Testing Lack of Symmetry in Spatial-Temporal Processes
Man Sik Park and Montserrat Fuentes1
Institute of Statistics Mimeo Series# 2585
SUMMARY
Symmetry is one of the main assumptions that are frequently taken for granted in most applications in the environmental research. However, many studies in environmental sciences show
that real data have so complex spatial-temporal dependency structures due to lack of symmetry
and other standard assumptions of the covariance function. In this study, we propose new formal
tests for lack of symmetry by using spectral representations of spatial-temporal covariance function.
The beauty of the tests is that classical analysis of variance (ANOVA) models are employed for
detecting lack of symmetry inherent in spatial-temporal processes. We evaluate the performance
of the tests by simulation study and, finally, apply to the PM2.5 daily concentration dataset.
Key Words: Symmetry; Separability; Spatial-temporal process; Spectral representation
1
M. S. Park is a graduate student in the Statistics Department at North Carolina State University (NCSU),
Raleigh, NC 27695-8203. Tel.: (919) 515-7748, Fax: (919) 515-1169, E-mail: [email protected]. M. Fuentes is
an Associate professor in the Statistics Department, NCSU, Raleigh, NC 27695-8203, E-mail: [email protected].
This research was sponsored by a National Science Foundation grant DMS 0353029, and by a US EPA cooperative
agreement.
Key words:
Air pollution; Asymmetry; nonseparability; Spatial-temporal process; Matérn covariance; Spectral
density function.
1
1
Introduction
Symmetry and separability are the main assumptions used in spatial statistics about a covariance
function. Symmetry and separability in spatial or spatial-temporal processes are highly related
to each other. Separability provides many advantages, such as the simplified representation of
the covariance matrix and, consequently, important computational benefits. Symmetry is related
to the spatial or spatial-temporal dependencies. This characteristic has been assumed because of
mathematical convenience, modeling parsimony or calculational efficiency. The common advantage
of symmetry and separability is the simplification attained for modeling purpose. However, many
studies in environmental sciences show that real data have such complex spatial-temporal dependency structures that are difficult to model and estimate by using just separability, symmetry or
other standard assumptions of the covariance function.
Lots of research about separability has been done so far while symmetry has not been in
the spotlight yet. Modeling nonseparable covariance functions is one of the keys for the more
reliable prediction in the environmental research fields. Cressie and Huang (1999) introduced a new
class of nonseparable, spatial-temporal stationary covariance functions with space-time interaction,
which have the separable covariance function as a special case. Gneiting (2002) also proposed
general classes of nonseparable, stationary spatial-temporal covariance functions which are directly
constructed in the space-time domain and are based on Fourier-free implementation. Fuentes at
al (2005) proposed a new class of nonseparable and nonstationary spatial-temporal covariance
models, which have a unique parameter indicating spatial-temporal dependency. In addition to the
modeling issue, many studys about testing lack of separability have been accomplished. Shitan and
Brockwell (1995) used an asymptotic χ2 test for stationary spatial autoregressive processes. Guo
and Billard (1998) proposed the Wald test for testing lack of a doubly-geometric process under
the temporal setting. A likelihood ratio test for lack of separability for i.i.d multivariate processes
2
was proposed by Mitchell (2002), and Mitchell et al. (2002). Fuentes (2006) developed a formal
test for lack of separability and lack of stationarity of spatial-temporal covariance functions by
applying a two-factor analysis of variance (ANOVA) procedure, which is applicable to more general
spatial-temporal covariance models.
The most relevant works about symmetry have been done by Scaccia and Martin (2005), and
Lu and Zimmerman (2005). Suppose that Z(s) : s = (s1 , s2 , · · · , sd )′ ∈ D ⊂ Rd denotes a spatial
process where s is a spatial site over a fixed domain D and Rd is a d-dimensional Euclidean space,
and the covariance function is defined as
C(h|θ) ≡ cov {Z(si + h), Z(si )} ,
where si = si1 , · · · , sid
′
∈ D, h = (h1 , · · · , hd )′ , and θ is a covariance parameter vector. For
two-dimensional rectangular lattice data, Scaccia and Martin (2005) developed new tests of axial
symmetry and separability which are, respectively, defined by, for all h1 and h2 ,
C(h1 , h2 |θ) = C(−h1 , h2 |θ),
and
C(h1 , h2 |θ) = C1 (h1 |θ 1 ) · C2 (h2 |θ 2 ),
where C1 and C2 are the positive-definite covariances of the corresponding spatial lags, h1 and h2 ,
and θ ′ = θ ′1 , θ ′2 . Their tests are performed in two stages: testing axial symmetry first and then, if
the hypothesis of axial symmetry is not rejected, testing separability. Under an n1 × n2 rectangular
lattice data, their tests are based on the periodogram denoted by
1
I(ω1 , ω2 ) =
(2π)2
nX
1 −1
nX
2 −1
h1 =−n1 +1 h2 =−n2 +1
C(h1 , h2 |θ) cos(h1 ω1 + h2 ω2 ).
Lu and Zimmerman (2005) also proposed diagnostic tests of axial symmetry and complete symmetry
3
which is defined by
C(h1 , h2 |θ) = C(−h1 , h2 |θ) = C(−h2 , h1 |θ) = C(h2 , h1 |θ),
for all h1 and h2 . Their tests of symmetries are also based on certain ratios of spatial periodograms.
However these noteworthy studies are only applicable for spatial processes, not spatial-temporal
ones and, therefore, no formal tests for lack of symmetry in spatial-temporal processes have been
developed yet although the modeling of asymmetric spatial-temporal processes has been researched
by Stein (2005). In this study, we propose new formal tests by using spectral representations of the
covariance function. The beauty of the tests is that classical analysis of variance (ANOVA) models
are employed for detecting lack of symmetry inherent in spatial-temporal processes.
This paper is organized as follows. In Section 2 we introduce the spectral representation under
the spatial-temporal setting. Based on the spectral representation, we propose new tests for lack of
symmetry in spatial-temporal processes in Section 3. The performances of the tests are evaluated
by simulation study in Section 4 and by the real application in Section 5. Finally, we present some
conclusions and final remarks in Section 6.
2
The Spectral Representation of Stationary Spatial-Temporal
Processes
In this section, we talk about the spectral representation of stationary spatial-temporal processes,
which is a major key for building new tests for lack of symmetry. Suppose that a spatial-temporal
process is denoted by Z(s; t) : s ∈ D ⊂ Rd , t ∈ [0, ∞) where t indicates measuring time. Then
the spatial-temporal process, {Z(s; t)} can be expressed in the spectral domain by sinusoidal forms
