on building valid space-time covariance modelsmateu/space-time-geostats.pdf- p. 1/68 on building...

- p. 1/68

On building valid space-time covariance models

Jorge Mateu

joint work with E. Porcu

Department of MathematicsUniversitat Jaume I, Castellón (Spain)

Avignon, 13-14 October 2005

Summary

Summary of the seminar

1 Some history ofspatio-temporal modelling

2 Background onspatio-temporal geostatistics

3 The DAGUM class for spatial(and ST) modelling

4 Building valid ST covariancemodels: Theoretical results

5 Building valid ST covariancemodels: Copulas

6 Applications

7 Conclusions

- p. 2/68

Summary of the seminar

1 Some history of spatio-temporal modelling

2 Background on spatio-temporal geostatistics

(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 Copulas.

3 The DAGUM class for spatial (and ST) modelling

4 Building valid ST covariance models: Theoretical results

(4.1) A new class of anisotropic space time covariances, (4.2) Mixed Forms, (4.3) New families of spectral densities.

5 Building valid ST covariance models: Copulas

(5.1) Stationary covariance functions through: Mixtures of copulas, Completely monotone functions and Copulas,

(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

Summary


1.1 Some history of STM






6 Applications

7 Conclusions

- p. 3/68


1989-1994 Separable Spatio-temporal Covariance Functions• Rohuani and Hall (1989)• Sampson and Guttorp (1992)• Dimitrakopoulos and Luo (1994)

1994-2005 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)

Summary








6 Applications

7 Conclusions

- p. 4/68


2000-2005 Nonstationary Nonseparable Spatio-temporal CovarianceFunctions• Christakos (2000)• Fuentes and Smith (2001)• Hristopoulos and Christakos (2001)• Ma (2002)• Fuentes (2002)

1999-2005 Stationary Nonseparable Anisotropic Spatio-temporalCovariance Functions• Shapiro and Botha (1999)• Fernández-Casal (2003)• Stein (2005)

Summary



2.1 (I) ST covariance functions

2.1 (II) ST covariance functions2.2 Separability and FullSymmetry2.3 Full Symmetry and ZonalAnisotropy

2.4 Mixed Forms2.5 (I) Some concepts onCopulas2.5 (II) Some concepts onCopulas2.5 (III) Some concepts onCopulas2.5 (IV) Some concepts onCopulas2.5 (V) Some concepts onCopulas2.5 (VI) Some concepts onCopulas2.5 (VII) Some concepts onCopulas




6 Applications

7 Conclusions- p. 5/68


• Spatio-temporal Random Fields:

Z(s, t) = m(s, t) + δ(s, t), s ∈

Summary









6 Applications


2.1 (II) ST covariance functions

• Recall...

1. Covariance Functions are Positive Definite, as stated inBochner’s Theorem (1949).

2. In the Isotropic Case, the Fourier transform can beexpressed as an integral of Bessel Functions (HankelTransform).

3. Variogram associated to Intrinsically Stationary RF areconditionally definite negative.

4. If dealing with an SRF, the properties and relationshipsbetween covariance and variogram are preserved.

• For instance, C(h, u) = C(0, 0) − γ(h, u)

Summary









6 Applications


2.2 Separability and Full Symmetry

1 Separability:

C(h, u) =C(h, 0)C(0, u)

C(0, 0)

2 Full Symmetry:

C(h, u) = C(−h, u) = C(h,−u) = C(−h,−u),

for every (h, u) ∈ Rd × R.

3 Relations: separable covariances are also fully symmetric,while viceversa is not necessarily true.

Summary









6 Applications


2.3 Full Symmetry and Zonal Anisotropy

n The great majority of contributions in literature regardscovariance models which are fully symmetric and isotropic.

n Unfortunately we do not dispose of such a large literaturefor the problem of zonal anisotropy, which is at least asimportant as the problem of full symmetry.

n Thus, there is a big need for models which are zonallyanisotropic in the spatial component.

Summary









6 Applications


2.4 Mixed Forms

n We are interested in removing some undesirable features of thepreviously proposed models, particularly following Stein’s remarkabout Gneiting’s approach and about some tensorial productcovariance models.

n Specifically, Stein (2003) observes that models of the typeexp(− |s| − |t|), obtained with a tensorial product of twoexponential covariance functions on space and time, are notdifferentiable at the origin and denote a lack of differentiabilityalong certain axis, which in turn implies discontinuities of theautocorrelation function away from the origin.

n Furthermore, Stein (2003), while emphasizing the need forspatio-temporal covariance functions which are sufficientlysmooth away from the origin, observes that Gneiting’s approachleads to some undesirable features, such as the fact thatwhatever the lack of smoothness of C(s, 0) for s near zero, it willbe shared by C(s, t) for t 6= 0 and s near zero, since C(s, t) isjust a rescaling of C(s, 0).

Summary









6 Applications


2.5 (I) Some concepts on Copulas

• Why Copulas?

