Testing Lack of Symmetry in Spatial-Temporal Processes. Man Sik Park and Montserrat Fuentes 1

Transcription

1 Testing Lack of Symmetry in Spatial-Temporal Processes Man Sik Park and Montserrat Fuentes 1 Institute of Statistics Mimeo Series# 2585 SUMMAY 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 PM 2.5 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), aleigh, NC Tel.: (919) , Fax: (919) , mspark@unity.ncsu.edu. M. Fuentes is an Associate professor in the Statistics Department, NCSU, aleigh, NC , fuentes@stat.ncsu.edu. This research was sponsored by a National Science Foundation grant DMS , and by a US EPA cooperative agreement. Key words: Air pollution; Asymmetry; nonseparability; Spatial-temporal process; Matérn covariance; Spectral density function. 1

2 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 χ 2 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

3 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 = (s 1, s 2,, s d ) D d} denotes a spatial process where s is a spatial site over a fixed domain D and d is a d-dimensional Euclidean space, and the covariance function is defined as C(h θ) cov {Z(s i + h), Z(s i )}, where s i = ( s i 1,, si d) D, h = (h 1,, h d ), 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 h 1 and h 2, C(h 1, h 2 θ) = C( h 1, h 2 θ), and C(h 1, h 2 θ) = C 1 (h 1 θ 1 ) C 2 (h 2 θ 2 ), where C 1 and C 2 are the positive-definite covariances of the corresponding spatial lags, h 1 and h 2, 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 n 1 n 2 rectangular lattice data, their tests are based on the periodogram denoted by I(ω 1, ω 2 ) = 1 (2π) 2 n 1 1 n 2 1 h 1 = n 1 +1 h 2 = n 2 +1 C(h 1, h 2 θ)cos(h 1 ω 1 + h 2 ω 2 ). Lu and Zimmerman (2005) also proposed diagnostic tests of axial symmetry and complete symmetry 3

4 which is defined by C(h 1, h 2 θ) = C( h 1, h 2 θ) = C( h 2, h 1 θ) = C(h 2, h 1 θ), for all h 1 and h 2. 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 epresentation 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 d, 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

5 frequency. If Z(s, t) is a stationary process with the corresponding covariance defined by C(h; u) cov {Z(s i + h, t k + u), Z(s i, t k )}, (1) then we can rewrite the process in the following Fourier-Stieltjes integral (Yaglom (1987)): Z(s, t) = d exp(is ω + iτt) dy (ω, τ), 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 C(h; u) = d exp(ih ω + iτu) G(dω; dτ), (2) where (h; u) = (s i s j ; t k t l ) for s i,s j D and t k, t l [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: g(ω; τ) = and the corresponding covariance function is given by 1 (2π) d+1 exp( ih ω iτu)c(h; u) dh du, (3) d C(h; u) = d exp(ih ω + iτu) g(ω; τ) dω dτ. (4) 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

6 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(s i, t k + u), Z(s i, t k )} = cov{z(s i, t l + u), Z(s i, t l )} 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(s i s j ; u) = C(s i s j ; u), (5) for any temporal lag u 0 and arbitrary four sites (i, j, i, j ) satisfying s i s j = s i s j. Under stationarity in space, (5) is reduced to C(h; u) = C(h; u), (6) where s i = s j + h and s i = s j + 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( h; u), (7) where h = (h 1,, h k 1, h k, h k+1,, h d ) 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

7 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) = If C is integrable, then (3) can be expressed as g(ω; τ) = (2π) (d+1) d exp{ih ω + iuτ}g(ω; τ) dω dτ. d exp{ ih ω iuτ}c(h; u) dh dτ = (2π) d d exp{ ih ω}f(h; τ) dh, 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: f(h; τ) = (2π) 1 where the complex conjugate of f(h; τ), f( h; τ) is represented as f( h; τ) = (2π) 1 = (2π) 1 exp{ iuτ}c(h; u) du = f c ( h; τ), (8) exp{ iuτ}c( h; u) du = (2π) 1 exp{iuτ}c(h; u) du. exp{ iuτ}c(h; u) du Without the stationarity in space, we can write the cross-spectral density function in (8) as f ab (τ) = (2π) 1 exp{ iuτ}cov{z(a, t), Z(b, t + u)} du = fba c (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: f ba (τ) = (2π) 1 exp{ iuτ}c(b a; u) du = (2π) 1 exp{ iuτ}c(b a; u) du = f ab (τ) (10) 7

