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Locally stationary spatio-temporal processes

  • Yasumasa Matsuda
  • Yoshihiro Yajima
Perspectives on data science for advanced statistics
  • 294 Downloads

Abstract

This paper proposes a locally stationary spatio-temporal process to analyze the motivating example of US precipitation data, which is a huge data set composed of monthly observations of precipitation on thousands of monitoring points scattered irregularly all over US continent. Allowing the parameters of continuous autoregressive and moving average (CARMA) random fields by Brockwell and Matsuda (J R Stat Soc Ser B 79(3):833–857, 2017) to be dependent spatially, we generalize locally stationary time series by Dahlhaus (Ann Stat 25:1–37, 1997) to spatio-temporal processes that are locally stationary in space. We develop Whittle likelihood estimation for the spatially dependent parameters and derive the asymptotic properties rigorously. We demonstrate that the spatio-temporal models actually work to account for nonstationary spatial covariance structures in US precipitation data.

Keywords

CARMA kernel Compound Poisson Locally stationary process Seasonal AR model Spatially dependent spectral density function Spatial nonstationarity Whittle likelihood estimation 

Introduction

Figure 1 shows the locations of monitoring stations scattered all over US continent on which monthly precipitation has been observed since 1895. The huge spatio-temporal data set of US precipitation is the motivating example in this paper to let us consider nonstationary spatio-temporal models. US precipitation data have the following features: First, thousands of monitoring points are scattered irregularly over US continent, while temporal observations are sampled in usual discrete time points. Second, the space time covariance is obviously nonstationary. More precisely, the covariance depends on space, although it may not critically on time. Third, data size of US precipitation is huge, namely, more than one hundred thousands even for 3 year period. Spatio-temporal models that account for the features are required for the analysis of US precipitation data.
Fig. 1

Locations of weather stations in US continent, on which monthly precipitation has been recorded since 1895

Continuous autoregressive and moving average (CARMA) random fields, which were proposed by Brockwell and Matsuda (2017) as stationary spatial model defined on \({\mathbf {R}}^d,d\ge 2\), shall be extended for the motivating example. Extensions to spatio-temporal random fields with stationary temporal and nonstationary spatial covariances are to be tried to describe spatially dependent behaviors in US precipitation data. Stationary temporal extension can be done easily by discrete ARMA time series models, while nonstationary spatial extension requires some careful considerations.

Nonstationary spatial models have been attracting great interests in spatial statistics areas, since it is usual to find nonstationary features in environmental data whose covariances depend not only on lags but also on locations (Sampson 2010). Kernel-based methods by Fuentes (2001), basis function approach by Nychka et al. (2002), convolution models by Higdon Higdon (1998), and spatial deformation methods by Guttorp and Sampson (1994) are the typical studies proposing nonstationary spatial models. Although all the approaches work well to express nonstationary spatial covariances in theoretically sophisticated ways, they have often difficulties in conducting estimation and kriging for huge spatial data sets, which are often the case recently because of rapid progress of data collecting technology such as remote sensing data by satellites. US precipitation is a typical case of huge spatio-temporal data set that requires nonstationary spatial covariance models.

Locally stationary processes, proposed by Dahlhaus (1997), are nonstationary time series by allowing parameters to be dependent on time. Dahlhaus (1997) succeeded in estimating the time dependence of parameters by a frequency domain-based method and derived the asymptotic properties rigorously. His essential idea that makes it possible to establish the asymptotic theories is in the expression of the time dependence of parameters \(\theta \), which is denoted as \(\theta (t/T)\) for sample size T. Similar studies in prior to his paper expressed time-dependent parameters as \(\theta (t)\), for which asymptotic arguments were difficult to formulate (Priestley 1971).

Extending locally stationary time series by Dahlhaus (1997) to random fields, we propose locally stationary spatio-temporal processes. CARMA random fields with spatially dependent parameters are special cases of locally stationary spatio-temporal processes with separable covariances given by the product of stationary temporal and locally stationary spatial covariances. Following Dahlhaus (1997) in estimation, we develop Whittle likelihood estimation for spatially dependent parameters in locally stationary spatio-temporal processes. To establish asymptotic theories for the estimation, we need to generalize a asymptotic scheme for time series to that for spatio-temporal data. Extending the so-called mixed asymptotics in spatial data (Stein 1999) to that for spatio-temporal data, in which sample size and sampling region jointly diverge, we derive the asymptotic properties rigorously.

The striking features of locally stationary spatio-temporal CARMA random fields are as follows: First, the parameters are efficiently estimated by minimizing Whittle likelihood which requires no matrix operations. Second, asymptotic theories for Whittle estimations are established under the asymptotic scheme regarded as an extension of mixed asymptotics in spatial statistics literature. Third, kriging and forecasting, which usually require huge matrix inversions for large spatial data set, are conducted with light computational burdens. Applying an approximation to the kriging procedure in Brockwell and Matsuda (2017), we conduct efficient kriging that does not require matrix inversions. Finally, locally stationary CARMA models provide an easy way of simulating spatio-temporal data with spatially nonstationary and temporally stationary covariances. Simulating spatial data with nonstationary covariances are also possible as a part of simulating spatio-temporal data.

We use the following notation. For \(A=(A_1,A_2), s=(s_1,s_2)\), \([0,A]=[0,A_1]\times [0,A_2]\), \(|A|=A_1\times A_2\), \(s/A=\left( s_1/A_1,s_2/A_2\right) \).

