Semiparametric estimation of spatiotemporal anisotropic long-range ...

Semiparametric estimation of spatiotemporal anisotropic long-range dependence M.P. Fr´ıas1 , M.D. Ruiz-Medina2 , J.M. Angulo2 and F.J. Alonso2 1

2

University of Jaén Campus Las Lagunillas s/n, E-23071 Jaén, Spain [email protected] University of Granada Campus de Fuente Nueva s/n, E-18071 Granada, Spain [email protected]; [email protected]; [email protected]

Summary. Spatiotemporal covariance models displaying anisotropic long-range dependence are introduced. The generation procedure proposed is based on the convolution with an anisotropic self-similar heavy-tailed kernel r. The fractal properties are also studied in connection with long-memory properties. Averaged estimators of the long-memory parameters, based on the partial integration of the periodogram with respect to each coordinate over a directional sequence of neighborhoods of the zero frequency, are computed to approximate such a behavior. A simulation study is carried out to illustrate the performance of the estimators proposed. Key words: long-range dependence, periodogram, semiparametric estimation, spatiotemporal anisotropy, spatiotemporal fractal models

1 Introduction Random field models play an important role in the statistical analysis of spatiotemporal data in environmental sciences, agriculture, climatology, meteorology, and hydrology, among many other applied disciplines (see, for example, [Chr00]; [CHB00]; [Wik03]). Methodologies in this context are usually developed by extension of Spatial Statistics techniques (see [BKRT00]; [Chr00]; [Haa02]; [RA02]; [RAAB03]; [WCN99], among others). An important aspect of such an extension is the introduction of new models of spatiotemporal variograms and covariance functions (see, for instance, [Gne02]; [Ma05]; [Ste05]). A common feature in geophysical and environmental applications is that data often present spatiotemporal long-range correlations. Covariance models representing this type of behavior have been introduced, for example, in [AAMR05]; [FRAA05]; [KLR05]. Most of these models are isotropic or display anisotropy between space and time; however, in real applications spatial anisotropy is also a common feature of strongly-dependent spatiotemporal data.

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M.P. Fr´ıas, M.D. Ruiz-Medina, J.M. Angulo and F.J. Alonso

In this paper, we introduce a procedure for generation of anisotropic spatiotemporal long-range dependence, based on the convolution of spatiotemporal covariance models with a separable self-similar kernel, which behaves like a Riesz kernel with different heavy-tail parameters for each direction. The associated class of Gaussian spatiotemporal processes can be introduced in the strong sense or in the weak sense (generalized random field framework) depending on the conditions assumed on the input random field involved in the generation procedure proposed (see Section 2). In Section 3, an estimation method, based on the partially integrated periodogram with respect to each coordinate, is designed considering averaged parameter estimators over directional sequences of neighborhoods of the zero frequency . In this sense, we refer to the method proposed as a semiparametric estimation method (see [Rob94], for the temporal case; [FARA05], for the spatial case) since it requires a parametric model only to represent the asymptotic functional form of the covariance model or, equivalently, the functional form of the spectral density in a neighborhood of the zero frequency. In Section 4, a simulation study is developed to illustrate the performance of the estimation method proposed. Final conclusions and open research directions are commented on in Section 5.

2 Model Generation In this section, a spatiotemporal model family displaying anisotropic long-range dependence is studied. We consider that a homogeneous random field X on Rd+1 displays anisotropic long-range dependence if the following limit holds: BX (t, x1 , . . . , xd ) −→ 1, d L(t, x1 , . . . , xd )|t|−1+2ν Πi=1 |xi |−1+2βi when |t| and |xi |, i = 1, . . . , d, tend to infinity. In the above definition, we can consider the parameterization Ht = ν + 1/2 and Hi = βi + 1/2, in terms of the temporal Hurst index Ht and the spatial Hurst indexes Hi , i = 1, . . . , d. Specifically, anisotropic spatiotemporal long-range dependence is generated in terms of the meansquare convolution of an input process Y with the spatiotemporal kernel r(t, z) = d |t|−1+ν Πi=1 |zi |−1+βi , (t, z) ∈ Rd+1 . Thus, we define the family of spatiotemporal covariance functions given by Z

