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Review of Analysis Methods for Correlations and Fluctuations

This review provides an analysis of various methods for studying correlations and fluctuations, including Pearson's normalized covariance, scale-dependent variance differences, and inverting fluctuations to correlations. The review also highlights the challenges and biases associated with certain statistical measures.

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Review of Analysis Methods for Correlations and Fluctuations

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  1. Review of Analysis MethodsforCorrelations and Fluctuations Tom Trainor Firenze July 7, 2006

  2. Agenda • Fluctuations on binned spaces • Pearson’s normalized covariance • Scale-dependent variance differences • Inverting fluctuations to correlations • Autocorrelations from pair counting • Relating n and pt fluctuations to physics • Trompe l’oeil and measure design

  3. Summary of Charge Fluctuation Measures Jeff Mitchell summary

  4. Summary of Event-by-event <pT> Fluctuation Measures FpT Goal of the observables: State a comparison to the expectation of statistically independent particle emission. Jeff Mitchell summary s2pT,dyn DspT,n FpT SpT

  5. those slides are nice catalogs, but… • Some similarity relations are violated for relevant conditions (e.g., low multiplicity) • Some statistical measures exhibit misleading behaviors and/or strong biases • Random variables in denominators lead to undesirable properties

  6. Fluctuations on Binned Spaces mixed-pair reference so, back to basics… D S nb-nb multiple events sb pt, n particle number n or scalar pt sum in bin 0 a b x bin size = scale single-particle space na-na 0 marginal variances variance xD xS sa x2 project 0 b na-na variances for bins a, b covariance a covariance betweena and b a x1 b two-particle space dependence on bin size or scale

  7. Pearson’s Normalized Covariance Karl Pearson, 1857-1936 covariance relative to marginal variances h2 e.g., forward-backward correlations geometric mean of marginal variances h1 correlated uncorrelated anticorrelated D S b a -1 rab = 1 > 0 0 < 0 normalized covariance – the basic correlation measure

  8. Variance Difference omit s2 factor in denominator to facilitate n vs pt comparisons Poisson Poisson CLT CLT normalized number covariance normalized pt covariance normalized (scaled) variances, covariances: Pearson CLT differential measure: a = b scale-dependent variance difference

  9. Correlations and Fluctuations a covariance distribution measures correlations the scale integral of a covariance distribution measures fluctuations variance difference autocorrelation each bin is a mean of normalized covariances increasing scale running integral macrobin size microbin separation angular autocorrelations contain averages of normalized covariances in 4D bins of (h1,h2,f1,f2)

  10. Fluctuation Inversion single point 2D scale integral data STAR acceptance statistical reference 2D scale inversion scale dependence autocorrelation Rosetta stone for fluctuation and correlation analysis

  11. Derivation – Short Form k macrobin average variance difference x2 xD xS x2 autocorrelation methods: M 0 Dx bin method a integration x1 0 dx pair counting x1 a+k kex xD xS x2 microbin average strip method x2,b covariance distribution space invariance? M 0 m k Dx often true ex average x1,a 0 dx kex x1

  12. Inversion Precision: Comparisons Pythia pair counting Jeff number autocorrelations CI CD inversion Duncan autocorrelations from pair counting or by inversion of fluctuation scale dependence

  13. Autocorrelations from Pair Counting autocorrelations the hard way average over a,b on kth, lth diagonals pt 200 GeV p-p n Dr/√rref Dr/√rref (GeV2/c2) k eh l ef STAR preliminary joint(h,f) autocorrelations ‘direct’ look at autocorrelation, equivalent ratios for n andpt computationally expensive:

  14. Projecting Two-point Momentum Space (yt1,yt2) correlations (h1,h2,f1,f2) correlations hD=h1-h2 (yt,h,f)1(yt,h,f)2 fD=f1-f2 Dr/√rref n - number Dr/√rref rapidity correlations with ‘hard’ pt cut not an autocorrelation away-side angular autocorrelations same-side Dr/√rref Dr/√rref pt Dr/√rref (GeV2/c2) inter-jet unlike-sign pairs intra-jet

  15. Two Correlation Types correlated and anticorrelated bins rare events D D S S nb-nb nb-nb hard exceptional events mixed reference 0 0 mixed reference common events soft na-na na-na 0 0 p-p Au-Au relative frequency Dr/√rref log Dr/√rref (GeV2/c2) amplitude 0 na-na

  16. The Physics Behind n Fluctuations 4D hypercube: (a,b,+,-) Au-Au CLT varies with bin sizedh, df CI (positive) p-p p-p Au-Au HBT CD (negative) fluctuation scale (bin size) dependence integrates these two-particlecorrelations

  17. The Physics Behind pt Fluctuations Au-Au CLT n varies with bin sized x p-p Au-Au p-p pt fluctuation scale (bin size) dependence integrates these two-particlecorrelations

  18. Trompe l’Oeil

  19. v2 and Elliptic Flow 1/Npart number correlations pt correlations 1/Npart 8 v2 1 same 11 points conventional abscissa and ordinate per-pair per-particle 1/Npart total variance difference 1 8 quadrupole n = mean participant path length multipoles extracted from pt (not number) autocorrelations elliptic flow compared directly to minijet correlations v2 gives a misleading impression of flow centrality and pt trends

  20. The Balance Function p-p 200 GeV Au-Au 130 GeV HBT project CD angular autocorrelations peak volume variation invert balance function BF rms width autocorrelation STAR acceptance width acceptance factor xy true width variation acceptance rms width HI STAR fixed amplitude 0 Dh 2 the BF width measures amplitudes, not correlation lengths

  21. Spt: the real DaVinci Code ebye B A C ~n3 STAR estruct CERES integral inversion A 130 GeV claim sNN assumes global thermalization 2.5 Dr/rref B 130 GeV Spt sNN direct 2-point correlations 8-12 centrality running average CERES 2.5 C STAR acceptance dependence sNN 2 SSC:

  22. Summary • Pearson’s normalized covariance provides the basis for fluctuation/correlation measure design • An integral equation connects fluctuation scale dependence to angular autocorrelations • Fluctuations are physically interpretable by means of two-particle correlations • Optimal projections of two-particle momentum space to lower-dimensional spaces are defined • Some F/C measures can be misleading

  23. s2 Dynamical – . Hijing 200 GeV Au-Au 65-85% central scale dependence fluctuations fluctuation inversion unphysical biased measure auto- correlations physical doesn’t tolerate low multiplicities correlations **Dr/rref** (GeV/c)2 Dr/rref (GeV/c)2 unphysical

  24. Fragmentation e+-e- 91 GeV p-p 200 GeV p-p 200 GeV a new view of fragmentation

  25. running average Ds2/dh CERES C Dr/rref sNN STAR dh, hD 0 2

  26. Fussing about Fragmentation e+-e- 91 GeV p-p 200 GeV p-p 200 GeV a new view of fragmentation

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