From Gram-Charlier Series to Orthogonal Polynomials

Transcription

1 From Gram-Charlier Series to Orthogonal Polynomials HC Eggers University of Stellenbosch Workshop on Particle Correlations and Fluctuations 2008 Kraków

2 Earlier work on Gram-Charlier Series (GCS) Work by S. Hegyi and T. Csörgő 1990 s, PLB 489,15 (2000) Typical hermite tensors for 3D Gram-Charlier Series: HCE, P Lipa PRD (1996) IJMPE 16,3205 (2007) Braz.J.Phys. 37,3A,877(2007)

3 Criteria for series expansions in femtoscopy Very general and basic talk about expansions Physics lays down some criteria: Kernel transform (source function) should be positive definite Symmetry of distribution for identical particles Choice of variables (governed by eg kernel, kinematics) Model independence Mathematics: Series should converge, at least asymptotically Orthogonality of Hilbert space basis {hn (x)} Preferably polynomials Statistics: Sensitivity to Statistical Independence Free parameters in fitting: Minimise no. of free parameters, preferably none Uncorrelated, stable best values Partial sums of expansion should be positive definite

4 Criteria for series expansions in femtoscopy Very general and basic talk about expansions Physics lays down some criteria: Kernel transform (source function) should be positive definite Symmetry of distribution for identical particles Choice of variables (governed by eg kernel, kinematics) Model independence Mathematics: Series should converge, at least asymptotically Orthogonality of Hilbert space basis {hn (x)} Preferably polynomials Statistics: Sensitivity to Statistical Independence Free parameters in fitting: Minimise no. of free parameters, preferably none Uncorrelated, stable best values Partial sums of expansion should be positive definite

5 Criteria for series expansions in femtoscopy Very general and basic talk about expansions Physics lays down some criteria: Kernel transform (source function) should be positive definite Symmetry of distribution for identical particles Choice of variables (governed by eg kernel, kinematics) Model independence Mathematics: Series should converge, at least asymptotically Orthogonality of Hilbert space basis {hn (x)} Preferably polynomials Statistics: Sensitivity to Statistical Independence Free parameters in fitting: Minimise no. of free parameters, preferably none Uncorrelated, stable best values Partial sums of expansion should be positive definite

6 Criteria for series expansions in femtoscopy Very general and basic talk about expansions Physics lays down some criteria: Kernel transform (source function) should be positive definite Symmetry of distribution for identical particles Choice of variables (governed by eg kernel, kinematics) Model independence Mathematics: Series should converge, at least asymptotically Orthogonality of Hilbert space basis {h n (x)} Preferably polynomials Statistics: Sensitivity to Statistical Independence Free parameters in fitting: Minimise no. of free parameters, preferably none Uncorrelated, stable best values Partial sums of expansion should be positive definite No expansion satisfies all criteria

7 Assumptions and notation in this talk Little reference to kernels, dual space, inversion No distinction between Gram-Charlier Series and Edgeworth x = (x, y, z) momentum or coordinate space f (x) measured distribution R 2 (q) κ i,j,k, measured cumulants of f (x) i,j,k {x, y, z} f 0 (x) λ i,j,k, analytical reference distribution (eg gaussian) cumulants of f 0 (x) Test ideas and convergence mostly in one dimension x x Symmetry: f ( x) = f (x) hence all odd orders zero κ 1 = κ 3 = = 0; offsets and asymmetry can be included (do later) Use analytical nongaussian distributions for f (x) replace later with data

8 Motivation for extensions beyond Gauss GCS Spherical/cartesian harmonics work well, so why look at other expansions? Gauss GCS has desirable properties: Its coefficients are cumulants, hence Sensitivity to statistical independence: GCS factorises if f (x) and f 0 (x) factorise No free parameters in case of fixed GCS Orthogonal system of hermite tensors/polynomials HOWEVER... Gram-Charlier series converge only asymptotically!

