Gram- Charlier and Edgeworth - PowerPoint Presentation

Slide1

Gram-Charlier and Edgeworth expansions for nongaussian correlations in femtoscopy

Michiel

de Kock University of StellenboschSouth Africa

Zimányi 2009 Winter School on Heavy Ion Physics

Slide2

Experimental

Femtoscopy

Fireball

Detector

Relative distance distribution

Wave function

Correlation function

Position

Momentum

Fourier Transform

Identical,non

-interacting particles

Slide3

First Approximation: Gaussian

Assume Gaussian

shape for correlator: Out, long and side Measuring Gaussian Radii through fitting

Slide4

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)Edgeworth and Gram-Charlier series Reference: T. Csörgő and S. Hegyi, Phys. Lett. B 489, 15 (2000).

High-Statistics Experimental Correlation functions: Not Gaussian!

Data:

http://drupl.star.bnl.gov/STAR/files/starpublications/50/data.htm

STAR

Au+Au

200

GeV

Slide5

Derivation of Gram-Charlier series

Assume one dimension,

with Moments: Cumulants: We want to use

cumulants to go beyond the Gaussian.

Slide6

First four Cumulants

Mean

VarianceSkewness

Kurtosis

Slide7

Why Cumulants

?

Cumulants are invariant under translation Cumulants are simpler than momentsOne-dimensional Gaussian:

Moments of a Gaussian

Cumulants

Slide8

Generating function

Moment generating function (Fourier Transform).

Cumulant

generating function (Log of Fourier Transform).Moments:

Cumulants

:

Moments to

Cumulants

:

Slide9

Reference function

Measured correlation function

Want to approximate g in terms of a reference functionGenerating functions of g and f: Start with a Taylor expansion in the Fourier Space

Slide10

Gram-Charlier Series

Coefficients are determined by the moments/cumulants

Useful property of Fourier transforms

Expansion in the derivatives of a reference function

Slide11

Determining the Coefficients

Taking logs on both sides and expanding

Coefficients in terms of Cumulant Differences:

Cumulant

differences to Coefficients

Slide12

Infinite Formal SeriesTruncate series to form a partial sum, from infinity to k

How good is this approximation in practice?

Partial Sums

Truncate to k terms

Slide13

We will now use analytical functions for the correlator

to test the Gram-Charlier expansion.

KurtosisNegative KurtosisZero KurtosisPositive KurtosisBeta Distribution

GaussianHypersecantStudent’s tNormal

Inverse Gaussian

Gaussian

Negative kurtosis

Positive kurtosis

Zero kurtosis

Slide14

Gram-Charlier Type A Series:Gaussian reference function

Gaussian gives Orthogonal Polynomials;

Rodrigues formula for Hermite polynomials.Gram-Charlier Series is not necessarily orthogonal!

Slide15

Negative-Kurtosis g(q)

Gaussian

Negative probabilities

Beta

Beta

Gram-

Charlier

(6

th

order)

Slide16

Positive-kurtosis g(q)

4

th Gram-Charlier

6th Gram-

Charlier

is worse

8

th

Gram-

Charlier

Gaussian

Hypersecant

Hypersecant

Hypersecant

Hypersecant

Slide17

Edgeworth Expansion

Same series; different truncationAssume 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

Slide18

Edgeworth

does better

Gram-Charlier (6 terms)

Edgeworth (6th order in n)

4

th

order

are

the same

Hypersecant

Gaussian

Hypersecant

Hypersecant

Hypersecant

Slide19

Interim SummaryAsymptotic SeriesEdgeworth and Gram-Charlier have the same convergenceGaussian 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...

Slide20

Hildebrandt Polynomials

Orthogonal polynomials:

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

distribtion:

Slide21

Orthogonality vs. Gram-CharlierPearson family: Orthogonal and Gram-CharlierChoose:

Either Gram-Charlier(derivatives of reference)

or Orthogonal PolynomialsGram-CharlierOrthogonal

PolynomialsPearson Family

Normal Inverse Gaussian

Finite moments and simple

cumulants

Construct polynomials or take derivatives

Slide22

Strategies for Positive kurtosis: Comparison

Gauss-

EdgeworthHildebrandt

NIG Gram-Charlier

NIG

Polynomials

Hypersecant

Hypersecant

Hypersecant

Hypersecant

Slide23

Strategies for Positive kurtosis: Difference

Gauss-

Edgeworth

Hildebrandt

Partial Sum

-

Hypersecant

NIG

Polynomials

NIG

Gram-

Charlier

Slide24

ConclusionsThe 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.

Slide25

Smoothness propertyAll derivatives should be zero at the endpoints of the reference functionNo “surface terms” in partial integration.Ensures coefficient are only dependent on the moments/cumulants

Slide26

Orthogonality?

Rodrigues formula: Orthogonal Polynomials

Correction function to ensure smooth contactSturm-Liouville Equation

Slide27

Pearson’s Differential Equation

If the degree of the correction function

w is greater than 2, the last equation would be impossible.

Slide28

Pearson Family

Impossible

BetaGammaInverse GammaF-Ratio

KurtosisSkewness

Student’s t

Gaussian

http://en.wikipedia.org/wiki/Pearson_distribution