8 because C(b a; u) = C(a b; u). From (9) and (10), the cross-spectral density function, f ab (τ) is real-valued. We can also show that, under axial symmetry in time, the phase, φ ab (τ) between Z(a; t) and Z(b; t) is represented as follows: { } φ ab (τ) tan 1 Im.fab (τ) = φ ba (τ) = 0, e.f ab (τ) where Im.f and e.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, J a (τ) as T 1 J a (τ) = K t=1 ( ) t Z (a; t)exp{ iτt}, 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 Ŵ(µ) = 1 B T t= ( ) [µ + 2πt] W, (11) where B T is a temporal bandwidth parameter. In the real application, following weight function is B T considered, ( ) 2πs Ŵ = T T 2π (2M + 1) 1, where M = B T T and s M. We can finally estimate the cross-spectral density function between Z(a; t) and Z(b; t) by f ab (τ) = 2π T T 1 Ŵ t=1 ( τ 2πt ) ( ) 2πt Î ab, (12) T T where the sample cross-periodogram Îab(τ) is defined by Here we introduce some assumptions: [ T 1 ( ) ] 1 t Î ab (τ) = 2π K 2 J a (τ)jb c T (τ). t=1 8

9 A.1 W(µ) is real-valued, even and of bounded variation such that, for < µ <, W(µ) dµ = 1 and W(µ) dµ <. A.2 For each h, u C(h; u) <, u= which implies that the temporal covariance is summable, that is, C(h; u) <. A.3 B T T and B T 0 as T. u= Under the assumptions A.1 through A.3, the expected value of the estimated cross-spectral density function, f ab (τ) can be obtained as { } E fab (τ) = 2π T 1 Ŵ T t=1 = where the error term is uniform in τ, and ( τ 2πt ) f ab (τ) T ( 2πt T ) + O(T 1 ) W(µ) f ab (τ B T µ) dµ + O(B 1 T T 1 ), f ab (τ) = q ρ (a s)q ρ (b s)f a+s,b+s (τ) ds, d for < τ <. Here we regard q ρ (s) as a tensor product of d one-dimensional filters, q ρ (s) = d i=1 q(s d), where q is of the form q(s) = 1/ρ I( s ρ/d), 9

10 where I( ) is an indicator function. fab (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 f ai b i (τ) and f aj b j (λ) is expressed by { ( ) lim B TTcov fai b i (τ), f aj b j (λ)} = 2π W 2 (µ) dµ T ( [ η {τ λ} fai a j (τ) f ] [ bi b j (τ) + η {τ + λ} fai b j (τ) f ]) aj b i (τ), (13) where 1 if τ 0(mod 2π) η{τ} = 0 otherwise. Fuentes (2006) says that, if we define the distance between pairs (a i,b i ) and (a j,b j ) 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, fai b i (τ) and f aj b j (λ), are approximately independent if either C.1 τ λ is sufficiently large so that W(µ + τ) 2 W(µ + λ) 2 dµ = 0, i.e. if τ λ bandwidth of W(µ) 2 or C.2 the distance between pairs (a i,b i ) and (a j,b j ) is greater than the bandwidth of q ρ (s). In practice, we can make the covariance in (13) almost zero by having the frequencies τ and λ and the pairs (a i,b i ) and (a j,b j ) 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, φ ab (τ), with mean 0 and covariance defined as lim B TT cov { φab (τ), φ ab (λ)} = π T ( ) W 2 (µ) dµ [η{τ λ} η{τ + λ}] [ ab (τ) 2 1 ], (14) 10