9 Motivation for extensions beyond Gauss GCS Spherical/cartesian harmonics work well, so why look at other expansions? Gauss GCS has desirable properties: Its coefficients are cumulants, hence Sensitivity to statistical independence: GCS factorises if f (x) and f 0 (x) factorise No free parameters in case of fixed GCS Orthogonal system of hermite tensors/polynomials HOWEVER... Gram-Charlier series converge only asymptotically!

10 Bad convergence of Fixed Gauss GCS Blue: f (x) = [4α cosh 2 (x/2α)] 1 to simulate nongaussian R 2 (q) Purple: f 0 (x) = exp( x 2 /2κ 2 )/ 2πκ 2 κ 2 = πα/ 3 so that η 2 = 0 f 2 (x), f 4 (x),... = partial sums of order r

11 Statistical independence and cumulants κ r,s,t Baseline: κ r,s,t = 0 for gaussian r+s+t > 2 Shape: κ r,0,0 = deviation from gauss r = 3, 4,... Independence: if f (x, y, z) factorises in any way then κ r,s,t = 0 eg if f (x, y, z) = f (x) f (y, z), then κ 1,1,0 = xy x y = 0 κ 2,1,0 = x 2 y x 2 y 2 x [ xy x y ] = 0 Tensors under affine transformation; in particular invariant under translation: κ r,s,t = κ r,s,t for x = x + const, r, s, t > 1 = easier to deal with offsets

12 Statistical independence and factorisation of GCS Coefficients of fixed Gram-Charlier series (GCS) [ f (x) = f 0 (x) ! η ijklh ijkl (x) + 1 6! η ijklmnh ijklmn (x) ] ( ) ηijklmnop + [35]η ijkl η mnop hijklmnop (x) ! are made up of cumulant differences η = κ λ with κ = cumulant of f (x), λ = cumulant of f 0 (x). If f (x, y, z) = f (x)f (y)f (z) and f 0 (x, y, z) = f 0 (x)f 0 (y)f 0 (z), all cross-term η are zero and the GCS factorises: f (x, y, z) = f 0 (x)[ ! η xxxx h xxxx (x) +...] f 0 (y)[ ! η yyyy h yyyy (y) +...] f 0 (z)[ ! η zzzz h zzzz (z) +...]

13 Again: Motivation for extensions beyond Gauss GCS Spherical/cartesian harmonics work well, so why look at other expansions? Gauss GCS has desirable properties: Its coefficients are cumulants, hence Sensitivity to statistical independence: GCS factorises if f (x) and f 0 (x) factorise No free parameters in case of fixed GCS Orthogonal system of hermite tensors/polynomials HOWEVER... Gram-Charlier series converge only asymptotically! HENCE look for compromises and extensions: Give up fixing λ 2 = free parameters Give up Gauss = Orthog. polynomials in Pearson system Give up polynomials = Transcendental functions Give up orthogonality = transform problems?

14 Formal convergence criteria Theorem by Cramér for f 0 = Gauss: If f (x) is a function with a continuous derivative such that + ( ) df 2 e x 2 /2 dx dx converges, and if lim x ± f (x) = 0, then the series f (x) = n=0 cn n! Dn f 0 (x), is absolutely and uniformly convergent for all x R. Effective convergence (Hegyi and Csörgő) when h(x) = f (x)/f 0 (x) R 2 (q)/f 0 (q) is in f 0 -Hilbert space ( ) f (x) 2 f 0 (x) dx is bounded f 0 (x) Theorems give no help regarding rate of convergence or asymptotic cases of GCS

15 From Fixed to Free Gram-Charlier series Can be done for ANY choice of f 0 (x) κ 2 = measured cumulant λ 2 = free parameter (eg gaussian σ 2 ) η 2 = κ 2 λ 2 Fixed GCS : Set λ 2 = κ 2 so η 2 = 0 f (x) = f 0 (x) [ ! η 4h 4 (x) + 1 6! η 6h 6 (x) + 1 8! (η η 2 4 )h 8(x) +... ] Free GCS : λ 2 remains free parameter = more terms and need to fit f (x) = f 0 (x) [ ! η 2h 2 (x) + 1 4! (η 4 + 3η 2 2 )h 4(x) + 1 6! (η η 4 η η 3 2 )h 6(x) + 1 8! (η η η 6η η 4 η η4 2 )h 8(x) +... ]