1 Modern literature emphasize the need for new models ofnonseparable covariance functions for spatial temporalphenomena

2 It would be very interesting to find some Link Functions allowingto build up nonseparable permissible closed forms starting fromthe margins, i.e. the spatial and the temporal covariance. ThisLink Functions should include the case of separability

3 The advantages of such construction would be considerable(estimation and inference)

4 We believe Copulas can be a good candidate in order to satisfythis purpose.

Summary









6 Applications


2.5 (II) Some concepts on Copulas

• Copulas: Literature

[1] Introduction: Nelsen (1999); Joe (1987)

[2] Archimedean Copulas: Genest and McKay (1986a,b); Genest(1987)

[3] Inferencial procedures for Archimedean Copulas: Genest andRivest (1993)

[4] Analysis of serial dependence (time series): Ferguson et al.(2000); Genest et al. (2002)

[5] Asymptotics for empirical copula process: Genest andRémillard (2004); Fermanian et al. (2002)

[6] Environmental Data: Drouet and Monbet (2004) for pollution inthe Atlantic Ocean; Favre et al. (2004) for a multivariatehydrological frequency analysis

Summary









6 Applications


2.5 (III) Some concepts on Copulas

• Copulas: Introduction

. Copulas have been largely used in several statistical contexts forthe study of dependence of two random variables and modelling ofrisk assessment in financial markets

. A desirable feature of copulas is given by the fact that the marginalstructures can be modelled separately and interaction between tworandom variables can further be modelled with copulas in order toimplement a bivariate structure

. Algorithms for choosing an appropriate copula for the interactionbetween two random variables can be found in Melchiori (2003).

Summary









6 Applications


2.5 (IV) Some concepts on Copulas

• Copulas: Definitions

Let I be the interval [0, 1] and let I2 be the cartesian productI × I ⊆

Summary









6 Applications


2.5 (V) Some concepts on Copulas

• Copulas: Important Theorems

Theorem (Sklar, 1959)

Let X, Y be random variables with distribution functions FX(x) andGY (y) respectively, and with joint distribution function HX,Y (x, y).Then, there exists a copula K(., .) such that ∀(x, y) ∈

Summary









6 Applications


2.5 (VI) Some concepts on Copulas

• Archimedean Copulas

Theorem

Let ϕ be a continuous strictly decreasing function from I2 to [0,∞]such that ϕ(1) = 0 and let ϕ−1 be the inverse of ϕ. Then, thefunction

K(u, v) = ϕ−1 {ϕ(u) + ϕ(v)} (1)

is a copula if and only if ϕ is convex.

Remarks:

1 The class given in equation (1) is called Archimedean Class

2 Depending on the choice of the convex function ϕ, some known classes can be obtained as particular cases. Forexample, 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−1 .

Summary









6 Applications


2.5 (VII) Some concepts on Copulas

• Multivariate Archimedean Copulas

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 theinverse of ϕ. The function

Kψ(u1, ..., un) = ϕ−1(ϕ(u1) + ...+ ϕ(un)) (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, theSpearman’s Rho and the Blest correlation coefficient (Genest and MacKay, 1987).

2 Archimedean copulas have an explicit relation with the Kendall’s τ . It can be shown that

τK = 1 + 4∫ 1

0

ϕ(t)

ϕ′(t)dt

Summary




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!



6 Applications

7 Conclusions

- p. 17/68


• The Dagum survival function (Zenga and Zini, 2001) has beenused 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 atthe origin and a good level of smoothing away from the origin. Thisfacts motivates our research.

• Our aim is to use ψ() as a radial function and inspect the range ofparameters allowing for its permissibility

Summary







3.4 Dagum: Remarks!





6 Applications

7 Conclusions

- p. 18/68


• How to show the positive definiteness of ψ(.)? The direct answercan be found with the so-called Hankel transforms, which areintegral 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 itis necessary to recur to sufficient conditions, such as:

1 Christakos sufficient conditions

2 Criteria of Pólya type

3 Gneiting’s criteria as extension of [2]

Summary







3.4 Dagum: Remarks!





6 Applications

7 Conclusions

- p. 19/68


• Using the criteria [3], we are able to state the following result:

TheoremConsider the function h 7→ ψ(‖h‖) where ψ(.) is defined as above. Ifθ < (7 − ε)/(1 + 5ε) and ε < 7, then ψ(‖h‖) is positive definite(indeed a permissible covariance function) in

Summary







3.4 Dagum: Remarks!





6 Applications

7 Conclusions

- p. 20/68

3.4 Dagum: Remarks!

1 As we deal with standard Gaussian weakly stationary isotropicRandom Fields, the corresponding variogram admits the formγ(.) = 1 − ψ(.)

2 The so obtained covariance function C(h) = ψ(‖h‖), is L1 andL2-integrable respectively under the conditions θ < d andθ < d/2 for d = 1, 2, 3 (dimension of the spatial domain)

3 In order to obtain spatio-temporal covariance and variogramstructures, we consider the following two alternatives:• A separable structure, obtained with the tensorial product ofC(h) = ψ(‖h‖) and C(u) = ψ(|u|), so that

C1(h, u) = ψ(‖h‖)ψ(|u|) = 1 − γ(h) − γ(u) + γ(h)γ(u)

• A nonseparable structure, obtained as a convex sum of C(h)and C(u), thus

C2(h, u) = ϑψ(‖h‖) + (1 − ϑ)ψ(|u|),

where ϑ ∈ [0, 1].

