# Spectral Methods in Spatial Statistics

Authors: Kun Chen; Lianmin Zhang; Maolin Pan
Journal: Discrete Dynamics in Nature and Society
 (2014)

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## Abstract

When the spatial location area increases becoming extremely large, it is very difficult, if not possible, to evaluate the covariance matrix determined by the set of location distance even for gridded stationary Gaussian process. To alleviate the numerical challenges, we construct a nonparametric estimator called periodogram of spatial version to represent the sample property in frequency domain, because periodogram requires less computational operation by fast Fourier transform algorithm. Under some regularity conditions on the process, we investigate the asymptotic unbiasedness property of periodogram as estimator of the spectral density function and achieve the convergence rate.


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## Body

## 1. Introduction

Recently, spatial statistics have attracted a number of researches from statisticians, geostatisticians, engineers, and econometricians [1–4]. The covariance clustering phenomena of neighborhood observations have always been the fundamental feature of spatial statistics, which means that closer observations have larger correlation and the correlation decays to zero for far separated ones. Therefore, the location information of spatial sited data plays a key role in constructing the probabilistic or statistical covariance function [5]. Using maximum likelihood estimation for the regularly spaced Gaussian data, the calculation of the covariance matrix inverse or determinant needs O(n3) operations with the sample size n. As n becomes large, it is difficult to implement statistical inference due to computational limitations. Fortunately, spectral method is a powerful tool to analyze this complex spatial structure using fewer calculations with O(nlog⁡n) operations [6, 7].This spectral method is widely applied in time series analysis [8], but little research has been done in spatial analysis. The spectral analysis, also known as harmonic analysis, is mainly based on the study of the spectral density function, which represents the underlying process property analog to covariance function. By spectral representation theorem, a positive definite covariance function in spatial domain is the Fourier transform of spectral density function in frequency domain. Operations performed on one domain have the corresponding operations on the other domain. In spatial domain, we usually use moment estimator for statistical inference. Under some regular assumptions, like ergodic or stationary, the nonparametric moment method provides asymptotically unbiased estimator. Because of the relationship between spatial and frequency domain, we can use the Fourier transform of moment estimator to make inference. The most important estimator in this sense is periodogram, which is asymptotically unbiased estimator of the spectral density function in time series analysis. Because the periodogram is the square of discrete Fourier transform (DFT), which decomposes the original sequence into components on different frequencies, in this case, the fast Fourier transform (FFT) provides a method to compute the DFT taking O(nlog⁡n) operations.However, it is unknown whether the periodogram estimators have asymptotic unbiased property for spatial process. In this paper, based on the spatial version periodogram, we prove that the result still holds in spatial analysis and obtain that the convergence rate is a function of the sampling domain.The periodogram method to study properties of regular stationary time series can be traced to Whittle [9], Ronsenblatt [10], and Priestly [11]. In some cases, the autocorrelation of time series decays very slowly, such that a pair of observations, when even separated far away, are still highly correlated. To overcome this difficulty, Heyde and Gay [12] introduced a smoothed version periodogram to estimate time series with long-range dependence. Geweke and Porter-Hudak [13] employed the log-periodogram as the response variable to construct a regression model for long memory time series.The paper is organized as follows. Section2 presents the basic definitions of periodogram and spectral density of spatial version. In Section 3, we give the main theorems on the asymptotic properties of periodogram. Concluding remarks and further research will be given in Section 4. Throughout the paper, the following notation will be adopted: O(1)(Op(1)) denotes a term (a random variable) that is bounded (in probability); o(1)(op(1)) denotes a term (a random variable) that converges to zero (in probability); and Rd denotes the Euclidean space with dimension d.


