On building valid space-time covariance models

Transcription

1 On building valid space-time covariance models Jorge Mateu joint work with E. Porcu Department of Mathematics Universitat Jaume I, Castellón (Spain) Avignon, October p. 1/68

2 of the seminar (2.1) Spatio-temporal covariance functions, (2.2) Separability and Full Symmetry, (2.3) Full Symmetry and Zonal Anisotropy, (2.4) Mixed forms, (2.5) Some concepts on. of the seminar models: 6 Applications (4.1) A new class of anisotropic space time covariances, (4.2) Mixed Forms, (4.3) New families of spectral. 7 Conclusions models: (5.1) Stationary covariance functions through: Mixtures of, Completely monotone functions and, (5.2) Nonstationary covariance functions and complete monotonicity, (5.3) Archimedean anisotropic covariance functions, (5.4) The Bernstein class. 6 Application: Indian Ocean wind speed data 7 Conclusions and further developments - p. 2/68

3 1. STM Separable Spatio-temporal Covariance Functions Rohuani and Hall (1989) Sampson and Guttorp (1992) Dimitrakopoulos and Luo (1994) Stationary Nonseparable Covariance Functions Jones and Zhang (1997) Cressie and Huang (1999) Kyriakidis and Journel (1999) Christakos (2000) Gneiting (2002) Ma (2002; 2003a; 2003b) Stein (2003) Fernández-Casal (2003) Kolovos et al (2004) Stein (2005) 1. STM 1.2 Some history of STM models: 6 Applications 7 Conclusions - p. 3/68

4 1.2 Some history of STM Nonstationary Nonseparable Spatio-temporal Covariance Functions Christakos (2000) Fuentes and Smith (2001) Hristopoulos and Christakos (2001) Ma (2002) Fuentes (2002) Stationary Nonseparable Anisotropic Spatio-temporal Covariance Functions Shapiro and Botha (1999) Fernández-Casal (2003) Stein (2005) 1. STM 1.2 Some history of STM models: 6 Applications 7 Conclusions - p. 4/68

5 2.1 (I) ST covariance functions Spatio-temporal Random Fields: Z(s, t) = m(s, t) + δ(s, t), s R d, t R, δ(s, t) SRF Covariance: Cov(δ(s + h, t + u), δ(s, t)) = C(h, u), s R d, t R Variogram: V ar(δ(s + h, t + u) δ(s, t)) = γ(h, u), s R d, t R 2.1 (I) ST covariance functions 2.1 (II) ST covariance functions 2.2 Separability and Full Symmetry 2.3 Full Symmetry and Zonal Anisotropy 2.4 Mixed Forms 2.5 (I) Some concepts on 2.5 (II) Some concepts on 2.5 (III) Some concepts on 2.5 (IV) Some concepts on 2.5 (V) Some concepts on 2.5 (VI) Some concepts on 2.5 (VII) Some concepts on models: 6 Applications 7 Conclusions - p. 5/68

6 2.1 (II) ST covariance functions Recall Covariance Functions are Positive Definite, as stated in Bochner s Theorem (1949). 2. In the Isotropic Case, the Fourier transform can be expressed as an integral of Bessel Functions (Hankel Transform). 3. Variogram associated to Intrinsically Stationary RF are conditionally definite negative. 4. If dealing with an SRF, the properties and relationships between covariance and variogram are preserved. For instance, C(h, u) = C(0, 0) γ(h, u) 2.1 (I) ST covariance functions 2.1 (II) ST covariance functions 2.2 Separability and Full Symmetry 2.3 Full Symmetry and Zonal Anisotropy 2.4 Mixed Forms 2.5 (I) Some concepts on 2.5 (II) Some concepts on 2.5 (III) Some concepts on 2.5 (IV) Some concepts on 2.5 (V) Some concepts on 2.5 (VI) Some concepts on 2.5 (VII) Some concepts on models: 6 Applications 7 Conclusions - p. 6/68

7 2.2 Separability and Full Symmetry 1 Separability: 2 Full Symmetry: C(h, u) = C(h, 0)C(0, u) C(0, 0) C(h, u) = C( h, u) = C(h, u) = C( h, u), for every (h, u) R d R. 2.1 (I) ST covariance functions 2.1 (II) ST covariance functions 2.2 Separability and Full Symmetry 2.3 Full Symmetry and Zonal Anisotropy 2.4 Mixed Forms 2.5 (I) Some concepts on 2.5 (II) Some concepts on 2.5 (III) Some concepts on 2.5 (IV) Some concepts on 2.5 (V) Some concepts on 2.5 (VI) Some concepts on 2.5 (VII) Some concepts on 3 Relations: separable covariances are also fully symmetric, while viceversa is not necessarily true. models: 6 Applications 7 Conclusions - p. 7/68

8 2.3 Full Symmetry and Zonal Anisotropy The great majority of contributions in literature regards covariance models which are fully symmetric and isotropic. Unfortunately we do not dispose of such a large literature for the problem of zonal anisotropy, which is at least as important as the problem of full symmetry. Thus, there is a big need for models which are zonally anisotropic in the spatial component. 2.1 (I) ST covariance functions 2.1 (II) ST covariance functions 2.2 Separability and Full Symmetry 2.3 Full Symmetry and Zonal Anisotropy 2.4 Mixed Forms 2.5 (I) Some concepts on 2.5 (II) Some concepts on 2.5 (III) Some concepts on 2.5 (IV) Some concepts on 2.5 (V) Some concepts on 2.5 (VI) Some concepts on 2.5 (VII) Some concepts on models: 6 Applications 7 Conclusions - p. 8/68

9 2.4 Mixed Forms We are interested in removing some undesirable features of the previously proposed models, particularly following Stein s remark about Gneiting s approach and about some tensorial product covariance models. Specifically, Stein (2003) observes that models of the type exp( s t ), obtained with a tensorial product of two exponential covariance functions on space and time, are not differentiable at the origin and denote a lack of differentiability along certain axis, which in turn implies discontinuities of the autocorrelation function away from the origin. Furthermore, Stein (2003), while emphasizing the need for spatio-temporal covariance functions which are sufficiently smooth away from the origin, observes that Gneiting s approach leads to some undesirable features, such as the fact that whatever the lack of smoothness of C(s, 0) for s near zero, it will be shared by C(s, t) for t 0 and s near zero, since C(s, t) is just a rescaling of C(s, 0). 2.1 (I) ST covariance functions 2.1 (II) ST covariance functions 2.2 Separability and Full Symmetry 2.3 Full Symmetry and Zonal Anisotropy 2.4 Mixed Forms 2.5 (I) Some concepts on 2.5 (II) Some concepts on 2.5 (III) Some concepts on 2.5 (IV) Some concepts on 2.5 (V) Some concepts on 2.5 (VI) Some concepts on 2.5 (VII) Some concepts on models: 6 Applications 7 Conclusions - p. 9/68