16 Distributions used for testing Consider only PDFs with < x < + : Name PDF f (x) Fixed Gauss 1 2πκ2 exp ( x 2 2κ 2 ) ( ) 1 Free Gauss σ exp x 2 2π 2σ 2 Logistic 1 4α cosh 2 (x/2α) Hypersecant 1 2α cosh(πx/2α) Laplace 1 2α exp( x /α) Normal Inverse Gauss αβ e αβ K 1 (α x 2 +β 2 ) π x 2 +β 2

17 Using Free Gauss for f 0 (x) f 0 (x) = 1 σ 2 /2σ2 e x 2π Both η 2 and all h n (x) now contain σ as free parameter: f (x) = f 0 (x) [ ! η 2(σ) h 2 (x σ) + 1 4! [η 4 + 3η 2 2 (σ)] h 4(x σ) +... ] [ = f 0 (x) ( x 2! (κ 2 σ 2 2 σ 2 ) ) σ ( x 4! [κ 4 + 3(κ 2 σ 2 ) 2 4 6x 2 σ 2 + 3σ 4 ) ] ] +... For every order of expansion r, determine σ by best fit: χ 2 = i [f (x i ) f n (x i σ)] 2 error 2 i σ 8 f 0 (x) [f (x) f n (x)] 2 dx

18 f (x) = Logistic distribution (set α = 1) f 0 (x) = Free Gauss: fitting σ 2 = λ 2

19 Logistic PDF vs Free Gauss: Results α = 1; best fits for σ {1.6, 1.8, 1.85, 2.1, 2.1}

20 Logistic PDF vs Free Gauss: tails

21 Logistic PDF vs Free Gauss: differences Best-fit value of σ increases with order of GCS partial sum Decrease of f f n with r is not monotonous

22 What property defines Gram-Charlier Series? Derivation of GCS uses cumulant-generating functions K (ξ) = ln e ξx f (x) dx K 0 (ξ) = ln e ξx f 0 (x) dx and multiple integration by parts (D = d/dx) ξ n e ξx f 0 (x) dx =... = e ξx ( D) n f 0 (x) dx yielding (1) cumulant coefficients and basis functions h n (x) = 1 f 0 (x) ( D)n f 0 (x), generalised by Romanovsky to (2) Rodrigues-formula functions h n (x) = 1 f 0 (x) ( D)n [f 0 (x) g(x) n ]

23 Properties of functions h(x) used in series expansions h n (x) = 1 f 0 (x) ( x) n [f 0 (x)g(x) n ] f 0 (x) constant h n (x) h s (x) f 0 (x) dx = δ ns Gauss GCS is HERE Y lm (Ω) HERE h n (x) = a 0 + a 1 x + + a n x n

24 Pearson System The Pearson System is defined by solutions to differential equation f 0 (x) f 0 (x) = (x a) b 0 + b 1 x + b 2 x 2 Solutions are classified in Pearson Types according to skewness κ 3 /κ 3/2 2 and kurtosis κ 4 /κ 2 2 First four moments/cumulants uniquely defined by (a, b 0, b 1, b 2 ) Measuring moments 1 4 tells us which Pearson type is appropriate = eliminate prejudice in choice of f 0 Includes some orthogonal systems, but also others (cf 2 slides later)

25 Orthogonal polynomials and the Pearson system Differential equation for orthogonal systems: g(x)h (x) + L(x)h (x) + λh(x) = 0 Define R(x) = exp[ (L/g)dx], then in the Rodrigues formula h n = f 1 0 ( D)n [f 0 g n ] g is exactly the coefficient of h while the weight function itself is given by f 0 (x) = R(x)/g(x) leading to f 0 (x) f 0 (x) = L(x) g (x) g(x) which is the same as the Pearson equation f 0 (x) f 0 (x) = (x a) b 0 + b 1 x + b 2 x 2