with different frequencies (ω, τ ), where ω is d-dimensional spatial frequency, and τ is temporal
4
frequency. If Z(s, t) is a stationary process with the corresponding covariance defined by
C(h; u) ≡ cov {Z(si + h, tk + u), Z(si , tk )} ,
(1)
then we can rewrite the process in the following Fourier-Stieltjes integral (Yaglom (1987)):
Z(s, t) =
Z
Z
Rd
exp(is′ ω + iτ t) dY (ω, τ ),
R
where Y is a random process with complex symmetry except for the constraint, dY (ω, τ ) =
dY c (−ω, −τ ), which ensures that Z(s; t) is real-valued. Here c stands for complex conjugate.
Using the spectral representation of Z, the covariance function C(h; u) can be represented as
Z
C(h; u) =
Rd
Z
exp(ih′ ω + iτ u) G(dω; dτ ),
(2)
R
where (h; u) = (si − sj ; tk − tl ) for si , sj ∈ D and tk , tl ∈ [0, ∞), and the function G is a positive
finite measure called the spectral measure or spectral distribution function for Z. The spectral
measure G is the expected squared modulus of the process Y denoted by
E |Y (ω, τ )|2 = G(ω; τ ).
We can easily see that C(h; u) in (2) is always positive-definite for any finite positive measure G. If
G has a density with respect to Lebesgue measure, the spectral density g is the Fourier transform
of the spatial-temporal covariance function:
1
g(ω; τ ) =
(2π)d+1
Z Z
R
Rd
exp(−ih′ ω − iτ u)C(h; u) dh du,
(3)
and the corresponding covariance function is given by
C(h; u) =
Z
Rd
Z
exp(ih′ ω + iτ u) g(ω; τ ) dω dτ.
(4)
R
The reason why we are interested in the spectral representation is that it is very easy to cast a
new spectral density function into the corresponding covariance function as long as we know the
spectral density function.
5
3
Tests for Lack of Symmetry In Spatial-Temporal Processes
We summarized the spectral representation of a stationary spatial-temporal processes in Section
2. Now we talk about new tests for lack of symmetry in spatial-temporal processes based on
the spectral representation. First, we define three types of symmetry under the spatial-temporal
setting. Provided that the covariance shown in (1) is assumed to be stationary in time, that is,
cov{Z(si , tk + u), Z(si , tk )} = cov{Z(si , tl + u), Z(si , tl )}
for arbitrary u, we define the three types of symmetry as following:
Definition 3.1 A process is called axially symmetric in time if
C(si − sj ; u) = C(si∗ − sj ∗ ; −u),
(5)
for any temporal lag u 6= 0 and arbitrary four sites (i, j, i∗ , j ∗ ) satisfying si − sj = si∗ − sj ∗ .
Under stationarity in space, (5) is reduced to
C(h; u) = C(h; −u),
(6)
where si = sj + h and si∗ = sj ∗ + h. What is important in (5) and (6) is that the directions and
the distances on spatial domain are the same, and the time lags have the same magnitudes but
different signs.
Definition 3.2 A process is called axially symmetric in space if
C(h; u) = C(h̊; u),
(7)
where h̊ = (h1 , · · · , hk−1 , −hk , hk+1 , · · · , hd )′ for k fixed.
As can be seen in (7), for temporal lag u fixed, all the spatial lags are the same except one spatial
lag, which has a different sign.
6
3.1
Test for Lack of Axial Symmetry in Time
Now we explain the analytical aspect of axial symmetry in time (Definition 3.1) in spatialtemporal process. By Bochner’s theorem, we can always write the positive-definite spatial-temporal
covariance in (4) in terms of the corresponding valid spectral density function, g in (3):
C(h; u) =
Z Z
Rd
R
exp{ih′ ω + iuτ }g(ω; τ ) dω dτ.
If C is integrable, then (3) can be expressed as
g(ω; τ ) = (2π)−(d+1)
= (2π)−d
Z Z
R
Z
Rd
exp{−ih′ ω − iuτ }C(h; u) dh dτ
exp{−ih′ ω}f (h; τ ) dh,
Rd
for h fixed and τ ∈ [0, ∞). Here f (h; τ ) is called the cross-spectral density function of Z(a, t) and
Z(a + h, t), and is defined as follows:
Z
f (h; τ ) = (2π)−1
R
exp{−iuτ }C(h; u) du = f c (−h; τ ),
(8)
where the complex conjugate of f (h; τ ), f (−h; τ ) is represented as
−1
f (−h; τ ) = (2π)
= (2π)−1
Z
Z
R
exp{−iuτ }C(−h; u) du = (2π)
R
exp{iuτ }C(h; u) du.
−1
Z
R
exp{−iuτ }C(h; −u) du
Without the stationarity in space, we can write the cross-spectral density function in (8) as
−1
fab (τ ) = (2π)
Z
R
c
exp{−iuτ }Cov{Z(a, t), Z(b, t + u)} du = fba
(τ ),
(9)
where a, b ∈ D. Under the axial symmtry in time, that is, if C(a − b; u) = C(a − b; −u), then the
cross-spectral density function is represented as following:
Z
exp{−iuτ }C(b − a; u) du
fba (τ ) = (2π)−1
R
−1
= (2π)
Z
R
exp{−iuτ }C(b − a; −u) du = fab (τ )
7
(10)
because C(b − a; −u) = C(a − b; u). From (9) and (10), the cross-spectral density function, fab (τ )
is real-valued. We can also show that, under axial symmetry in time, the phase, φab (τ ) between
Z(a; t) and Z(b; t) is represented as follows:
φab (τ ) ≡ tan
−1
Im.fab (τ )
Re.fab (τ )
= φba (τ ) = 0,
where Im.f and Re.f are, respectively, the imaginary part and the real part of f .
Now we propose a new test for lack of axial symmetry in time by using the asymptotic properties
of the cross-spectral density function and the phase. For an arbitrary site a, we can define the
tapered Fourier transform, Ja (τ ) as
Ja (τ ) =
T
−1
X
t=1
t
K
Z (a; t) exp{−iτ t},
T
where K is a tapering function and, in this study, is considered constant, i.e. K(x) = 1 for all x.
The spectral window, W (µ) can be estimated by
∞
1 X
[µ + 2πt]
c
W (µ) =
W
,
BT t=−∞
BT
(11)
where BT is a temporal bandwidth parameter. In the real application, following weight function is
considered,
c
W
2πs
T
=
T
(2M + 1)−1 ,
2π
where M = BT T and s ≤ M . We can finally estimate the cross-spectral density function between
Z(a; t) and Z(b; t) by
T −1
2πt b
2π X c
2πt
b
W τ−
fab (τ ) =
,
Iab
T
T
T
t=1
where the sample cross-periodogram Ibab (τ ) is defined by
" T −1
#−1
X
t
2
Ja (τ )Jbc (τ ).
K
Ibab (τ ) = 2π
T
t=1
Here we introduce some assumptions:
8
(12)
A.1 W (µ) is real-valued, even and of bounded variation such that, for −∞ < µ < ∞,
Z
W (µ) dµ = 1
R
and
Z
R
|W (µ)| dµ < ∞.
A.2 For each h,
∞
X
u=−∞
|u||C(h; u)| < ∞,
which implies that the temporal covariance is summable, that is,
∞
X
u=−∞
|C(h; u)| < ∞.
A.3 BT T → ∞ and BT → 0 as T → ∞.
Under the assumptions A.1 through A.3, the expected value of the estimated cross-spectral density
function, fbab (τ ) can be obtained as
−1
o 2π TX
n
2πt
2πt e
b
c
W τ−
fab (τ )
+ O(T −1 )
E fab (τ ) =
T
T
T
t=1
Z
W (µ)feab (τ − BT µ) dµ + O(BT−1 T −1 ),
=
R
where the error term is uniform in τ , and
feab (τ ) =
Z
Rd
qρ (a − s)qρ (b − s)fa+s,b+s (τ ) ds,
for −∞ < τ < ∞. Here we regard qρ (s) as a tensor product of d one-dimensional filters, qρ (s) =
Qd
i=1 q(sd ),
where q is of the form
q(s) = 1/ρ · I(|s| ≤ ρ/d),
9
where I(·) is an indicator function. feab (τ ) is the smoothed cross-spectral density function within
a band of frequencies in the region of τ and a region in space in the neighborhood of a and b, and
the covariance between fbai bi (τ ) and fbaj bj (λ) is expressed by
Z
o
n
2
b
b
W (µ) dµ
lim BT T cov fai bi (τ ), faj bj (λ) = 2π
T →∞
where
R
i
i
h
h
× η {τ − λ} feai aj (τ )febi bj (τ ) + η {τ + λ} feai bj (τ )feaj bi (τ ) ,
η{τ } =
(13)