10 2.5 (I) Some concepts on Why? 1 Modern literature emphasize the need for new models of nonseparable covariance functions for spatial temporal phenomena 2 It would be very interesting to find some Link Functions allowing to build up nonseparable permissible closed forms starting from the margins, i.e. the spatial and the temporal covariance. This Link Functions should include the case of separability 3 The advantages of such construction would be considerable (estimation and inference) 4 We believe can be a good candidate in order to satisfy this purpose. 2.1 (I) ST covariance functions 2.1 (II) ST covariance functions 2.2 Separability and Full Symmetry 2.3 Full Symmetry and Zonal Anisotropy 2.4 Mixed Forms 2.5 (I) Some concepts on 2.5 (II) Some concepts on 2.5 (III) Some concepts on 2.5 (IV) Some concepts on 2.5 (V) Some concepts on 2.5 (VI) Some concepts on 2.5 (VII) Some concepts on models: 6 Applications 7 Conclusions - p. 10/68

11 2.5 (II) Some concepts on : Literature [1] Introduction: Nelsen (1999); Joe (1987) [2] Archimedean : Genest and McKay (1986a,b); Genest (1987) [3] Inferencial procedures for Archimedean : Genest and Rivest (1993) [4] Analysis of serial dependence (time series): Ferguson et al. (2000); Genest et al. (2002) [5] Asymptotics for empirical copula process: Genest and Rémillard (2004); Fermanian et al. (2002) [6] Environmental Data: Drouet and Monbet (2004) for pollution in the Atlantic Ocean; Favre et al. (2004) for a multivariate hydrological frequency analysis 2.1 (I) ST covariance functions 2.1 (II) ST covariance functions 2.2 Separability and Full Symmetry 2.3 Full Symmetry and Zonal Anisotropy 2.4 Mixed Forms 2.5 (I) Some concepts on 2.5 (II) Some concepts on 2.5 (III) Some concepts on 2.5 (IV) Some concepts on 2.5 (V) Some concepts on 2.5 (VI) Some concepts on 2.5 (VII) Some concepts on models: 6 Applications 7 Conclusions - p. 11/68

12 2.5 (III) Some concepts on : Introduction have been largely used in several statistical contexts for the study of dependence of two random variables and modelling of risk assessment in financial markets A desirable feature of is given by the fact that the marginal structures can be modelled separately and interaction between two random variables can further be modelled with in order to implement a bivariate structure Algorithms for choosing an appropriate copula for the interaction between two random variables can be found in Melchiori (2003). 2.1 (I) ST covariance functions 2.1 (II) ST covariance functions 2.2 Separability and Full Symmetry 2.3 Full Symmetry and Zonal Anisotropy 2.4 Mixed Forms 2.5 (I) Some concepts on 2.5 (II) Some concepts on 2.5 (III) Some concepts on 2.5 (IV) Some concepts on 2.5 (V) Some concepts on 2.5 (VI) Some concepts on 2.5 (VII) Some concepts on models: 6 Applications 7 Conclusions - p. 12/68

13 2.5 (IV) Some concepts on : Definitions Let I be the interval [0, 1] and let I 2 be the cartesian product I I R 2. A two-dimensional copula is a function K : I 2 I having the following properties K(w, 0) = K(0, v) = 0, w, v I. K(w, 1) = w; K(1, v) = v. K(w 2, v 2 ) K(w 2, v 1 ) K(w 1, v 2 ) + K(w 1, v 1 ) 0, for every w 1, w 2, v 1, v 2 I such that w 1 w 2 and v 1 v 2. Formally, a bivariate copula is a cumulative probability function with uniform marginal distributions over I. 2.1 (I) ST covariance functions 2.1 (II) ST covariance functions 2.2 Separability and Full Symmetry 2.3 Full Symmetry and Zonal Anisotropy 2.4 Mixed Forms 2.5 (I) Some concepts on 2.5 (II) Some concepts on 2.5 (III) Some concepts on 2.5 (IV) Some concepts on 2.5 (V) Some concepts on 2.5 (VI) Some concepts on 2.5 (VII) Some concepts on models: 6 Applications 7 Conclusions - p. 13/68

14 2.5 (V) Some concepts on : Important Theorems Theorem (Sklar, 1959) Let X, Y be random variables with distribution functions F X (x) and G Y (y) respectively, and with joint distribution function H X,Y (x, y). Then, there exists a copula K(.,.) such that (x, y) R 2 H X,Y (x, y) = K(F X (x), G Y (y)) If F X (x) and G Y (y) are continuous, K(.,.) is unique. Corollary Let H X,Y (x, y), (x, y) R 2, be a joint distribution function with (continuous) marginal distribution functions F X (x) and G Y (y) admitting inverse functions F 1 X (w) and G 1 Y (v), respectively. Then, a unique copula exists such that, u, v I, K(w, v) = H X,Y (F 1 X (w), G 1 Y (v)) 2.1 (I) ST covariance functions 2.1 (II) ST covariance functions 2.2 Separability and Full Symmetry 2.3 Full Symmetry and Zonal Anisotropy 2.4 Mixed Forms 2.5 (I) Some concepts on 2.5 (II) Some concepts on 2.5 (III) Some concepts on 2.5 (IV) Some concepts on 2.5 (V) Some concepts on 2.5 (VI) Some concepts on 2.5 (VII) Some concepts on models: 6 Applications 7 Conclusions - p. 14/68

15 2.5 (VI) Some concepts on Archimedean Theorem Let ϕ be a continuous strictly decreasing function from I 2 to [0, ] such that ϕ(1) = 0 and let ϕ 1 be the inverse of ϕ. Then, the function is a copula if and only if ϕ is convex. Remarks: 1 The class given in equation (1) is called Archimedean Class K(u, v) = ϕ 1 {ϕ(u) + ϕ(v)} (1) 2 Depending on the choice of the convex function ϕ, some known classes can be obtained as particular cases. For example, the families of Clayton, Gumbel or Frank are particular cases of the Archimedean class 3 The function ϕ is said to be the generator of the Archimedean copula 4 In this work we only refer to functions admitting proper inverse ϕ (I) ST covariance functions 2.1 (II) ST covariance functions 2.2 Separability and Full Symmetry 2.3 Full Symmetry and Zonal Anisotropy 2.4 Mixed Forms 2.5 (I) Some concepts on 2.5 (II) Some concepts on 2.5 (III) Some concepts on 2.5 (IV) Some concepts on 2.5 (V) Some concepts on 2.5 (VI) Some concepts on 2.5 (VII) Some concepts on models: 6 Applications 7 Conclusions - p. 15/68

