Scalable Semiparametric Spatio-temporal Regression for Large Data Analysis

Abstract

With the rapid advances of data acquisition techniques, spatio-temporal data are becoming increasingly abundant in a diverse array of disciplines. Here, we develop spatio-temporal regression methodology for analyzing large amounts of spatially referenced data collected over time, motivated by environmental studies utilizing remotely sensed satellite data. In particular, we specify a semiparametric autoregressive model without the usual Gaussian assumption and devise a computationally scalable procedure that enables the regression analysis of large datasets. We estimate the model parameters by maximum pseudolikelihood and show that the computational complexity can be reduced from cubic to linear of the sample size. Asymptotic properties under suitable regularity conditions are further established that inform the computational procedure to be efficient and scalable. A simulation study is conducted to evaluate the finite-sample properties of the parameter estimation and statistical inference. We illustrate our methodology by a dataset with over 2.96 million observations of annual land surface temperature, and comparison with an existing state-of-the-art approach to spatio-temporal regression highlights the advantages of our method.

Supplementary materials accompanying this paper appear online.

Appendix A: Notation and Assumptions

We first introduce some notation and conventions. Given an $n\times n$ matrix ${\varvec{P}}= (p_{ij})_{n\times n}$, we use $\mathrm{{tr}}({\varvec{P}})$ and $\mathrm{{det}}({\varvec{P}})$ to denote the trace and determinant of a square matrix ${\varvec{P}}$, and we let $vec_D({\varvec{P}})$ denote the column vector formed by the diagonal elements of ${\varvec{P}}$. The (i, j)th element of a matrix ${\varvec{P}}$ is denoted by $\mathrm{{ent}}_{ij}({\varvec{P}})$. We define $\Vert {\varvec{P}}\Vert _{1} = \max _{1\le j\le n}\sum _{i=1}^n|p_{ij}|$ and $\Vert {\varvec{P}}\Vert _{\infty } = \max _{1\le i\le n}\sum _{j=1}^n|p_{ij}|$. We also let $\Vert {\varvec{P}}\Vert _2=\{\lambda _{\max }({\varvec{P}}'{\varvec{P}})\}^{1/2}$ and $\Vert {\varvec{P}}\Vert _F=\{\mathrm{{tr}}({\varvec{P}}'{\varvec{P}})\}^{1/2}$ denote the spectral norm and the Frobenius norm, respectively. Let $abs({\varvec{P}}) = (|p_{i,j}|)_{n\times n}$. A sequence of $n\times n$ matrix ${\varvec{P}}_n$ is said to be uniformly bounded in row and column sums (UB), if $\sup _{n\ge 1}\Vert {\varvec{P}}_n\Vert _{1} <\infty$ and $\sup _{n\ge 1}\Vert {\varvec{P}}_n\Vert _{\infty } <\infty$. We also use ${{\varvec{0}}}$ and ${{\varvec{1}}}$ to denote a matrix or a vector with all elements equal zero and one, respectively. For a real-valued function $f({\varvec{x}})$, ${\varvec{x}}= ({\varvec{X}}_1, \ldots , x_k)'\in {\mathbb {R}}^k$, we let $\nabla f({\varvec{x}})$ denote the gradient vector and let $\nabla ^2 f({\varvec{x}})$ denote the Hessian matrix. The partial derivative of f with respect to $x_j$ is denoted by $\partial _{x_j} f({\varvec{x}})$ or $\frac{\partial f({\varvec{x}})}{\partial x_j}$, whereas the second partial derivative with respect to $x_j$ is denoted as $\partial _{x_jx_j} f({\varvec{x}})$ (or $\frac{\partial ^2 f({\varvec{x}})}{\partial x_j^2}$).

In the following, we provide the regularity conditions for establishing the large sample properties of the PMLE ${\widehat{{\varvec{\delta }}}}$.

A.1 The spatial weight matrix ${\varvec{W}}$ is nonstochastic and symmetric, with zero diagonal elements.

A.2 The parameter space $\mathbf {\Theta }_{{\varvec{\delta }}}$ of ${\varvec{\delta }}= ({\varvec{\beta }}', {\varvec{\theta }}', \sigma ^2)'$ is compact and is the product space of $\mathbf {\Theta }_{{\varvec{\beta }}}$, $\mathbf {\Theta }_{{\varvec{\theta }}}$, and $[{\underline{\sigma }}^2, {\bar{\sigma }}^2]$, where $\mathbf {\Theta }_{{\varvec{\theta }}}$ is a compact set such that the matrix ${\varvec{{\mathcal {I}}}}_N -\lambda {\varvec{W}}$ is nonsingular and the eigenvalues of $A({\varvec{\theta }})$ are less than one in magnitude, while $\mathbf {\Theta }_{{\varvec{\beta }}}$ is a compact subset of ${\mathbb {R}}^k$. The true value ${\varvec{\delta }}_0= ({\varvec{\beta }}_0', {\varvec{\theta }}_0', \sigma _0^2)'$ lies in the interior of $\mathbf {\Theta }_{{\varvec{\delta }}}$.