11 where the coherency between between Z(a; t) and Z(b; t), ab (τ) is defined as / ab (τ) = f ab (τ) f aa (τ) f bb (τ). From (14), the asymptotic variance is simply denoted as ( ) lim B TT Var { φab (τ)} = π W 2 (µ) dµ [1 η{2τ}] [ ab (τ) 2 1 ]. (15) T Unfortunately, we can not use the asymptotic result of φ ab (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 φ ab (τ) to φ ab (τ) given by φ ab (τ) = φ ab (τ)/ [ ab (τ) 2 1 ] 1/2. (16) Then, from (15) and (16), we derive the asymptotic normal distribution of φ ab (τ) with mean 0 and variance given by lim B TTVar [ φab (τ)] = [ ab (τ) 2 1 ] 1 lim B TTVar { φab (τ)} T T ( ) = π W 2 (µ) dµ [1 η{2τ}]. However, ab (τ) is unknown in practice, so we newly define φ ab (τ) as φ ab (τ) = φ ab (τ)/ [ ab (τ) 2 1] 1/2. (17) By the Slutsky s theorem, we can obtain the same asymptotic normal distribution of φ ab (τ) in (16) as the one of φ ab (τ) in (17). Based on the assumptions, C.1 and C.2, we implicitly know that, under the null hypothesis H 0 : φ ab (τ) = φ ab(τ)/ [ ab (τ) 2 1 ] 1/2 = 0, φ evaluated at different pairs and different frequencies can be treated independent approximately (see Appendix). With the information of asymptotic distribution of the adjusted phase, φ ab (τ) in (17), we propose a formal test for lack of axial symmetry in time by employing analysis of variance (ANOVA) 11

12 procedure. First we compute φ a i b i (τ j ) at arbitrary two sites, {(a i,b i )} m i=1 and a set of temporal frequencies, {τ j } n j=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, a i b i = h = (h 1, h 2,, h d ). In order to apply to two-way ANOVA procedure, we rewrite φ a i b i (τ j ) as follows: φ a i b i (τ j ) = φ a i b i (τ j ) + ǫ ai b i (τ j ), (18) where φ a i b i (τ j ) = φ ai b i (τ j )/ [ ai b i (τ j ) 2 1 ] 1/2. Here ǫai b i (τ j ) asymptotically has the following assumptions; E{ǫ ai b i (τ j )} = 0, i, j, Var{ǫ ai b i (τ j )} = σ 2 ǫ, i, j, and Cov{ǫ ai b i (τ j ), ǫ ak b k (τ l )} = 0, i, j, k, l satisfying C.1 and C.2. We also express (18) as φ a i b i (τ j ) = α i + β j + ǫ ai b i (τ j ), (19) where the parameters {α i } and {β j } 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, α i is not significant. So, lack of the stationarity in space can be detected by the classical ANOVA technique to test the null hypothesis: φ a i b i (τ j ) = β j + ǫ ai b i (τ j ) against the alternative hypothesis shown in (19). Under axial symmetry in time, the phase is zero. So, Temporal Frequency effect, β j is statistically zero. The classical ANOVA technique is employed to check lack of axial symmetry in time by testing the null hypothesis: φ a i b i (τ j ) = α i + ǫ ai b i (τ j ) 12

13 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 α i = β j = 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( h; u), where h = (h 1,, h k 1, h k, h k+1,, h d ) 0 for k fixed. For the simplification of developing the test, we only consider h = ( h 1, h 2 ) for d = 2. Then we introduce a new version of the cross-spectral density function between Z(a 1, a 2, t) and Z(a 1, a 2 + h 2, t + u), k(ω 1 ; h 2, u) given by k(ω 1 ; h 2, u) = for fixed a 2, h 2, t and u. If C is integrable, then g(ω; τ) = (2π) 3 = (2π) 2 exp{ih 2 ω 2 + iuτ}g(ω; τ) dω 2 dτ, (20) exp{ ih ω iuτ}c(h; u) dh dτ 2 exp{ ih 2 ω 2 iuτ}k(ω 1 ; h 2, u) du dh 2. 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 k(ω 1 ; h 2, u) = (2π) 1 exp{ ih 1 ω 1 }C(h 1, h 2 ; u) dh 1 = k c ( ω 1, h 2 ; u). If a process is axially symmetric in space, that is, C(h 1, h 2 ; u) = C( h 1, h 2 ; u), then k(ω 1 ; h 2, u) = (2π) 1 = (2π) 1 exp{ ih 1 ω 1 }C(h 1, h 2 ; u) dh 1 exp{ ih 1 ω 1 }C( h 1, h 2 ; u) dh 1 = k( ω 1 ; h 2, u). 13

