Influence diagnostics in Gaussian spatial–temporal linear models with separable covariance

Abstract

In recent decades, there has been a growing interest in modeling spatial–temporal data, which can be found in many fields including geoscience, meteorology and ecology, among many others. The spatial–temporal dependence structure modeling, using a random field approach, is an indispensable tool to estimate the parameters that define this structure. However, this estimation may be greatly affected by the presence of atypical observations in the sampled data. Our proposal is to extend the results of Uribe-Opazo et al. (J Appl Stat 39:615–630, 2012) and De Bastiani et al. (Test 24:322–340, 2015) in the studies of diagnostic techniques to assess the sensitivity of the maximum likelihood estimators to small perturbations in the response variable for the spatial–temporal linear models with separable covariance. The method’s viability is illustrated in a simulation study, and in an application to eggs anchovy (Engraulis ringens) abundance data in ichthyoplankton surveys from the northern zone of Chile. The results show that the proposed methodology allows to detect influential observations in a spatial-temporal data set when their covariances are separable.

Appendix A: The observed information matrix

The observed information matrix for Gaussian spatial–temporal linear models is obtained from the second derivative of the loglikelihood function (9) and defined by $\mathrm {I(\varvec{\theta })}=\partial ^2{\mathcal {L}}(\varvec{\theta })/\partial \varvec{\theta }\partial \varvec{\theta }^T=-{{\textbf{L}}}(\varvec{\theta })$ evaluated at $\varvec{\theta }=\hat{\varvec{\theta }}$, where ${{\textbf{L}}}(\varvec{\theta })$ is the Hessian matrix given by

$$\begin{aligned} {{\textbf{L}}}(\varvec{\theta }) =-\frac{\partial ^2{\mathcal {L}}(\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^\textrm{T}} =\begin{pmatrix} {{\textbf{L}}}_{\beta \beta } &{} {{\textbf{L}}}_{\beta \phi }\\ {{\textbf{L}}}_{\phi \beta } &{} {{\textbf{L}}}_{\phi \phi } \end{pmatrix}, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} {{\textbf{L}}}_{\beta \beta }&= \frac{\partial ^2}{\partial \varvec{\beta }\partial \varvec{\beta }^\textrm{T}}\big ({{\textbf{X}}}^\textrm{T}\varvec{\Sigma }^{-1}\pmb {\epsilon }\big ) = -{{\textbf{X}}}^\textrm{T}\varvec{\Sigma }^{-1}{{\textbf{X}}},\\ {{\textbf{L}}}_{\beta \phi }&= \big (\mathbf {{L}}_{\beta \phi _1}, {{\textbf{L}}}_{\beta \phi _2}, {{\textbf{L}}}_{\beta \phi _3}\big )^T,\\ {{\textbf{L}}}_{\beta \phi _1}&= -{{\textbf{X}}}^\textrm{T}\varvec{\Sigma }^{-1}\frac{\partial \varvec{\Sigma }}{\partial \phi _1}\varvec{\Sigma }^{-1}\varvec{\epsilon } = -{{\textbf{X}}}^\textrm{T}\varvec{\Sigma }^{-1} \big ( {{\textbf{I}}}_s \otimes \mathbf {R_t}(\phi _3) \big ) \varvec{\Sigma }^{-1}\varvec{\epsilon },\\ {{\textbf{L}}}_{\beta \phi _2}&= -{{\textbf{X}}}^\textrm{T}\varvec{\Sigma }^{-1}\frac{\partial \varvec{\Sigma }}{\partial \phi _2}\varvec{\Sigma }^{-1}\varvec{\epsilon } = -{{\textbf{X}}}^\textrm{T}\varvec{\Sigma }^{-1} \big (\mathbf {R_s}(\tau ) \otimes \mathbf {R_t}(\phi _3) \big ) \varvec{\Sigma }^{-1}\varvec{\epsilon },\\ {{\textbf{L}}}_{\beta \phi _3}&= -{{\textbf{X}}}^\textrm{T}\varvec{\Sigma }^{-1}\frac{\partial \varvec{\Sigma }}{\partial \phi _3}\varvec{\Sigma }^{-1}\varvec{\epsilon } = -{{\textbf{X}}}^\textrm{T}\varvec{\Sigma }^{-1} \left( \big (\phi _1 {{\textbf{I}}}_s + \phi _2\mathbf {R_s}(\tau )\big ) \otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial \phi _3}\right) \varvec{\Sigma }^{-1}\varvec{\epsilon },\\ {{\textbf{L}}}_{\phi \phi }&= \begin{pmatrix} {L}_{\phi _1} &{} {L}_{\phi _1\phi _2} &{} {L}_{\phi _1\phi _3}\\ {L}_{\phi _1\phi _2} &{} {L}_{\phi _2} &{} {L}_{\phi _2\phi _3}\\ {L}_{\phi _1\phi _3} &{} {L}_{\phi _2\phi _3} &{} {L}_{\phi _3},\\ \end{pmatrix}, \end{aligned} \end{aligned}$$