Z

BX (t1 , t2 ; y, z) = Rd+1

Rd+1

d |t1 − s1 |−1+ν Πi=1 |yi − xi |−1+βi

d × BY (s1 , s2 ; x, v)|s2 − t2 |−1+ν Πi=1 |vi − zi |−1+βi ds1 dxds2 dv,

where BY denotes the covariance function of the input process Y. We are interested in the case where BY is non separable with respect to time and space; the separable case has been already studied in [FRAA05]. The covariance function BX can be defined in the strong sense or in the weak sense depending on the conditions assumed on process Y, namely, on its covariance function BY . In the case where Y is a zero-mean second-order spatiotemporal process, satisfying suitable second-order regularity and moment conditions that ensure the integral (??) exists, BX is defined in the strong sense. That is, there exists a Gaussian random field X, given by Z

X(t, x) =

m.s.

Rd+1

d |t − s|−1+ν Πi=1 |xi − yi |−1+βi Y (s, y)dsdy,

(1)

Semiparametric estimation of anisotropic LRD

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with covariance function BX , where the parameter space containing (ν, β1 , . . . , βd ) coincides with or is included in (0, 1)d+1 . Otherwise, X must be introduced in the weak sense considering a suitable space of test functions that allows to define (??) as the covariance function of a (second-order) generalized random field (see, for example, [RAA03]). In both cases, BX provides a non-separable covariance model representing anisotropic long-range dependence in space and time. Note that in the case where the input random field Y is a fractal random field, that is, Y is mean-square H¨ older continuous of order γ (i.e. its fractal dimension is D = d + 1 − γ), the output random field X has fractal dimension at each coordinate Dt = 5/2−ν −γ/d, and Di = 5/2−βi −γ/d, i = 1, . . . , d. Thus, the following linear relationship holds between fractal dimensions and Hurts indexes (i.e. long-memory parameters) in the case where Y displays short-range dependence: Dt + γ/d = 3 − Ht , Di + γ/d = 3 − Hi , i = 1, . . . , d, where Dt and Di , i = 1, . . . , d, respectively denote the fractal dimensions at temporal and spatial coordinates.

2.1 Examples We introduce some special cases of (??)-(1), which include the ordinary and generalized definitions of X in (1) respectively corresponding to the strong-sense and weak-sense formulations of (??), depending on the conditions satisfied by the process Y considered. Example 1. Let Y1 be a stationary spatiotemporal process having spectral density 1 given by fY1 (ω, λ) = (ρ+|ω|2 )γ +(̺+kλk γ + ν > 1/2, 2 )α , ρ, ̺ > 0, with ν < 1/2, d X

βi < d/2,

α+

i=1

d X

βi > d/2, where, as before, (ν, β1 , . . . , βd ) is the parameter

i=1

vector defining the self-similar convolution kernel r. Under the conditions assumed on the parameter space, process X can be defined in the strong sense as a secondorder random field. Example 2. The parametric covariance model of the input process Y2 considered here includes, as a particular case, the spatiotemporal covariance model considered in Example 1. Specifically, the spectral density fY2 is given by fY2 (ω, λ) = 1 [(ρ+|ω|2 )γ +(̺+kλk2 )α ]δ d X

, ρ, ̺ > 0, with ν < 1/2,

δγ +ν > 1/2,

d X

βi < d/2,

δα+

i=1

βi > d/2. Such restrictions on the parameter space defining the corresponding

i=1

covariance models BX and BY2 (i.e. the spectral densities fX and fY2 ) are considered in order to get the strong-sense identity in equation (??). Example 3. Let X be a spatiotemporal process given by Z Z

X(t, z) =

m.s

|y − x|2 |t − s|−1+ν d −1+βi − Π |z − y | exp i i i=1 (2πs)d/2 2s

ε(t, x)dxdsdy, (2)

where ν, βi ∈ (0, 1), i = 1, . . . , d. Here ε represents a zero-mean spatiotemporal white noise. The covariance function BX is then non separable and presents anisotropic heavy-tail behavior in space and in time, for ν, βi ∈ (0, 1), i = 1, . . . , d. Random field X must be defined in the weak sense as a second-order generalized random

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M.P. Fr´ıas, M.D. Ruiz-Medina, J.M. Angulo and F.J. Alonso N

field on H −ν (R) di=1 H −βi (R), where H s (R) denotes the fractional Sobolev space of order s on R (see, [RAA03]).