Summary







3.4 Dagum: Remarks!





6 Applications

7 Conclusions

- p. 21/68


1 It is interesting to note that the spatial variogramγ(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 calculatedfrom the expression

xp =

(

p−1/ε − 1

λ

)−1/θ

2 Observe that the structure C2(., .) is somehow similar to theso-called Product-sum model of De Cesare et al. (2001), even ifthey do not impose any restriction on the parameters of thelinear combination (they only need to be positive). On the otherhand, Ma (2003) proposed a linear combination of the sametype as our C2(., .), imposing a larger range for ϑ, which can benegative under some conditions. In this case Ma (2003) theoremcan not be applied, as it is easy to show that C2(., .) is notcompletely monotone.

Summary







3.4 Dagum: Remarks!





6 Applications

7 Conclusions

- p. 22/68


3 The non-differentiability of the spatio-temporal structure C2(., .) isnot surprising, as we are working with linear combinations. But,neither are the structures proposed by De Iaco et al. (2001) andMa (2003).

4 It is trivial to show that the covariance function C1(., .) is L1 andL2-integrable under the same conditions considered in thespatial case. Observe that the integrability condition is notsatisfied for C2(., .). In fact a function defined on

Summary




4 Building valid ST covariancemodels: Theoretical results4.1 (I) A new class ofAnisotropic ST Cov.4.1 (II) A new class ofAnisotropic ST Cov.4.1 (III) A new class ofAnisotropic ST Cov.4.1 (IV) A new class ofAnisotropic ST Cov.

4.2 (I) Mixed Forms

4.2 (II) Mixed Forms

4.2 (III) Mixed Forms

4.2 (IV) Mixed Forms4.2 (V) Mixed Forms: AnExample4.2 (VI) Mixed Forms: AnExample4.2 (VII) Mixed Forms: AnExample4.3 (I) New families of spectraldensities4.3 (II) New families of spectraldensities4.3 (III) New families of spectraldensities4.3 (IV) New families of spectraldensities


6 Applications

7 Conclusions

- p. 23/68

4.1 (I) A new class of Anisotropic ST Cov.

Our strategy is to create opportune partitions of the spatiallag vector h ∈ Rd in the following way.If d = (d1, d2, . . . , dn) and h ∈ Rd we can always write

h = (h1,h2, . . . ,hn) ∈ Rd1 × Rd2 × · · · × Rdn

so that

(i) C(h) = C(k) for any h,k ∈ Rd if and only if ‖hi‖ = ‖ki‖ forall i = 1, 2, . . . , n.

(ii) The resulting covariance admits the representation

C(h) = C(‖h1‖, . . . , ‖hn‖)

=∫∞

0. . .∫∞

0

∏ni=1 Ωdi(‖hi‖ri)dF (r1, . . . , rn)

with Ωd(t) = Γ(d/2)(

2t

)

J(d−2)/2(t), Jd(.) the Bessel functionof the first kind of order d.

Summary





4.2 (I) Mixed Forms





6 Applications

7 Conclusions

- p. 24/68

4.1 (II) A new class of Anisotropic ST Cov.

Theorem Let ψ1, ψ2 be either:

(i) Bernstein functions or(ii) Intrinsically stationary variograms (ψ ≡ γ).

Let L be the bivariate Laplace transform of a nonnegativerandom vector (X1, X2) with distribution function F . Then

C(h1,h2,h3,h4) =σ2

ψ1(‖h1‖2)d32 ψ2(‖h2‖2)

d42

L(

‖h3‖2

ψ1(‖h1‖2), ‖h4‖

2

ψ2(‖h2‖2)

)

is a covariance function in Rd1 × Rd2 × Rd3 × Rd4 .

Summary





4.2 (I) Mixed Forms





6 Applications

7 Conclusions

- p. 25/68

4.1 (III) A new class of Anisotropic ST Cov.

RemarkThe previous theorem represents the generalization ofTheorem 2 in Gneiting (2002) and has been presented in thesimplest and general form. Starting from previous result it iseasy to obtain a large variety of closed forms.

Corollary: SPACE TIME ZONALLY ANISOTROPICCOVARIANCESLet ψ1, ψ2 be either Bernstein functions, variograms orincreasing and concave functions on [0,∞). Then,

C(h1, h2, h3, u) =σ2

ψ1(|h1|2)1/2ψ2(|u|2)1/2L(

|h2|2

ψ1(|h1|2), |h3|

2

ψ2(|u|2)

)

with hi, u ∈ R, i = 1, 2, 3, is a stationary nonseparable spacetime covariance function with spatially anisotropiccomponents, defined on R3 × R.

Summary





4.2 (I) Mixed Forms





6 Applications

7 Conclusions

- p. 26/68

4.1 (IV) A new class of Anisotropic ST Cov.

Example

Consider the bivariate Laplace transform

L(θ1, θ2) =1−e−θ1−θ2θ1+θ2

,

for θ1 6= 0 or θ2 6= 0, and where L(0, 0) = 1.

Applying previous Corollary, we easily obtain

C(h, u) =

(

1 − e−

|h2|2ψ1(|h1|2)

−|h3|2

ψ2(|u|2)

)

× ψ1(|h1|2)1/2ψ2(|u|

2)1/2

|h2|2ψ2(|u|2)+|h3|2ψ1(|h1|2)

where h = (h1, h2, h3) ∈ R3 and u ∈ R.

