Positive Definite Functions on Spheres

Transcription

1 Positive Definite Functions on Spheres Tilmann Gneiting Institute for Applied Mathematics Heidelberg University, Germany International Conference Ars Conjectandi Basel, Switzerland, October 15,

2 Prelude: Ars Conjectandi, Bernoulli, and Gegenbauer Jakob Bernoulli studied the equation m 1 k=0 k n 1 = 1 n (B n(m) B n (0)) where m and n are positive integers, leading to what we now know as the Bernoulli polynomials the Bernoulli polynomials can be defined via the generating function where t,x (0,1) te tx e t 1 = n=0 B n (x) tn n!, similarly, the Gegenbauer polynomials can be defined via the generating function where t,x (0,1) and λ > 0 1 (1 2xt+t 2 ) λ = n=0 C λ n(x)t n, 2

3 Motivation: Global data Isotropic correlation functions on Euclidean spaces: The classes Φ d Isotropic correlation functions on spheres: The classes Ψ d Some parametric families Open problems 3

4 Isotropic correlation functions on spheres in a wealth of geophysical, meteorological, climatological and other applications, spatial data need to be modeled on the surface of the Earth this motivates the study of random fields { Z(x) : x S d }, where S d = {x R d+1 : x = 1} is the unit sphere in the Euclidean space R d+1 in a simplifying special case, the process has unit variance and mean zero and is isotropic, i.e., there exists a function such that ψ : [0,π] R corr(z(x),z(y)) = ψ(θ(x,y)) for x,y S d, where θ(x, y) = arccos( x, y ) denotes the great circle, spherical or geodesic distance on S d 4

5 Positive definiteness any such function ψ : [0,π] R is positive definite on S d, in the sense that n n i=1j=1 a i a j ψ(θ(x i,x j )) = var for all a 1,...,a n R and x 1,...,x n S d n i=1 a i Z(x i ) 0 conversely, every positive definite function ψ on S d with ψ(0) = 1 is the correlation function of an isotropic Gaussian process class Ψ d Ψ d = { ψ : [0,π] R ψ is continuous, ψ(0) = 1, and θ ψ(θ) is positive definite on S d } the members of the class Ψ 2 are the basic building blocks for covariance models for global data 5

6 Motivation: Global data Isotropic correlation functions on Euclidean spaces: The classes Φ d Isotropic correlation functions on spheres: The classes Ψ d Some parametric families Open problems 6

7 Isotropic correlation functions on Euclidean spaces the traditional Flat Earth approximation motivates the study of random fields { Z(x) : x R d } in a simplifying special case, the process is standardized, stationary, and isotropic, i.e., there exists a function such that ϕ : [0, ) R corr(z(x),z(y)) = ϕ( x y )) for x,y R d Purely Spatial Correlation Function Correlation Distance in km 7

8 Positive definiteness any such function ϕ : [0, ) R is positive definite on R d, in the sense that n n i=1j=1 a i a j ϕ( x i x j ) = var for all a 1,...,a n R and x 1,...,x n R d n i=1 a i Z(x i ) 0 conversely, every positive definite function ϕ on R d with ϕ(0) = 1 is the correlation function of a stationary and isotropic Gaussian process class Φ d Φ d = { ϕ : [0, ) R ϕ is continuous, ϕ(0) = 1, and t ϕ(t) is positive definite on R d } the members of the class Φ 2 and Φ 3 are the basic building blocks for covariance models for local and regional spatial data 8

9 The classes Φ d in a classical paper, Schoenberg (1938) showed that Φ 1 Φ 2 and Φ d Φ = Φ d d=1 ϕ Φ d if and only if ϕ(t) = [0, ) Ω d(rt)df(r) for t 0, where F is a probability measure on [0, ) and Ω d (t) = Γ(d/2) ( 2 t ) (d 2)/2 J (d 2)/2 (t) ϕ Φ if and only if ϕ(t) = [0, ) e r2 t 2 df(r) for t 0, where F is a probability measure on [0, ) 9