3 Estimation Method Let Y be a stationary spatiotemporal random field with spectral density fY , satisfying the conditions assumed in Section 2. We address the problem of estimation of the parameter vector (ν, β1 , . . . , βd ) appearing in equations (??)-(1) characterizing the convolution kernel r. The directional estimation method proposed here is based on the local regularity of the spectral density function partially integrated with respect to each coordinate in a sequence of neighborhoods of the zero frequency, considering different directions. We here refer to the special case where Y displays short-range dependence for each fixed direction. The general case where Y may display longrange dependence at certain directions can be formulated in a similar way to the one presented in this section, by adding the long-memory parameters of process Y, in1+i volved at each direction, to the order of magnitude of FX , the partially integrated spectral density at direction 1 + i, for i = 0, . . . , d. In this case, the joint estimation of the long-memory parameters involved in the definition of the convolution kernel r and the spectral density fY is obtained. d Since the spectral density is fX (ω, λ) = fY (ω, λ)|ω|−2ν Πi=1 |λi |−2βi , we have Z

1 FX (ω, λ0 ) =

ω

0

fX (υ, λ0 )dυ ∼ KfY (0, λ0 )

|ω|−2ν+1 d Πi=1 |λ0i |−2βi , −2ν + 1

as ω goes to zero, for a fixed λ0 . The approximation

1 FX (qω,λ 0 ) 1 (ω,λ 0 ) FX

(3)

∼ q −2ν+1 , with

q ∈ (0, 1), then allows the definition of an estimator νˆ of ν. Similarly, we have Z

1+i FX (ω 0 , . . . , λi , . . . , λ0d ) =

λi

fX (ω 0 , . . . , θi , . . . , λ0d )dθi

0

∼ KfY (ω 0 , . . . , 0, . . . , λ0d )

(λi )−2βi +1 0 −2ν d |ω | Πj6=i |λ0j |−2βj , −2βi + 1

as λi → 0, for the estimation of βi , i = 1, . . . , d. The estimators νˆ and βî , i = 1, . . . , d, are then given by log νˆ = −

ˆ 1 (qω,λ 0 ) F X ˆ 1 (ω,λ 0 ) F X

2 log q

1 + , 2

βî = −

log

ˆ 1+i (ω 0 ,...,qλi ,...,λ0 ) F d X ˆ 1+i (ω 0 ,...,λi ,...,λ0 ) F X

d

2 log q

+

1 2

(4)

1+i respectively, where FˆX , i = 1, . . . , d, are computed from the partially integrated periodogram at each direction considered. In the estimation procedure proposed we first compute different values of νˆ for a finite set of fixed values of the spatial frequencies, and different values of βî , i = 1, . . . , d, for a finite set of the temporal and remaining spatial frequencies. Then, we calculate a directional estimate of each parameter based on the average of the computed estimates. Finally, we repeat the above steps for a finite sequence of fixedvalue frequency sets for each parameter. A finite sequence of parameter estimates is then obtained which provides information about the true parameter values from a directional sample approximation through neighborhoods of the zero frequency.


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4 Simulations A simulation study has been developed to show the performance of the estimation method, based on the partially integrated periodogram, described in the previous section. We consider here a particular case of Example 3 with d = 2. Let X be a stationary spatiotemporal random field having spectral density fX (ω, λ1 , λ2 ) =

C|λ|2 |ω|−2ν |λ1 |−2β1 |λ2 |−2β2 . π(|λ|4 + |ω|2 )

Equation (3) then holds with fY (0, λ0 ) = i+1 FX (ω 0 , λi , λ0j ) ∼

C |λ0 |−2 , π

and equation (??) with

C|λ0j |2 |ω 0 |−2ν |λi |−2βi +1 |λ0j |−2βj , + |ω 0 |2 )

π(|λ0j |4

i 6= j, i, j ∈ {1, 2}.

