Semiparametric estimation of spatiotemporal
anisotropic long-range dependence
M.P. Frı́as1 , 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
mpfrias@ujaen.es
University of Granada
Campus de Fuente Nueva s/n, E-18071
Granada, Spain
mruiz@ugr.es; jmangulo@ugr.es; falonso@ugr.es
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.
1202
M.P. Frı́as, 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
1203
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 Hölder 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
1204
M.P. Frı́as, 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 ν̂ 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 ν̂ and β̂i , i = 1, . . . , d,
are then given by
log
ν̂ = −
1 (qω,λ 0 )
F̂X
1 (ω,λ 0 )
F̂X
2 log q
1
+ ,
2
log
β̂i = −
1+i
F̂X
(ω 0 ,...,qλi ,...,λ0
d)
1+i
F̂X
(ω 0 ,...,λi ,...,λ0
)
d
2 log q
+
1
2
(4)
1+i
respectively, where F̂X
, 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 ν̂ for
a finite set of fixed values of the spatial frequencies, and different values of β̂i ,
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.
Semiparametric estimation of anisotropic LRD
1205
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 o2
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 o2 ,
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 o2 ,
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
1206
M.P. Frı́as, M.D. Ruiz-Medina, J.M. Angulo and F.J. Alonso
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 ν̂ considering λ0j1 =
2π/128, . . . , 2π6/128, λ0j2 = 2π/128, . . . , 2π6/128.
ν̂ (σ̂)
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 β̂1 considering ω 0 =
2π/512, . . . , 2π6/512, λ0j2 = 2π/512, . . . , 2π6/512.
β̂1 (σ̂)
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
Semiparametric estimation of anisotropic LRD
1207
Table 3. Average and standard deviation of the values of β̂2 , considering ω 0 =
2π/128, . . . , 2π6/128, λ0j1 = 2π/128, . . . , 2π6/128.
β̂2 (σ̂)
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.
1208
M.P. Frı́as, M.D. Ruiz-Medina, J.M. Angulo and F.J. Alonso
References
[AR99]
Angulo, J.M., Ruiz-Medina, M.D.: Multiresolution approximation to
the stochastic inverse problem. Adv. App. Prob., 31, 1039–1057 (1999)
[AAMR05] Angulo, J.M., Anh, V.V., McVinish, R., Ruiz-Medina, M.D.: Fractional
kinetic equations driven by Gaussian or infinitely divisible noise. Adv.
Appl. Prob., 37, 366–392 (2005)
[BKRT00] Brown, P.E., Karesen, K.F., Roberts, G.O., Tonellato, S.: Blurgenerated non-separable space-time models. J. Roy. Stat. Soc. B, 62,
847–860 (2000)
[Chr00]
Christakos, G.: Modern Spatiotemporal Geostatistics. Oxford University Press, New York (2000)
[CHB00]
Christakos, G., Hristopoulos, D.T., Bogaert, P.: On the physical geometric concept at the basis of space/time geostatistical hydrology. Adv.
Water Res., 23, 799–810 (2000)
[FARA05] Frı́as, M.P., Alonso, F.J., Ruiz-Medina, M.D., Angulo, J.M.: Semiparametric estimation of spatial long-range dependence. Submitted
[FRAA05] Frı́as, M.P., Ruiz-Medina, M.D., Alonso, F.J., Angulo, J.M.: Spatiotemporal generation of long-range dependence models and estimation. Environmetrics, 17, 139–146 (2006)
[Gne02]
Gneiting, T.: Nonseparable, stationary covariance functions for space
time data. J. Amer.Stat. Assoc., 97, 590–600 (2002)
[Haa02]
Haas, T.C.: New systems for modeling, estimating, and predicting
a multivariate spatiotemporal process. Environmetrics, 13, 311–332
(2002)
[KLR05]
Kelbert, M., Leonenko, N.N., Ruiz-Medina, M.D.: Fractional random
fields associated with stochastic fractional heat equation. Adv. Appl.
Prob., 37, 108–133 (2005)
[Ma05]
Ma, C.: Spatio-temporal variograms and covariance models. Adv. Appl.
Prob., 37, 706–725 (2005)
[Rob94]
Robinson, P.M.: Semiparametric analysis of long memory time series.
Ann. Stat., 22, 515–539 (1994)
[Rob03]
Robinson, P.M.: Time Series with Long Memory. Oxford University
Press, Oxford (2003)
[RAAB03] Ruiz-Medina, M.D., Alonso, F.J., Angulo, J.M., Bueso, M.C.: Functional stochastic modeling and prediction of spatio-temporal processes.
J. Geoph. Res. – Atmosph., 108(D24), 9003, doi:10.1029/2003JD003416
(2003)
[RA02]
Ruiz-Medina, M.D., Angulo, J.M.: Spatio-temporal filtering using
wavelets. Stoch. Environ. Res. Risk Asses., 16, 241–266 (2002)
[RAA03]
Ruiz-Medina, M.D., Angulo, J.M., Anh, V.V.: Fractional generalized
random fields on bounded domains. Stoch. Anal. Appl., 21, 465–492
(2003)
[Ste05]
Stein, M.L.: Space-time covariance functions. J. Amer. Stat. Assoc.,
100, 310–321 (2005).
[Wik03]
Wikle, C.K.: Spatio-temporal methods in climatology. In: Encyclopedia
of Life Support Systems. EOLSS, Paris (2003)
[WCN99]
Wikle, C.K., Cressie, N.: A dimension-reduction approach to space-time
Kalman filtering. Biometrika, 86, 815-829 (1999)