Locally stationary random fields

Extension of stationary CARMA random fields

CARMA random fields were introduced by Brockwell and Matsuda (2017) as stationary models over \({\mathbf {R}}^d,d\ge 2\). We shall extend them to spatio-temporal models with spatially nonstationary and temporarily stationary covariances. Consider CARMA random fields driven by a compound Poisson sheet on \({\mathbf {R}}^2\):
$$\begin{aligned} X(s)=\sum _j g(\theta , s-e_j)Z_j,\quad s\in {\mathbf {R}}^2, \end{aligned}$$
(1)
where \(g(\theta ,s)\) is a CARMA kernel with parameters \(\theta \), \(e_j\)s are knot points distributed randomly over \({\mathbf {R}}^2\), and \(Z_j\)s are independent and identical random variables with mean 0 and variances \(\tau ^2\). Here, we normalize the CARMA kernel g(s) to satisfy \(g(0)=1\). Let n(dx) be the number of knot points contained in the region \(dx\in {\mathbf {R}}^2\). Then, we normalize them to satisfy \(E(n(dx))=\mathrm{var}(n(dx)=dx\). The two normalizations are necessary to guarantee the identifiability for \(\tau ^2\).
We shall begin from the stationary temporal extension of CARMA random fields by a discrete ARMA model. Extending iid variables \(Z_j\)s to stationary time series \(Z_{jt}\) by discrete ARMA models, which is defined by the following:
$$\begin{aligned} \phi \left( B\right) Z_{jt}=\psi \left( B\right) \sigma \varepsilon _{jt}, \end{aligned}$$
(2)
where \(\varepsilon _{jt}\)s are mutually independent and identically distributed random variables with mean 0 and variance 1, and \(\phi \) and \(\psi \) are autoregressive and moving average polynomials given by the following:
$$\begin{aligned} \phi \left( B\right)&=1-\phi _1B-\cdots -\phi _pB^p,\nonumber \\ \psi \left( B\right)&=1+\psi _1B+\cdots +\psi _qB^q, \end{aligned}$$
where B is the backward shift operator defined by \(BZ_{jt}=Z_{j,t-1}\), we have temporally extended CARMA random fields expressed by the following:
$$\begin{aligned} X(s,t)=\sum _j g\left( \theta , s-e_j\right) Z_{jt},\quad s\in {\mathbf {R}}^2,\quad t=1,2,\ldots , \end{aligned}$$
which provides separable space time covariances that are stationary both in space and time.
Next, let us try a nonstationary extension. Allowing the parameters \(\theta \), \(\phi ,\psi \), and \(\sigma \) to depend spatially on s, we have the spatially nonstationary model denoted as follows:
$$\begin{aligned} X(s,t)=\sum _j g\left( \theta (s), s-e_j\right) Z_{jt}(s),\quad s\in {\mathbf {R}}^2,\quad t=1,2,\ldots , \end{aligned}$$
(3)
where \(Z_{jt}(s)\) is the stationary ARMA time series generated by (2) with spatially dependent parameters \(\phi (s), \psi (s)\) and \(\sigma (s)\).
Inference for the spatially dependent parameters \(\theta (s),\phi (s),\psi (s),\sigma (s)\) in (3) are inconsistent, since the domains of the parameter diverge as the observation regions [0, A] for s in X(st) tend to be large. Consistent estimation for the spatially dependent parameters requires finer samples over the domain as the sample size tends to be large. Following Dahlhaus (1997), we replace the spatial dependencies for the parameters with the local dependencies defined by \(\theta (s/A),\phi (s/A),\psi (s/A),\sigma (s/A)\), which leads to the expression:
$$\begin{aligned} X_A(s,t)=\sum _j g\left( \theta \left( \frac{s}{A}\right) , s-e_j\right) Z_{jt}\left( \frac{s}{A}\right) ,\quad s\in [0,A],\quad t=1,2,\ldots . \end{aligned}$$
(4)
We call it the locally stationary spatio-temporal CARMA processes in the followings. We shall apply the spatio-temporal model to US precipitation data later in “Empirical studies” to check empirically if it can actually catch the spatially nonstationary behaviors.

Locally stationary spatio-temporal processes

Here, we generalize the locally stationary spatio-temporal CARMA processes in (4) to locally stationary spatio-temporal processes. Dahlhaus (1997) proposed locally stationary processes to express nonstationarity with valid asymptotic theories. Here, we extend the one for nonstationary time series to that for spatio-temporal data. We consider the cases when locally stationary in space but stationary in time that include (4) as a special case.

Definition 1

A spatio-temporal process \(X_A(s,t), s\in [0,A]\subset {\mathbf {R}}^2, t=1,2,\ldots \) is called a temporally stationary and spatially locally stationary process with transfer function K, if there exists a representation:
$$\begin{aligned} X_A(s,t)=\int _\mathbf{{R^2}}\int _{-\pi }^{\pi }K\left( \frac{s}{A},\omega ,\lambda \right) \exp (i\omega 's+i\lambda t)\mathrm{d}\xi (\omega )\mathrm{d}\zeta (\lambda ), \end{aligned}$$
(5)
where \(\xi (\omega )\) and \(\zeta (\lambda )\) are mutually independent stochastic processes on \({\mathbf {R}}^2\) and \([-\pi ,\pi ]\) with \(\overline{\xi (\omega )}=\xi (-\omega )\) and \(\overline{\zeta (\lambda )}=\zeta (-\lambda )\) , respectively, and satisfy
$$\begin{aligned} cum\left( \mathrm{d}\xi (\omega _1),\ldots ,\mathrm{d}\xi (\omega _k)\right)&=\eta \left( \sum _{i=1}^k\omega _{i1}\right) \eta \left( \sum _{i=1}^k\omega _{i2}\right) a_k(\omega _1,\ldots ,\omega _{k-1})\mathrm{d}\omega _1\cdots \mathrm{d}\omega _{k-1},\\ cum\left( \mathrm{d}\zeta (\lambda _1),\ldots ,\mathrm{d}\zeta (\lambda _k)\right)&=\eta \left( \sum _{i=1}^k\lambda _i\right) b_k(\lambda _1,\ldots ,\lambda _{k-1})\mathrm{d}\lambda _1\cdots \mathrm{d}\lambda _{k-1}, \end{aligned}$$
where cum is the cumulant function, \(a_1=b_1=0, a_2=b_2=1\), \(|a_k(\omega _1,\ldots ,\omega _{k-1})|\le \mathrm{const}_k\), and \(|b_k(\lambda _1,\ldots ,\lambda _{k-1})|\le \mathrm{const}_k\) for \(k\ge 3\) and \(\eta (x)=\sum _{j=-\infty }^{\infty } \delta (x+2\pi j)\) for the Dirac delta function \(\delta \).
Let us define spatially dependent spectral density function by, for \(u=s/A\):
$$\begin{aligned} f(u,\omega ,\lambda )&= \lim _{A\rightarrow (\infty ,\infty )}(2\pi )^{-3}\\&\quad\times \int _{{\mathbf {R}}^2}\sum _{k=-\infty }^{\infty }cov(X_A(s+h/2,t),X_A(s-h/2,t-k))\exp (-ih'\omega -ik\lambda )\mathrm{d}h\\&=|K(u,\omega ,\lambda )|^2. \end{aligned}$$
Then, the spatio-temporal CARMA model in (4) is a special case in (5) in the sense that the spatially dependent spectral density is expressed by the following:
$$\begin{aligned} \left| \tilde{g}_{\mathrm{sp}}(u,\omega )\right| ^2\times \left| \tilde{g}_{\mathrm{tmp}}(u,\lambda )\right| ^2, \end{aligned}$$
where
$$\begin{aligned} \tilde{g}_{\mathrm{sp}}(u,\omega )&=\frac{1}{2\pi }\int _{{\mathbf {R}}^2}g(\theta (u),s)\exp (-i\omega 's)\mathrm{d}s,\\ \tilde{g}_{\mathrm{tmp}}(u,\lambda )&=\frac{\sigma (u)}{\sqrt{2\pi }}\frac{\psi (u,\exp (-i\lambda ))}{\phi (u,\exp (-i\lambda ))}. \end{aligned}$$
In other words, the model in (4) is regarded as the separable case when the transfer function is given by the following:
$$\begin{aligned} K(u,\omega ,\lambda )=\tilde{g}_{\mathrm{sp}}(u,\omega )\times \tilde{g}_{\mathrm{tmp}}(u,\lambda ). \end{aligned}$$