Summary





4.2 (I) Mixed Forms





6 Applications

7 Conclusions

- p. 27/68

4.2 (I) Mixed Forms

Result 1. Let ϕ(t) be a completely monotone function (t ≥ 0) andlet ψ(t) be a positive function whose derivative is completelymonotone. Then

C(h,u) =σ2

ψ(‖h‖2)l/2ϕ

(

‖u‖2

ψ(‖h‖2)

)

, (h,u) ∈ Rd × Rl (3)

is a space-time covariance function.

Result 2. Let C1(h, u), C2(h, u), (h, u) ∈ Rd × R be validnonseparable spatio-temporal covariance functions and b > 0. Thenboth C1(h, u) + C2(h, u) and bC1(h, u) are valid spatio-temporalcovariance functions in Rd × R.

Result 3. Let C1(h, u), C2(h, u), (h, u) ∈ Rd × R be validspatio-temporal covariance functions. Then their tensorial productC1(h, u) ·C2(h, u) is still a valid spatio-temporal covariance function.

Summary





4.2 (I) Mixed Forms





6 Applications

7 Conclusions

- p. 28/68


Result 4. (Mixed Forms). Let C1(h, u), C2(h, u), (h, u) ∈ Rd×Rbe valid spatio-temporal covariance functions. Let C3(h), C4(u) berespectively valid spatial and temporal covariance functions. Then

C(h, u) = Ci(h, u) + C3(h) + C4(u), i = 1, 2 (4)

C(h, u) = C1(h, u)C2(h, u) + C3(h) + C4(u) (5)

C(h, u) = C1(h, u) + C2(h, u) + C3(h) + C4(u) (6)

C(h, u) = (λ12C1(h, u)C2(h, u) + λ3C3(h) + λ4C4(u))ξ (7)

and

C(h, u) = (λ1C1(h, u) + λ2C2(h, u) + λ3C3(h) + λ4C4(u))ξ (8)

are valid spatio-temporal covariance functions with constantsλ12, λi, i = 1, ..., 4 nonnegative weights and the external smoothingparameter ξ a natural number.

Summary





4.2 (I) Mixed Forms





6 Applications

7 Conclusions

- p. 29/68


It is interesting to note that following Gneiting, above expressionswould take the forms

C(h, u) =σ2

ψ(‖h‖2)l/2ϕi

|u|2

ψ(‖h‖2)

+ ϕ3

(

‖h‖2)

+ ϕ4

(

|u|2)

, i = 1, 2

C(h, u) =σ2

ψ2(‖h‖2)l/2ψ1(|u|2)d/2ϕ1

‖h‖2

ψ1(|u|2)

ϕ2

|u|2

ψ2(‖h‖2)

+ ϕ3

(

‖h‖2)

+ ϕ4

(

|u|2)

C(h, u) =σ2

ψ1(|u|2)d/2ϕ1

‖h‖2

ψ1(|u|2)

+σ2

ψ2(‖h‖2)l/2ϕ2

|u|2

ψ2(‖h‖2)

+

ϕ3

(

‖h‖2)

+ ϕ4

(

|u|2)

Summary





4.2 (I) Mixed Forms





6 Applications

7 Conclusions

- p. 30/68

4.2 (IV) Mixed Forms

C(h, u) = (λ12σ2

ψ2(‖h‖2)l/2ψ1(|u|2)d/2ϕ1

‖h‖2

ψ1(|u|2)

ϕ2

|u|2

ψ2(‖h‖2)

+λ3ϕ3

(

‖h‖2)

+ λ4ϕ4

(

|u|2)

)ξ

C(h, u) = (λ1σ2

ψ1(|u|2)d/2ϕ1

‖h‖2

ψ1(|u|2)

+ λ2σ2

ψ2(‖h‖2)l/2ϕ2

|u|2

ψ2(‖h‖2)

λ3ϕ3

(

‖h‖2)

+ λ4ϕ4

(

|u|2)

)ξ

where ϕi (t) , t ≥ 0, i = 1, ..., 4 are completely monotonefunctions and ψj(t), t ≥ 0, j = 1, 2 are positive functions withcompletely monotone derivative.

Summary





4.2 (I) Mixed Forms





6 Applications

7 Conclusions

- p. 31/68

4.2 (V) Mixed Forms: An Example

C1(h, u) =k

Bυβ/2+εu,h

(a1a2 ‖h‖ |u|)υ

×

Kυ

a2‖h‖(

a1|u|2α1+1)β/2

Kυ

a1|u|(

a2‖h‖2α2+1)β/2

, (h, u) ∈ 0 global smoothing parameters.

k = σ4

22(υ−1)(Γ(υ))2

Bu,h =(

1 + a1 |u|2α1 + a2 ‖h‖2α2 + a1a2 |u|2α1 ‖h‖2α2)

Kν (.) : Modified Bessel Function of the second kind of orden ν

Summary





4.2 (I) Mixed Forms





6 Applications

7 Conclusions

- p. 32/68

4.2 (VI) Mixed Forms: An Example

C1(h, u) C2(h, u)