 1 if τ ≡ 0(mod 2π)


 0
otherwise.
Fuentes (2006) says that, if we define the distance between pairs (ai , bi ) and (aj , bj ) as the minimum
distance between any of the two sites in the first pair and any of the two sites in the second pair,
then the the estimated cross-spectral density functions, fbai bi (τ ) and fbaj bj (λ), are approximately
independent if either
C.1 kτ − λk is sufficiently large so that
Z
R
|W (µ + τ )|2 |W (µ + λ)|2 dµ = 0,
i.e. if kτ − λk ≫ bandwidth of |W (µ)|2 or
C.2 the distance between pairs (ai , bi ) and (aj , bj ) is greater than the bandwidth of qρ (s).
In practice, we can make the covariance in (13) almost zero by having the frequencies τ and λ and
the pairs (ai , bi ) and (aj , bj ) sufficiently apart. As mentioned above, the phase is zero in the case
of axial symmetry in time. So we can get asymptotic normality of the estimated phase, φbab (τ ),
with mean 0 and covariance defined as
Z
o
n
2
b
b
W (µ) dµ
lim BT T cov φab (τ ), φab (λ) = π
T →∞
R
× [η{τ − λ} − η{τ + λ}] |Rab (τ )|−2 − 1 ,
10
(14)
where the coherency between between Z(a; t) and Z(b; t), Rab (τ ) is defined as
Rab (τ ) = feab (τ )
q
feaa (τ )febb (τ ).
From (14), the asymptotic variance is simply denoted as
Z
o
n
2
b
W (µ) dµ [1 − η{2τ }] |Rab (τ )|−2 − 1 .
lim BT T Var φab (τ ) = π
T →∞
(15)
R
Unfortunately, we can not use the asymptotic result of φbab (τ ) for the development of a testing
method because the asymptotic variance in (15) depends on the relative position of a and b. So an
appropriate transformation is needed. To stabilize the asymptotic variance, we transform φbab (τ )
to φeab (τ ) given by
1/2
|Rab (τ )|−2 − 1
.
φeab (τ ) = φbab (τ )
(16)
Then, from (15) and (16), we derive the asymptotic normal distribution of φeab (τ ) with mean 0 and
variance given by
o
i n
h
−1
lim BT T Var φbab (τ )
lim BT T Var φeab (τ ) = |Rab (τ )|−2 − 1
T →∞
T →∞
Z
W 2 (µ) dµ [1 − η{2τ }] .
=π
R
However, Rab (τ ) is unknown in practice, so we newly define φb∗ab (τ ) as
h
i1/2
∗
b
b
bab (τ )|−2 − 1
.
|R
φab (τ ) = φab (τ )
(17)
By the Slutsky’s theorem, we can obtain the same asymptotic normal distribution of φeab (τ ) in (16)
as the one of φb∗ab (τ ) in (17). Based on the assumptions, C.1 and C.2, we implicitly know that,
1/2
= 0, φe∗ evaluated at different
under the null hypothesis H0 : φ∗ab (τ ) = φab (τ )/ |Rab (τ )|−2 − 1
pairs and different frequencies can be treated independent approximately (see Appendix).
With the information of asymptotic distribution of the adjusted phase, φb∗ab (τ ) in (17), we
propose a formal test for lack of axial symmetry in time by employing analysis of variance (ANOVA)
11
procedure. First we compute φb∗ai bi (τj ) at arbitrary two sites, {(ai , bi )}m
i=1 and a set of temporal
frequencies, {τj }nj=1 that cover the space-time domain. What is important here is that arbitrary
two sites should be selected based on the condition given by, for h fixed,
ai − bi = h = (h1 , h2 , · · · , hd )′ .
In order to apply to two-way ANOVA procedure, we rewrite φb∗ai bi (τj ) as follows:
φb∗ai bi (τj ) = φ∗ai bi (τj ) + ǫai bi (τj ),
(18)
1/2
. Here ǫai bi (τj ) asymptotically has the following
where φ∗ai bi (τj ) = φai bi (τj )/ |Rai bi (τj )|−2 − 1
assumptions; E{ǫai bi (τj )} = 0, ∀i, j, Var{ǫai bi (τj )} = σǫ2 , ∀i, j, and Cov{ǫai bi (τj ), ǫak bk (τl )} =
0, ∀i, j, k, l satisfying C.1 and C.2. We also express (18) as
φb∗ai bi (τj ) = αi + βj + ǫai bi (τj ),
(19)
where the parameters {αi } and {βj } are “Location” effect and “Temporal Frequency” effect, respectively. Suppose that the spatial-temporal process is stationary in space, then its covariance
does not depend on the relative position of the sites, which implies that, under the stationarity in
space, “Location” effect, αi is not significant. So, lack of the stationarity in space can be detected
by the classical ANOVA technique to test the null hypothesis:
φb∗ai bi (τj ) = βj + ǫai bi (τj )
against the alternative hypothesis shown in (19). Under axial symmetry in time, the phase is
zero. So, “Temporal Frequency” effect, βj is statistically zero. The classical ANOVA technique is
employed to check lack of axial symmetry in time by testing the null hypothesis:
φb∗ai bi (τj ) = αi + ǫai bi (τj )
12
against the alternative hypothesis shown in (19). In addition, we can also check both lack of
axial symmetry in time and lack of stationarity in space simultaneously by examining whether
αi = βj = 0 or not.
3.2
Test for Lack of Axial Symmetry in Space
Now we talk about the second type of symmetry, axial symmetry in space (Definition 3.2). A
process is called axially symmetric in space provided that the following condition is satisfied:
C(h; u) = C(h̊; u),
where h̊ = (h1 , · · · , hk−1 , −hk , hk+1 , · · · , hd )′ 6= 0 for k fixed. For the simplification of developing
the test, we only consider h̊ = (−h1 , h2 )′ for d = 2. Then we introduce a new version of the
cross-spectral density function between Z(a1 , a2 , t) and Z(a1 , a2 + h2 , t + u), k(ω1 ; h2 , u) given by
k(ω1 ; h2 , u) =
Z Z
R
R
exp{ih2 ω2 + iuτ }g(ω; τ ) dω2 dτ,
(20)
for fixed a2 , h2 , t and u. If C is integrable, then
−3
g(ω; τ ) = (2π)
−2
= (2π)
Z Z
R
R2
R
R
Z Z
exp{−ih′ ω − iuτ }C(h; u) dh dτ
exp{−ih2 ω2 − iuτ }k(ω1 ; h2 , u) du dh2 .
Since the function k in (20) is the Fourier transform of the spatial-temporal covariance function
with respect to one of the spatial frequencies, we can also write k in an alternative form denoted
by
−1
k(ω1 ; h2 , u) = (2π)
Z
R
exp{−ih1 ω1 }C(h1 , h2 ; u) dh1 = k c (−ω1 , h2 ; u).
If a process is axially symmetric in space, that is, C(h1 , h2 ; u) = C(−h1 , h2 ; u), then
k(ω1 ; h2 , u) = (2π)−1
−1
= (2π)
Z
Z
R
exp{−ih1 ω1 }C(h1 , h2 ; u) dh1
R
exp{−ih1 ω1 }C(−h1 , h2 ; u) dh1 = k(−ω1 ; h2 , u).
13
So, k is always real-valued and the following result is obtained:
−1
ψ(ω1 ; h2 , u) = tan
Im.k(ω1 ; h2 , u)
Re.k(ω1 ; h2 , u)
−1
= tan
Im.k(−ω1 ; h2 , u)
Re.k(−ω1 ; h2 , u)
= ψ(−ω1 ; h2 , u) = 0,
where ψ(ω1 ; h2 , u) is a new version of the phase between Z(a1 , a2 , t) and Z(b1 , a2 + h2 , t + u) for
fixed a2 , h2 , t and u.
Now we propose a new testing method for the asymptotic properties of the functions k and ψ.