14 So, k is always real-valued and the following result is obtained: { } { } ψ(ω 1 ; h 2, u) = tan 1 Im.k(ω1 ; h 2, u) = tan 1 Im.k( ω1 ; h 2, u) e.k(ω 1 ; h 2, u) e.k( ω 1 ; h 2, u) = ψ( ω 1 ; h 2, u) = 0, where ψ(ω 1 ; h 2, u) is a new version of the phase between Z(a 1, a 2, t) and Z(b 1, a 2 + h 2, t + u) for fixed a 2, h 2, 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(= N 1 N 2 ) sites on regular grids and at the measuring times T, then, for a 2 and t fixed, we can define J 1 (ω 1 ; a 2, t), N 1 1 J 1 (ω 1 ; a 2, t) = 1 n 1 =1 K ( n1 N 1 ) Z ( 1 n 1, a 2 ; t)exp { i 1 n 1 ω 1 }, where 1 is the unit distance of the first spatial coordinate. We also define the sample spectral window Ŵ(µ) by Ŵ(µ) = 1 B N1 j= ( ) [µ + 2πj] W, (21) B N1 where where < µ <. Then k 1 (ω 1 ; h 2, u) is represented by k 1 (ω 1, h 2 ; u) = 2π N 1 1 Ŵ N 1 Î 1 (ω 1 ; h 2, u) = [ n 1 =1 2π N 1 1 n 1 =1 ( ω 1 2πn ) ( ) 1 2πn1 Î 1 ; h 2, u, (22) N 1 N 1 { ( ) } ] 1 K 2 n1 1 N 1 J 1 (ω 1 ; a 2, t)j c 1 (ω 1 ; a 2 + h 2, t + u). Here we introduce some additional assumptions: A.4 for fixed h 2 and u, h 1 C(h 1, h 2 ; u) dh 1 <, 14

15 which also implies that the spatial covariance is summable, that is, C(h 1, h 2 ; u) dh 1 <. A.5 B N1 N 1 and B N1 0 as N 1. Under the assumptions A.1, A.4 and A.5, we can obtain the asymptotic properties of the esimated phase, ψ 1 (ω 1 ; h 2, u) with mean ψ 1 (ω 1 ; h 2, u) and the variance defined as { lim B N 1 N 1 Var ψ 1 (ω 1 ; h 2, u)} N 2 ( ) = π W 2 (µ) dµ [1 η{2ω 1 }] [ Q 1 (ω 1 ; h 2, u) 2 1 ], (23) where a new version of the coherency, Q(ω 1 ; h 2, u) between two arbitrary points in two-dimensional space, (a 2, t) and (a 2 + h 2, t + u), is defined by / Q(ω 1 ; h 2, u) = k(ω 1 ; h 2, u) k(ω 1 ; 0, 0). In general, we can not directly use the asymptotic result of ψ 1 (ω 1 ; h 2, u) in order to make a new test for lack axial symmetry in space because the asymptotic variance in (23) depends on h 2 and u. In order to make the asymptotic variance independent of h 2 and u we transform ψ 1 (ω 1 ; h 2, u) to ψ 1 (ω 1 ; h 2, u) defined by ψ 1 (ω 1 ; h 2, u) = ψ 1 (ω 1 ; h 2, u)/ [ Q 1 (ω 1; h 2, u) 2 1 ] 1/2. (24) In practice, however, Q 1 (ω 1 ; h 2, u) is a unknown parameter, so, by using the estimated coherency, Q 1 (ω 1 ; h 2, u), we newly define ψ 1 (ω 1 ; h 2, u) as ψ 1 (ω 1 ; h 2, u) = ψ 1 (ω 1 ; h 2, u)/ [ Q 1 (ω 1; h 2, u) 2 1] 1/2. (25) If we use an appropriate Q 1 (ω 1 ; h 2, u) as an estimate of Q 1 (ω 1 ; h 2, u), then we can get the same asymptotic distribution of ψ 1 (ω 1 ; h 2, u) in (25) as the one of ψ 1 (ω 1 ; h 2, u) in (24). 15

