • The expansions are not based on fitting; this might be an advantage in higher dimensions. • For measured distributions g(q) close to Gaussian, the Edgeworth expansion performs better than Gram-Charlier. • For highly nongaussian distributions g(q), both series expansions fail. • Choosing nongaussian reference functions f(q) can significantly improve description. • Negative kurtosis g(q): use Beta distribution for f(q) • Positive kurtosis g(q): choose reference f(q) to closely resemble g(q) • Cumulants and Moments are only a good idea if the shape is nearly Gaussian.