−2 −1 0 1 2

−2−1

01

2

−2 −1 0 1 2

−2−1

01

2

x

y

z

x

y

v

Summary





4.2 (I) Mixed Forms





6 Applications

7 Conclusions

- p. 33/68

4.2 (VII) Mixed Forms: An Example

C1(h, u) C2(h, u)

0.0 0.5 1.0 1.5 2.0 2.5

−2−1

01

2

0.0 0.5 1.0 1.5 2.0 2.5

−2−1

01

2

x

y

z

x

y

v

Summary





4.2 (I) Mixed Forms





6 Applications

7 Conclusions

- p. 34/68

4.3 (I) New families of spectral densities

The aim of this proposal is to build families of spectral densitieswhose associated covariance function is differentiable at theorigin

Starting from Stein (2005), we find new families of spectraldensities associated to spatial temporal processes whoseinverse Fourier transform is differentiable at the origin

• Define a spectral density obtained as tensorial product of twospectral densities, in the following way. Consider

f1, f2 : R→ R

spectral density functions.

• Define the function f(., .) as:

f(w, τ) = f1(α1|τ |a + β1||w||b)f2(α2|τ |a + β2||w||b)

w ∈ R2, τ ∈ R and α1, α2, β1, β2, a, b nonnegative parameters.

Summary





4.2 (I) Mixed Forms





6 Applications

7 Conclusions

- p. 35/68

4.3 (II) New families of spectral densities

NOTATION:Consider a spectral density defined over 0, andwhere C is the autocovariance function associated to f .

Summary





4.2 (I) Mixed Forms





6 Applications

7 Conclusions

- p. 36/68

4.3 (III) New families of spectral densities

Theorem:For j = 1, 2, 3, suppose ∂

l

∂ωljg1 and ∂

l

∂ωljg2 exist and are

integrable for l ≤ k and that |ω|n ∂l

∂ωljg1 and |ω|

n ∂l

∂ωljg2 are

integrable. Then for x 6= 0, DmC exists form = (m1,m2,m3) : m1 +m2 +m3 ≤ n.

Remark:B The above theorems indicate the criteria to follow in orderto choose spectral densities whose associated covariance isdifferentiable at the origin.B It is over our scope to obtain an analytic expression of thecovariance.

Summary





4.2 (I) Mixed Forms





6 Applications

7 Conclusions

- p. 37/68

4.3 (IV) New families of spectral densities

An example

0246

8

100 24 6

8 100.05

0.10

0.15

0.20

0 2 4 6 8 10

02

46

810

0246

8

100 24 6

8 100.05

0.10

0.15

0.20

0 2 4 6 8 10

02

46

810

vtau

0246

8

100 24 6

8 100.02

0.04

0.06

0 2 4 6 8 10

02

46

810

f(w, τ) = φ1φ2(α21 + α

211|τ|

2)−ν1−

d12 (α22 + (β

221||w||

2))−ν2−

d22

Summary





5 Building valid ST covariancemodels: Copulas5.1 (I) Stat. STCF: Mixtures ofcopulas5.1 (II) Stat. STCF: Mixtures ofcopulas5.1 (III) Stat. STCF: Mixtures ofcopulas5.1 (IV) Stat. STCF: CMF andcopulas5.1 (V) Stat. STCF: CMF andcopulas5.1 (VI) Stat. STCF: CMF andcopulas5.1 (VII) Stat. STCF: CMF andcopulas5.2 (I) Nonstationary STCF andCM5.2 (II) Nonstationary STCF andCM5.3 (I) Archimedean anisotropicSTCF5.3 (II) Archimedean anisotropicSTCF5.3 (III) Archimedeananisotropic STCF5.3 (IV) Archimedeananisotropic STCF5.3 (V) Archimedeananisotropic STCF5.3 (VI) Archimedeananisotropic STCF5.3 (VII) Archimedeananisotropic STCF5.3 (VIII) Archimedeananisotropic STCF5.4 (I) STCF: The Bernsteinclass5.4 (II) STCF: The Bernsteinclass5.4 (III) STCF: The Bernsteinclass

6 Applications

7 Conclusions

- p. 38/68

5.1 (I) Stat. STCF: Mixtures of copulas

Theorem

Let γs(h) and γt(u) be spatial and temporal valid variograms,respectively, with h ∈

Summary






6 Applications

7 Conclusions

- p. 39/68

5.1 (II) Stat. STCF: Mixtures of copulas

An Example

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 copulas. Applying previous theorem, we obtain thefollowing nonseparable stationary structure

Cs,t(h, u) = δ

(

1 − e−γs(h)−γt(u))

γs(h) + γt(u)+ (1 − δ)

(

1 − e−γs(h)) (

1 − e−γt(u))

γs(h)γt(u)

Summary






6 Applications

7 Conclusions

- p. 40/68

5.1 (III) Stat. STCF: Mixtures of copulas

An Example (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 familyproposed in Ma (2003b, Example 3, page 103).In this case the spatial margin becomes

Cs(h, 0) =

1 if h = 0

‖h‖−α(

1 − e−‖h‖α)

elsewhere

Summary






6 Applications

7 Conclusions

- p. 41/68

5.1 (IV) Stat. STCF: CMF and copulas

Theorem

Let ϕ(t) be a completely monotone function with t > 0. Let γs(h)and γt(u) be spatial and temporal valid variograms, respectively,with h ∈

