Gram-Charlier and Edgeworth expansions for nongaussian correlations in femtoscopy Michiel de Kock...
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Gram-Charlier and Edgeworth expansions for nongaussian correlations in femtoscopy Michiel de Kock University of Stellenbosch South Africa Zimányi 2009 Winter School on Heavy Ion Physics
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Transcript of Gram-Charlier and Edgeworth expansions for nongaussian correlations in femtoscopy Michiel de Kock...
Gram-Charlier and Edgeworth expansions for nongaussian correlations in
femtoscopy
Michiel de Kock University of Stellenbosch
South Africa
Zimányi 2009 Winter School on Heavy Ion Physics
Experimental Femtoscopy
][ 1),( )( 1)(23 rqrrq SdC
1)( qC
),( rq
)(rS
r
Fireball Detector
21 ppq
Relative distance distribution
Wave function
Correlation function
Position Momentum
r q
2p
1p
Fourier Transform
Identical,non-interacting particles
2longlongout
2side
longout2out
12
0
00
0
qqq
q
qqq
Rij
First Approximation: Gaussian
ijjiji qRqC 2exp1)( q
• Assume Gaussian shape for correlator:
• Out, long and side
• Measuring Gaussian Radii through fitting
),,( sidelongout qqqq
• 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!
)GeV/c(outq)(1 qC
Data: http://drupl.star.bnl.gov/STAR/files/starpublications/50/data.htm
STAR Au+Au 200 GeV
Derivation of Gram-Charlier series
2221 ]E[][][ qqq
)(1)( qgC q• Assume one dimension,
with
• Moments:
• Cumulants:
We want to use cumulants to go beyond the Gaussian.
1)( dqqg
rrrr qdqqgqq )(][
2longlongout
2side
longout2out
12
0
00
0
qqq
q
qqq
RK ijij
First four CumulantsMean Variance
Skewness Kurtosis
1 2
3 4
Why Cumulants?
036
03
44224
4
323
3
22
222
11
• Cumulants are invariant under translation • Cumulants are simpler than moments
One-dimensional Gaussian:
Moments of a Gaussian Cumulants
2
)(exp
2
1)(
2qqg
cqq
0
)()(
x
r
r xGixd
d
0
)(log)(
x
r
r xGixd
d222
21
413144
312133
2122
11
364
3
Generating function
dqeqgj
ixixixxG
iqx
jj
j
)(!
)()()(1)(
02
2!2
11
Moment generating function (Fourier Transform).
Cumulant generating function (Log of Fourier Transform).
Moments:
Cumulants:
1
33
!31
22
!21
1 !
)()()()()](log[
jj
j
j
ixixixixxG
Moments to Cumulants:
)(xG
Reference function
)(
)(
xF
xG 33221
!3!2!11 x
cx
cx
c
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
)(qf
)(qg
1
*
0
*
!
)(exp
!
)()()(
jj
j
jj
jixq
j
ix
j
ixdqeqfxF
10 !
)(exp
!
)()()(
jj
j
jj
jixq
j
ix
j
ixdqeqgxG
Gram-Charlier Series
)(!3
)(''!2
)('!1
)()( )3(321 qfc
qfc
qfc
qfqg
Coefficients are determined by the moments/cumulants
Useful property of Fourier transforms
Expansion in the derivatives of a reference function
)()()()()()(''
)()()(')()(3)3(2 xFixqfxFixqf
xFixqfxFxf
33221
!3!2!11
)(
)(x
cx
cx
c
xF
xG
222
21
413144
312133
2122
11
364
3
c
c
c
c
Determining the Coefficients
0 !
)(log)(logj
j
j
j
xxGxF
*jjjjc
33221
!3!2!11
)(
)(x
cx
cx
c
xF
xG
Taking logs on both sides and expanding
Coefficients in terms of Cumulant Differences:
Cumulant differences to Coefficients
Infinite Formal Series
Truncate series to form a partial sum, from infinity to k
How good is this approximation in practice?
)(!