If Z is observed only at N (= N1 N2 ) sites on regular grids and at the measuring times T , then, for
a2 and t fixed, we can define J∆1 (ω1 ; a2 , t),
J∆1 (ω1 ; a2 , t) = ∆1
NX
1 −1
K
n1 =1
n1
N1
Z (∆1 n1 , a2 ; t) exp {−i∆1 n1 ω1 } ,
where ∆1 is the unit distance of the first spatial coordinate. We also define the sample spectral
c (µ) by
window W
where
c (µ) =
W
∞
[µ + 2πj]
1 X
W
,
BN1
BN1
−∞ < µ < ∞. Then b
k∆1 (ω1 ; h2 , u) is represented by
N1 −1
2π X
b
c ω1 − 2πn1 Ib∆ 2πn1 ; h2 , u ,
k∆1 (ω1 , h2 ; u) =
W
1
N1
N1
N1
n1 =1
where
"
Ib∆1 (ω1 ; h2 , u) = 2π
NX
1 −1 K2
n1 =1
n1
N1
∆1
#−1
c
× J∆1 (ω1 ; a2 , t) J∆
(ω1 ; a2 + h2 , t + u) .
1
Here we introduce some additional assumptions:
A.4 for fixed h2 and u,
Z
R
(21)
j=−∞
|h1 ||C(h1 , h2 ; u)| dh1 < ∞,
14
(22)
which also implies that the spatial covariance is summable, that is,
Z
R
|C(h1 , h2 ; u)| dh1 < ∞.
A.5 BN1 N1 → ∞ and BN1 → 0 as N1 → ∞.
Under the assumptions A.1, A.4 and A.5, we can obtain the asymptotic properties of the esimated
phase, ψb∆1 (ω1 ; h2 , u) with mean ψ∆1 (ω1 ; h2 , u) and the variance defined as
o
n
lim BN1 N1 Var ψb∆1 (ω1 ; h2 , u)
N2 →∞
=π
Z
W (µ) dµ [1 − η{2ω1 }] |Q∆1 (ω1 ; h2 , u)|−2 − 1 ,
(23)
2
R
where a new version of the coherency, Q(ω1 ; h2 , u) between two arbitrary points in two-dimensional
space, (a2 , t) and (a2 + h2 , t + u), is defined by
Q(ω1 ; h2 , u) = k(ω1 ; h2 , u) |k(ω1 ; 0, 0)|.
In general, we can not directly use the asymptotic result of ψb∆1 (ω1 ; h2 , u) in order to make a new
test for lack axial symmetry in space because the asymptotic variance in (23) depends on h2 and
u. In order to make the asymptotic variance independent of h2 and u we transform ψb∆1 (ω1 ; h2 , u)
to ψe∆1 (ω1 ; h2 , u) defined by
1/2
|Q∆1 (ω1 ; h2 , u)|−2 − 1
.
ψe∆1 (ω1 ; h2 , u) = ψb∆1 (ω1 ; h2 , u)
(24)
In practice, however, Q∆1 (ω1 ; h2 , u) is a unknown parameter, so, by using the estimated coherency,
b ∆ (ω1 ; h2 , u), we newly define ψb∗ (ω1 ; h2 , u) as
Q
1
∆1
∗
ψb∆
(ω1 ; h2 , u)
1
h
i1/2
b ∆ (ω1 ; h2 , u)|−2 − 1
.
|Q
= ψb∆1 (ω1 ; h2 , u)
1
(25)
b ∆ (ω1 ; h2 , u) as an estimate of Q∆ (ω1 ; h2 , u), then we can get the same
If we use an appropriate Q
1
1
∗ (ω ; h , u) in (25) as the one of ψ
e∆ (ω1 ; h2 , u) in (24).
asymptotic distribution of ψb∆
1 2
1
1
15
Now we propose a formal test for axial symmetry in space for spatial-temporal processes. For
the m pairs,
ai2 , tai ; bi2 , tbi
m
i=1
, in two-dimensional space consisting of the second spatial and
the temporal coordinates, and a set of first spatial frequencies, {ωj }nj=1 , we can get ψbi⋆ (ωj ) ≡
∗
ωj ; (ai2 , tai ; bi2 , tbi ) . Arbitrarily, the pairs of two points in two-dimensional space are selected
ψb∆
1
based on the conditions given by
ai2 − bi2 = h2 ,
and tai − tbi = tci − tdi = u,
for i = 1, · · · , m and for the given first spatial lag h2 and time lag u. In order to apply to traditional
two-way ANOVA procedure, we rewrite ψbi⋆ (ωj ) as follows:
ψbi⋆ (ωj ) = ψi⋆ (ωj ) + ei (ωj ),
(26)
where
ψi⋆ (ωj )
= ψ ∆1
"
#1/2
−2
i
a
i
b
Q∆ ωj ; (a2 , ti ; b2 , ti ) − 1
,
1
ωj ; (ai2 , tai ; bi2 , tbi )
E{ei (ωj )} = 0 and Var{ei (ωj )} = σe2 , asymptotically, and Cov{ei (ωj ), ek (ωl )} = 0, ∀i, j, k, l approximately. We also express (26) as
ψbi⋆ (ωj ) = γi + δj + ei (ωj ),
(27)
where the parameters {γi } and {δj } are “Space-Time Interaction” effect and “Spatial Frequency”
effect, respectively. Since, under the stationarity in space-time, the covariance does not rely on their
relative postion, it is quite reasonable that “Space-Time Interaction” effect, {γi } is not significant.
If axial symmetry in space is in a spatial-temporal process, the phase in (25) is zero, which means
that “Spatial Frequency” effect is also zero. Therefore, we can detect lack of stationarity in spacetime as well as lack of axial symmetry in space from the two main effects in the classical two-way
ANOVA model.
16
In this section, we defined two types of symmetry inherent in spatial-temporal processes and
developed the formal tests, which are based on some useful functions in spectral-domain analysis.
One of the advantages of our methods is that the classical ANOVA model is easily employed and,
therefore, the interpretation can be more persuadable.
4
Simulation Study
In Section 3, we proposed new formal tests for lack of axial symmetry in time and for lack of axial
symmetry in space in spatial-temporal processes. In this section, we evaluate the performance
of these tests by simulation study where the underlying covariance is an asymmetric exponential
stationary spatial-temporal one. Now we introduce the simulation steps for checking the behaviours
of the new tests. Here is the instruction for testing lack of axial symmetry in time.
1) Choose m pairs of sites which are far from each other by the given spatial lags, h = (h1 , h2 )′ .
Keep the within-pair distance (khk) much smaller than gρ (s) in C.2 or the effective range, but
the between-pair distance greater than or equal to them in order to maintain the cross-spectral
densities of each pair asymptotically independent.
2) Compute the test statistic shown in (17) for each pair.
3) Apply this statistic to the traditional ANOVA procedure by considering “Temporal Frequency” effect and “Location” effect.
4) Repeat 1) through 3) under the different directions to search for the specific directions causing
lack of axial symmetry in time.
In case of axial symmetry in space, one big difference from the previous case occurs in step 1).
1) Find m pairs of points which are far from each other as specified by the second spatial lag
17
(latitudinal lag) and the temporal lag, (h2 , u)′ . Keep the between-pair distance larger than
the effective ranges for space and for time.
2) Compute the test statistic proposed in (25) for each pair.
3) Apply this statistic to the traditional ANOVA procedure by considering “Spatial (Longitudinal) Frequency” effect and “Space-Time Interaction” effect.
4) Repeat 1) through 3) under the different directions to search for the specific directions causing
lack of axial symmetry in space.
For the simplification of the simulation setup, we consider the spatial bandwidth gρ (0), that is, we