16 2.5 (VII) Some concepts on Multivariate Archimedean Theorem (Kimberling, 1974) Let ϕ be a continuous strictly decreasing function from [0, 1] to [0, ] such that ϕ(0) = and ϕ(1) = 0. Denote with ϕ 1 the inverse of ϕ. The function K ψ (u 1,..., u n ) = ϕ 1 (ϕ(u 1 ) ϕ(u n )) (2) is a copula, for n 2, if and only if ϕ 1 is completely monotone on [0, ). How to choose the generating Function? 1 The generating function can be choosen through some measure of concordance, such as the Kendall s Tau, the Spearman s Rho and the Blest correlation coefficient (Genest and MacKay, 1987). 2 Archimedean have an explicit relation with the Kendall s τ. It can be shown that 1 τ K = ϕ(t) dt ϕ (t) 2.1 (I) ST covariance functions 2.1 (II) ST covariance functions 2.2 Separability and Full Symmetry 2.3 Full Symmetry and Zonal Anisotropy 2.4 Mixed Forms 2.5 (I) Some concepts on 2.5 (II) Some concepts on 2.5 (III) Some concepts on 2.5 (IV) Some concepts on 2.5 (V) Some concepts on 2.5 (VI) Some concepts on 2.5 (VII) Some concepts on models: 6 Applications 7 Conclusions - p. 16/68

17 3.1 Dagum: Introduction The Dagum survival function (Zenga and Zini, 2001) has been used in various economical applications: ψ(t) = 1, if t = 0 ( ) 1 1 (1+λt θ ) ε, if t > 0 This function has some desirable properties of differentiability at the origin and a good level of smoothing away from the origin. This facts motivates our research. 3.1 Dagum: Introduction 3.2 Dagum: Instruments 3.3 Dagum: Solution! 3.4 Dagum: Remarks! 3.5 Dagum: Other Remarks! 3.6 Dagum: Final Remarks! models: 6 Applications Our aim is to use ψ() as a radial function and inspect the range of parameters allowing for its permissibility 7 Conclusions - p. 17/68

18 3.2 Dagum: Instruments How to show the positive definiteness of ψ(.)? The direct answer can be found with the so-called Hankel transforms, which are integral of mixtures of Bessel functions of the second kind This calculus is often impossible (Gneiting, 2001) and unfortunately this was the case. If the direct calculus of the Hankel transform is not possible, then it is necessary to recur to sufficient conditions, such as: 1 Christakos sufficient conditions 2 Criteria of Pólya type 3.1 Dagum: Introduction 3.2 Dagum: Instruments 3.3 Dagum: Solution! 3.4 Dagum: Remarks! 3.5 Dagum: Other Remarks! 3.6 Dagum: Final Remarks! models: 6 Applications 7 Conclusions 3 Gneiting s criteria as extension of [2] - p. 18/68

19 3.3 Dagum: Solution! Using the criteria [3], we are able to state the following result: Theorem Consider the function h ψ( h ) where ψ(.) is defined as above. If θ < (7 ε)/(1 + 5ε) and ε < 7, then ψ( h ) is positive definite (indeed a permissible covariance function) in R 3. Remarks: From Lemma ( ) of Sasvári (1994) it can be deduced that the Dagum class is not valid if θ < 2/ε. However, this condition does not clash with our condition. The Dagum class coincides with the so-called Cauchy class (Gneiting and Schlather, 2004) ONLY if we set ε = Dagum: Introduction 3.2 Dagum: Instruments 3.3 Dagum: Solution! 3.4 Dagum: Remarks! 3.5 Dagum: Other Remarks! 3.6 Dagum: Final Remarks! models: 6 Applications 7 Conclusions The validity on R 3 implies validity on lower dimensions, so that this covariance can be used even for temporal models. - p. 19/68

20 3.4 Dagum: Remarks! 1 As we deal with standard Gaussian weakly stationary isotropic Random Fields, the corresponding variogram admits the form γ(.) = 1 ψ(.) 2 The so obtained covariance function C(h) = ψ( h ), is L 1 and L 2 -integrable respectively under the conditions θ < d and θ < d/2 for d = 1, 2, 3 (dimension of the spatial domain) 3 In order to obtain spatio-temporal covariance and variogram structures, we consider the following two alternatives: A separable structure, obtained with the tensorial product of C(h) = ψ( h ) and C(u) = ψ( u ), so that C 1 (h, u) = ψ( h )ψ( u ) = 1 γ(h) γ(u) + γ(h)γ(u) 3.1 Dagum: Introduction 3.2 Dagum: Instruments 3.3 Dagum: Solution! 3.4 Dagum: Remarks! 3.5 Dagum: Other Remarks! 3.6 Dagum: Final Remarks! models: 6 Applications 7 Conclusions A nonseparable structure, obtained as a convex sum of C(h) and C(u), thus C 2 (h, u) = ϑψ( h ) + (1 ϑ)ψ( u ), where ϑ [0, 1]. - p. 20/68

21 3.5 Dagum: Other Remarks! 1 It is interesting to note that the spatial variogram γ(h) = 1 ψ( h ) never reaches the sill, but its practical range (i.e. the quantile of order p, 0 p 1) can be easily calculated from the expression ( p 1/ε 1 x p = λ ) 1/θ 2 Observe that the structure C 2 (.,.) is somehow similar to the so-called Product-sum model of De Cesare et al. (2001), even if they do not impose any restriction on the parameters of the linear combination (they only need to be positive). On the other hand, Ma (2003) proposed a linear combination of the same type as our C 2 (.,.), imposing a larger range for ϑ, which can be negative under some conditions. In this case Ma (2003) theorem can not be applied, as it is easy to show that C 2 (.,.) is not completely monotone. 3.1 Dagum: Introduction 3.2 Dagum: Instruments 3.3 Dagum: Solution! 3.4 Dagum: Remarks! 3.5 Dagum: Other Remarks! 3.6 Dagum: Final Remarks! models: 6 Applications 7 Conclusions - p. 21/68

22 3.6 Dagum: Final Remarks! 3 The non-differentiability of the spatio-temporal structure C 2 (.,.) is not surprising, as we are working with linear combinations. But, neither are the structures proposed by De Iaco et al. (2001) and Ma (2003). 4 It is trivial to show that the covariance function C 1 (.,.) is L 1 and L 2 -integrable under the same conditions considered in the spatial case. Observe that the integrability condition is not satisfied for C 2 (.,.). In fact a function defined on R d is not integrable on R d R. Thus, the same conclusions can be achieved for the Product-sum model of De Cesare et al. (2001). 3.1 Dagum: Introduction 3.2 Dagum: Instruments 3.3 Dagum: Solution! 3.4 Dagum: Remarks! 3.5 Dagum: Other Remarks! 3.6 Dagum: Final Remarks! models: 6 Applications 7 Conclusions - p. 22/68

23 4.1 (I) A new class of Our strategy is to create opportune partitions of the spatial lag vector h R d in the following way. If d = (d 1, d 2,..., d n ) and h R d we can always write so that h = (h 1, h 2,..., h n ) R d 1 R d 2 R d n (i) C(h) = C(k) for any h, k R d if and only if h i = k i for all i = 1, 2,..., n. (ii) The resulting covariance admits the representation C(h) = C( h 1,..., h n ) =... n 0 0 i=1 Ω d i ( h i r i )df (r 1,..., r n ) with Ω d (t) = Γ(d/2) ( 2 t ) J(d 2)/2 (t), J d (.) the Bessel function of the first kind of order d. 4.1 (I) A new class of 4.1 (II) A new class of 4.1 (III) A new class of 4.1 (IV) A new class of 4.2 (I) Mixed Forms 4.2 (II) Mixed Forms 4.2 (III) Mixed Forms 4.2 (IV) Mixed Forms 4.2 (V) Mixed Forms: An 4.2 (VI) Mixed Forms: An 4.2 (VII) Mixed Forms: An 4.3 (I) New families of spectral 4.3 (II) New families of spectral 4.3 (III) New families of spectral 4.3 (IV) New families of spectral - p. 23/68 models:

24 4.1 (II) A new class of Theorem Let ψ 1, ψ 2 be either: (i) Bernstein functions or (ii) Intrinsically stationary variograms (ψ γ). Let L be the bivariate Laplace transform of a nonnegative random vector (X 1, X 2 ) with distribution function F. Then C(h 1, h 2, h 3, h 4 ) = σ 2 ψ 1 ( h 1 2 ) d 32 ψ 2 ( h 2 2 ) is a covariance function in R d 1 R d 2 R d 3 R d 4. d 42 L ( ) h3 2, h 4 2 ψ 1 ( h 1 2 ) ψ 2 ( h 2 2 ) 4.1 (I) A new class of 4.1 (II) A new class of 4.1 (III) A new class of 4.1 (IV) A new class of 4.2 (I) Mixed Forms 4.2 (II) Mixed Forms 4.2 (III) Mixed Forms 4.2 (IV) Mixed Forms 4.2 (V) Mixed Forms: An 4.2 (VI) Mixed Forms: An 4.2 (VII) Mixed Forms: An 4.3 (I) New families of spectral 4.3 (II) New families of spectral 4.3 (III) New families of spectral 4.3 (IV) New families of spectral - p. 24/68 models:

25 4.1 (III) A new class of Remark The previous theorem represents the generalization of Theorem 2 in Gneiting (2002) and has been presented in the simplest and general form. Starting from previous result it is easy to obtain a large variety of closed forms. Corollary: SPACE TIME ZONALLY ANISOTROPIC COVARIANCES Let ψ 1, ψ 2 be either Bernstein functions, variograms or increasing and concave functions on [0, ). Then, ( ) σ C(h 1, h 2, h 3, u) = 2 ψ 1 ( h 1 2 ) 1/2 ψ 2 L h2 2 ( u 2 ) 1/2 ψ 1 ( h 1 2 ), h 3 2 ψ 2 ( u 2 ) with h i, u R, i = 1, 2, 3, is a stationary nonseparable space time covariance function with spatially anisotropic components, defined on R 3 R. 4.1 (I) A new class of 4.1 (II) A new class of 4.1 (III) A new class of 4.1 (IV) A new class of 4.2 (I) Mixed Forms 4.2 (II) Mixed Forms 4.2 (III) Mixed Forms 4.2 (IV) Mixed Forms 4.2 (V) Mixed Forms: An 4.2 (VI) Mixed Forms: An 4.2 (VII) Mixed Forms: An 4.3 (I) New families of spectral 4.3 (II) New families of spectral 4.3 (III) New families of spectral 4.3 (IV) New families of spectral - p. 25/68 models:

26 4.1 (IV) A new class of Consider the bivariate Laplace transform L(θ 1, θ 2 ) = 1 e θ 1 θ 2 θ 1 +θ 2, for θ 1 0 or θ 2 0, and where L(0, 0) = 1. Applying previous Corollary, we easily obtain C(h, u) = ( 1 e h 2 2 ψ 1 ( h 1 2 h 3 2 ) ψ 2 ( u 2 ) where h = (h 1, h 2, h 3 ) R 3 and u R. ) ψ 1( h 1 2 ) 1/2 ψ 2 ( u 2 ) 1/2 h 2 2 ψ 2 ( u 2 )+ h 3 2 ψ 1 ( h 1 2 ) 4.1 (I) A new class of 4.1 (II) A new class of 4.1 (III) A new class of 4.1 (IV) A new class of 4.2 (I) Mixed Forms 4.2 (II) Mixed Forms 4.2 (III) Mixed Forms 4.2 (IV) Mixed Forms 4.2 (V) Mixed Forms: An 4.2 (VI) Mixed Forms: An 4.2 (VII) Mixed Forms: An 4.3 (I) New families of spectral 4.3 (II) New families of spectral 4.3 (III) New families of spectral 4.3 (IV) New families of spectral - p. 26/68 models:

27 4.2 (I) Mixed Forms Result 1. Let ϕ(t) be a completely monotone function (t 0) and let ψ(t) be a positive function whose derivative is completely monotone. Then C(h, u) = ( σ 2 u 2 ) ψ( h 2 ) ϕ l/2 ψ( h 2, (h, u) R d R l (3) ) is a space-time covariance function. Result 2. Let C 1 (h, u), C 2 (h, u), (h, u) R d R be valid nonseparable spatio-temporal covariance functions and b > 0. Then both C 1 (h, u) + C 2 (h, u) and bc 1 (h, u) are valid spatio-temporal covariance functions in R d R. Result 3. Let C 1 (h, u), C 2 (h, u), (h, u) R d R be valid spatio-temporal covariance functions. Then their tensorial product C 1 (h, u) C 2 (h, u) is still a valid spatio-temporal covariance function. 4.1 (I) A new class of 4.1 (II) A new class of 4.1 (III) A new class of 4.1 (IV) A new class of 4.2 (I) Mixed Forms 4.2 (II) Mixed Forms 4.2 (III) Mixed Forms 4.2 (IV) Mixed Forms 4.2 (V) Mixed Forms: An 4.2 (VI) Mixed Forms: An 4.2 (VII) Mixed Forms: An 4.3 (I) New families of spectral 4.3 (II) New families of spectral 4.3 (III) New families of spectral 4.3 (IV) New families of spectral - p. 27/68 models:

28 4.2 (II) Mixed Forms Result 4. (Mixed Forms). Let C 1 (h, u), C 2 (h, u), (h, u) R d R be valid spatio-temporal covariance functions. Let C 3 (h), C 4 (u) be respectively valid spatial and temporal covariance functions. Then C(h, u) = C i (h, u) + C 3 (h) + C 4 (u), i = 1, 2 (4) C(h, u) = C 1 (h, u)c 2 (h, u) + C 3 (h) + C 4 (u) (5) C(h, u) = C 1 (h, u) + C 2 (h, u) + C 3 (h) + C 4 (u) (6) C(h, u) = (λ 12 C 1 (h, u)c 2 (h, u) + λ 3 C 3 (h) + λ 4 C 4 (u)) ξ (7) and C(h, u) = (λ 1 C 1 (h, u) + λ 2 C 2 (h, u) + λ 3 C 3 (h) + λ 4 C 4 (u)) ξ (8) are valid spatio-temporal covariance functions with constants λ 12, λ i, i = 1,..., 4 nonnegative weights and the external smoothing parameter ξ a natural number. 4.1 (I) A new class of 4.1 (II) A new class of 4.1 (III) A new class of 4.1 (IV) A new class of 4.2 (I) Mixed Forms 4.2 (II) Mixed Forms 4.2 (III) Mixed Forms 4.2 (IV) Mixed Forms 4.2 (V) Mixed Forms: An 4.2 (VI) Mixed Forms: An 4.2 (VII) Mixed Forms: An 4.3 (I) New families of spectral 4.3 (II) New families of spectral 4.3 (III) New families of spectral 4.3 (IV) New families of spectral - p. 28/68 models:

