expansions for nongaussian correlations in femtoscopy Michiel de Kock University of Stellenbosch South Africa Zimányi 2009 Winter School on Heavy Ion Physics Experimental ID: 933367
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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
Slide2Experimental
Femtoscopy
Fireball
Detector
Relative distance distribution
Wave function
Correlation function
Position
Momentum
Fourier Transform
Identical,non
-interacting particles
Slide3First Approximation: Gaussian
Assume Gaussian
shape for correlator: Out, long and side Measuring Gaussian Radii through fitting
Slide4Measured 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
Slide5Derivation of Gram-Charlier series
Assume one dimension,
with Moments: Cumulants: We want to use
cumulants to go beyond the Gaussian.
Slide6First four Cumulants
Mean
VarianceSkewness
Kurtosis
Slide7Why Cumulants
?
Cumulants are invariant under translation Cumulants are simpler than momentsOne-dimensional Gaussian:
Moments of a Gaussian
Cumulants
Slide8Generating function
Moment generating function (Fourier Transform).
Cumulant
generating function (Log of Fourier Transform).Moments:
Cumulants
:
Moments to
Cumulants
:
Slide9Reference 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
Slide10Gram-Charlier Series
Coefficients are determined by the moments/cumulants
Useful property of Fourier transforms
Expansion in the derivatives of a reference function
Slide11Determining the Coefficients
Taking logs on both sides and expanding
Coefficients in terms of Cumulant Differences:
Cumulant
differences to Coefficients
Slide12Infinite 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
Slide13We 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
Slide14Gram-Charlier Type A Series:Gaussian reference function
Gaussian gives Orthogonal Polynomials;
Rodrigues formula for Hermite polynomials.Gram-Charlier Series is not necessarily orthogonal!
Slide15Negative-Kurtosis g(q)
Gaussian
Negative probabilities
Beta
Beta
Gram-
Charlier
(6
th
order)
Slide16Positive-kurtosis g(q)
4
th Gram-Charlier
6th Gram-
Charlier
is worse
8
th
Gram-
Charlier
Gaussian
Hypersecant
Hypersecant
Hypersecant
Hypersecant
Slide17Edgeworth 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
Slide18Edgeworth
does better
Gram-Charlier (6 terms)
Edgeworth (6th order in n)
4
th
order
are
the same
Hypersecant
Gaussian
Hypersecant
Hypersecant
Hypersecant
Slide19Interim 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...
Slide20Hildebrandt 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:
Slide21Orthogonality 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
Slide22Strategies for Positive kurtosis: Comparison
Gauss-
EdgeworthHildebrandt
NIG Gram-Charlier
NIG
Polynomials
Hypersecant
Hypersecant
Hypersecant
Hypersecant
Slide23Strategies for Positive kurtosis: Difference
Gauss-
Edgeworth
Hildebrandt
Partial Sum
-
Hypersecant
NIG
Polynomials
NIG
Gram-
Charlier
Slide24ConclusionsThe 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.
Slide25Smoothness 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
Slide26Orthogonality?
Rodrigues formula: Orthogonal Polynomials
Correction function to ensure smooth contactSturm-Liouville Equation
Slide27Pearson’s Differential Equation
If the degree of the correction function
w is greater than 2, the last equation would be impossible.
Slide28Pearson Family
Impossible
BetaGammaInverse GammaF-Ratio
KurtosisSkewness
Student’s t
Gaussian
http://en.wikipedia.org/wiki/Pearson_distribution