with elements,

$$\begin{aligned} \begin{aligned} L_{\phi _i\phi _j}&= \frac{1}{2}\text {tr}\Bigg [\varvec{\Sigma }^{-1}\left( \frac{\partial \varvec{\Sigma }}{\partial \phi _i} \varvec{\Sigma }^{-1} \frac{\partial \varvec{\Sigma }}{\partial \phi _j} - \frac{\partial ^2\varvec{\Sigma }}{\partial \phi _i\partial \phi _j}\right) \Bigg ]\\&\quad +\frac{1}{2}\varvec{\epsilon }^\textrm{T}\varvec{\Sigma }^{-1} \left( \frac{\partial ^2\varvec{\Sigma }}{\partial \phi _i\partial \phi _j} - \frac{\partial \varvec{\Sigma }}{\partial \phi _i} \varvec{\Sigma }^{-1} \frac{\partial \varvec{\Sigma }}{\partial \phi _j} - \frac{\partial \varvec{\Sigma }}{\partial \phi _j} \varvec{\Sigma }^{-1} \frac{\partial \varvec{\Sigma }}{\partial \phi _i} \right) \varvec{\Sigma }^{-1}\varvec{\epsilon }, \end{aligned} \end{aligned}$$

where $\pmb {\epsilon }={{\textbf{Y}}}-{{\textbf{X}}}\varvec{\beta }$ and considering $\varvec{\Sigma } = \varvec{\Sigma _s} \otimes \varvec{\Sigma _t} = \big (\phi _1 {{\textbf{I}}}_{n_s} + \phi _2 \mathbf {R_s}(\tau )\big ) \otimes \mathbf {R_t}(\phi _3)$ and

$$\begin{aligned}{} & {} \frac{\partial \varvec{\Sigma }}{\partial \phi _1} \\{} & {} \quad = {{\textbf{I}}}_s \otimes \mathbf {R_t}(\phi _3),\,\,\, \frac{\partial \varvec{\Sigma }}{\partial \phi _2} = \mathbf {R_s}(\tau ) \otimes \mathbf {R_t}(\phi _3),\,\,\, \frac{\partial \varvec{\Sigma }}{\partial \phi _3} = \left( \phi _1 {{\textbf{I}}}_s + \phi _2\mathbf {R_s}(\tau )\right) \otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial \phi _3}, \end{aligned}$$