As in the previous section, an estimator of the parameter vector (ν, β1 , β2 ) can be defined (see equation (4)) in terms of the partially integrated periodogram with respect to each frequency coordinate, where {X(t, z) : t = 1, . . . , n, z1 = 1, . . . , n1 , z2 = 1, . . . , n2 } represent the data from process X on a grid in R3 , and with N = n× n1 × n2 . We have generated 20 realizations of the spatiotemporal process introduced in (2), with parameter values ν = 0.3, β1 = 0.375 and β2 = 0.45. Each value of the process X is approximated by the Monte Carlo method as X(t, z1 , z2 ) =

128 128 128 1 XX X |t−s|−1+ν |z1 −j1 |−1+β1 |z2 −j2 |−1+β2 Y (s, j1 , j2 ), 3 129 s=0 j =0 j =0 1

2

where t ∈ (42.5, 105.5), z1 ∈ (32.5, 95.5), z2 ∈ (32.5, 95.5), and the process Y is approximated by Y (t, z1 , z2 ) =

128 128 1 1 X X |(z1 , z2 ) − (j1 , j2 )|2 exp − ε(t, j1 , j2 ), 1292 j =0 j =0 (2πt)1/2 2t 1

2

with t ∈ (10, 138), z1 ∈ (0, 128), z2 ∈ (0, 128), and ε(t, j1 , j2 ), j1 , j2 = 0, . . . , 128, being independent zero-mean Gaussian random values. The parameter estimators of ν and βi , i = 1, 2, are all computed considering the values q = 0.45, q = 0.5 and q = 0.55 in equation (4), and the following three arguments in the logarithmic functions involved: 2 P[qm] Pn Pn1 Pn2 0 0 2πs s=1 t=1 z1 =1 z2 =1 X(t, z1 , z2 ) exp i t n + z1 λ1 + z2 λ2 2 , Pm Pn Pn1 Pn2 2πs 0 0 s=1 t=1 z1 =1 z2 =1 X(t, z1 , z2 ) exp i t n + z1 λ1 + z2 λ2 n o 2 P[qm1 ] Pn Pn1 Pn2 2πj1 0 0 j1 =1 t=1 z1 =1 z2 =1 X(t, z1 , z2 ) exp i tω + z1 n1 + z2 λ2 n o 2 , Pm1 Pn Pn1 Pn2 2πj1 0 0 j1 =1 t=1 z1 =1 z2 =1 X(t, z1 , z2 ) exp i tω + z1 n1 + z2 λ2 n o P[qm2 ] Pn Pn1 Pn2 2 2πj2 0 0 j2 =1 t=1 z1 =1 z2 =1 X(t, z1 , z2 ) exp i tω + z1 λ1 + z2 n2 n o 2 , Pm2 Pn Pn1 Pn2 2πj2 0 0 j2 =1 t=1 z1 =1 z2 =1 X(t, z1 , z2 ) exp i tω + z1 λ1 + z2 n2

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where m, m1 , m2 ∈ N, satisfying the conditions established in [FARA05]. The directional approximation to the behavior of fX close to the zero frequency is given by a sequence of averaged estimates which are computed, respectively, from the sets of fixed values

Nk1 = Nk2 = Nk3 =

λ0j1 =

ω0 =

ω0 =

2π 2π6 2π 2π6 ,..., , λ0j2 = ,..., kn1 kn1 kn2 kn2 2π 2π6 2π6 0 2π ,..., ,..., , λj2 = kn kn kn2 kn2

,

2π6 2π 2π6 2π ,..., ,..., , λ0j1 = kn kn kn1 kn1

,

and

,

with k = 2, 4, 8, 16, 32, 64. Tables 1,2 and 3 show some estimates of the long-memory parameters for particular choices of q (the perturbation of the frequency) and m (the truncation point). In particular, Tables 1 and 3 respectively show the averaged estimates for ν and β2 with k = 2 and m, m2 = 6, 7, 8, 9, 10, 11. The averaged estimates for β1 are showed in Table 2 for k = 8 and m1 = 17, 18, 19, 20, 21, 22. Table 1. Average and standard deviation of the values of νˆ considering λ0j1 = 2π/128, . . . , 2π6/128, λ0j2 = 2π/128, . . . , 2π6/128. νˆ (ˆ σ)

q = 0.45

q = 0.5

m=6 m=7 m=8 m=9 m = 10 m = 11

0.1277 0.2444 0.2116 0.2845 0.2588 0.2364

0.2501 0.2056 0.2866 0.2518 0.3002 0.2744

(0.0772) (0.0509) (0.0545) (0.0319) (0.0366) (0.0387)

q = 0.55 (0.0572) (0.0587) (0.0376) (0.0368) (0.0334) (0.0330)