## 2. Spectral Domain

We consider a weakly stationary Gaussian process with mean zero in which data are drawn from the regularly spaced grid ond-dimensional Euclidean space, Z={Z(s);s∈Rd}∈R (see Figure 1). Here, we also call Z a random field on Rd. A random field Z is weakly stationary if its mean is constant, and it has finite covariance function C, such that C(h)=Cov(Z(s+h),Z(s)), for any single location h belonging to domain D⊂Rd. Let X(ω) be an orthogonal process on a regular gridded lattice, which is mean zero and has independent increments
(1)E[(X(ω4)-X(ω3))(X(ω2)-X(ω1))]=0,
for ω4>ω3>ω2>ω1, with variance E(|dX(ω)|2)=f(ω)dω, where f(ω) is the spectral density function. By spectral representation theorem [14], the real-valued weakly stationary random field Z(s) with mean 0 has representation
(2)Z(s)=∫Rd‍exp⁡{iω′s}dX(ω).
Then the covariance function can be represented as
(3)Cov(Z(s),Z(s+h))=Cov(Z(0),Z(h))=E[∫R2‍exp⁡⁡{iω′0}dX(ω)∫R2‍exp⁡⁡{iω′h}dX(ω)]=∫Rd‍exp⁡{iω′h}E(|dX(ω)|2)=∫Rd‍exp⁡{iω′h}f(ω)dω.Figure 1
Simulation of regular spaced lattice data.If the covariance function satisfies∫R2‍|C(x)|dx<∞, the spectral density f is the Fourier transform of the covariance function
(4)f(ω)=1(2π)d∫Rd‍exp⁡{-iω′h}C(h)dh.
If we further assume an additional assumption as in Fuentes [7], (a)
∑s∈D‍[1+∥s∥]|C(s)|<∞, where ∥·∥ is the Euclidean norm given by ∥s∥=s12+⋯+sd2, then the spectral density of Z has uniformly bounded first-order derivative. To simplify our discussion, we illustrate the idea by random fields on a two-dimensional grid; that is, Z={Z(s);s=(s1,s2),s1=1,…,n1;s2=1,…,n2}, in what follows.We estimate the spectral density function on the lattice by periodogram of spatial version(5)IN(ω)=1(2π)21n1n2|∑s1=1n1‍∑s2=1n2‍Z(s)exp⁡{-is′ω}|2,
where N=n1n2 is the total sample size; i=-1; and ω=(ω1,ω2) is the spatial frequencies, computed at ω1=2πl1/n1 and ω2=2πl2/n2 for
(6)(l1,l2)={⌊-n1-12⌋,…,n1-⌊n12⌋}×{⌊-n2-12⌋,…,n2-⌊n22⌋}.In time series study, under some regular conditions on the stationary process, the periodogram is asymptotically unbiased estimator of spectral density; that is,In(ω)→f(ω) as sample size n→∞, where the bias is of order O(n-1). However, the periodogram is not a consistent estimator of the spectral density [8].