only focus on the cross spectral density functions at the selected pairs.
Before presenting the simulation study, we briefly explain the asymmetric spatial-temporal
stationary covariance given by
n p
o
C(h; u) = σ1 exp − β 2 (u − h′ v)2 + α2 khk2 + σ0 I(khk = u = 0),
(28)
where σ0 is the nugget, σ1 is the partial sill, and α and β are the decaying rates of spatial correlation
and of temporal correlation. Here, the asymmetry parameter vector, v ≡ (v1 , v2 )′ ∈ R2 controls
(lack of) symmetry realized in spatial-temporal processes. For example, v = 0 yields the covariance
satisfying axial symmetry in time. If only one element in v is zero, then axial symmetry in space
is satisfied. We call asymmetry in space and time, otherwise.
Now we explain the fundamental simulation setup for realizing the tests. The number of iterations is set to 100 and, at each iteration, the observations are generated from the multivariate
normal distribution with the mean 0 and the variance-covariance matrix in (28). The covariance
parameters are preassigned as follows; σ0 = 0.01, σ1 = 1, α = 0.02, β = 0.75. Here the spacing
unit for space is 10 and the unit for time is 1.
18
0
10
20
30
40
50
60
60
50
0
10
20
30
40
50
40
30
20
10
0
0
10
20
30
40
50
60
Testing Lack of Axial Symmetry in Time
60
4.1
0
10
30
40
50
60
20
30
40
(d) NNE - SSW
50
60
20
30
40
50
60
50
60
60
0
10
20
30
40
50
60
50
30
20
10
0
10
10
(c) ENE - WSW
40
50
40
30
20
10
0
0
0
(b) East - West
60
(a) ESE - WNW
20
0
10
20
30
40
(e) North - South
50
60
0
10
20
30
40
(f) NNW - SSE
Figure 1: The Six Different Ways of Choosing Pairs of Two Sites for Testing Lack of Axial Symmetry
in Time. Note that the x-axis is the easting and the y-axis is the northing.
For the test for lack of axial symmetry in time, we consider the spatial domain with 16 pairs of
two sites as shown in Figure 1 and we generate 51 observations over time at each selected site.
The within-pair distance is set to 20 for North-South and East-West directions and 10 ×
√ 5 for
the other directions. We also set the between-pair distance greater than or equal to the spatial
effective range, 3/α = 150 (15 spacing units). The temporal frequencies, {τj } are selected as follows;
τj = πj/25 with j = 3 (5) 23. Then we construct the test statistic for lack of axial symmetry in
time, φb∗ai bi (τj ) in (19) at the following temporal frequencies; 3π/25, 8π/25, 13π/25, · · · , 23π/25.
19
As can be seen in Figure 1, we also consider the six different directions determined by the pairs on
−0.05
−0.025
0
0.025
0.05
0.05
−0.05
−0.025
0
0.025
0.05
0.025
0
−0.025
−0.05
−0.05
−0.025
0
0.025
0.05
the spatial domain.
−0.05
0
0.025
0.05
0
0.025
(d) NNE - SSW
0.05
0
0.025
0.05
0.05
−0.05
−0.025
0
0.025
0.05
0.025
−0.025
−0.05
−0.025
−0.025
(c) ENE - WSW
0
0.025
0
−0.025
−0.05
−0.05
−0.05
(b) East - West
0.05
(a) ESE - WNW
−0.025
−0.05
−0.025
0
0.025
(e) North - South
0.05
−0.05
−0.025
0
0.025
0.05
(f) NNW - SSE
Figure 2: The Contour Plots of Empirical Power of “Temporal Frequency” Effect Under Asymmetry
in Space and Time. Note that the null hypothesis is located on the origin (v = 0).
Figure 2 displays the contour plots of empirical powers for ”Temporal Frequency” effect under
asymmetry in space and time (v 6= 0) for each direction shown in Figure 1. From Figure 2, we
can see that the direction of pair is directly related to the detectability of this test for lack of axial
symmetry in time, for example, in case of ESE direction, the empirical power increases as v1 and
20
v2 change in the ESE-WSW direction (see Figure 1(a), Figure 2(a)) and lack of axial symmetry
is not detected well when v1 and v2 are along the direction which is exactly perpendicular to the
direction of pair. So, it is necessary to test lack of axial symmetry in time for several directions.
0
0
2
5
4
10
6
15
8
20
10
25
12
The empirical powers of “Location” effect are inside the range from 0.02 to 0.11 (Figure 3(a)). This
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.00
(a) “Location” effect
0.05
0.10
0.15
0.20
0.25
(b) empirical rejection probability
Figure 3: The Histograms of Empirical Power of “Location” Effect and empirical probability of
rejecting the normality assumption by Pearson’s χ2 Normality Test of the residuals. Note that
these histograms are based on all the directions combined.
result is quite reasonable in that the covariance in (28) is (second-order) stationary in space as well
as in time. So we can conclude that there does not exist any apparent evidence against stationarity
in space. Pearson’s χ2 test was employed to check the normality condition for the residuals. The
empirical probabilities of rejecting the normality assumption are inside the range from 0.05 to 0.22
(Figure 3(b)), which is quite bigger than expected, but we don’t think that this affects the validity
of this testing method seriously.
21
4.2
Testing Lack of Axial Symmetry in Space
What we have to consider next is to check whether lack of axial symmetry in space exists in the
spatial-temporal process or not. For testing lack of axial symmetry in space, we consider 16 pairs
of two points where the position of each point is represented by a spatial index and a time index
(Figure 4). The two points are apart from each other by 2 spacing units in latitude and by 1 unit
in time. The spatial frequencies, ωj are selected as follows; ωj = πj/30 with j = 2 (9) 29. Then
the test statistic, ψbi⋆ (ωj ) in (26) is computed at the following spatial frequencies; 2π/30, 11π/30,
20π/30, and 29π/30. After then, Two-way ANOVA model is employed for each direction shown in
50
40
30
20
10
0
0
10
20
30
40
50
Figure 4.
0
10
20
30
40
50
60
0
(a) D.1
10
20
30
40
50
60
(b) D.2
Figure 4: The two different ways of choosing pairs of two points for testing lack of axial symmetry
in space. Note that the x-axis is the northing and the y-axis is the time.
Now we explain the results obtained from the ANOVA approach including the two main effects;
“Spatial Frequency” and “Space-Time Interaction”. Figure 5 displays the empirical power of “Spatial (Longitudinal) Frequency” effect under asymmetry in space and time with v 6= 0. As one can