16 Now we propose a formal test for axial symmetry in space for spatial-temporal processes. For the m pairs, {( a i 2, ta i ; bi 2, m i)} tb, in two-dimensional space consisting of the second spatial and i=1 the temporal coordinates, and a set of first spatial frequencies, {ω j } n j=1, we can get ψ i (ω j) ψ 1 ( ωj ; (a i 2, ta i ; bi 2, tb i )). Arbitrarily, the pairs of two points in two-dimensional space are selected based on the conditions given by a i 2 b i 2 = h 2, and t a i t b i = t c i t d i = u, for i = 1,, m and for the given first spatial lag h 2 and time lag u. In order to apply to traditional two-way ANOVA procedure, we rewrite ψ i (ω j) as follows: ψ i (ω j ) = ψ i (ω j ) + e i (ω j ), (26) where ) ψi (ω j ) = ψ 1 (ω / [ ( ) j ; (a i 2, t a i ; b i 2, t b i) Q 1 ω j ; (a i 2, t a i ; b i 2, t b 2 1/2 i) 1], E{e i (ω j )} = 0 and Var{e i (ω j )} = σ 2 e, asymptotically, and Cov{e i (ω j ), e k (ω l )} = 0, i, j, k, l approximately. We also express (26) as ψ i (ω j ) = γ i + δ j + e i (ω j ), (27) where the parameters {γ i } and {δ j } 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, {γ i } 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

17 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 = (h 1, h 2 ). Keep the within-pair distance ( h ) much smaller than g ρ (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) epeat 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

18 (latitudinal lag) and the temporal lag, (h 2, 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) epeat 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 g ρ (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 C(h; u) = σ 1 exp { } β 2 (u h v) 2 + α 2 h 2 + σ 0 I( h = u = 0), (28) where σ 0 is the nugget, σ 1 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 (v 1, v 2 ) 2 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; σ 0 = 0.01, σ 1 = 1, α = 0.02, β = Here the spacing unit for space is 10 and the unit for time is 1. 18

19 Testing Lack of Axial Symmetry in Time (d) NNE - SSW (c) ENE - WSW (b) East - West 60 (a) ESE - WNW (e) North - South (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

20 As can be seen in Figure 1, we also consider the six different directions determined by the pairs on the spatial domain (a) ESE - WNW (b) East - West (c) ENE - WSW (d) NNE - SSW (e) North - South (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 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 v 1 and 20

21 v 2 change in the ESE-WSW direction (see Figure 1(a), Figure 2(a)) and lack of axial symmetry is not detected well when v 1 and v 2 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. The empirical powers of Location effect are inside the range from 0.02 to 0.11 (Figure 3(a)). This (a) Location effect (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 χ 2 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 χ 2 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

22 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, ω j are selected as follows; ω j = πj/30 with j = 2 (9) 29. Then the test statistic, ψ i (ω j ) 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 Figure (a) D.1 (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 0. As one can 22

23 (a) D1 (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 χ 2 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

24 (a) Space-Time Interaction effect (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 χ 2 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 eal 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 PM 2.5 concentrations which were the averages of hourly values, which are obtained from the Models- 3/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, The main reason why we are interested in PM

25 latitude 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 PM 2.5 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 PM 2.5 anomaly concentrations subtracted by the spatial and temporal trends. Here the spatial bandwidth g ρ (0) is considered for the simplicity for analyzing the data. 5.1 Testing Lack of Axial Symmetry in Time We obtain the estimates of phase and coherency, φ ab (τ) and ab (τ) by calculating the estimated cross-spectrum in (12) where the spectral window, Ŵ(α) in (11) has a bandwidth of 2πB T with B T = 1/28. In order to make the estimates uncorrelated approximately, we choose the temporal 25

26 frequencies τ j for j = 1,, n = B T T satisfying that the spacings between the τ j are at least π/14, the between-pair distance, (a i,b i ) and (a j,b j ) for i 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 Direction ESE - WNW East - West ENE - WSW NNE - SSW North - South NNW - SSE Item Degree of Sum of Freedom Squares F value Pr(F) Pr ( χ 2) Frequencies <.01 sites <.01 esiduals Frequencies <.01 sites <.01 esiduals Frequencies <.01 sites <.01 esiduals Frequencies sites esiduals Frequencies sites <.01 esiduals <.01 Frequencies sites <.01 esiduals The temporal frequencies, τ j are selected as follows; τ j = 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, φ a i b i (τ j ) in (19) at the following temporal frequencies; τ 1 = 6π/181, τ 2 = 19π/181,, τ 13 = 175π/181. We consider the 16 pairs, {a i,b i }, 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

27 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) 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 χ 2 test presents that the residuals are satisfied with normality assumption for most of the directions. 5.2 Testing Lack of Axial Symmetry in Space (a) D.3 (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. The estimates of phase and coherency, ψ 1 and Q 1 in (25) can be obtained from cross-spectrum in (22) where the spectral window, Ŵ(α) in (21) has a bandwidth of 2πB N 1 with B N1 = 1/12. The 27