## 3. Main Results

Analogous to one-dimensional equally spaced time series observations, define the DFT on two-dimensional grid as(7)J(ω1,ω2)=12πn1n2∑s1=1n1‍∑s2=1n2‍Z(s1,s2)exp⁡{-i(ω1s1+ω2s2)}=JR(ω1,ω2)+iJϕ(ω1,ω2),
where JR(ω1,ω2) and Jϕ(ω1,ω2) denote the real and imaginary parts, respectively, and
(8)JR(ω1,ω2)=12πn1n2∑s1=1n1‍∑s2=1n2‍Z(s1,s2)cos⁡(ω1s1+ω2s2),Jϕ(ω1,ω2)=12πn1n2∑s1=1n1‍∑s2=1n2‍Z(s1,s2)sin(ω1s1+ω2s2).Since it is known that(9)E(IN(ω))=E(J(ω))2=E(JR(ω))2+E(Jϕ(ω))2,
the expectation of periodogram is equal to the variance of real and imaginary parts of DFT. To achieve this aim, we first show some asymptotic properties of Dirichlet kernel DN(ω)=sin(Nω/2)/sin(ω/2) in time series. By simple calculation, we have
(10)12πN∫-ππ‍DN(u-ω)DN(v+ω)dω=0,hhhhhhhhhhhhhhhifor0<|u-v|<2π,12πN∫-ππ‍DN(u-ω)DN(u+ω)dω=O(1N),hhhhhhhhhhhhhhhhhhhhhhhhifor0<|u|<π.
Also, Anderson [15] shows that
(11)12πN∫-ππ‍DN(u-ω)DN(u+ω)f(ω)dω=o(1)hhhhhhhhhhhhhhhhhhhhhhhhhfor0<|u|<π.Also, the Fej´er kernel defined as KN(ω)=DN2(ω) holds an equation
(12)12πN∫-ππ‍KN(ω)dω=1.Using the above formula (10), we obtain the next theorem on asymptotic properties of real and imaginary parts of DFT.Theorem 1.
Under Assumption (a), the variance value of real and imaginary parts of DFT is given by(13)E(JR2(ω1,ω2))=A+cos⁡(ω1(n1+1)+ω2(n2+1))B,E(Jϕ2(ω1,ω2))=A-cos⁡(ω1(n1+1)+ω2(n2+1))B,
where
(14)A=12(2π)2N×∬Π2Kn1(ω1-λ1)Kn2(ω2-λ2)f(λ1,λ2)dλ1dλ2,B=12(2π)2N×∬Π2∏j=12Dnj(ωj+λj)Dnj(ωj-λj)hhhhhhhhh×f(λ1,λ2)dλ1dλ2,
where Π2=(-π,π)×(-π,π).Proof.
Consider(15)E(JR2(ω1,ω2))+iEJR(ω1,ω2)Jϕ(ω1,ω2)=EJR2(ω1,ω2)(JR(ω1,ω2)+iJϕ(ω1,ω2))=1(2π)2NE{∑s1=1n1‍∑s2=1n2‍∑t1=1n1‍∑t2=1n2‍Z(s1,s2)Z(t1,t2)hhhhhhhhhhhhhhhhhhhhhhh×cos⁡⁡(ω1s1+ω2s2)hhhhhhhhhhhhhhhhhhhhhhh×cos⁡(ω1t1+ω2t2)∑s1=1n1‍}hhh+i1(2π)2NE{∑s1=1n1‍∑s2=1n2‍∑t1=1n1‍∑t2=1n2‍Z(s1,s2)Z(t1,t2)hhhhhhhhhhhhhhhhhhhhhhhhihhh×sin⁡(ω1s1+ω2s2)hhhhhhhhhhhhhhhhhhhhhhhhihhh×sin(ω1t1+ω2t2)∑t2=1n2‍}=1(2π)2N∑s1,t1‍∑s2,t2‍C(s1-t1,s2-t2)ei(ω1t1+ω2t2)hhhhhhhhhhhhh×cos⁡(ω1s1+ω2s2)=1(2π)2Nhhh×∬Π2‍∑s1,t1‍∑s2,t2‍f(λ1,λ2)ei[(s1-t1)λ1+(s2-t2)λ2]+i(ω1t1+ω2t2)hhhhhhhhhhihhh×cos⁡⁡(ω1s1+ω2s2)dλ1dλ2=1(2π)2Nhhh×∬Π2‍∑s1,s2‍cos⁡⁡(ω1s1+ω2s2)ei(s1λ1+s2λ2)hhhhhhhhhhhh×∑t1,t2‍ei(ω1-λ1)t1+i(ω2-λ2)t2f(λ1,λ2)dλ1dλ2=12(2π)2N∬Π2‍∑s1,s2‍(ei(ω1s1+ω2s2)+e-i(ω1s1+ω2s2))hhhhhhhhhhhhhhih×ei(s1λ1+s2λ2)hhhhhhhhhhhhhhih×ei(n1+1)(ω1-λ1)/2+i(n2+1)(ω2-λ2)/2hhh×sinn1(ω1-λ1)/2sin(ω1-λ1/2)sinn2(ω2-λ2)/2sin(ω2-λ2)/2hhh×f(λ1,λ2)dλ1dλ2=12(2π)2Nhhh×∬Π2‍{×(sinn2(ω2-λ2)/2sin(ω2-λ2)/2)2ei(n1+1)ω1+i(n2+1)ω2hhhhhhhhhh×∏j=12‍sinnj(ωj+λj)/2sin(ωj+λj)/2sinnj(ωj-λj)/2sin(ωj-λj)/2hhhhhhhhhh+(sinn1(ω1-λ1)/2sin(ω1-λ1)/2)2hhhhhhhhhh×(sinn2(ω2-λ2)/2sin(ω2-λ2)/2)2}hhh×f(λ1,λ2)dλ1dλ2=12(2π)2Nhhh×∬Π2‍Kn1(ω1-λ1)Kn2(ω2-λ2)hhhhhhhh×f(λ1,λ2)dλ1dλ2hhhhhhhh+12(2π)2Ncos⁡[(n1+1)ω1+(n2+1)ω2]hhh×∬Π2‍∏j=12‍Dnj(ωj+λj)Dn1(ωj-λj)f(λ1,λ2)dλ1dλ2hhh+i12(2π)2Nsin[(n1+1)ω1+(n2+1)ω2]hhh×∬Π2‍∏j=12‍Dnj(ωj+λj)Dn1(ωj-λj)f(λ1,λ2)dλ1dλ2.