29 4.2 (III) Mixed Forms It is interesting to note that following Gneiting, above expressions would take the forms C(h, u) = σ 2 ψ( h 2 ) l/2 ϕ u 2 ( i ψ( h 2 + ϕ 3 h 2) + ϕ 4 ( u 2), i = 1, 2 ) C(h, u) = 2 ϕ 2 σ ψ 2 ( h 2 ) l/2 ψ 1 ( u 2 ) d/2 ϕ 1 ψ 1 ( u 2 ) ψ 2 ( h 2 ) + ( ϕ 3 h 2) ( + ϕ 4 u 2) C(h, u) = σ 2 ψ 1 ( u 2 ) d/2 ϕ h 2 σ 2 1 ψ 1 ( u 2 + ) ψ 2 ( h 2 ) l/2 ϕ u 2 2 ψ 2 ( h 2 + ) ( ϕ 3 h 2) ( + ϕ 4 u 2) 4.1 (I) A new class of 4.1 (II) A new class of 4.1 (III) A new class of 4.1 (IV) A new class of 4.2 (I) Mixed Forms 4.2 (II) Mixed Forms 4.2 (III) Mixed Forms 4.2 (IV) Mixed Forms 4.2 (V) Mixed Forms: An 4.2 (VI) Mixed Forms: An 4.2 (VII) Mixed Forms: An 4.3 (I) New families of spectral 4.3 (II) New families of spectral 4.3 (III) New families of spectral 4.3 (IV) New families of spectral - p. 29/68 models:

30 4.2 (IV) Mixed Forms C(h, u) = σ 2 (λ 12 ψ 2 ( h 2 ) l/2 ψ 1 ( u 2 ) d/2 ϕ h 2 1 ψ 1 ( u 2 ϕ u 2 2 ) ψ 2 ( h 2 ) ( +λ 3 ϕ 3 h 2) ( + λ 4 ϕ 4 u 2) ) ξ C(h, u) = σ 2 (λ 1 ψ 1 ( u 2 ) d/2 ϕ h 2 σ 2 1 ψ 1 ( u 2 + λ 2 ) ψ 2 ( h 2 ) l/2 ϕ u 2 2 ψ 2 ( h 2 ) ( λ 3 ϕ 3 h 2) ( + λ 4 ϕ 4 u 2) ) ξ where ϕ i (t), t 0, i = 1,..., 4 are completely monotone functions and ψ j (t), t 0, j = 1, 2 are positive functions with completely monotone derivative. 4.1 (I) A new class of 4.1 (II) A new class of 4.1 (III) A new class of 4.1 (IV) A new class of 4.2 (I) Mixed Forms 4.2 (II) Mixed Forms 4.2 (III) Mixed Forms 4.2 (IV) Mixed Forms 4.2 (V) Mixed Forms: An 4.2 (VI) Mixed Forms: An 4.2 (VII) Mixed Forms: An 4.3 (I) New families of spectral 4.3 (II) New families of spectral 4.3 (III) New families of spectral 4.3 (IV) New families of spectral - p. 30/68 models:

31 4.2 (V) Mixed Forms: An C 1 (h, u) = Kυ C 2 (h, u) = C 1 (h, u)+ k B υβ/2+ε u,h a 2 h ( a 1 u 2α 1 +1 (a 1 a 2 h u ) υ ) β/2 Kυ a 1 u ( a 2 h 2α 2 +1 ) β/2, (h, u) R d R σ α 2 1 Γ(α 2 ) (a 2 h )α σ 2 2 Kα (a 2 2 h )+ 1 2 α 1 1 Γ(α 1 ) (a 1 u )α 1 Kα (a 2 1 u ) a 1, a 2 0 scale parameters. α 1, α 2 (0, 1] local smoothing parameters. υ, ε > 0 global smoothing parameters. k = σ 4 2 2(υ 1) (Γ(υ)) 2 B u,h = (1 + a 1 u 2α 1 + a 2 h 2α 2 + a 1 a 2 u 2α 1 h 2α 2 ) Kν (.) : Modified Bessel Function of the second kind of orden ν 4.1 (I) A new class of 4.1 (II) A new class of 4.1 (III) A new class of 4.1 (IV) A new class of 4.2 (I) Mixed Forms 4.2 (II) Mixed Forms 4.2 (III) Mixed Forms 4.2 (IV) Mixed Forms 4.2 (V) Mixed Forms: An 4.2 (VI) Mixed Forms: An 4.2 (VII) Mixed Forms: An 4.3 (I) New families of spectral 4.3 (II) New families of spectral 4.3 (III) New families of spectral 4.3 (IV) New families of spectral - p. 31/68 models:

32 4.2 (VI) Mixed Forms: An C 1 (h, u) C 2 (h, u) (I) A new class of 4.1 (II) A new class of 4.1 (III) A new class of 4.1 (IV) A new class of 4.2 (I) Mixed Forms 4.2 (II) Mixed Forms 4.2 (III) Mixed Forms 4.2 (IV) Mixed Forms 4.2 (V) Mixed Forms: An z x y v x y 4.2 (VI) Mixed Forms: An 4.2 (VII) Mixed Forms: An 4.3 (I) New families of spectral 4.3 (II) New families of spectral 4.3 (III) New families of spectral 4.3 (IV) New families of spectral - p. 32/68 models:

33 4.2 (VII) Mixed Forms: An C 1 (h, u) C 2 (h, u) (I) A new class of 4.1 (II) A new class of 4.1 (III) A new class of 4.1 (IV) A new class of 4.2 (I) Mixed Forms 4.2 (II) Mixed Forms 4.2 (III) Mixed Forms 4.2 (IV) Mixed Forms 4.2 (V) Mixed Forms: An z x y v x y 4.2 (VI) Mixed Forms: An 4.2 (VII) Mixed Forms: An 4.3 (I) New families of spectral 4.3 (II) New families of spectral 4.3 (III) New families of spectral 4.3 (IV) New families of spectral - p. 33/68 models:

34 4.3 (I) New families of spectral The aim of this proposal is to build families of spectral whose associated covariance function is differentiable at the origin Starting from Stein (2005), we find new families of spectral associated to spatial temporal processes whose inverse Fourier transform is differentiable at the origin Define a spectral density obtained as tensorial product of two spectral, in the following way. Consider f 1, f 2 : R R spectral density functions. Define the function f(.,.) as: f(w, τ) = f 1 (α 1 τ a + β 1 w b )f 2 (α 2 τ a + β 2 w b ) w R 2, τ R and α 1, α 2, β 1, β 2, a, b nonnegative parameters. 4.1 (I) A new class of 4.1 (II) A new class of 4.1 (III) A new class of 4.1 (IV) A new class of 4.2 (I) Mixed Forms 4.2 (II) Mixed Forms 4.2 (III) Mixed Forms 4.2 (IV) Mixed Forms 4.2 (V) Mixed Forms: An 4.2 (VI) Mixed Forms: An 4.2 (VII) Mixed Forms: An 4.3 (I) New families of spectral 4.3 (II) New families of spectral 4.3 (III) New families of spectral 4.3 (IV) New families of spectral - p. 34/68 models:

35 4.3 (II) New families of spectral NOTATION: Consider a spectral density defined over R 3 : where g 1, g 2 : R 3 R, g 1 = f 1 h 1, g 2 = f 2 h 2, f(w) = g 1 (w)g 2 (w) h 1 (w) = α 1 w 3 a + β 1 (w 1, w 2 ) b, h 2 (w) = α 2 w 3 a + β 2 (w 1, w 2 ) b. w = (w 1, w 2, w 3 ), x = (x 1, x 2, x 3 ) R 3 and denote with D m the partial differential operator Theorem: m x m 1 1 x m 2 2 x m 3 3, with m = m 1 + m 2 + m 3. Suppose that D l g 1 and D l g 2 exist and are integrable for l k, and ω m 1 1 ω m 2 2 ω3 m3 D l g 1 (w) and ω m 1 1 ω m 2 2 ω3 m3 D l g 2 (w) are integrable for l k. Then D m C exists, for x j 0, for every j s.t. k j > 0, and where C is the autocovariance function associated to f. 4.1 (I) A new class of 4.1 (II) A new class of 4.1 (III) A new class of 4.1 (IV) A new class of 4.2 (I) Mixed Forms 4.2 (II) Mixed Forms 4.2 (III) Mixed Forms 4.2 (IV) Mixed Forms 4.2 (V) Mixed Forms: An 4.2 (VI) Mixed Forms: An 4.2 (VII) Mixed Forms: An 4.3 (I) New families of spectral 4.3 (II) New families of spectral 4.3 (III) New families of spectral 4.3 (IV) New families of spectral - p. 35/68 models:

36 4.3 (III) New families of spectral Theorem: For j = 1, 2, 3, suppose l g ωj l 1 and l integrable for l k and that ω n ω l j g 2 exist and are l g ωj l 1 and ω n integrable. Then for x 0, D m C exists for m = (m 1, m 2, m 3 ) : m 1 + m 2 + m 3 n. l ω l j g 2 are Remark: The above theorems indicate the criteria to follow in order to choose spectral whose associated covariance is differentiable at the origin. It is over our scope to obtain an analytic expression of the covariance. 4.1 (I) A new class of 4.1 (II) A new class of 4.1 (III) A new class of 4.1 (IV) A new class of 4.2 (I) Mixed Forms 4.2 (II) Mixed Forms 4.2 (III) Mixed Forms 4.2 (IV) Mixed Forms 4.2 (V) Mixed Forms: An 4.2 (VI) Mixed Forms: An 4.2 (VII) Mixed Forms: An 4.3 (I) New families of spectral 4.3 (II) New families of spectral 4.3 (III) New families of spectral 4.3 (IV) New families of spectral - p. 36/68 models:

37 4.3 (IV) New families of spectral An example vtau f(w, τ) = φ 1 φ 2 (α α2 11 τ 2 ) ν 1 d 1 2 (α (β2 21 w 2 )) ν 2 d (I) A new class of 4.1 (II) A new class of 4.1 (III) A new class of 4.1 (IV) A new class of 4.2 (I) Mixed Forms 4.2 (II) Mixed Forms 4.2 (III) Mixed Forms 4.2 (IV) Mixed Forms 4.2 (V) Mixed Forms: An 4.2 (VI) Mixed Forms: An 4.2 (VII) Mixed Forms: An 4.3 (I) New families of spectral 4.3 (II) New families of spectral 4.3 (III) New families of spectral 4.3 (IV) New families of spectral - p. 37/68 models:

38 5.1 (I) Stat. : Mixtures of Theorem Let γ s (h) and γ t (u) be spatial and temporal valid variograms, respectively, with h R d and u R. Let K(v, w) be an Archimedean copula, where (v, w) [0, 1] 2. Then, C s,t (h, u) = exp( vγ s (h) wγ t (u))dk(v, w) is a valid nonseparable spatio-temporal covariance function. models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 38/68 anisotropic

39 5.1 (II) Stat. : Mixtures of An Consider the family B11 in Joe (1997, p.148) K(v, w) = δmin {v, w} + (1 δ)vw, where δ [0, 1], so that it can be seen as a convex sum of two elementary. Applying previous theorem, we obtain the following nonseparable stationary structure (1 e γ s(h) γ t (u) ) (1 e γ s(h) ) ( 1 e γ t (u)) C s,t (h, u) = δ + (1 δ) γs(h) + γ t (u) γs(h)γ t (u) models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 39/68 anisotropic

40 5.1 (III) Stat. : Mixtures of An (continues) Note that setting δ = 1, and γ(x) = x α, with x real, 0 α 2, we obtain as a particular case the Ma family proposed in Ma (2003b, 3, page 103). In this case the spatial margin becomes C s (h, 0) = 1 if h = 0 h α (1 e h α) elsewhere models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 40/68 anisotropic

41 5.1 (IV) Stat. : F and Theorem Let ϕ(t) be a completely monotone function with t > 0. Let γ s (h) and γ t (u) be spatial and temporal valid variograms, respectively, with h R d and u R. Let k be a positive natural number. Then, the following forms (i) ϕ(γ s (h) + γ t (u)) (ii) ϕ( h 2 + γ t (u)) (iii) (iv) ϕ (2k) (γ s (h))ϕ( h 2 + γ t (u)) ϕ (2k) (γ s (h)v)ϕ( h 2 w + γ t (u)z)dk(v, w, z) are valid nonseparable stationary spatio-temporal covariance functions on R d R. Particularly, (iii) and (iv) are stationary covariances if and only if ϕ 2k (0) is finite. models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 41/68 anisotropic

42 5.1 (V) Stat. : F and Consider now the completely monotone function in Gneiting (2002) ϕ(t) = [ e ct1/2 + e ct1/2] ν, (9) where c, ν are positive parameters and t > 0. Writing c = 1, without loss of generality, we obtain that the second derivative admits the expression ϕ (2) (t) = 1 4 ν {(ν + 1) 1 t t 1/2 ( e t + e t ) 2 + [( e t + e t )] ν+2 (e 2 t + e 2 t 2 ) + t 3/2 ( e 2 t e 2 t )} [( e t + e t )] ν+2 which also admits a finite limit at the origin, whatever the value of ν is. models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 42/68 anisotropic

43 5.1 (VI) Stat. : F and (Cont.) Applying point (iii) in previous theorem, it is possible to obtain the following nonseparable covariance function: C s,t (h, u) = 1 4 ν ( (ν + 1) h 2 + u 2) 1 e 2 h 2 + u 2 + e 2 h 2 + u ( h 2 + u 2) 3/2 e 2 h 2 + u 2 e 2 h 2 + u 2 ( h 2 + u 2) 1/2 h e 2 + u 2 h + e 2 + u 2 2 L 1 and L 2 integrable for any value of ν h e 2 + u 2 h + e 2 + u 2 ν+2 (10) models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 43/68 anisotropic