Example 1

Consider an example for (4) when CARMA(2,1) is temporally extended by a seasonal AR polynomial \((1-\phi _1(u)B)(1-\phi _2(u)B^{12})\). CARMA(2,1) kernel (see Eq. (31) in Brockwell and Matsuda 2017) is expressed as follows:
$$\begin{aligned} g(u,s)=(1-\theta _3(u))\exp (\theta _1(u)||s||)+\theta _3(u)\exp (\theta _2(u)||s||), \theta _1(u)<\theta _2(u)<0. \end{aligned}$$
Hence, the transfer function in (5) is expressed as the product of
$$\begin{aligned} \tilde{g}_{\mathrm{sp}}(u,\omega )= (1-\theta _3(u))\theta _1(u)\left\{ ||\omega ||^2+\theta _1(u)^2\right\} ^{-3/2}+\theta _3(u)\theta _2(u)\left\{ ||\omega ||^2+\theta _2(u)^2\right\} ^{-3/2} \end{aligned}$$
and
$$\begin{aligned} \tilde{g}_{\mathrm{tmp}}(u,\lambda )=\frac{\sigma (u)}{\sqrt{2\pi }}\frac{1}{(1-\phi _1(u)\exp (-i\lambda ))(1-\phi _2(u)\exp (-12i\lambda ))}. \end{aligned}$$
Finally, we comment on the method to simulate spatio-temporal data with the spectral density proportional to \(|K(\omega,\lambda)|^2\). Simulate bivariate standard normal numbers \(\omega _j\) over \({\mathbf {R}}^2\) and uniform random numbers \(\lambda _k\) over \([-\pi ,\pi ]\). Let \(\phi (\cdot )\) be the bivariate standard normal density function and suppose that the transfer function is expressed by \(K(\cdot )=K_1(\cdot )+iK_2(\cdot )\) for real-valued functions \(K_1,K_2\). Simulate zero-mean iid variables \(\varepsilon _j\) and \(z_k\) with variances \(1/\sqrt{\phi(\omega_j)}\) and one, respectively. Then, generate spatio-temporal data by
$$\begin{aligned} X_A(s,t)=&2\sum _j\sum _k\left\{ K_1(s/A,\omega _j,\lambda _k)\cos (\omega _j's+\lambda _kt) -K_2(s/A,\omega _j,\lambda _k)\sin (\omega _j's+\lambda _kt)\right\} \\&\times \varepsilon _jz_k,\quad s\in {\mathbf {R}}^2,\quad t=1,2,\ldots , \end{aligned}$$
which are simulated spatio-temporal data with the spatially dependent spectral density proportional to \(|K(u,\omega ,\lambda )|^2\).

Estimation of parameters

Whittle likelihood

Suppose that we have observed spatio-temporal data \(X_A(s_p,t), p=1,\ldots ,N,t=1,\ldots ,T\) that follow locally stationary models in (5) with the spatially dependent spectral density function \(f(u,\omega ,\lambda )=|K(u,\omega ,\lambda )|^2\), which is expressed as \(f(\theta _u,\omega ,\lambda )\) with spatially dependent parameters \(\theta _u\). Our aim is to estimate \(\theta _u\) for a fixed \(u\in [0,1]^2\) in a nonparametric way that would not specify any parametric form for the dependence of \(\theta \) on u. In other words, we assume parametric function for the spectral density with parameter \(\theta \) that may depend u, but do not give any parametric form for the functional form of \(\theta (u)\). We assume that all the observation points \(\{s_p\}\subset [0,A]\).