Summary






6 Applications

7 Conclusions

- p. 42/68

5.1 (V) Stat. STCF: CMF and copulas

Example

Consider now the completely monotone function in Gneiting (2002)

ϕ(t) =[

ect1/2

+ e−ct1/2]−ν

, (9)

where c, ν are positive parameters and t > 0. Writing c = 1, withoutloss of generality, we obtain that the second derivative admits theexpression

ϕ(2)

(t) =

14ν

{

(ν + 1) 1t

(

e2√t + e−2

√t − 2

)

+ t−3/2(

e2√t − e−2

√t)}

[(

e√t + e−

√t)]ν+2

+

−t−1/2(

e√t + e−

√t)2

[(

e√t + e−

√t)]ν+2

which also admits a finite limit at the origin, whatever the value of νis.

Summary






6 Applications

7 Conclusions

- p. 43/68

5.1 (VI) Stat. STCF: CMF and copulas

Example (Cont.)

Applying point (iii) in previous theorem, it is possible to obtain thefollowing nonseparable covariance function:

Cs,t(h, u) =

14ν

(ν + 1)(

‖h‖2 + |u|2)−1

e2

√

‖h‖2+|u|2+ e

−2√

‖h‖2+|u|2 − 2

+

(

‖h‖2 + |u|2)−3/2

e2

√

‖h‖2+|u|2 − e−2√

‖h‖2+|u|2

−(

‖h‖2 + |u|2)−1/2

e

√

‖h‖2+|u|2+ e

−√

‖h‖2+|u|2

2

e

√

‖h‖2+|u|2+ e

−√

‖h‖2+|u|2

ν+2

(10)

L1 and L2 integrable for any value of ν

Summary






6 Applications

7 Conclusions

- p. 44/68

5.1 (VII) Stat. STCF: CMF and copulas

Example (Cont.)

Summary






6 Applications

7 Conclusions

- p. 45/68

5.2 (I) Nonstationary STCF and CM

Theorem

Let γ(s, t) be an intrinsically stationary variogram such thatγ(0, 0) = 0. Let ϕ(t), t > 0, be a completely monotone function.

Then

ϕ(Ψ(s1, s2, t1, t2))

with

Ψ(s1, s2, t1, t2) = 1/2 {γ(2s1, 2t1) + γ(2s2, 2t2)}

−{γ(s1 + s2, t1 + t2) − γ(s1 − s2, t1 − t2)}

is a valid nonstationary covariance function on

Summary






6 Applications

7 Conclusions

- p. 46/68

5.2 (II) Nonstationary STCF and CM

Theorem

Under the same conditions of the previous theorem, we have that

ϕ(Ψ∗(s1, s2, t1, t2))

with

Ψ∗(s1, s2, t1, t2) = 1 + 1/2 {γ(2s1, 2t1) + γ(2s2, 2t2)}

−

{

1 + γ(s1 + s2, t1 + t2)1 + γ(s1 − s2, t1 − t2)

}

is a valid nonstationary covariance in

Summary






6 Applications

7 Conclusions

- p. 47/68

5.3 (I) Archimedean anisotropic STCF

Theorem

Assume that ψ(x) is a completely monotone function on [0,∞). IfC1(x), . . . , Cd(x), CT (x) are stationary covariance functions on thereal line, and the functions ψ−1(CT (x)), ψ−1(Ck(x)), k = 1, . . . , d,are continuous, increasing and concave on [0,∞), then

C(s; t) = ψ

{

d∑

k=1

ψ−1(Ck(|sk|)) + ψ−1(CT (|t|))

}

, s ∈

Summary






6 Applications

7 Conclusions

- p. 48/68

5.3 (II) Archimedean anisotropic STCF

A simple example

Consider the completely monotone function

ψ(x) = (1 + x)−1/θ1 , s > 0

with θ1 positive parameter, whose inverse is ψ−1(x) = x−θ1 − 1.Also consider the covariance functions:

Ck(x) = (1 + |x|)−1/θk , k = 1, ..., d

CT (x) = (1 + |x|)−1/θT

It is easy to verify that ψ−1 ◦Ci(x) = (1 + x)θ1/θi − 1, i = 1, ..., d, iscontinuous, increasing and concave iff θ1 < θi. Then, we obtain:

C(x, t) =

[

d∑

k=1

(1 + |sk|)ρi + (1 + |t|)ρT − d

]−1/θ1

is a valid nonseparable stationary isotropic spatio-temporalcovariance function, with ρi = θ1/θi, i = 1, ..., d, and ρT = θ1/θT .

Summary






6 Applications

7 Conclusions

- p. 49/68

5.3 (III) Archimedean anisotropic STCF

h1

0.0

0.2

0.4

0.6

0.8

1.0

h2

0.0

0.2

0.4

0.6

0.8

1.0

Gam

ma(h1,h2)

0.0

0.2

0.4

0.6

0.8

Summary






6 Applications

7 Conclusions

- p. 50/68

5.3 (IV) Archimedean anisotropic STCF

Remark:

1 The criteria allows to build nonseparable spatial temporalcovariance functions which are not necessarily Lp norms.

2 This class can be used for spatial-anisotropic temporal-isotropicphenomena.

3 The function ψ is a link function allowing to build a nonseparablestructure starting from the temporal and spatial margins.