28 spatial (longitudinal) frequencies, {ω j } for j = 1,, n = B N1 N 1 are chosen by the way that the spacings between the ω j are at least π/6, and the distance between any pairs, {(a i 2, ta i ), (bi 2, tb i )} and {(a j 2, ta j ), (bj 2, tb j )} for i j is set large enough, and the within each pair, {(ai 2, ta i ), (bi 2, tb i )} is set small enough. Here, we consider the following spatial frequencies, {ω j }; ω j = 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, ψ i (ω j ) in (26) is constructed at the following temporal frequencies; ω 1 = π/10, ω 2 = 3π/10,, ω 5 = 9π/10. We consider the 16 pairs, {(a i 2, ta i ), (bi 2, tb i )}, 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 Direction D.3 D.4 Item Degree of Sum of Freedom Squares F value Pr(F) Pr ( χ 2) Frequencies Interactions esiduals Frequencies Interactions esiduals 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(h 1, h 2 ; u) C( h 1, h 2 ; u) for h 2 = 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

29 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 PM 2.5 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 = (h 1,, h k 1, h l, h k+1,, h l 1, h k, h l+1,, h d ) for k l. This test could also be 29

30 approached by the spectral representation that we have used in this study. 7 Appendix We will show the asymptotic normality of φ ab (τ) in (17). Fuentes (2006) provides the asymptotic normality and the approximate independence of the cross-spectral density function, fab (τ) evaluated at different frequencies and at different sites. The approximate independence between f ai b i (τ) and f aj b j (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 φ ab (τ) in (17). Suppose that φ ik φ a i b i (τ k ), Θ ( φ ik, φ jl), and θ ( φik, ik, φ jl, ik ), where φik and ik are the phase and the coherency at the i th pair at the temporal frequency τ k. Then, by Taylor-series expansion, where Θ = Θ = Θ + Θ θ ( θ θ ) + o p (B 1 T T 1 ), ( φ ik, φ jl), and θ = (φik, ik, φ jl, ik ). Under the null hypothesis that φ ik = 0 and φ jl = 0, we can reexpress the previous equation as Θ = Θ 0 + Θ θ 0 ( θ θ0 ) + o p (B 1 T T 1 ), where Θ 0 = (0, 0), θ 0 = (0, ik, 0, jl ), and Θ θ 0 = 1 [ ik 2 1] 1/ jl 2 1 1/2 0. Under the assumptions A.1 through A.3, we asymptotically obtain the mean and the variance 30

31 Σ Θ, which is denoted by ( Σ Θ = (B T T) Θ ( )Σ Θ ) θ θ 0 θ 0 ) B T TVar ( φik B T Tcov ( φik, φ ) jl [ ] ik 2 [ ] 1/2 [ 1 = ik 2 1 jl 2 1 B T Tcov ( φjl, φ ) ) ik B T TVar ( φjl [ ] 1/2 [ ] [ ] 1/2 ik 2 1 jl 2 1 jl 2 1 [ ) ) ] where Σ θ = E ( θ θ0 ( θ θ0. If either of the conditions C.1 and C.2 is satisfied, then ] 1/2, φ ai b i (τ k ) is approximately independent of φ aj b j (τ l ) if and only if f ai b i (τ k ) is apprximately independent of f aj b j (τ l ). Therefore, we finally compute the following asymptotic variance: Σ Θ = eferences ( ) π W 2 (α) dα [1 η{2τ k }] 0 ( ) 0 π W 2 (α) dα [1 η{2τ l }]. [1] Cressie, N., Huang, H.-C., Classes of nonseparable, spatio-temporal siteary covariance functions. J. Amer. Statist. Assoc. 94, [2] Fuentes, M., Chen, L., Davis, J., Lackmann, G., Modeling and predicting complex space-time structures and patterns of coastal wind fields. Environmetrics 16, [3] Fuentes, M., Testing for separability of spatial-temporal covariance functions. J. Statist. Plan. Inference 136, [4] Gneiting, T., Nonseparable, siteary covariance functions for space-time data. J. Amer. Statist. Assoc. 97, [5] Guo, J. H., Billard, L., Some inference results for causual autoregressive processes on a plane. J. Time Ser. Anal. 19,

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