22
0.1
−0.1
−0.05
0
0.05
0.1
0.05
0
−0.05
−0.1
−0.1
−0.05
0
0.05
0.1
−0.1
(a) D1
−0.05
0
0.05
0.1
(b) D2
Figure 5: The Contour Plots of Empirical Power of “Spatial Frequency” Effect Under Asymmetry
in Space and Time.
see, the null hypothesis is located on the point which is nothing but the solution of u − h′ v = 0
for h and u given, for instance, v = (0, −0.05)′ for D.1 and v = (0, 0.05)′ for D.2. The empirical
power increases as v moves from the corresponding null hypothesis, especially along the dashed
lines, which can be regarded as the actual alternative hypothesis. Figure 6 illustrates the empirical power of “Space-Time Interaction” effect under asymmetry in space and time, and empirical
probability of rejecting the normality assumption by Pearson’s χ2 Normality Test of the residuals.
From Figure 6(a), stationarity in space and in time can be somehow guaranteed. As can be seen in
Figure 6(b), the normality condition is too much rejected, but is not so inappropriate as to affect
the application to the simple ANOVA model.
In this section, we evaluated the performance of new tests for lack of axial symmetry in time
and for lack of axial symmetry in space. By simulation study, we see that lacks of symmetry are
well detected under general asymmetry in space and time, and stationarity in space, or in space
and time is properly maintained under asymmetric stationary covariance structure. In addition
23
12
10
8
15
4
6
10
0
2
5
0
0.02
0.04
0.06
0.08
0.10
0.05
(a) “Space-Time Interaction” effect
0.10
0.15
0.20
(b) empirical rejection probability
Figure 6: The Histograms of Empirical Power of “Space-Time Interaction” Effect and empirical
probability of rejecting the normality assumption by Pearson’s χ2 Normality Test of the residuals.
Note that these histograms are based on all the directions combined.
to the two main effects, we can also consider the direction effect in the ANOVA model, but it is
probably easier to run the model for each direction for the comfortable interpretation.
5
Real Application
In Section 4, we evaluated the performances of the two tests for lack of symmetry proposed in
Section 3. As the results from the simulation study, the proposed testing methods detect the
corresponding lack of symmetry under general asymmetry in space and time. In this section, we
apply the new testing methods to the real air-pollution dataset. Here we consider the daily PM2.5
concentrations which were the averages of hourly values, which are obtained from the Models3/Community Multiscale Air Quality (CMAQ) modeling system with the spatial resolution of
36km × 36km. These data were provided by the U.S. Environmental Protection Agency (EPA).
The spatial domain of our interest is the eastern U.S and the southern Canada, and the time domain
is January 1st through December 29th, 2001. The main reason why we are interested in PM2.5
24
50
1
5
10
15
20
25
30
35
40
45
50
55
60
45
60
55
50
40
40
35
30
35
latitude
45
25
20
30
15
10
5
25
1
−100
−95
−90
−85
−80
−75
−70
longitude
Figure 7: The map of the sites of 3721 (= 61 × 61) centroids of grid cells where each cell is size of
36km × 36km. Note that the numbers on the right of grid cells are row indice and the ones on the
top are column indice.
concentrations is that this air-pollutant is one of the important factors in the public health problem
and, according to many environmenal studies, has complex spatial or spatial-temporal dependency
structure (Zidek (1997) and Golam Kibria et al. (2002)).
Before applying the test for lack of axial symmetry in time, we remove the spatial and the
temporal trends. For a PM2.5 concentration at site s and time t, Z(s, t), we remove the average
over time at each site and the average over space at each time. Then we employ our tests for lack
of symmetry to the PM2.5 anomaly concentrations subtracted by the spatial and temporal trends.
Here the spatial bandwidth gρ (0) is considered for the simplicity for analyzing the data.
5.1
Testing Lack of Axial Symmetry in Time
bab (τ ) by calculating the estimated
We obtain the estimates of phase and coherency, φbab (τ ) and R
c (α) in (11) has a bandwidth of 2πBT with
cross-spectrum in (12) where the spectral window, W
BT = 1/28. In order to make the estimates uncorrelated approximately, we choose the temporal
25
frequencies τj for j = 1, · · · , n = ⌊BT T ⌋ satisfying that the spacings between the τj are at least
π/14, the between-pair distance, (ai , bi ) and (aj , bj ) for i 6= j is set large enough and the within-pair
distance is also set much small enough. Here ⌊a⌋ means the integer nearest to a.
Table 1: Analysis of variance
Degree of
Freedom
Sum of
Squares
Frequencies
sites
Residuals
13
15
180
Frequencies
sites
Residuals
Direction
Item
ESE - WNW
East - West
ENE - WSW
NNE - SSW
North - South
NNW - SSE
F value
Pr(F )
5.31
6.07
17.91
4.11
4.07
< .01
< .01
13
15
180
6.14
3.90
15.57
5.46
3.00
Frequencies
sites
Residuals
13
15
180
6.10
2.99
13.46
6.28
2.67
Frequencies
sites
Residuals
13
15
180
1.96
1.95
12.79
2.13
1.83
Frequencies
sites
Residuals
13
15
180
1.42
3.69
13.58
1.45
3.26
Frequencies
sites
Residuals
13
15
180
2.06
5.29
17.56
1.63
3.62
Pr χ2
0.73
< .01
< .01
0.29
< .01
< .01
0.40
0.01
0.03
0.33
0.14
< .01
< .01
0.08
< .01
0.17
The temporal frequencies, τj are selected as follows; τj = πj/181 with j = 6 (13) 175, where the
uniform spacing of 13π/181 is slightly longer than π/14. We then construct the test statistic, for
lack of axial symmetry in time, φb∗ai bi (τj ) in (19) at the following temporal frequencies; τ1 = 6π/181,
τ2 = 19π/181,· · · , τ13 = 175π/181. We consider the 16 pairs, {ai , bi }, i = 1, · · · , 16 shown in Figure
1. It can be seen, from Figure 1, that between-pair distance is at least 15 spacing units (unit=36km)
and the within-pair distance is set to 2 units for East-West direction and North-South direction,
and
√
5 units for the other directions. We also take into account the effect of the direction of pair.
26
Now we talk about the result of the test for lack of axial symmety in time. Table 1 displays
the output from Two-way ANOVA analysis for checking lack of axial symmetry as well as lack of