Summary






6 Applications

7 Conclusions

- p. 51/68

5.3 (V) Archimedean anisotropic STCF

Theorem

Let C1(x), . . . C3(x), CT (x) be stationary covariance functions onthe real line, such that both are completely monotone. Let ψ1, ψ2above defined such that:• ψ−11 ◦ ψ2

• ψ−12 ◦ Ci(x), i = 1, 2, 3• ψ−11 ◦ CT (x)

are positive functions with completely monotonic derivative. Then

C(s, t) = ψ1

[

ψ−11 ◦ ψ2

(

3∑

i=1

ψ−12 (Ci(si))

)

+ ψ−11 (C(t))

]

is a stationary isotropic spatio-temporal covariance function.

Summary






6 Applications

7 Conclusions

- p. 52/68

5.3 (VI) Archimedean anisotropic STCF

Theorem

Let C1(x), . . . C3(x), CT (x) be stationary covariance functions onthe real line, such that both are completely monotonic. Let ψi,i = 1, ..., 3 above defined such that :• ψ−11 ◦ ψ2, ψ

−12 ◦ ψ3

• ψ−13 ◦ Ci(x), i = 1, 2• ψ−12 ◦ C3(x)

• ψ−11 ◦ CT (x)

are positive functions with completely monotonic derivative. Then

C(x, t) = ψ1

ψ−11 ◦ ψ2

ψ−12 ◦ ψ3

2∑

i=1

ψ−13 (Ci(si))

+ ψ−12 (C3(s3))

+ ψ−11 (CT (t))

is a stationary anisotropic spatio-temporal covariance function.

Summary






6 Applications

7 Conclusions

- p. 53/68

5.3 (VII) Archimedean anisotropic STCF

An example

Consider the completely monotone functions

ψi(x) = (1 + x)−1/θi , s > 0, i = 1, 2

with θi positive parameters, whose inverse is ψ−1i (x) = x−θi − 1.

Also consider the covariance functions:

Ck(x) = (1 + |x|)−1/θk , k = 1, ..., 3

CT (x) = (1 + |x|)−1/θT

It is easy to verify that conditions of previous theorem are satisfiediff θ1 < θ2, θ1 < θT and θ2 < θk, k = 1, ..., 3. Under this constraint,applying previous theorem we obtain:

C(x, t) =

[

(

3∑

k=1

(1 + |sk|)ρk − 2)ρ2 + (1 + |t|)ρT − 1

]−1/θ1

is a valid nonseparable stationary isotropic spatio-temporalcovariance function, with ρ1 = θ1/θ2, ρk = θ2/θk, i = 1, ..., d, andρT = θ1/θT .

Summary






6 Applications

7 Conclusions

- p. 54/68

5.3 (VIII) Archimedean anisotropic STCF

h2

0.0

0.2

0.4

0.6

0.8

1.0

u

0.0

0.2

0.4

0.6

0.8

1.0

Gam

ma(h2,u)

0.0

0.1

0.2

0.3

0.4

0.5

Summary






6 Applications

7 Conclusions

- p. 55/68

5.4 (I) STCF: The Bernstein class

Open Problems

1 Great majority of contributions in space-time covariance regardsspatially isotropic covariance functions.

2 Thus, there is a big need of more models which are zonallyanisotropic in the spatial component and not necessarilyLp-norms.

Recall a positive function ψ(.) defined on the positive real line iscalled a Bernstein function if its first derivative is completelymonotone, viz.

(−1)k−1ψ(t)(k) ≥ 0,

for any positive integer k and t > 0.

Summary






6 Applications

7 Conclusions

- p. 56/68

5.4 (II) STCF: The Bernstein class

Theorem

Let L(., .) be a bivariate Laplace transform of a nonnegative randomvector (X1, X2). Let ψi(.), i = 1, ..., d and ψT (.) be positiveBernstein functions defined on the positive real line. Then, ford = 1, 2, ...

C(h, u) = L(∑di=1 ψi(|h|i), ψT (|u|))

is a stationary nonseparable spatio-temporal covariance function in

Summary






6 Applications

7 Conclusions

- p. 57/68

5.4 (III) STCF: The Bernstein class

An example

Let L(θ1, θ2) be the Laplace transform for the Frechet-Hoeffdinglower bound of bivariate copulas, whose equation is

L(θ1, θ2) =exp(−θ1)−exp(−θ2)

θ1−θ2

Now, consider the following Bernstein functions

ψ1(t) = (a1tα1 + 1)β

ψ2(t) =(a2t

α2+b)b(a2t

α2+1)

ψT (t) = tρ,

where a1, a2 are positive scale parameters, α1, α2 ∈ (0, 1], β ∈ [0, 1]b > 1 and ρ ∈ [0, 1).