Let \(B=(B_1,B_2)\) and let \(w_{\mathrm{sp}}(x)\) and \(w_{\mathrm{tmp}}(x)\) be tapers defined on \([-1/2,1/2]^2\) and [0, 1], respectively. Then, the local discrete Fourier transform and periodogram for \(u\in [0,1]^2\) is defined by the following:
$$\begin{aligned} d_B(u,\omega ,\lambda )&=\frac{(2\pi )^{-3/2}|A|}{N\sqrt{T|B|}}\sum _{p=1}^N\sum _{t=1}^T X_A(s_p,t)\exp (-i\omega 's_p-i\lambda t)w_{\mathrm{sp}}\left( \frac{s_p-Au}{B}\right) w_{\mathrm{tmp}}\left( \frac{t}{T}\right) ,\\ I_B(u,\omega ,\lambda )&=|d_B(u,\omega ,\lambda )|^2. \end{aligned}$$
Let h(x) be a probability density function over \([0,1]^2\). We assume that \(s_p\)s are independently and identically distributed over [0, A] with the density \(|A|^{-1}h(s/A)\). Under conditions that will be clarified later, we find that \(I_B(u,\omega ,\lambda )\) is biased unlike discrete time series case, and that
$$\begin{aligned} E I_B(u,\omega ,\lambda )\rightarrow C_uf(u,\omega ,\lambda )+\tilde{C}_uk(u,\lambda ), \end{aligned}$$
as \(A=(A_1,A_2)\rightarrow (\infty ,\infty )\), where the second term is the bias term and
$$\begin{aligned} C_u&=h(u)^2\int _{[-1/2,1/2]^2}w_{sp}(x)^2 \mathrm{d}x\int _{[0,1]}w_{\mathrm{tmp}}(x)^2 \mathrm{d}x,\\ \tilde{C}_u&=(2\pi )^{-2}N^{-1}|A|h(u)\int _{[-1/2,1/2]^2}w_{\mathrm{sp}}(x)^2\mathrm{d}x\int _{[0,1]}w_{\mathrm{tmp}}(x)^2\mathrm{d}x,\\ k(u,\lambda )&=\int _{{\mathbf {R}}^2}f(u,\alpha ,\lambda )\mathrm{d}\alpha . \end{aligned}$$
Noticing that \(f(u,\omega ,\lambda )=f(\theta _u,\omega ,\lambda ),k(u,\lambda )=k(\theta _u,\lambda )\), we propose to estimate \(\theta _u\) by minimizing Whittle likelihood function with respect to \(\theta \), which is defined by the following:
$$\begin{aligned}l_w(\theta )=\int _D\int _{-\pi }^{\pi }\left\{ \frac{I_B(u,\omega ,\lambda )}{C_uf(\theta ,\omega ,\lambda )+\tilde{C}_uk(\theta ,\lambda )} +\log (C_uf(\theta ,\omega ,\lambda )+\tilde{C}_uk(\theta ,\lambda ))\right\} \mathrm{d}\omega \mathrm{d}\lambda , \end{aligned}$$
(6)
where D is a compact and symmetric region on \({\mathbf {R}}^2\), such that \(-\omega \in D\) whenever \(\omega \in D\). Regarding \(C_u,\tilde{C}_u\) as nuisance parameters and concentrating out \(C_u\) from the function, we have the concentrated likelihood
$$\begin{aligned} l_c(\theta )=&\log \left\{ \frac{1}{2\pi |D|}\int _{D}\int _{-\pi }^{\pi }\frac{I_B(u,\omega ,\lambda )}{f(\theta ,\omega ,\lambda )+ck(\theta ,\lambda )}\mathrm{d}\omega \mathrm{d}\lambda \right\} \nonumber \\&+\frac{1}{2\pi |D|}\int _{D}\int _{-\pi }^{\pi }\log \left\{ f(\theta ,\omega ,\lambda )+ck(\theta ,\lambda )\right\} \mathrm{d}\omega \mathrm{d}\lambda , \end{aligned}$$
(7)
where \(c=\tilde{C}_u/C_u>0\), the nuisance parameter. Minimizing \(l_c(\theta )\) with respect to \(\theta \) and c for a fixed u, we estimate \(\theta _u\) by \(\hat{\theta }\), which means that the dependencies of \(\theta _u\) on u are estimated in the nonparametric way.

Notice that \(l_c(\theta )\) cannot identify the scale parameter \(\sigma _u^2\) when \(f(u,\omega ,\lambda )\) is given by \(\sigma _u^2f_0(u,\omega ,\lambda )\), as it is seen easily that \(l_c(\theta )\) does not depend on \(\sigma _u^2\). Hence, Whittle estimation proposed here just provides the estimators only for the parameters included in \(f_0(u,\omega ,\lambda )\). In addition, \(l_c\) in which the periodogram is replaced with the modified one multiplied with any constant would provide exactly the same values of the estimators by the same reason. In Example 1, all the parameters except for \(\sigma (u)^2\) are identifiable by the likelihood \(l_c\) and can be estimated by minimizing it.

Remark 1

In practice, the integration in (7) should be replaced with the Riemannian summation. Let \(\omega _j,\lambda _j\) be jth element in the set of Fourier frequency:
$$\begin{aligned} \left\{ (2\pi p_1/A_1,2\pi p_2/A_2), 2\pi q/T|p_1,p_2,q= 0,\pm 1,\pm 2,\ldots \right\} . \end{aligned}$$
Namely, the parameter \(\theta \) is estimated practically by minimizing
$$\begin{aligned} \tilde{l}_c(\theta )=&\log \left\{ M^{-1}\sum _{(\omega _j,\lambda _j)\in D\times [-\pi ,\pi ]} \frac{I_B(u,\omega _j,\lambda _j)}{f(\theta ,\omega _j,\lambda _j)+ck(\theta ,\lambda _j)} \right\} \nonumber \\&+M^{-1}\sum _{(\omega _j,\lambda _j)\in D\times [-\pi ,\pi ]}\log \left\{ f(\theta ,\omega _j,\lambda _j)+ck(\theta ,\lambda _j)\right\} , \end{aligned}$$
with respect to \(\theta \), where M is the cardinality of the Fourier frequencies included in \(D\times [-\pi ,\pi ]\).

Assumptions

One of the advantages of employing locally stationary models is in making it possible to establish the asymptotic properties of the Whittle estimator. The followings are the assumptions required to derive them. The first and second assumptions specify the asymptotic scheme in which the estimator is consistent and asymptotic normal, which is the mixed asymptotics where the sample size and sampling region diverge jointly.
  1. (A1)
    Suppose that \(X_A(s,t)\) follows locally stationary processes in (5) with the spatially dependent spectral density function \(f(u,\omega ,\lambda )=|K(u,\omega ,\lambda )|^2\), and is observed on \((s_p,t),p=1,\ldots ,N,t=1,\ldots ,T,s_p\in [0,A]\). \(s_p,p=1,\ldots ,N\) are written as follows:
    $$\begin{aligned} s_p=(A_1\varepsilon _{p1},A_2\varepsilon _{p2})', \end{aligned}$$
    where \(\varepsilon _p=(\varepsilon _{p1},\varepsilon _{p2})\) is a sequence of independently and identically distributed random vectors with a probability density function h(x) over the compact region \([0,1]^2\).
     