10 Examples in Φ Matérn family ϕ(t) = 21 ν Γ(ν) ( ) t ν ( ) t Kν s s (ν > 0; s > 0) member of the class Φ studied by Whittle (1954), Matérn (1960) and Tatarski (1961) ν ϕ(t) = exp( t 2 ) when s 1 = 2 ν 0.4 ν = 5 2 ϕ(t) = ( 1+t+t 2 /3 ) exp( t) 0.2 ν = 3 2 ν = 1 ϕ(t) = (1+t) exp( t) ϕ(t) = tk 1 (t) t ν = 1 2 ϕ(t) = exp( t) ν 0 nugget effect 10

11 Examples in Φ Powered exponential family ϕ(t) = e (t/s)α (α (0,2]; s > 0) member of the class Φ α = 2 Gaussian ϕ(t) = exp( t 2 ) α = 1 exponential ϕ(t) = exp( t) α 0 nugget effect Cauchy family ϕ(t) = ( 1+ ( t s ) α ) β/α (α (0,2]; β > 0; s > 0) member of the class Φ studied by Gneiting and Schlather (2004) 11

12 Compactly supported functions covariance structures with compact support allow for computationally efficient prediction and simulation thus, parametric families of compactly supported members of the class Φ 3 are of particular interest Family Analytic Expression Smoothness Spherical ϕ(t) = Askey ϕ(t) = Wendland ϕ(t) = ( 1+ 1 )( t 1 t 2s s ( 1 t s ( ) 2+τ + )( ) 2 1+(4+τ) t 1 t s s + ) 4+τ + k = 0 α = 1 k = 0 α = 1 k = 1 α = 2 support parameter s > 0 and shape parameter τ 0 12

13 Statistical modeling the sample path properties of a stationary and isotropic Gaussian random field depend on the analytic properties of ϕ Φ d with the behavior at the origin being of particular interest if ϕ has fractal index α (0,2], in the sense that 1 ϕ(t) t α as t 0, then the graph {(x,z(x)) : x R d } R d+1 of a sample path has fractal or Hausdorff dimension D = d +1 α 2 if the fractal index is α = 2 then u ϕ( u ) is at least twice differentiable and the following holds: u ϕ( u ) is 2k times differentiable a Gaussian sample path is k times differentiable Matérn family: α = 2min(ν,1); k = ν ; powered exponential and Cauchy families: k = 0 if α (0,2) and k = if α = 2 13

14 Motivation: Global data Isotropic correlation functions on Euclidean spaces: The classes Φ d Isotropic correlation functions on spheres: The classes Ψ d Some parametric families Open problems 14

15 The classes Ψ d class Ψ d Ψ d = { ψ : [0,π] R ψ is continuous, ψ(0) = 1, and θ ψ(θ) is positive definite on S d } the class of the continuous and isotropic correlation functions: if ψ Ψ d then there exists a Gaussian process {Z(x) : x S d } such that corr(z(x),z(y)) = ψ(θ(x,y)) for x,y S d some long-standing questions if we restrict a function ϕ : [0, ) R in the class Φ d to a function on [0,π], do we obtain a member of the class Ψ d? in particular, can we use the standard models, such as the Matérn, powered exponential, Cauchy, spherical, Askey and Wendland families, with the spherical distance substituting for the Euclidean distance? 15

16 A natural and useful but restrictive construction construction: if ϕ Φd+1 for some d 1, then the function ψ : [0, π] R, θ 7 ϕ 2 sin 2θ corresponds to the restriction of an isotropic correlation function in Rd+1 to the sphere Sd, whence ψ Ψd preserves the fractal index in the following sense: if ϕ has fractal index α (0, 2] then ψ(0) ψ(θ) = O(θ α) as θ 0 α = 1.9 D = 2.05 α = 1.5 D = 2.25 α = 1.0 D =

17 Gegenbauer polynomials to characterize the class Ψ d, we consider orthogonal polynomials with argument cos θ given λ > 0 and an integer k 0, the function Ck λ (cosθ) is defined by the expansion 1 (1 2tcosθ +t 2 ) λ = k=0 C λ k (cosθ)tk for θ [0,π], where t ( 1,1) and Ck λ is the ultraspherical or Gegenbauer polynomial of order λ and degree k in particular, C0 λ(cosθ) 1 and Cλ 1 (cosθ) cosθ for all λ > 0 17