Then,

C(h1,h2, u) =exp

(

−(a1‖h1‖α1+1)β−a2‖h2‖α2+bb(a2‖h2‖α2+1)

)

−exp(−|u|ρ)

(a1‖h1‖α1+1)β+a2‖h2‖α2+bb(a2‖h2‖α2+1)

−|u|ρ

Summary






6 Applications6.1 Application: Wind SpeedData

6.2 Presentation of Data6.3 (I) VariogramNonparametric Estimation6.3 (II) VariogramNonparametric Estimation

6.4 Estimation via WLS6.5 (I) Comparison amongstfour Models6.5 (II) Comparison among fourModels6.6 (I) Parameters Estimation:diagnostics6.6 (II) Cross-validation:Diagnostics

6.7 Movies

7 Conclusions

- p. 58/68

6.1 Application: Wind Speed Data

. data regard wind speed (mt/sec)

. Zone: Western Pacific Ocean, between Australia e NewGuinea (145◦E-175◦E, 14◦N-16◦N).

. Data sampled on a regular grid (17 × 17 nods, at 210 kmdistance).

. Data are obtained every 6 hours.

Summary









6.7 Movies

7 Conclusions

- p. 59/68

6.2 Presentation of Data

Summary









6.7 Movies

7 Conclusions

- p. 60/68

6.3 (I) Variogram Nonparametric Estimation

. Local Linear Estimator:

N−1∑

i=1

N∑

j=i+1

(

Z(si, ti) − Z(sj , tj))2 − (β0, β10, β01)

1∥

∥

∥si − sj∥

∥

∥ − r∣

∣

∣ti − tj

∣

∣

∣− u

2

·

KH

∥

∥

∥si − sj∥

∥

∥ − r∣

∣

∣ti − tj

∣

∣

∣− u

.KH (v) =1

|H|K(

H−1v)

.K (·): bidimensional kernel

. H = h2S

S = Cov({

(∥

∥

∥si − sj

∥

∥

∥,∣

∣

∣ti − tj

∣

∣

∣)′; 1 ≤ i < j ≤ N

})

Summary









6.7 Movies

7 Conclusions

- p. 61/68

6.3 (II) Variogram Nonparametric Estimation

Summary









6.7 Movies

7 Conclusions

- p. 62/68

6.4 Estimation via WLS

. Link Function:

∑

i,j ωi,j(θ)(

∧γ(hi, uj) − γ(hi, uj | θ)

)2

. Weights:

ωi,j(θ) =|N(ri,uj)|γ(hi,uj |θ)

2

. Algorithm: Levemberg-Marquandt

Summary









6.7 Movies

7 Conclusions

- p. 63/68

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) = c0 + σ2

[

1 −1

(au+ 1)2exp

(

−b2 ‖h‖2

au+ 1

)]

. Model corresponding to example 4 of Cressie and Huang(SVCH4) having equation :

γ(h, u | θ) = c0 + σ2

[

1 −au+ 1

[

(au+ 1)2 + b2 ‖h‖2](d+1)/2

]

. Mixed Forms (Porcu) having equation:

C(h, u | υ = 1/2) ∝k

Bεu,hexp

(

−a1 |u|

(

a2 ‖h‖2α2 + 1

)β/2−

a2 ‖h‖(

a1 |u|2α1 + 1

)β/2

)

Summary









6.7 Movies

7 Conclusions

- p. 64/68

6.5 (II) Comparison among four Models

Porcu SVEXPS

SVCH2 SVCH4

Summary









6.7 Movies

7 Conclusions

- p. 65/68

6.6 (I) Parameters Estimation: diagnostics

Model Nugget Sill MQP

Porcu 0.0864 13.6402 0.0004

SVEXPS 0.4018 13.7093 0.0033

SVCH2 0.4581 1.7021 0.0036

SVCH4: 0.7958 12.7849 0.0029

. MQP = 1∑i,j nij

∑

i,j nij(

γ̂(ri,uj)

γ̄(ri,uj)− 1)2

Summary









6.7 Movies

7 Conclusions

- p. 66/68

6.6 (II) Cross-validation: Diagnostics

Model MES EQM EQMA

Porcu0.00777 0.17863 0.40501

SVEXPS 0.00656 0.31348 0.55903

SVCH2 0.01788 0.35235 0.42639

SVCH4 0.01369 0.19856 0.38807

Summary









6.7 Movies

7 Conclusions

- p. 67/68

6.7 Movies

Summary






6 Applications

7 Conclusions

7 Conclusions

- p. 68/68

7 Conclusions

There is a real and motivating need of good statisticalmodels...

Here we have presented several options: Mixed forms,Spectral densities, Dagum covariances, Copulas-basedcovariances.

Each model is driven by data so that there is no model thatbest fits to all the data, each model serves to eachparticular situation.

Bookmark for section 0Summary of the seminar

Bookmark for section 11.1 Some history of STM1.2 Some history of STM

Bookmark for section 22.1 (I) ST covariance functions2.1 (II) ST covariance functions2.2 Separability and Full Symmetry2.3 Full Symmetry and Zonal Anisotropy2.4 Mixed Forms2.5 (I) Some concepts on Copulas2.5 (II) Some concepts on Copulas2.5 (III) Some concepts on Copulas2.5 (IV) Some concepts on Copulas2.5 (V) Some concepts on Copulas2.5 (VI) Some concepts on Copulas2.5 (VII) Some concepts on Copulas

Bookmark for section 33.1 Da

on building valid space-time covariance modelsmateu/space-time-geostats.pdf- p. 1/68 on building...

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