  2. (A2)

    We assume that \(A_j,B_j,j=1,2\), N and T are the functions of k, such that \(A_j=A_j(k),B_j=B_j(k)\rightarrow \infty \), \(N=N_k\rightarrow \infty \) and \(T=T_k\rightarrow \infty \) as \(k\rightarrow \infty \). \(N_k^{-1}|A_k|\rightarrow 0\), \(B_j(k)/A_j(k)\rightarrow 0\), \(\sqrt{T_k|B_k}|B_j(k)^{-2}\rightarrow 0\), \(\sqrt{T_k^{-3}|B_k}|\rightarrow 0\), and \(\sqrt{T_k|B_k|}B_j(k)/A_j(k)\rightarrow 0\) for \(j=1,2\) as \(k\rightarrow \infty \).

     
  3. (A3)

    The spatially dependent spectral density function \(f(u,\omega ,\lambda )\) is an integrable, bounded, and twice partially differentiable function with respect to \(\omega \in {\mathbf {R}}^2\), \(\lambda \in [-\pi ,\pi ]\), and partially differentiable with respect to \(u\in [0,1]^2\).

     
  4. (A4)

    The tapers \(w_{sp}(x),x\in [-.5,.5]^2\) and \(w_{tmp}(x),x\in [0,1]\) are twice partially differentiable functions when they are regarded as functions over \({\mathbf {R}}^2\) and \({\mathbf {R}}\), respectively.

     
  5. (A5)

    We fit, for a fixed \(u\in [0,1]^2\), the parametric spectral density \(f(\theta _u,\omega ,\lambda )\), \(\theta _u\in \Theta \), a compact subset in \({\mathbf {R}}^d\). \(f(\theta _u,\omega ,\lambda )\) is positive on \(\Theta \times D\times [-\pi ,\pi ]\) and twice differentiable with respect to \(\theta _u\) for \((\omega ,\lambda )\in D\times [-\pi ,\pi ]\). \(\theta _1(u)\ne \theta _2(u)\) implies that \(f(\theta _1(u),\omega ,\lambda )\ne f(\theta _2(u),\omega ,\lambda )\) on a subset of \(D\times [-\pi ,\pi ]\) with positive Lebesgue measure. The true parameter denoted by \(\theta _0(u)\) lies in the interior of \(\Theta \), namely \(f(\theta _0(u),\omega ,\lambda )=f(u,\omega ,\lambda )\).

     

Asymptotic results

Consider the asymptotic results under the scheme in (A1) and (A2). Let \(\hat{\theta }_k(u)\) be the estimator minimizing \(l_c(\theta )\) in (7) for a fixed \(u\in [0,1]^2\) under the asymptotic scheme in (A1) and (A2) for \(k=1,2,\ldots \).

Theorem 1

Under Assumptions A1–A5,
  1. 1.

    For a fixed \(u\in [0,1]^2\), such that \(h(u)>0\), \(\hat{\theta }_k(u)\) converges to \(\theta _0(u)\) in probability as \(k\rightarrow \infty \).

     
  2. 2.
    For a fixed \(u\in [0,1]^2\), such that \(h(u)>0\),
    $$\begin{aligned} \sqrt{T_k|B_k|}\left( \hat{\theta }_k(u)-\theta _0(u)\right) \rightarrow N\left( 0,b_w\left( \Gamma _{0u}-\Phi _{0u}\right) ^{-1} (2\Gamma _{0u}+\Delta _{0u}) \left( \Gamma _{0u}-\Phi _{0u}\right) ^{-1} \right) , \end{aligned}$$
    in distribution as \(k\rightarrow \infty \), where \(\Gamma _{0u}=\Gamma (\theta _0(u)), \Phi _{0u}=\Phi (\theta _0(u)), \Delta _{0u}=\Delta (\theta _0(u))\) with
    $$\begin{aligned} b_w&= \left\{ \int \int |w_{\mathrm{sp}}(x)|^4|w_{\mathrm{tmp}}(y)|^4\mathrm{d}x\mathrm{d}y \right\} \left\{ \int \int |w_{\mathrm{sp}}(x)|^2|w_{\mathrm{tmp}}(y)|^2\mathrm{d}x\mathrm{d}y \right\} ^{-2},\\ \Gamma (\theta )&=(2\pi )^{-3}\int _D\int _{-\pi }^{\pi } \left( \frac{\partial \log f(\theta ,\omega ,\lambda )}{\partial \theta } \right) \left( \frac{\partial \log f(\theta ,\omega ,\lambda )}{\partial \theta } \right) ' \mathrm{d}\omega \mathrm{d}\lambda ,\\ \Phi (\theta )&=(2\pi )^{-3}(2\pi |D|)^{-1}\int _D\int _{-\pi }^{\pi } \left( \frac{\partial \log f(\theta ,\omega ,\lambda )}{\partial \theta } \right) \mathrm{d}\omega \mathrm{d}\lambda \int _D\int _{-\pi }^{\pi } \left( \frac{\partial \log f(\theta ,\omega ,\lambda )}{\partial \theta } \right) ' \mathrm{d}\omega \mathrm{d}\lambda ,\\ \Delta (\theta )&=(2\pi )^{-3}\int _D\int _{-\pi }^{\pi }\int _D\int _{-\pi }^{\pi } \left( \frac{\partial \log f(\theta ,\omega _1,\lambda _1)}{\partial \theta } \right) \left( \frac{\partial \log f(\theta ,\omega _2,\lambda _2)}{\partial \theta } \right) '\\&\quad\times a_4(\omega _1,-\omega _1,\omega _2)b_4(\lambda _1,-\lambda _1,\lambda _2) \mathrm{d}\omega _1 \mathrm{d}\lambda _1\mathrm{d}\omega _2 \mathrm{d}\lambda _2. \end{aligned}$$
     

The asymptotic variance is different from the popular one in discrete time series models (Dunsmuir 1979). Precisely, \(\Phi (\theta )\) in the asymptotic variance disappears in the cases of discrete time series models, since the integration of logged spectral density is the constant, i.e., the logged innovation variance (see Theorem 5.8.1 in Brockwell and Davis 1991). It is different also from the one in Matsuda and Yajima 2009, which employs the non-concentrated Whittle likelihood in (6). The non-concentrated likelihood estimator does not include \(\Phi (\theta )\) in the asymptotic variance. Hessian matrices between (6) and (7) correspond in the cases of discrete time series, while they do not in our cases, which is the reason for the difference.

