1. Zimányi 2009 Winter School on Heavy Ion Physics
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
matriks=Kij ;

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,
Edgeworth and Gram-Charlier series
Reference: T. Csörgő and S. Hegyi, Phys. Lett. B 489, 15 (2000).
STAR Au+Au 200 GeV
Data:
Check Best Fit

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? Cumulants are 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:
konnekteer met vorige prentjie
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?
Beklemtoon die infinity
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
Log-ln plots
Negative Kurtosis
Zero Kurtosis
Positive Kurtosis
Beta Distribution
Gaussian
Hypersecant
Student’s t
Normal Inverse Gaussian

14. Gram-Charlier Type A Series: Gaussian reference function
Gaussian gives Orthogonal Polynomials;
Rodrigues formula for Hermite polynomials.
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
Same line in

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

19. Different reference function for different measured kurtosis
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...
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20. Hildebrandt Polynomials
Student’s t distribtion:
Orthogonal polynomials:
te swart
Student’s t distribution has limited number of moments (2m-1).
Hildebrandt polynomials don’t exist for higher orders.

21. Orthogonality vs. Gram-Charlier
Pearson family: Orthogonal and Gram-Charlier
Choose: Either Gram-Charlier
(derivatives of reference)
or Orthogonal Polynomials
Gram-Charlier
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
Sit f by almal en g

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.

25. Smoothness property
All derivatives should be zero at the endpoints of the reference function
No “surface terms” in partial integration.
Ensures coefficient are only dependent on the moments/cumulants
Elimimating surface terms, each coefficient is fixed by a moment, (essential measured quantity). Coefficients are functions of moments/Cumulants

26. Orthogonality? Rodrigues formula: Orthogonal Polynomials
Correction function to ensure smooth contact
Reference function must obey a Sturm-Liouvill equation so that the polynomials are orthogonal
Sturm-Liouville Equation

27. Pearson’s Differential Equation
If the degree of the correction function w is greater than 2, the last equation would be impossible.

28. Pearson Family Gaussian Impossible Kurtosis Beta Student’s t F-Ratio
Inverse Gamma
Gamma
Skewness