44 5.1 (VII) Stat. : F and (Cont.) models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 44/68 anisotropic

45 5.2 (I) Nonstationary and Theorem Let γ(s, t) be an intrinsically stationary variogram such that γ(0, 0) = 0. Let ϕ(t), t > 0, be a completely monotone function. Then ϕ(ψ(s 1, s 2, t 1, t 2 )) with Ψ(s 1, s 2, t 1, t 2 ) = 1/2 {γ(2s 1, 2t 1 ) + γ(2s 2, 2t 2 )} {γ(s 1 + s 2, t 1 + t 2 ) γ(s 1 s 2, t 1 t 2 )} is a valid nonstationary covariance function on R d R. models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 45/68 anisotropic

46 5.2 (II) Nonstationary and Theorem Under the same conditions of the previous theorem, we have that with ϕ(ψ (s 1, s 2, t 1, t 2 )) Ψ (s 1, s 2, t 1, t 2 ) = 1 + 1/2 {γ(2s 1, 2t 1 ) + γ(2s 2, 2t 2 )} { } 1 + γ(s1 + s 2, t 1 + t 2 ) 1 + γ(s 1 s 2, t 1 t 2 ) is a valid nonstationary covariance in R d R. models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 46/68 anisotropic

47 5.3 (I) Archimedean anisotropic Theorem Assume that ψ(x) is a completely monotone function on [0, ). If C 1 (x),..., C d (x), C T (x) are stationary covariance functions on the real line, and the functions ψ 1 (C T (x)), ψ 1 (C k (x)), k = 1,..., d, are continuous, increasing and concave on [0, ), then { d } C(s; t) = ψ ψ 1 (C k ( s k )) + ψ 1 (C T ( t )), s R d, t R, k=1 is a stationary covariance function on R d R. models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 47/68 anisotropic

48 5.3 (II) Archimedean anisotropic A simple example Consider the completely monotone function ψ(x) = (1 + x) 1/θ 1, s > 0 with θ 1 positive parameter, whose inverse is ψ 1 (x) = x θ 1 1. Also consider the covariance functions: C k (x) = (1 + x ) 1/θ k, k = 1,..., d C T (x) = (1 + x ) 1/θ T It is easy to verify that ψ 1 C i (x) = (1 + x) θ 1/θ i 1, i = 1,..., d, is continuous, increasing and concave iff θ 1 < θ i. Then, we obtain: C(x, t) = [ d (1 + s k ) ρ i + (1 + t ) ρ T d k=1 ] 1/θ1 is a valid nonseparable stationary isotropic spatio-temporal covariance function, with ρ i = θ 1 /θ i, i = 1,..., d, and ρ T = θ 1 /θ T. models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 48/68 anisotropic

49 5.3 (III) Archimedean anisotropic Gamma(h1,h2) h h1 models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 49/68 anisotropic

50 5.3 (IV) Archimedean anisotropic Remark: 1 The criteria allows to build nonseparable spatial temporal covariance functions which are not necessarily L p norms. 2 This class can be used for spatial-anisotropic temporal-isotropic phenomena. 3 The function ψ is a link function allowing to build a nonseparable structure starting from the temporal and spatial margins. models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 50/68 anisotropic

51 5.3 (V) Archimedean anisotropic Theorem Let C 1 (x),... C 3 (x), C T (x) be stationary covariance functions on the real line, such that both are completely monotone. Let ψ 1, ψ 2 above defined such that: ψ 1 1 ψ 2 ψ 1 2 C i (x), i = 1, 2, 3 ψ 1 1 C T (x) are positive functions with completely monotonic derivative. Then C(s, t) = ψ 1 [ ψ 1 1 ψ 2 ( 3 i=1 ψ 1 2 (C i (s i )) ) + ψ 1 1 (C(t)) is a stationary isotropic spatio-temporal covariance function. ] models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 51/68 anisotropic

52 5.3 (VI) Archimedean anisotropic Theorem Let C 1 (x),... C 3 (x), C T (x) be stationary covariance functions on the real line, such that both are completely monotonic. Let ψ i, i = 1,..., 3 above defined such that : ψ 1 1 ψ 2, ψ 1 2 ψ 3 ψ 1 3 C i (x), i = 1, 2 ψ 1 2 C 3 (x) ψ 1 1 C T (x) are positive functions with completely monotonic derivative. Then C(x, t) = ψ 1 ψ 1 1 ψ 2 ψ ψ 3 ψ 1 3 (C i (s i )) + ψ 1 2 (C 3 (s 3 )) + ψ 1 1 (C T (t)) i=1 is a stationary anisotropic spatio-temporal covariance function. models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 52/68 anisotropic

53 5.3 (VII) Archimedean anisotropic An example Consider the completely monotone functions ψ i (x) = (1 + x) 1/θ i, s > 0, i = 1, 2 with θ i positive parameters, whose inverse is ψ 1 i (x) = x θ i 1. Also consider the covariance functions: C k (x) = (1 + x ) 1/θ k, k = 1,..., 3 C T (x) = (1 + x ) 1/θ T It is easy to verify that conditions of previous theorem are satisfied iff θ 1 < θ 2, θ 1 < θ T and θ 2 < θ k, k = 1,..., 3. Under this constraint, applying previous theorem we obtain: C(x, t) = [ ( 3 (1 + s k ) ρ k 2) ρ 2 + (1 + t ) ρ T 1 k=1 is a valid nonseparable stationary isotropic spatio-temporal ] 1/θ1 covariance function, with ρ 1 = θ 1 /θ 2, ρ k = θ 2 /θ k, i = 1,..., d, and models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 53/68 anisotropic

54 5.3 (VIII) Archimedean anisotropic 0.5 Gamma(h2,u) u h2 models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 54/68 anisotropic

55 5.4 (I) : The Bernstein class Open Problems 1 Great majority of contributions in space-time covariance regards spatially isotropic covariance functions. 2 Thus, there is a big need of more models which are zonally anisotropic in the spatial component and not necessarily L p -norms. Recall a positive function ψ(.) defined on the positive real line is called a Bernstein function if its first derivative is completely monotone, viz. ( 1) k 1 ψ(t) (k) 0, for any positive integer k and t > 0. models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 55/68 anisotropic

56 5.4 (II) : The Bernstein class Theorem Let L(.,.) be a bivariate Laplace transform of a nonnegative random vector (X 1, X 2 ). Let ψ i (.), i = 1,..., d and ψ T (.) be positive Bernstein functions defined on the positive real line. Then, for d = 1, 2,... C(h, u) = L( d i=1 ψ i( h i ), ψ T ( u )) is a stationary nonseparable spatio-temporal covariance function in R d R. How to create Bernstein Covariances 1 The method is very simple and allows for very flexible forms of spatio temporal covariances which are zonally anisotropic. 2 Choose a convenient partition of the spatial lag vector h R d, say (h 1,..., h k ), k d. 3 Choose d + 1 Bernstein functions and a Bivariate Laplace transform models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 56/68 anisotropic