)1()(0
qfdq
d
j
cqg
j
j
jj
Partial Sums
)(
)(!
)1()(0
qf
qfdq
d
j
cqg
k
jk
j
jj
Truncate to k terms
We will now use analytical functions for the correlator to test the Gram-Charlier expansion.
Kurtosis
Negative Kurtosis Zero Kurtosis Positive Kurtosis
Beta Distribution Gaussian Hypersecant
Student’s t
Normal Inverse Gaussian
Gaussian
Negative kurtosis Positive kurtosisZero kurtosis
qqf vs.)(log
Gram-Charlier Type A Series:Gaussian reference function
Gaussian gives Orthogonal Polynomials;Rodrigues formula for Hermite polynomials.Gram-Charlier Series is not necessarily orthogonal!
154515)(
36)(
1)(
2466
244
22
qqqqH
qqqH
qqH
)]([)()( )(1 qfqH r
dqd
qfr
])(!3
)(!2
)(!1
1)[()( 33
22
11 qH
cqH
cqH
cqfqg
2
)(exp
2
1)(
2qqf
)()(!
)1()(0
qfqHj
cqf j
k
j
jjk
Negative-Kurtosis g(q)
Gaussian
q
Negative probabilitiesq
q
q
)(qf
Beta)(qg
)(6 qf
)()(6 qgqf
BetaGram-Charlier (6th order) )(qg
)()(!
)1()(0
qfqHj
cqf j
k
j
jjk
Positive-kurtosis g(q)
4th Gram-Charlier
6th Gram-Charlier is worse 8th Gram-Charlier
q q
q q
Gaussian)(qf Hypersecant
)(qg
)(4 qf
)(6 qf )(8 qf
Hypersecant
)(qgHypersecant Hypersecant
)(qg
)(qg
)()10(
)(
)(
)(
6236!6
1
55!51
44!41
33!31
qH
xH
qH
qH
Edgeworth Expansion
• Same series; different truncation• Assume that unknown correlator g(q) is the sum of n
variables.
1 !exp)|(
j
jjn
jn
ixn
n
xGnxG
Truncate according to order in n instead of a number of terms (Reordering of terms).
...)(3556)(
)(280)(35)(
)(10)(
)(
82453!8
16!6
16
9!913
37!71
435!51
5
6!612
34!41
4
33!31
qHqH
qHqHqH
qHqH
qH
Gram-Charlier Edgeworth
Edgeworth does better
Gram-Charlier (6 terms) Edgeworth (6th order in n)
4th order arethe same
Hypersecant)(qg
Gaussian)(qf
q q
q q
)(6 qf )(6 qf
Hypersecant)(qg
Hypersecant)(qg
Hypersecant)(qg
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)1. Solves negative probabilities.2. 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...
Hildebrandt Polynomials2
1
2
2
21
1),(
1)(
m
a
q
maBqf
Orthogonal polynomials:
)4(61
)4(32
)6(454
6
212
43
24
22
246
)(
)1()2)(1(2
)3)(2)(1(
)1(2
mq
mq
mqmS
mmqmmm
qmmmmS
mqmmS
• Student’s t distribution has limited number of moments (2m-1).• Hildebrandt polynomials don’t exist for higher orders.
Student’s t distribtion:
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 22
221)(
q
qaKexf
Normal Inverse Gaussian• Finite moments and simple cumulants• Construct polynomials or take derivatives
Strategies for Positive kurtosis: ComparisonGauss-Edgeworth Hildebrandt
NIG Gram-Charlier NIG Polynomials
q
)(6 qf
q
q
Hypersecant)(qg
Hypersecant
q
Hypersecant Hypersecant
)(qg
)(qg)(qg
)(6 qf
)(6 qf)(6 qf
Strategies for Positive kurtosis: Difference
)()(6 qgqf
02.0Gauss-Edgeworth Hildebrandt 004.0
001.0
Partial Sum-Hypersecant
005.0 NIG PolynomialsNIG Gram-Charlier
q
q
q
q
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.