18 The classes Ψ d in another classical paper, Schoenberg (1941) showed that Ψ 1 Ψ 2 and Ψ d Ψ = Ψ d d=1 ψ Ψ d if and only if C (d 1)/2 ψ(θ) = b k (cos θ) d,k k=0 C (d 1)/2 k (1) with coefficients b d,k 0 that sum to 1 ψ Ψ if and only if ψ(θ) = k=0 b,k (cosθ) k with coefficients b,k 0 that sum to 1 18

19 The classes Ψ 1 and Ψ 2 ψ Ψ 1 if and only if ψ(θ) = k=0 b 1,k cos(kθ) with Fourier coefficients b 1,k 0 that sum to 1, where b 1,k = 2 π π 0 cos(kθ)ψ(θ)dθ for k 1 ψ Ψ 2 if and only if ψ(θ) = k=0 b 2,k k +1 P k(cosθ), with Legendre coefficients b 2,k 0 that sum to 1, where P k is the Legendre polynomial of degree k 0 and b 2,k = 2k +1 2 π 0 P k(cosθ) sinθ ψ(θ)dθ 19

20 Dimension walks generally, if d 2 the coefficient b d,k in the Gegenbauer expansion of a function ψ Ψ d equals b d,k = 2k +d d π (Γ( d 1 2 ))2 Γ(d 1) π 0 C(d 1)/2 k (cosθ)(sinθ) d 1 ψ(θ)dθ the Fourier cosine and Gegenbauer coefficients of a continuous function ψ : [0,π] R relate to each other in dimension walks, in that (k +d 1)(k +d) (k +1)(k +2) b d+2,k = b d,k d(2k +d 1) d(2k +d+3) b d,k+2 for integers d 1 and k 1 in particular, b 3,k = 1 2 (k +1)( b 1,k b 1,k+2 ), whence ψ Ψ 3 if and only if ψ Ψ 1 and the sequences of both the even and the odd Fourier cosine coefficients are nonincreasing 20

21 Motivation: Global data Isotropic correlation functions on Euclidean spaces: The classes Φ d Isotropic correlation functions on spheres: The classes Ψ d Some parametric families Open problems 21

22 The Φ construction Theorem A: If ϕ Φ then the function belongs to the class Ψ ψ : [0,π] R, θ ϕ(θ 1/2 ) Family Analytic Expression Valid Parameter Range Matérn Matérn Powered exponential ψ(θ) = 2ν 1 Γ(ν) Cauchy ψ(θ) = ( ) θ 1/2 ν ( ) θ 1/2 K ν ( ) ν ( ) ψ(θ) = 2ν 1 θ θ K ν Γ(ν) c c ( ( ) α ) θ ψ(θ) = exp c ( 1+ c c c > 0; ν > 0 c > 0; ν (0, 1 2 ] c > 0; α (0,1] ( ) α ) τ/α θ c > 0; α (0,1]; τ > 0 c 22

23 Compact support Theorem B: Suppose that d = 1 or d = 3. If the function ϕ Φ d satisfies ϕ(t) = 0 for t π, the function defined by ψ : [0,π] R, θ ϕ(θ) belongs to the class Ψ d in particular, families of compactly supported functions ϕ Φ 3 induce families of locally supported functions ψ Ψ 3 Family Analytic Expression Spherical ψ(θ) = Askey ψ(θ) = Wendland ψ(θ) = ( 1+ 1 )( θ 1 θ 2c c ( 1 θ c ( ) 2+τ + )( 1+τ θ 1 θ c c ) 2 + ) 4+τ support parameter c (0,π] and shape parameter τ