we have,

$$\begin{aligned} \frac{\partial ^2\varvec{\Sigma }}{\partial \phi _i\partial \phi _j}= \begin{pmatrix} \frac{\partial ^2\varvec{\Sigma }}{\partial \phi _1^2} &{} \frac{\partial ^2\varvec{\Sigma }}{\partial \phi _1\partial \phi _2} &{} \frac{\partial ^2\varvec{\Sigma }}{\partial \phi _1\partial \phi _3}\\ \frac{\partial ^2\varvec{\Sigma }}{\partial \phi _1\partial \phi _2} &{} \frac{\partial ^2\varvec{\Sigma }}{\partial \phi _2^2} &{} \frac{\partial ^2\varvec{\Sigma }}{\partial \phi _2\partial \phi _3}\\ \frac{\partial ^2\varvec{\Sigma }}{\partial \phi _1\partial \phi _3} &{} \frac{\partial ^2\varvec{\Sigma }}{\partial \phi _2\partial \phi _3} &{} \frac{\partial ^2\varvec{\Sigma }}{\partial \phi _3^2} \\ \end{pmatrix}, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \frac{\partial ^2\varvec{\Sigma }}{\partial \phi _1^2}&= \frac{\partial ^2\varvec{\Sigma }}{\partial \phi _2^2} = \frac{\partial ^2\varvec{\Sigma }}{\partial \phi _1\partial \phi _2} = {{\textbf{0}}},\\ \frac{\partial ^2\varvec{\Sigma }}{\partial \phi _3^2}&= \frac{\partial ^2}{\partial \phi _3^2}\bigg (\Big (\phi _1 {{\textbf{I}}}_s + \phi _2\mathbf {R_s}(\tau )\Big ) \otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial \phi _3}\bigg )\\&= \Big (\phi _1 {{\textbf{I}}}_s + \phi _2\mathbf {R_s}(\tau )\Big ) \otimes \frac{\partial ^2\mathbf {R_t}(\phi _3)}{\partial \phi _3^2},\\ \frac{\partial ^2\varvec{\Sigma }}{\partial \phi _1\partial \phi _3}&= \frac{\partial ^2}{\partial \phi _1\partial \phi _3}\Bigl ({{\textbf{I}}}_s \otimes \mathbf {R_t}(\phi _3)\Bigr ) = {{\textbf{I}}}_s \otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial \phi _3},\\ \frac{\partial ^2\varvec{\Sigma }}{\partial \phi _2\partial \phi _3}&= \frac{\partial ^2}{\partial \phi _2\partial \phi _3}\Bigl (\mathbf {R_s}(\tau ) \otimes \mathbf {R_t}(\phi _3)\Bigr ) = \mathbf {R_s}(\tau ) \otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial \phi _3}.\\ \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned}{} & {} L_{\phi _1} = \frac{1}{2}\text {tr}\bigg [\varvec{\Sigma }^{-1}\Big ({{\textbf{I}}}_s \otimes \mathbf {R_t}(\phi _3)\Big ) \varvec{\Sigma }^{-1} \Big ({{\textbf{I}}}_s \otimes \mathbf {R_t}(\phi _3)\Big ) \bigg ] \\{} & {} \quad +\frac{1}{2}\varvec{\epsilon }^\textrm{T}\varvec{\Sigma }^{-1} \left( -2\left( {{\textbf{I}}}_s \otimes \mathbf {R_t}(\phi _3)\right) \varvec{\Sigma }^{-1}\left( {{\textbf{I}}}_s \otimes \mathbf {R_t}(\phi _3)\right) \varvec{\Sigma }^{-1}\right) \varvec{\Sigma }^{-1}\varvec{\epsilon },\\{} & {} L_{\phi _2} = \frac{1}{2}\text {tr}\bigg [\varvec{\Sigma }^{-1} \Big (\mathbf {R_s}(\tau ) \otimes \mathbf {R_t}(\phi _3)\Big ) \varvec{\Sigma }^{-1} \Big (\mathbf {R_s}(\tau ) \otimes \mathbf {R_t}(\phi _3)\Big ) \bigg ] \\{} & {} \quad +\frac{1}{2}\varvec{\epsilon }^\textrm{T}\varvec{\Sigma }^{-1} \bigg (-2 \Big (\mathbf {R_s}(\tau ) \otimes \mathbf {R_t}(\phi _3)\Big )\varvec{\Sigma }^{-1}\Big (\mathbf {R_s}(\tau ) \otimes \mathbf {R_t}(\phi _3)\Big ) \bigg )\varvec{\Sigma }^{-1}\varvec{\epsilon },\\{} & {} L_{\phi _3} = \frac{1}{2}\text {tr}\Bigg [\varvec{\Sigma }^{-1}\bigg (\Big (\phi _1 {{\textbf{I}}}_s + \phi _2\mathbf {R_s}(\tau )\Big ) \otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial \phi _3}\bigg )\\{} & {} \quad \varvec{\Sigma }^{-1} \bigg (\Big (\phi _1 {{\textbf{I}}}_s + \phi _2\mathbf {R_s}(\tau )\Big ) \otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial \phi _3}\bigg ) \\{} & {} \quad -\bigg (\Big (\phi _1 {{\textbf{I}}}_s + \phi _2\mathbf {R_s}(\tau )\Big ) \otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial ^2\phi _3^2}\bigg )\Bigg ]\\{} & {} \quad + \frac{1}{2}\varvec{\epsilon }^\textrm{T}\varvec{\Sigma }^{-1} \Bigg (\bigg (\Big (\phi _1 {{\textbf{I}}}_s + \phi _2\mathbf {R_s}(\tau )\Big ) \otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial ^2\phi _3^2}\bigg ) \\{} & {} \quad 2 -\bigg (\Big (\phi _1 {{\textbf{I}}}_s + \phi _2\mathbf {R_s}(\tau )\Big ) \otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial \phi _3}\bigg ) \varvec{\Sigma }^{-1} \bigg (\Big (\phi _1 {{\textbf{I}}}_s + \phi _2\mathbf {R_s}(\tau )\Big ) \otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial \phi _3}\bigg )\Bigg )\varvec{\Sigma }^{-1}\varvec{\epsilon },\\{} & {} L_{\phi _1\phi _3} = \frac{1}{2}\text {tr}\Bigg [\varvec{\Sigma }^{-1}\bigg (\Big ({{\textbf{I}}}_s \otimes \mathbf {R_t}(\phi _3)\Big )\varvec{\Sigma }^{-1}\bigg (\Big (\phi _1 {{\textbf{I}}}_s + \phi _2\mathbf {R_s}(\tau )\Big ) \otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial \phi _3}\bigg ) \\{} & {} \quad -\bigg ({{\textbf{I}}}_s \otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial \phi _3}\bigg ) \Bigg ] + \frac{1}{2}\varvec{\epsilon }^\textrm{T}\varvec{\Sigma }^{-1} \Bigg (\bigg ({{\textbf{I}}}_s \otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial \phi _3}\bigg ) \\{} & {} \quad -\bigg ({{\textbf{I}}}_s \otimes \mathbf {R_t}(\phi _3)\bigg )\varvec{\Sigma }^{-1}\bigg (\Big (\phi _1 {{\textbf{I}}}_s + \phi _2\mathbf {R_s}(\tau )\Big ) \otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial \phi _3}\bigg ) \\{} & {} \quad -\bigg ({{\textbf{I}}}_s \otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial \phi _3}\bigg )\varvec{\Sigma }^{-1}\bigg ({{\textbf{I}}}_s \otimes \mathbf {R_t}(\phi _3)\bigg )\Bigg )\varvec{\Sigma }^{-1}\varvec{\epsilon },\\{} & {} L_{\phi _2\phi _3} = \frac{1}{2}\text {tr}\Bigg [\varvec{\Sigma }^{-1}\bigg ( \Big (\mathbf {R_s}(\tau ) \otimes \mathbf {R_t}(\phi _3)\Big ) \varvec{\Sigma }^{-1} \bigg (\Big (\phi _1 {{\textbf{I}}}_s + \phi _2\mathbf {R_s}(\tau )\Big ) \otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial \phi _3}\bigg ) \\{} & {} \quad -\bigg (\mathbf {R_s}(\tau )\otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial \phi _3}\bigg )\Bigg ] + \frac{1}{2}\varvec{\epsilon }^\textrm{T}\varvec{\Sigma }^{-1} \Bigg ( \bigg (\mathbf {R_s}(\tau )\otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial \phi _3}\bigg ) \\{} & {} \quad -\bigg (\mathbf {R_s}(\tau ) \otimes \mathbf {R_t}(\phi _3)\bigg ) \varvec{\Sigma }^{-1} \bigg (\Big (\phi _1 {{\textbf{I}}}_s + \phi _2\mathbf {R_s}(\tau )\Big ) \otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial \phi _3}\bigg ) \\{} & {} \quad -\bigg (\Big (\phi _1 {{\textbf{I}}}_s + \phi _2\mathbf {R_s}(\tau )\Big ) \otimes \frac{\partial \mathbf {R_t}(\phi _3)}{\partial \phi _3}\bigg ) \varvec{\Sigma }^{-1} \bigg (\mathbf {R_s}(\tau ) \otimes \mathbf {R_t}(\phi _3)\bigg ) \Bigg )\varvec{\Sigma }^{-1}\varvec{\epsilon }, \end{aligned}$$