57 5.4 (III) : The Bernstein class An example Let L(θ 1, θ 2 ) be the Laplace transform for the Frechet-Hoeffding lower bound of bivariate, whose equation is L(θ 1, θ 2 ) = exp( θ 1) exp( θ 2 ) θ 1 θ 2 Now, consider the following Bernstein functions ψ 1 (t) = (a 1 t α 1 + 1) β ψ 2 (t) = (a 2t α 2 +b) b(a 2 t α 2 +1) ψ T (t) = t ρ, where a 1, a 2 are positive scale parameters, α 1, α 2 (0, 1], β [0, 1] b > 1 and ρ [0, 1). Then, C(h 1, h 2, u) = exp ( (a 1 h 1 α 1 +1) β a 2 h 2 α 2 +b b(a 2 h 2 α 2 +1) (a 1 h 1 α 1 +1) β + a 2 h 2 α 2 +b b(a 2 h 2 α 2 +1) u ρ ) exp( u ρ ) models: 5.1 (I) Stat. : Mixtures of 5.1 (II) Stat. : Mixtures of 5.1 (III) Stat. : Mixtures of 5.1 (IV) Stat. : F and 5.1 (V) Stat. : F and 5.1 (VI) Stat. : F and 5.1 (VII) Stat. : F and 5.2 (I) Nonstationary and 5.2 (II) Nonstationary and 5.3 (I) Archimedean anisotropic 5.3 (II) Archimedean anisotropic 5.3 (III) Archimedean anisotropic 5.3 (IV) Archimedean- p. 57/68 anisotropic

58 6.1 Application: Wind Speed Data data regard wind speed (mt/sec) Zone: Western Pacific Ocean, between Australia e New Guinea (145 E-175 E, 14 N-16 N). Data sampled on a regular grid (17 17 nods, at 210 km distance). Data are obtained every 6 hours. models: 6 Applications 6.1 Application: Wind Speed Data 6.2 Presentation of Data 6.3 (I) Variogram Nonparametric Estimation 6.3 (II) Variogram Nonparametric Estimation 6.4 Estimation via WLS 6.5 (I) Comparison amongst four Models 6.5 (II) Comparison among four Models 6.6 (I) Parameters Estimation: diagnostics 6.6 (II) Cross-validation: Diagnostics 6.7 Movies 7 Conclusions - p. 58/68

59 6.2 Presentation of Data models: 6 Applications 6.1 Application: Wind Speed Data 6.2 Presentation of Data 6.3 (I) Variogram Nonparametric Estimation 6.3 (II) Variogram Nonparametric Estimation 6.4 Estimation via WLS 6.5 (I) Comparison amongst four Models 6.5 (II) Comparison among four Models 6.6 (I) Parameters Estimation: diagnostics 6.6 (II) Cross-validation: Diagnostics 6.7 Movies 7 Conclusions - p. 59/68

60 6.3 (I) Variogram Nonparametric Estimation Local Linear Estimator: N 1 i=1 N ( 2 Z(s i, t i ) Z(s j, t j )) (β0, β 10, β 01 ) j=i+1 K H s i s j r t i t j u 1 s i s j r t i t j u 2 models: K H (v) = 1 ( K H 1 ) v H K ( ): bidimensional kernel H = h 2 ({ S }) ti ) S = Cov ( s i s j, t j ; 1 i < j N 6 Applications 6.1 Application: Wind Speed Data 6.2 Presentation of Data 6.3 (I) Variogram Nonparametric Estimation 6.3 (II) Variogram Nonparametric Estimation 6.4 Estimation via WLS 6.5 (I) Comparison amongst four Models 6.5 (II) Comparison among four Models 6.6 (I) Parameters Estimation: diagnostics 6.6 (II) Cross-validation: Diagnostics 6.7 Movies 7 Conclusions - p. 60/68

61 6.3 (II) Variogram Nonparametric Estimation models: 6 Applications 6.1 Application: Wind Speed Data 6.2 Presentation of Data 6.3 (I) Variogram Nonparametric Estimation 6.3 (II) Variogram Nonparametric Estimation 6.4 Estimation via WLS 6.5 (I) Comparison amongst four Models 6.5 (II) Comparison among four Models 6.6 (I) Parameters Estimation: diagnostics 6.6 (II) Cross-validation: Diagnostics 6.7 Movies 7 Conclusions - p. 61/68

62 6.4 Estimation via WLS Link Function: i,j ω i,j(θ) ( γ(hi, u j ) γ(h i, u j θ) ) 2 Weights: ω i,j (θ) = N(r i,u j ) γ(h i,u j θ) 2 Algorithm: Levemberg-Marquandt models: 6 Applications 6.1 Application: Wind Speed Data 6.2 Presentation of Data 6.3 (I) Variogram Nonparametric Estimation 6.3 (II) Variogram Nonparametric Estimation 6.4 Estimation via WLS 6.5 (I) Comparison amongst four Models 6.5 (II) Comparison among four Models 6.6 (I) Parameters Estimation: diagnostics 6.6 (II) Cross-validation: Diagnostics 6.7 Movies 7 Conclusions - p. 62/68

63 6.5 (I) Comparison amongst four Models Exponential Separable Model (SVEXPS) Model corresponding to example 2 of Cressie and Huang (SVCH2) having equation : γ(h, u θ) = c 0 + σ 2 [1 1 ( (au + 1) 2 exp b2 h 2 )] au + 1 Model corresponding to example 4 of Cressie and Huang (SVCH4) having equation : γ(h, u θ) = c 0 + σ 2 [1 Mixed Forms (Porcu) having equation: C(h, u υ = 1/2) k B ε u,h ] au + 1 [ (au + 1) 2 + b 2 h 2] (d+1)/2 models: ( ) a 1 u exp ( a2 h 2α ) a 2 h β/2 ( a1 u 2α ) β/2 6 Applications 6.1 Application: Wind Speed Data 6.2 Presentation of Data 6.3 (I) Variogram Nonparametric Estimation 6.3 (II) Variogram Nonparametric Estimation 6.4 Estimation via WLS 6.5 (I) Comparison amongst four Models 6.5 (II) Comparison among four Models 6.6 (I) Parameters Estimation: diagnostics 6.6 (II) Cross-validation: Diagnostics 6.7 Movies 7 Conclusions - p. 63/68

64 6.5 (II) Comparison among four Models Porcu SVEXPS models: SVCH2 SVCH4 6 Applications 6.1 Application: Wind Speed Data 6.2 Presentation of Data 6.3 (I) Variogram Nonparametric Estimation 6.3 (II) Variogram Nonparametric Estimation 6.4 Estimation via WLS 6.5 (I) Comparison amongst four Models 6.5 (II) Comparison among four Models 6.6 (I) Parameters Estimation: diagnostics 6.6 (II) Cross-validation: Diagnostics 6.7 Movies 7 Conclusions - p. 64/68