24 Covariance localization in atmospheric data assimilation, locally supported members of the class Ψ 3 are used for the distance-dependent filtering of spatial covariance estimates on planet Earth (Hamill 2001) the standard approach relies on Gaspari and Cohn (1999) s function ϕ GC Φ 3, where ϕ GC (t) = t2 +5t 3 +8t 4 8t 5, 0 t 2 1, 1 3 t 1( 8t 2 +8t 1 ) (1 t) 4, 1 2 t 1, 0, t 1, and the aforementioned natural construction, which yields the function ψ GC Ψ 2, where ψ GC (θ) = ϕ sin 2 θ GC sin 2 c and c (0,π] is a support parameter for θ [0,π] 24

25 Covariance localization Theorem B suggests a simple alternative, in that ψ TG Ψ 2, where ( θ ψ TG (θ) = ϕ GC for θ [0,π] c and c (0,π] is a support parameter ) PSI psi_tg psi_gc THETA the fact that ψ TG (θ) > ψ GC (θ) at all distances θ (0,c) suggests that ψ TG is the more effective localization function similar comments apply to covariance tapers in spatial statistics 25

26 Motivation: Global data Isotropic correlation functions on Euclidean spaces: The classes Φ d Isotropic correlation functions on spheres: The classes Ψ d Some parametric families Open problems 26

27 Open problems Conjecture 1. The statement of Theorem B holds in any dimension d 1. Problem 2. For d 1 and c (0,π), find the infimum of the curvature at the origin among the functions ψ Ψ d with ψ(θ) = 0 for θ c. Problem 3. For d 1 and c (0,π), find the associated Turán constant, i.e., find the supremum of S d ψ(θ(y o,y))dy among the functions ψ Ψ d with ψ(θ) = 0 for θ c. Problem 4. For d 1 and 0 < b < c < π, find the associated pointwise Turán constant, that is, find the supremum of ψ(b) among the functions ψ Ψ d with ψ(θ) = 0 for θ c. 27

28 Open problems Conjecture 5. Every ψ Ψ d admits a derivative of order [(d 1)/2] on (0,π), and this is the strongest such statement possible. Conjecture 6. Every ψ Ψ d admits a real-valued and isotropic convolution root. Conjecture 7. If d = 2l + 1 and c (0, π) the truncated power function, ( ψ(θ) = 1 θ ) τ c belongs to the class Ψ d if and only if τ l+1. + Problem 8. For what values of α (0,2] and c (0,π] does the truncated sine power function, ( ( ψ(θ) = 1 sin π ) ) θ α ½(θ c), 2 c belong to the class Ψ d? 28

29 Open problems Problem 9. The Berkeley Earth Surface Temperature project initially used an isotropic correlation model, ψ(θ) = exp 4 i=1 c i θ i. For what values of the parameter vector (c 1,c 2,c 3,c 4 ) does ψ belong to the class Ψ 2? Conjecture 10. If ψ Ψ d has fractal index α (0,2] the associated regular Gaussian particle has fractal or Hausdorff dimension d + 1 α 2 almost surely. Problem 11. Find Abelian and Tauberian theorems: What type of asymptotic behavior of the Gegenbauer coefficients, b d,k, corresponds to a given fractal index, α (0,2]? 29

30 Open problems Problem 12. Develop and study covariance models for multivariate isotropic random fields on Sd. Problem 13. Develop and study covariance models for non-isotropic random fields on Sd. Problem 14. Develop and study covariance models for spatiotemporal random fields on the domain Sd R. 30

31 Selected references Gneiting, T. (2013). Strictly and non-strictly positive definite functions on spheres. Bernoulli, 19, Huang, C., Zhang, H. and Robeson, S. M. (2011). On the validity of commonly used covariance and variogram functions on the sphere. Mathematical Geosciences, 43, Jun, M. and Stein, M. L. (2007). An approach to producing space-time covariance functions on spheres. Technometrics, 49, Jun, M. (2011). Non-stationary cross-covariance models for multivariate processes on a globe. Scandinavian Journal of Statistics, 38, Schoenberg, I. J. (1938). Metric spaces and completely monotone functions. Annals of Mathematics, 39, Schoenberg, I. J. (1942). Positive definite functions on spheres. Duke Mathematical Journal, 9, Ziegel, J. (2013). Convolution roots and differentiability of isotropic positive definite functions on spheres. Proceedings of the American Mathematical Society, in press. 31