where $L_{\phi _1\phi _2} = {{\textbf{0}}}$. Considering the Matérn model to describe the temporal variability given in (8), we have that $\partial \mathbf {R_t}(\phi _3)/\partial \phi _3 = \left[ \partial r_{ij}/\partial \phi _3\right]$ and $\partial ^2\mathbf {R_t}(\phi _3)/\partial \phi _3^2 = \left[ \partial ^2 r_{ij}/\partial \phi _3^2\right]$, where for

$$\begin{aligned}{} & {} K_\kappa ^{'}(u) = \partial K_\kappa (u)/\partial u = -(1/2)\left( K_{\kappa -1}(u) + K_{\kappa +1}(u)\right) ,\\{} & {} \frac{\partial r_{ij}}{\partial \phi _3} = -\frac{1}{\phi _3}\left( \kappa r_{ij} + \frac{1}{2^{\kappa -1}\Gamma (\kappa )}\left( \frac{\delta _{ij}}{\phi _3}\right) ^{\kappa +1}K^{'}_\kappa \left( \frac{\delta _{ij}}{\phi _3}\right) \right) , \end{aligned}$$

and

$$\begin{aligned}{} & {} K_\kappa ^{''}(u) = \partial ^2 K_\kappa (u)/\partial u^2 = (1/4)\left( K_{\kappa -2}(u) + 2K_{\kappa }(u) + K_{\kappa +2}(u)\right) ,\\{} & {} \quad \begin{aligned} \frac{\partial ^2 r_{ij}}{\partial \phi _3^2} =&\frac{\kappa (\kappa +1) r_{ij}}{\phi _3^2} + \\ {}&\left( \frac{1}{\phi _3^2 2^{\kappa - 1}\Gamma (\kappa )} \left( \frac{\delta _{ij}}{\phi _3}\right) ^{\kappa +1} \left( 2(\kappa +1) K^{'}_\kappa \left( \frac{\delta _{ij}}{\phi _3}\right) + \left( \frac{\delta _{ij}}{\phi _3}\right) K^{''}_\kappa \left( \frac{\delta _{ij}}{\phi _3}\right) \right) \right) , \end{aligned} \end{aligned}$$

with $i \ne j, i,j=1,\ldots ,n$.