0.2103 0.1587 0.2525 0.2122 0.2684 0.2998

(0.0663) (0.0680) (0.0436) (0.0426) (0.0388) (0.0290)

Table 2. Average and standard deviation of the values of βˆ1 considering ω 0 = 2π/512, . . . , 2π6/512, λ0j2 = 2π/512, . . . , 2π6/512. βˆ1 (ˆ σ)

q = 0.45

q = 0.5

m1 m1 m1 m1 m1 m1

0.3591 0.3894 0.3854 0.4092 0.4066 0.4046

0.3772 0.4039 0.3993 0.4205 0.4174 0.4348

= 17 = 18 = 19 = 20 = 21 = 22

(0.1054) (0.0762) (0.0790) (0.0555) (0.0567) (0.0575)

q = 0.55 (0.0839) (0.0584) (0.0617) (0.0438) (0.0453) (0.0349)

0.3940 0.3885 0.4124 0.4306 0.4270 0.4414

(0.0633) (0.0678) (0.0480) (0.0376) (0.0393) (0.0353)

Convergence theorems ensure good properties for the estimators considered (see, for instance, [Rob03]). In practice, the values of parameter q are selected close to


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Table 3. Average and standard deviation of the values of βˆ2 , considering ω 0 = 2π/128, . . . , 2π6/128, λ0j1 = 2π/128, . . . , 2π6/128. βˆ2 (ˆ σ)

q = 0.45

q = 0.5

m2 m2 m2 m2 m2 m2

0.3433 0.4452 0.4425 0.4569 0.4552 0.4519

0.4470 0.4368 0.4561 0.4503 0.4721 0.4683

=6 =7 =8 =9 = 10 = 11

(0.0367) (0.0111) (0.0124) (0.0048) (0.0053) (0.0054)

q = 0.55 (0.0135) (0.0128) (0.0055) (0.0056) (0.0055) (0.0052)

0.4386 0.4268 0.4491 0.4424 0.4677 0.4713

(0.0157) (0.0149) (0.0064) (0.0064) (0.0063) (0.0027)

√ 0.5, for solving stability problems, and the values of m close to k × n (minimizing 1/m + 1/kn), although other criteria can be adopted for large n.

5 Conclusion The procedure proposed for generation of anisotropic spatiotemporal long-range dependence covariance models can be connected with the introduction of random field models through the linear filter theory. Since the anisotropic fractal filter considered is self-similar, certain fractional moment conditions on the second-order structure of the input random field must be required in order to establish a strong-sense definition of such covariance models. However, in some practical cases the conditions assumed on the input covariance model might be too restrictive. The generalized random field theory then provides a suitable framework for the introduction of a wider class of spatiotemporal covariance models in terms of self-similar fractal filters (see, for example, [RAA03]). In particular, non-continuous covariance models can also be introduced within this framework (e.g. covariance models with a discontinuity at the origin, i.e. a nugget effect in the corresponding variogram). In this case, suitable bases, like wavelet bases, allow in practice to work with such models (see, for instance, [AR99]). The methodology followed can be extended to more general families of anisotropic filters and connected with the theory of fractional pseudodifferential equations (see [FRAA05]). Different extensions can be considered regarding the estimation procedure. For instance, partial integration of the periodogram can be achieved in terms of test function with suitable supports and moment conditions. The extension of the semiparametric estimation procedure proposed in [Rob94], and its spatial formulation in [FARA05], to the anisotropic context is now being addressed by the authors in a subsequent paper, considering the local regularity of the marginal spectral densities. Acknowledgements. This work has been supported in part by projects MTM2005-08597 of the DGI and P05-FQM-00990 of the Andalousian CICE, Spain.

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