26 Selected Pearson Distributions Derivatives of Pearson... include Jacobi/Gegenbauer/Chebychev, Hermite, Laguerre exclude Legendre include some transcendental functions Type Name κ 3 κ 3/2 2 κ 4 κ 2 2 f 0 (x) Support N Gauss 0 0 e x 2 /2 (, + ) II Gegenbauer 0 < 0 [1 x 2 )] m a 2 Chebychev 1,2 VII Student s t 0 > 0 [1 + x 2 a 2 ] m ( a, +a) (, + ) V Gamma > 0 x m e αx [0, ) X Exponential > 0 e αx [0, ) XI Pareto > 0 x m (0, )

27 Erdélyi Theorem (1953) If g(x) has degree higher than 2, it is not possible to satisfy both orthogonality and Rodrigues simultaneously Note that f 0 is not restricted by this: there is room for orthogonal transcendental functions which are GCS

28 GCS with nongaussian f 0 Symmetric Normal Inverse Gaussian as toy model for nongaussian f 0 f 0 (x σ, τ) = στ e στ K 1(σ x 2 + τ 2 ) π x 2 + τ 2 Two free parameters σ, τ > 0 Continuous deformation from gaussian Each order of GCS must be minimised separately (very slow) Derivatives are not polynomials but transcendental functions, eg h 2 (x σ, τ) = x 2 (( x 2 + τ 2) σ ) 2τ 2 + (x 2 + τ 2 ) 2 σ ( 3x 2 τ 2) K 0 ( σ x 2 + τ 2 ) (x 2 + τ 2 ) 3/2 K 1 ( σ x 2 + τ 2 )

29 Laplace vs Symmetric Normal Inverse Gaussian: min[f 0 (f f 2 ) 2 ] Rough best fit for second order for σ = 0.6, τ = 1.4 Blue = Laplace = f (x α) = 1 2α exp( x /α) Mauve = f n (x) of Normal Inverse Gaussian

30 Laplace vs Symmetric Normal Inverse Gaussian: min[f 0 (f f 4 ) 2 ] Best fit for 4th order: σ = 0.9, τ = 2.3 ± 0.1

31 Laplace vs Symmetric Normal Inverse Gaussian: min[f 0 (f f 6 ) 2 ] Best fit for 6th order: σ = 0.5 ± 0.1, τ = 4.0 ± 0.1

32 GCS and Gram-Schmidt orthogonalisation Orthogonalisation of system of independent functions {f n (x)} always possible Not restricted to polynomial systems Wronskian of f 0, f 0, f 0,... must be nonzero: avoid eigenvectors of D n

33 GCS and Gram-Schmidt orthogonalisation Orthogonalisation of system of independent functions {f n (x)} always possible Not restricted to polynomial systems Wronskian of f 0, f 0, f 0,... must be nonzero: avoid eigenvectors of D n BUT this takes you out of GCS Wronskian = det f 0 f 0 f 0 f (n) 0 f 0 f 0 f (3) 0 f (n+1) 0 f 0 f (3) 0 f (4) 0 f (n+2) f (n) 0 f (n+1) 0 f (n+2) 0 f (2n) 0

34 Comparing expansion types Fixed/Free Gauss GCS Nongauss GCS Harmonics description of shape bad better good radius of convergence small better large transform to dept on dept on coordinate space kernel kernel good statist. independence good dept bad handling offsets good dept bad parameters in 1D 0/1 1 2 n/a parameters in 3D 3/6 6??

35 Conclusions and Summary If only shape is interesting, harmonics are a good, versatile, well-developed tool While non-gcs expansions are more versatile, they cannot be interpreted statistically No Free Lunch Gram-Charlier series have advantages (statistical independence, transformation properties), but their convergence must be tested case by case GCS expansions with transcendental functions very promising, but orthogonality may have to be sacrificed: No Free Lunch At a minimum, statistical independence should be tested by direct measurement of the appropriate cumulants Extension to 3D is mathematically easy, but fitting may give problems Asymmetric GCS (eg for Q inv ) can be extended in the same way Offsets will result in inclusion of odd-order terms, but cumulants for r>1 are invariant