A.3 The vector of innovations ${\varvec{V}}_t = (v_{1,t}, \ldots , v_{N,t})'$ $\sim iid(0, \sigma _0^2 {\varvec{{\mathcal {I}}}}_{N})$ and $E(|v_{j,t}|^{4+\eta }) <\infty$ for some $\eta >0$ for all j, t.

A.4 The precision matrix, infinite sum of power of ${\varvec{A}}({\varvec{\theta }}_0)$, and the design matrix are UB. Namely,

• (i) ${\varvec{\Sigma }}({\varvec{\theta }})^{-1}={\varvec{B}}({\varvec{\theta }})'({\varvec{\Omega }}({\varvec{\theta }}))^{-1}{\varvec{B}}({\varvec{\theta }})$ and ${\varvec{S}}(\lambda )^{-1}$ are UB, $\forall {\varvec{\theta }}\in \mathbf {\Theta }$.

• (ii) $\sum _{h=1}^{\infty }abs({\varvec{A}}({\varvec{\theta }}_0)^h)$ is UB.

• (iii) The $N\times k$ design matrix ${\varvec{X}}_{t}$ is nonstochastic with elements UB in N and t.

A.5 $\lim _{N\rightarrow \infty } \frac{1}{N} {\varvec{X}}'{\varvec{\Sigma }}({\varvec{\theta }})^{-1}{\varvec{X}}=\lim _{N\rightarrow \infty } \frac{1}{N} {\varvec{X}}'{\varvec{B}}({\varvec{\theta }})'({\varvec{\Omega }}({\varvec{\theta }}))^{-1}{\varvec{B}}({\varvec{\theta }}){\varvec{X}}$ is nonsingular, $\forall {\varvec{\theta }}\in \mathbf {\Theta }$.

A.6 $\liminf _{N \rightarrow \infty } N^{-1}\sum _{j=1}^{NT} \nabla ^2 f_j({\varvec{\alpha }})$ is nonsingular, where $f_j({\varvec{\alpha }})$ $=$ $-\log (\lambda _j({\varvec{\theta }})\sigma ^{-2}\sigma _0^2)$ $+$ $\lambda _j({\varvec{\theta }})\sigma ^{-2}\sigma _0^2$, ${\varvec{\alpha }}= ({\varvec{\theta }}', \sigma ^2)'$, and $\lambda _j({\varvec{\theta }})$, $j=1, \ldots , NT$, are the distinct eigenvalues of ${\varvec{\Sigma }}({\varvec{\theta }})^{-1}{\varvec{\Sigma }}({\varvec{\theta }}_0)$ in nonincreasing order.

A.7 ${\varvec{\Sigma }}({\varvec{\theta }})$, $\partial _{\theta _i} ({\varvec{\Sigma }}({\varvec{\theta }})^{-1})$, $\partial _{\theta _i\theta _j}^2 ({\varvec{\Sigma }}({\varvec{\theta }})^{-1})$, and $\partial _{\theta _i\theta _j\theta _k}^3 ({\varvec{\Sigma }}({\varvec{\theta }})^{-1})$ are UB in ${\varvec{\theta }}= (\theta _1, \theta _2, \theta _3)' \in \mathbf {\Theta }$.

A.8 $\lim _{N\rightarrow \infty }N^{-1}{\varvec{\Omega }}_{N}$ is nonsingular, where

$$\begin{aligned} {\varvec{\Omega }}_{N} =&\left( \begin{array}{cccc} \mathrm{{tr}}( {\varvec{m}}_{\lambda }^2 ) &{} \mathrm{{tr}}( {\varvec{m}}_{\lambda } {\varvec{m}}_{\gamma } ) &{} \mathrm{{tr}}( {\varvec{m}}_{\lambda } {\varvec{m}}_{\rho } ) &{} -\frac{1}{\sigma _0^2} \mathrm{{tr}}({\varvec{m}}_{\lambda }) \\ * &{} \mathrm{{tr}}( {\varvec{m}}_{\gamma }^2 ) &{} \mathrm{{tr}}( {\varvec{m}}_{\gamma } {\varvec{m}}_{\rho } ) &{} -\frac{1}{\sigma _0^2} \mathrm{{tr}}({\varvec{m}}_{\gamma })\\ * &{} * &{} \mathrm{{tr}}( {\varvec{m}}_{\rho }^2 ) &{} -\frac{1}{\sigma _0^2} \mathrm{{tr}}({\varvec{m}}_{\rho }) \\ * &{} * &{} * &{} \frac{NT}{ \sigma _0^4 } \end{array} \right) , \end{aligned}$$

(11)

with ${\varvec{m}}_{\lambda }$, ${\varvec{m}}_{\gamma }$, and ${\varvec{m}}_{\rho }$ defined in (S.14) in Supplementary Materials.