Applying the similar calculation procedure toE(Jϕ2(ω1,ω2))+iEJR(ω1,ω2)Jϕ(ω1,ω2), we can get the result that E(Jϕ2(ω1,ω2))=A-cos⁡(ω1(n1+1)+ω2(n2+1))B.By the conclusion of Theorem1 and formula (9), it follows that
(16)E(IN(ω))=E(JR2(ω))+E(Jϕ2(ω))=1(2π)2N∬Π2‍Kn1(ω1-λ1)Kn2(ω2-λ2)hhhhhhhhhhhhi×f(λ1,λ2)dλ1dλ2.Therefore, the bias of periodogramIN(ω) to spectral density is
(17)b(ω)=E[IN(ω)-f(ω)]=1(2π)2N∬Π2‍Kn1(ω1-λ1)Kn2(ω2-λ2)hhhhhhhhhhhhihi×f(λ1,λ2)dλ1dλ2-f(ω).The next theorem evaluates the bias in (17), such that we can prove the asymptotic unbiasedness property of periodogram for regular lattice random fields.Theorem 2.
Under Assumption(a), for ω∈Π2=(-π,π)×(-π,π), we have
(18)b(ω1,ω2)=min⁡{O(n1-1),O(n2-1)}.Proof.
By formula (17), it follows that
(19)b(ω1,ω2)=1(2π)2n1n2∬-ππKn1(ω1-λ1)Kn2(ω2-λ2)hhhhhhhhhhhhhhhhh×(f(λ1,λ2)-f(ω1,ω2))dλ1dλ2=1(2π)2n1n2∬-ππKn1(ω1-λ1)Kn2(ω2-λ2)hhhhhhhhhhhhh×{12(λ-ω)T∇f(ω)hhhhhhhhhhhhhhhhh+12(λ-ω)T∇2f(ω*)(λ-ω)}dλ1dλ2,
where ∇f(ω)=(fω1,fω2)T and ∇2f(ω*) denote the first- and second-order derivatives of f(ω) at ω and ω*, respectively, where ∥ω*-ω∥<∥λ-ω∥. By the fact that ∥∇2f(ω*)∥<K and by integration range ∥λ-ω∥2=(λ1-ω1)2+(λ2-ω2)2<K, we have
(20)∥b(ω)∥≤K(2π)2n1n2∫-π-ω1π-ω1‍∫-π-ω2π-ω2‍Kn1(λ1)Kn2(λ2)hhh×(fω1λ1+fω2λ2)dλ1dλ2hhh+K(2π)2n1n2∫-π-ω1π-ω1‍∫-π-ω2π-ω2‍Kn1(λ1)Kn2hhhhhhhhhhhhhhhhhhhhhhhhi×(λ2)∥λ∥2dλ1dλ2≤K(2π)2n1n2hhh×{|fω1|∫-π-ω1π-ω1‍Kn1(λ1)λ1dλ1∫-π-ω2π-ω2‍Kn2(λ2)dλ2hhhhhh+|fω2|∫-π-ω2π-ω2‍Kn2(λ2)λ2dλ2∫-π-ω1π-ω1‍Kn1(λ1)dλ1}hhh+O(1n1n2)≤K(2π)n1{∫-π-ω1-η‍+∫-ηη‍+∫ηπ-ω1‍Kn1(λ1)λ1dλ1}hhh+K(2π)n2{∫-π-ω2-η‍+∫-ηη‍+∫ηπ-ω2‍Kn2(λ2)λ2dλ2}hhh+O(1n1n2).
Also,
(21)K(2π)n1∫-ηη‍Kn1(λ1)λ1dλ1+K(2π)n2∫-ηη‍Kn2(λ2)λ2dλ2=0,
because the integrand is odd, and the remaining terms hold as
(22)K(2π)n1{∫-π-ω1-η‍+∫ηπ-ω1‍}Kn1(λ1)λ1dλ1=O(n1-1),K(2π)n2{∫-π-ω2-η‍+∫ηπ-ω2‍}Kn2(λ2)λ2dλ2=O(n2-1),
because KN(u)<1/sin2(u/2), for 0<u<π.The idea of Theorem2 can be directly extended to random fields on d-dimensional regular gridded lattice. In conclusion, the periodogram of spatial version is asymptotically unbiased estimator of the spectral density function, in which the convergence rate is the minimum of sampling dimension, that is, min⁡{n1-1,…,nd-1}.