Empirical studies

We apply locally stationary spatio-temporal CARMA models in (4) to US precipitation data, the motivating example for the temporal and nonstationary extensions of CARMA random fields, to check empirical properties of Whittle likelihood estimation and forecasting performances based on the identified model. US precipitation data are monthly precipitation observed at weather stations all over US from 1895 through 1997, which is available in the web page of Institute for Mathematics Applied to Geosciences (IMAG): http://www.image.ucar.edu/Data/US.monthly.met/USmonthlyMet.shtml.

We downloaded monthly total precipitation observed in the weather stations for 48 months period from January, 1994 till December, 1997. They are regarded as a spatio-temporal data, namely monthly observations of spatial data. Precisely, total millimeters of precipitation during the month in the weather station were recorded with the longitude and latitude. See Fig. 1 in “Introduction” for locations of weather stations, which are seen to be irregularly spaced all over US continent. We transformed the longitudes and latitudes to rectangular coordinates with one unit of 100 km to identify the locations of weather stations. All the observations z are transformed by
$$\begin{aligned} y=\log (1+z), \end{aligned}$$
for which our analysis is applied.

We fit the locally stationary spatio-temporal CARMA(2,1) model introduced in Example 1, where \(\theta _3\), the smoothness parameter and \(\phi ,\psi \), the AR parameters, are designed to be dependent spatially. The other two of \(\theta _1\) and \(\theta _2\) were fixed as 3.63 and 0.53 to guarantee the identifiability, which are obtained by minimizing (7) in which the periodogram was modified with the one for the spatial weight \(w_{\mathrm{sp}}=1\), namely by the usual Whittle likelihood estimation. The samples for 36 months from Jan. in 1994 to Dec. in 1996 were used for conducting the estimation minimizing the Whittle likelihood function in (7), where the weight \(w_{\mathrm{sp}}(x)=\exp (-x^2/8^2) and w_{\mathrm{tmp}}(x)=1\) were employed.

Before introducing the estimation results, we shall state the reason why we focus on CARMA(2,1) kernels in the empirical study. First, for general higher order CARMA kernels, the parameters that govern smoothness of covariances have low identifiability for discretely observed data. In other words, fit of continuous models to discrete data usually results in low identifiability. Second, CARMA(2,1) kernels are general enough to cover practical behaviors of covariance functions including CAR(1), which reduces to Ornstein-Uhlenbeck process in one dimension, and express even negative covariances (see Example 2.1 of Brockwell and Matsuda 2017). Finally, model selection criteria such as AIC do not work for CARMA model selection, since Whittle estimators do not have standard asymptotic results that justify the use of the criteria.

The estimated spatial dependence of the CARMA(2,1) parameters, which were smoothed figures of the estimators in 27 mesh points over \([0,1]^2\), are depicted with their standard errors in Figs. 2, 3, and 4.
Fig. 2

Point estimators and standard errors for \(\theta _3(u)\) in the CARMA(2,1) kernel

Fig. 3

Point estimators and standard errors for the autoregressive parameter \(\phi _1(u)\) in the seasonal AR model

Fig. 4

Point estimators and standard errors for the seasonal autoregressive parameter \(\phi _2(u)\) in the seasonal AR model

We find that the estimators caught the spatial dependencies of the three parameters well in the nonparametric way. Figure 2 of the smoothness parameter shows that smoothness of covariances decreases over the range in the Rocky mountains in comparison with that in plain fields, which appeals to our intuitive observations of spiky behaviors of precipitations in mountainous areas. Figures 3 and 4 show that the seasonal coefficient depends spatially in the way of gradually decreasing from the west to east, while the autoregressive coefficient is nearly constant of 0.40. In the east coast area, even negative estimator is found for the seasonal parameter.

Finally, we conduct 1, 2, and 3 months ahead forecasts for the samples from Jan. till Dec. in 1997 by the identified spatio-temporal CARMA(2,1) model based on the samples till Dec. 1996. Table 1 shows the MSEs of the forecasts for precipitation in 100 randomly selected stations from the ones in 1997, in comparison with the two benchmarks given by the averages of precipitation on the previous and same months in the past 3 years. For example, the two benchmark forecasts in March, 1997 are 3 year averages of precipitation in February, 1995–1997 and those of March, 1994–1996.

The forecasts by the identified spatio-temporal CARMA model are constructed as follows. Suppose that we construct one step ahead forecast for \(X_A(w,t+1)\) at a location w by the samples \(X_A(s_j,k), j=1,\ldots ,N,k\le t\). By (4), we have, for \(I_t\) being the information generated by \(X_A(s,k),s\in [0,A],k\le t\):
$$\begin{aligned} E(X_A(w,t+1)|I_{t})&=\sum _jg(w/A,w-e_j)E(Z_{j,t+1}(w/A)|I_{t})\nonumber \\&=\sum _j\phi _1(w/A)g(w/A,w-e_j)Z_{j,t}(w/A)\nonumber \\&\quad +\sum _j\phi _2(w/A)g(w/A,w-e_j)Z_{j,t-11}(w/A)\nonumber \\&=\phi _1(w/A)X_A(w,t)+\phi _2(w/A) X_A(w,t-11). \end{aligned}$$
(8)
When \(Z_{jt}(w/A)\) is not zero mean, the forecast should be modified with
$$\begin{aligned} \mu (w/A)+\phi _1(w/A)(X_A(w,t)-\mu (w/A))+\phi _2(w/A) (X_A(w,t-11)-\mu (w/A)), \end{aligned}$$
where \(\mu (w/A)\) is the temporal mean of \(X_A(w,t)\). \(X_{A}(w,t)\) and \(\mu (w/A)\), when there are no observations at w, should be estimated with
$$\begin{aligned} \hat{X}_A(w,t)=\sum _jc_j(w)X_{A}(s_j,t)\quad { \text{ and } } \quad\hat{\mu }(w/A)=\frac{1}{36}\sum _{k=1}^{36}\hat{X}_A(w,t+1-k), \end{aligned}$$
respectively, where \(c_j(w)=\{\sum _k g(w/A,w-s_k)\}^{-1}g(w/A,w-s_j)\), the identified CARMA kernel normalized to let the total summation to be 1. It is found by (4) that \(\hat{X}_A(w,t)\) is the estimator obtained by replacing the knots \(\{e_j\}\) with \(\{s_j\}\) and \(Z_{jt}(w/A)\) with \(X_A(s_j,t)\), which is an computationally feasible approximation for the exact kriging \(E(X_A(w,t)|X_A(s_j,t),j=1,2,\ldots )\) that can work for huge data set. Multi-step (h-step, say) ahead forecasts for \(X_A(w,t+h)\) are constructed recursively for \(h=2,3,\ldots \) by replacing the unobserved values with the predicted values in the previous step.
We found from Table 1 that the forecast for each month by the spatio-temporal CARMA(2,1) model is not always better than those of the benchmarks in terms of MSE, although it is better on average. There are some months such as Jan. and Aug. when the CARMA forecast works poorly than the benchmark 1, which suggests that the stationary assumption of temporal correlation should be relaxed, namely, the seasonal ARMA parameters may depend on month as well as locations. Spatio-temporal models that are nonstationary not only in space but also in time can improve the fit for US precipitation data.
Table 1