stationarity in space for each direction. “Location” effect is significant under 5% significance level
for every direction, which implies that this spatial-temporal process is nonstationary in space. However, “Temporal Frequency” effects are not significant for North-South direciton, and NNW-SSE
direction. This means that C(h; u) 6= C(h; −u) for the other directions and, therefore, covariance (or correlation) between aribitrary two sites with the fixed spatial difference changes as time
changes, especially in the Northeastern-Southwestern direction. Pearson’s χ2 test presents that the
residuals are satisfied with normality assumption for most of the directions.
300
240
180
120
60
0
0
60
120
180
240
300
360
Testing Lack of Axial Symmetry in Space
360
5.2
0
10
20
30
40
50
60
0
(a) D.3
10
20
30
40
50
60
(b) D.4
Figure 8: The two different ways of choosing pairs of two points for testing lack of axial symmetry
in space. Note that the x-axis is the northing and the y-axis is the time.
b ∆ in (25) can be obtained from cross-spectrum
The estimates of phase and coherency, ψb∆1 and Q
1
c (α) in (21) has a bandwidth of 2πBN with BN = 1/12. The
in (22) where the spectral window, W
1
1
27
spatial (longitudinal) frequencies, {ωj } for j = 1, · · · , n = ⌊BN1 N1 ⌋ are chosen by the way that
the spacings between the ωj are at least π/6, and the distance between any pairs, {(ai2 , tai ), (bi2 , tbi )}
and {(aj2 , taj ), (bj2 , tbj )} for i 6= j is set large enough, and the within each pair, {(ai2 , tai ), (bi2 , tbi )}
is set small enough. Here, we consider the following spatial frequencies, {ωj }; ωj = πj/10 with
j = 1 (2) 9, where the uniform spacing of π/5 is slightly longer than π/6. The test statistic for lack
of axial symmetry in space, ψbi⋆ (ωj ) in (26) is constructed at the following temporal frequencies;
ω1 = π/10, ω2 = 3π/10, · · · , ω5 = 9π/10. We consider the 16 pairs, {(ai2 , tai ), (bi2 , tbi )}, i = 1, · · · , 16
shown in Figure 8. The temporal between-pair distance is set to at least 100 units (unit=1 day)
and the spatial between-pair distance is set to more than 15 units (unit=36km). We also take the
two different directions into account.
Table 2: Analysis of variance
Degree of
Freedom
Sum of
Squares
Frequencies
Interactions
Residuals
5
15
180
Frequencies
Interactions
Residuals
5
15
180
Direction
Item
D.3
D.4
F value
Pr(F )
38.83
59.00
185.63
2.51
1.27
0.04
0.25
13.75
83.25
176.00
0.94
1.89
Pr χ2
0.30
0.46
0.04
0.89
Now we explain the output from the ANOVA model for testing lack of axial symmetry in space.
Table 2 shows that (lack of) axial symmetry in time and, even, “Space-Time Interaction” effect are
deeply dependent on how we make the pairs. In case of direction D.3, only “Spatial Frequency”
effect is significant under 5% significance level, that is, C(h1 , h2 ; u) 6= C(−h1 , h2 ; u) for h2 = 72(km)
and u = 1 fixed. This implies that covariance (or correlation) between aribitrary two sites with one
spatial lag and time lag fixed as the other spatial lag changes, especially in direction D.3, and lack
of axial symmetry in space is inherent in this spatial-temporal process. However, when direction
28
D.4 is considered, only “Space-Time Interaction” effect is significant. Since the covariance between
any two measurements depends on their relative position under nonstationarity in space, or in space
and time, this nonzero effect can be one evidence against stationarity in space and time. For both
directions, normality assumption for the residuals is satisfied.
In this section, we applied the formal tests to the real Air-pollution dataset. Based on the results
from the tests for lack of axial symmetry in time (Table 1) and lack of axial symmety in space (Table
2), we finally reach the conclusion that the spatial-temporal process of PM2.5 anomaly concentration has apparent evidences for lack of axial symmetry in time for the Northeastern-Southwestern
direction and for lack of axial symmetry in space. Some of the main factors causing these lacks of
symmetry can be external meteorological conditions, for instance, air pressure, temperature, wind
direction, and so on. These factors tend to make spatial-temporal processes to look moving toward
some direction.
6
Discussion
In this study, we introduced new concepts of symmetry in spatial-temporal processes and proposed
new formal tests for lack of axial symmetry in time and for lack of axial symmetry in space. We
evaluated the performances of the tests by simulation study and the real application. The main
advantage of the tests is that we can easily check not only the existence of lack of symmetry but
also the potential direction causing asymmetry besides the existence of nonstationarity.
As part of our further research, we will be developing a formal test for lack of diagonal symmetry
in space defined by, under stationarity in space,
C(h; u) = C(ḧ; u)
where ḧ = (h1 , · · · , hk−1 , hl , hk+1 , · · · , hl−1 , hk , hl+1 , · · · , hd )′ for k 6= l. This test could also be
29
approached by the spectral representation that we have used in this study.
7
Appendix
We will show the asymptotic normality of φb∗ab (τ ) in (17). Fuentes (2006) provides the asymptotic normality and the approximate independence of the cross-spectral density function, fbab (τ )
evaluated at different frequencies and at different sites. The approximate independence between
fbai bi (τ ) and fbaj bj (λ) is also obtained under either of the conditions, C.1 and C.2. Based on the
information from Fuentes (2005), we try to find the asymptotic distribution of φb∗ab (τ ) in (17).
′
′
b ≡ φbik , R
b ≡ φb∗ , φb∗ , and θ
bik , φbjl , R
bik , where φbik and R
bik
Suppose that φb∗ik ≡ φb∗ai bi (τk ), Θ
ik
jl
are the phase and the coherency at the ith pair at the temporal frequency τk . Then, by Taylor-series
expansion,
b b − θ + op (B −1 T −1 ),
b = Θ + ∂Θ θ
Θ
T
∂θ
′
where Θ = φ∗ik , φ∗jl , and θ = (φik , Rik , φjl , Rik )′ . Under the null hypothesis that φik = 0 and
φjl = 0, we can reexpress the previous equation as
b b − θ 0 + op (B −1 T −1 ),
b = Θ0 + ∂ Θ θ
Θ
T
∂θ 0
where Θ0 = (0, 0)′ , θ 0 = (0, Rik , 0, Rjl )′ , and