## 4. Conclusions and Future Research

In this paper, we consider the nonparametric method to estimate the spectral density function. The spectral density function represents the population property and describes the behavior of underlying process. By constructing a moment estimator periodogram in frequency domain, which is a Fourier transform of covariance function in space domain, the spectral density can be estimated to be asymptotically unbiased. The spectral method outperforms traditional spatial method, because it provides significant computational advantages. In addition, we obtain the convergence rate as the minimum of spatial sampling dimension in the lattice. However, the consistency and asymptotic normality has not been established in this paper, which is important in statistical inference, hypothesis testing, and interval forecasts. On the other hand, by using the Whittle likelihood [9], we can approximate the maximum likelihood method [16] in spatial domain by Whittle estimator in frequency domain and thus estimate the specific parameter in the spectral density.However, our asymptotic results are based on increasing domain asymptotics, allowing the fact that, as the sample size increases, the distance increases becoming sufficiently large, and thus the observations are nearly independent. However, if one is concerned to predict the value of a point in a bounded area, it is necessary to consider fixed domain asymptotics rather than increasing domain asymptotics. Specifically, even for regularly sited data, results from maximum likelihood estimate only exist in very limited cases. The difficulty under the fixed domain asymptotics is that there is at least one function of parameters that cannot be consistently estimated even for regularly gridded Gaussian process. This is indeed the case for Gaussian random fields with isotropic exponential covariance functionC(h)=σ2exp⁡(-|h|/α), where σ2 is the sill parameter and α is the range parameter. Ying [17] proved that when d=1, the maximum likelihood estimator of σ2/α is strongly consistent, but not possible for either σ2 or α. Therefore, it needs more researches on the fixed domain asymptotics in spatial statistics.



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