Comparisons of MSEs among the forecasts for US monthly precipitation in 1997 by CARMA(2,1) and two benchmarks evaluated by the samples from Jan. 1994 till Dec. 1996, where 100 stations were randomly chosen for the precipitation to be predicted

Month

MSE

bmrk 1

bmrk 2

Step 1

Step 2

Step 3

Jan.

0.228

0.321

0.331

0.592

0.598

Feb.

0.811

0.698

0.628

0.592

0.529

Mar.

0.615

0.559

0.558

0.557

0.546

Apr.

0.559

0.451

0.532

0.402

0.444

May

0.263

0.346

0.286

0.352

0.321

Jun.

0.296

0.291

0.327

0.391

0.455

Jul.

0.552

0.526

0.513

0.495

0.490

Aug.

0.260

0.449

0.327

0.362

0.361

Sep.

0.408

0.355

0.321

0.322

0.342

Oct.

0.461

0.366

0.350

0.363

0.357

Nov.

0.341

0.317

0.394

0.373

0.366

Dec.

0.468

0.625

0.397

0.471

0.481

Average

0.439

0.442

0.414

0.439

0.441

By the identified CARMA, we conducted the 1, 2, and 3 step forecasts. The benchmarks are the 3 year averages of precipitation on the previous and same months in the past 3 years. For example, the two benchmark forecasts in March, 1997 are 3 year averages of precipitation in February, 1995–1997 and those of March, 1994–1996

Discussion

This paper has proposed locally stationary spatio-temporal processes to describe the empirical properties of US precipitation data, the huge set of spatio-temporal data. Extending stationary CARMA random fields on \({\mathbf {R}}^2\) to spatio-temporal models with spatially nonstationary and temporarily stationary covariances, we have locally stationary spatio-temporal CARMA processes, which are, moreover, generalized to locally stationary spatio-temporal processes. Following Dahlhaus (1997), we estimate the spatially dependent parameter by minimizing Whittle likelihood and derive the asymptotic properties rigorously. Applications to US precipitation data demonstrate that the nonstationary spatial behaviors are accounted well by the locally stationary CARMA(2,1) model.

The critical restriction of spatio-temporal CARMA processes is that the covariances are confined to separable ones given by the products of spatial and temporal covariances. Nonseparable extensions that can express fruitful class of covariance structures are our next target. One more interesting extension is to allow Lévy sheets that drive CARMA random fields to have infinite variances, which makes it possible to express several varieties of spiky behaviors in spatial data. New parameter estimation method over Whittle estimation, which may not work for the infinite variance cases, is required. Their asymptotic properties are important issues that attract empirical as well as mathematical interests.

Sketch of the proof

This section shows the outline of the proof for Theorem 1.

Proof of Theorem 1(a)

Let \(\theta _1(u)\ne \theta _0(u)\) for a fixed \(u\in [0,1]^2\), such that \(h(u)>0\). By Lemmas 3 and 6 in Matsuda and Yajima (2009), we have
$$\begin{aligned} l_c(\theta _1(u))&\rightarrow \log \left\{ \frac{1}{2\pi |D|} \int _D\int _{-\pi }^{\pi } \frac{f(\theta _0(u),\omega ,\lambda )}{f(\theta _1(u),\omega ,\lambda )} \mathrm{d}\omega \mathrm{d}\lambda \right\} \\&\quad +\frac{1}{2\pi |D|}\int _D\int _{-\pi }^{\pi }\log \left\{ f(\theta _1(u),\omega ,\lambda ) \right\} \mathrm{d}\omega \mathrm{d}\lambda \\&\quad +\mathrm{Const}(u) :=l_{\infty }(\theta _1(u)), \end{aligned}$$
in probability as \(k\rightarrow \infty \). By the identifiability condition in (A5) and Jensen’s inequality, we have
$$\begin{aligned} l_{\infty }(\theta _1(u))-l_{\infty }(\theta _0(u))&= \log \left\{ \frac{1}{2\pi |D|} \int _D\int _{-\pi }^{\pi } \frac{f(\theta _0(u),\omega ,\lambda )}{f(\theta _1(u),\omega ,\lambda )} \mathrm{d}\omega \mathrm{d}\lambda \right\} \\&\quad -\frac{1}{2\pi |D|}\int _D\int _{-\pi }^{\pi }\log \left\{ \frac{f(\theta _0(u),\omega ,\lambda )}{f(\theta _1(u),\omega ,\lambda )} \right\} \mathrm{d}\omega \mathrm{d}\lambda \\&>0. \end{aligned}$$
It follows that, for any positive constant \(K(\theta _0(u),\theta _1(u))\) that is less than \(l_{\infty }(\theta _1(u))-l_{\infty }(\theta _0(u))\):
$$\begin{aligned} \lim _{k\rightarrow \infty } P\left\{ l_c(\theta _0(u))-l_c(\theta _1(u))<-K(\theta _0(u),\theta _1(u)) \right\} =1. \end{aligned}$$
For any \(\delta >0\), there is an \(H_{k,\delta }\) of the form:
$$\begin{aligned} \delta \left( C_1\int _D\int _{-\pi }^{\pi }I_B(u,\omega ,\lambda )\mathrm{d}\omega \mathrm{d}\lambda \right) ^{-1}\left\{ C_2\int _D\int _{-\pi }^{\pi }I_B(u,\omega ,\lambda )\mathrm{d}\omega \mathrm{d}\lambda +C_3\right\} \end{aligned}$$
such that, for any \(\theta _1(u)\) and \(\theta _2(u)\) that satisfy \(||\theta _1(u)-\theta _2(u)||<\delta \):
$$\begin{aligned} |l_c(\theta _2(u))-l_c(\theta _1(u))|<H_{k,\delta }, \end{aligned}$$
because, letting \(a(\theta ,\lambda )=bk(\theta ,\lambda )\) and \(\theta ^{*}\) be the mean value between \((\theta _1(u)\) and \(\theta _2(u)\), we have
$$\begin{aligned} |l_c&(\theta _2(u))-l_c(\theta _1(u))|\le \left( \int _D\int _{-\pi }^{^\pi } \frac{I_B(u,\omega ,\lambda )}{f(\theta ^{*},\omega ,\lambda )+a(\theta ^{*},\lambda )}\mathrm{d}\omega \mathrm{d}\lambda \right) ^{-1}\\&\times \int _D\int _{-\pi }^{\pi }I_B(u,\omega ,\lambda )\left| \frac{1}{f(\theta _2(u),\omega ,\lambda )+a(\theta _2(u),\lambda )}-\frac{1}{f(\theta _1(u),\omega ,\lambda )+a(\theta _1(u),\lambda )} \right| \mathrm{d}\omega \mathrm{d}\lambda \\&+\left| \log \{ f(\theta _2(u),\omega ,\lambda )+a(\theta _2(u),\lambda )\}- \log \{ f(\theta _1(u),\omega ,\lambda )+a(\theta _1(u),\lambda )\} \right| . \end{aligned}$$
In addition, it is easily seen from the form of \(H_{k,\delta }\) that there exists a \(\delta >0\), such that
$$\begin{aligned} \lim _{k\rightarrow \infty }P\left( H_{k,\delta }<K(\theta _0,\theta _1) \right) =1. \end{aligned}$$
Applying Lemma 2 of Walker (1964), we have the consistency.