1
 [|R |−2 −1]1/2
b
∂Θ
ik

=
∂θ 0 
0
0
0

0
h
1
|Rjl |
−2
i1/2
−1
0 

.

0
Under the assumptions A.1 through A.3, we asymptotically obtain the mean and the variance
30
ΣΘ , which is denoted by
ΣΘ
!′
b
∂Θ
Σθ
= (BT T )
∂θ 0

BT T Var φbik

h
i


|Rik |−2 − 1

=

BT T cov φbjl , φbik


i1/2 h
i1/2
 h
|Rik |−2 − 1
|Rjl |−2 − 1
b
∂Θ
∂θ 0
where Σθ = E
b − θ0
θ
!
b − θ0
θ
BT T cov φbik , φbjl
h
i1/2 h
i1/2
|Rik |−2 − 1
|Rjl |−2 − 1
BT T Var φbjl
h
i
|Rjl |−2 − 1





,




′ . If either of the conditions C.1 and C.2 is satisfied, then
φbai bi (τk ) is approximately independent of φbaj bj (τl ) if and only if fbai bi (τk ) is apprximately inde-
pendent of fbaj bj (τl ). Therefore, we finally compute the following asymptotic variance:
ΣΘ

 π

=

Z
R
W 2 (α) dα [1 − η{2τk }]
0
0
π
Z
R
W 2 (α) dα [1 − η{2τl }]



.

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