Proof of Theorem 1(b)

By Taylor series expansion
$$\begin{aligned} 0=\sqrt{T_k|B_k|}\frac{\partial l_c(\theta _0(u))}{\partial \theta }+\frac{\partial ^2 l_c(\theta ^{*})}{\partial \theta \partial \theta '} \sqrt{T_k|B_k|}\left( \hat{\theta }_u-\theta _0(u)\right) , \end{aligned}$$
where \(\theta ^{*}\) is the mean value between \(\theta _0(u)\) and \(\hat{\theta }_u\). Hence
$$\begin{aligned} \sqrt{T_k|B_k|}\left( \hat{\theta }_u-\theta _0(u)\right) =\left( -\frac{\partial ^2 l_c(\theta ^{*})}{\partial \theta \partial \theta '}\right) ^{-1}\times \left\{ \sqrt{T_k|B_k|}\frac{\partial l_c(\theta _0(u))}{\partial \theta }\right\} . \end{aligned}$$
Then, by Lemma 7, in Matsuda and Yajima (2009):
$$\begin{aligned}\sqrt{T_k|B_k|}\frac{\partial l_c(\theta _0(u))}{\partial \theta }&=\sqrt{T_k|B_k|}(2\pi |D|)^{-1}\\&\quad \times \int _D\int _{-\pi }^{\pi }C_u^{-1}\left\{ I_B(u,\omega ,\lambda )-EI_B(u,\omega ,\lambda )\right\} \frac{\partial f^{-1}(\theta _0(u),\omega ,\lambda )}{\partial \theta }+o_p(1)\\&\rightarrow N\left\{ 0,(2\pi )^6(2\pi |D|)^{-2}b_w\left( 2\Gamma _{0u}+\Delta _{0u}\right) \right\} . \end{aligned}$$
By the consistency of \(\hat{\theta }_u\)
$$\begin{aligned} -\frac{\partial ^2 l_c(\theta ^{*})}{\partial \theta \partial \theta '} &=-(2\pi |D|)^{-1}\int _D\int _{-\pi }^{\pi }\left\{ C_u^{-1}I_B(u,\omega ,\lambda )\frac{\partial ^2}{\partial \theta \partial \theta '}f^{-1}(\theta ^{*},\omega ,\lambda )\right\} \mathrm{d}\omega \mathrm{d}\lambda \\&\quad +(2\pi |D|)^{-2} \int _D\int _{-\pi }^{\pi } \left\{ C_u^{-1}I_B(u,\omega ,\lambda )\frac{\partial }{\partial \theta }f^{-1}(\theta ^{*},\omega ,\lambda )\right\} \mathrm{d}\omega \mathrm{d}\lambda \\&\quad \times \int _D\int _{-\pi }^{\pi } \left\{ C_u^{-1}I_B(u,\omega ,\lambda )\frac{\partial }{\partial \theta }f^{-1}(\theta ^{*},\omega ,\lambda )\right\} '\mathrm{d}\omega \mathrm{d}\lambda \\&\quad+(2\pi |D|)^{-1}\int _D\int _{-\pi }^{^\pi }\left\{ f(\theta ^{*},\omega ,\lambda )\frac{\partial ^2}{ \partial \theta \partial \theta '}f^{-1}(\theta ^{*},\omega ,\lambda )\right. \\&\quad \left. +\frac{\partial }{\partial \theta } f(\theta ^{*},\omega ,\lambda )\frac{\partial }{\partial \theta '}f^{-1}(\theta ^{*},\omega ,\lambda ) \right\} \mathrm{d}\omega \mathrm{d}\lambda +o_p(1)\\&\rightarrow -(2\pi )^3(2\pi |D|)^{-1}\left( \Gamma _{0u}-(2\pi |D|)^{-1}\Phi _{0u}\right) . \end{aligned}$$

Notes

Acknowledgements

The research was supported by the Grants-in-Aid for Scientific Research, 17H01701, 17H02508.

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Copyright information

© Japanese Federation of Statistical Science Associations 2018

Authors and Affiliations

  1. 1.Graduate School of Economics and ManagementTohoku UniversitySendaiJapan

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