A composite spatial predictor via local criteria under a misspecified model

Abstract

Spatial prediction and variable selection for the study area are both important issues in geostatistics. If spatially varying means exist among different subareas, globally fitting a spatial regression model for observations over the study area may be not suitable. To alleviate deviations from spatial model assumptions, this paper proposes a methodology to locally select variables for each subarea based on a locally empirical conditional Akaike information criterion. In this situation, the global spatial dependence of observations is considered and the local characteristics of each subarea are also identified. It results in a composite spatial predictor which provides a more accurate spatial prediction for the response variables of interest in terms of the mean squared prediction errors. Further, the corresponding prediction variance is also evaluated based on a resampling method. Statistical inferences of the proposed methodology are justified both theoretically and numerically. Finally, an application of a mercury data set for lakes in Maine, USA is analyzed for illustration.

Appendix

Proof of Theorem 1 Taking the expectation on both sides of (14), we have the following results for any $M\in \mathcal {M}$ and $a=1,\dots ,A$.

$$\begin{aligned} E\left[ LCAIC(M;D_a;\varvec{\theta })\right]= & {} E\left\{ \sum _{\{\varvec{s}_i\in D_a\}}\left( Z(\varvec{s}_i)-\hat{S}_M(\varvec{s}_i;\varvec{\theta })\right) ^2+2\sigma _{\varepsilon }^2\sum _{\{i:\varvec{s}_i\in D_a\}} \left[ \varvec{H}_M(\varvec{\theta })\right] _{ii}\right\} \nonumber \\=\, & {} E\left\{ \sum _{\{\varvec{s}_i\in D_a\}}\left( Z(\varvec{s}_i)-\hat{S}_M(\varvec{s}_i;\varvec{\theta })\right) ^2\right\} +2\sigma _{\varepsilon }^2E\left\{ \sum _{\{i:\varvec{s}_i\in D_a\}}\left[ \varvec{H}_M(\varvec{\theta })\right] _{ii}\right\} \nonumber \\=\, & {} E\left\{ \sum _{\{\varvec{s}_i\in D_a\}}\left( Z(\varvec{s}_i)-S(\varvec{s}_i)+S(\varvec{s}_i)-\hat{S}_M(\varvec{s}_i;\varvec{\theta })\right) ^2\right\} +2\sigma _{\varepsilon }^2k_a\nonumber \\= & {} \sum _{\{\varvec{s}_i\in D_a\}}E\left[ \left( Z(\varvec{s}_i)-S(\varvec{s}_i)\right) ^2\right] +\sum _{\{\varvec{s}_i\in D_a\}}E\left[ \left( S(\varvec{s}_i)-\hat{S}_M(\varvec{s}_i; \varvec{\theta })\right) ^2\right] \nonumber \\&+2\sum _{\{\varvec{s}_i\in D_a\}}E\left[ \left( Z(\varvec{s}_i)-S(\varvec{s}_i)\right) \left( S(\varvec{s}_i)-\hat{S}_M(\varvec{s}_i;\varvec{\theta })\right) \right] +2\sigma _{\varepsilon }^2k_a\nonumber \\= & {} \sum _{\{\varvec{s}_i\in D_a\}}E\left[ \left( \varepsilon (\varvec{s}_i)\right) ^2\right] +\sum _{\{\varvec{s}_i\in D_a\}}E\left[ \left( S(\varvec{s}_i)-\hat{S}_M(\varvec{s}_i; \varvec{\theta })\right) ^2\right] \nonumber \\&+2\sum _{\{\varvec{s}_i\in D_a\}}E\left[ \varepsilon (\varvec{s}_i)S(\varvec{s}_i)\right] -2\sum _{\{\varvec{s}_i\in D_a\}}E\left[ \varepsilon (\varvec{s}_i)\hat{S}_M(\varvec{s}_i; \varvec{\theta })\right] +2\sigma _{\varepsilon }^2k_a\nonumber \\= & {} E\left[ \sum _{\{\varvec{s}_i\in D_a\}}\left( \hat{S}_M(\varvec{s}_i;\varvec{\theta })-S(\varvec{s}_i)\right) ^2\right] +n_a\sigma ^2_{\varepsilon }\nonumber \\&-2\sum _{\{\varvec{s}_i\in D_a\}}E\left[ \varepsilon (\varvec{s}_i)\hat{S}_M(\varvec{s}_i;\varvec{\theta })\right] +2\sigma _{\varepsilon }^2k_a\,, \end{aligned}$$

(34)

where $[\varvec{H}_M(\varvec{\theta })]_{ii}$ denotes the ith diagonal element of matrix $\varvec{H}_M(\varvec{\theta })$, $k_a\equiv \sum\nolimits _{\{i:\varvec{s}_i\in D_a\}}\left[ \varvec{H}_M(\varvec{\theta })\right] _{ii}$ is associated with subarea $D_a$ and is a constant when model parameters are known, the fifth equality is based on $Z(\varvec{s}_i)=S(\varvec{s}_i)+\varepsilon (\varvec{s}_i)$, and the sixth equality follows from $\varepsilon (\varvec{s}_i)\sim N(0,\sigma ^2_{\varepsilon })$ and $S(\varvec{s}_i)$ are independent and $n_a>0$ represents the number of observations in the subarea $D_a$. Therefore, the proof of (15) remains to show

$$\begin{aligned} \sum _{\{\varvec{s}_i\in D_a\}}E\left[ \varepsilon (\varvec{s}_i)\hat{S}_M(\varvec{s}_i;\varvec{\theta })\right] =\sigma _{\varepsilon }^2k_a\,. \end{aligned}$$

Because $Z(\varvec{s}_i)=S(\varvec{s}_i)+\varepsilon (\varvec{s}_i)$, we have

$$\begin{aligned} \sum _{\{\varvec{s}_i\in D_a\}}E\left[ \varepsilon (\varvec{s}_i)\hat{S}_M(\varvec{s}_i;\varvec{\theta })\right]= & {} \sum _{\{\varvec{s}_i\in D_a\}}EE\left[ \varepsilon (\varvec{s}_i)\hat{S}_M(\varvec{s}_i;\varvec{\theta })\big |\varvec{S}\right] \nonumber \\= & {} \sum _{\{\varvec{s}_i\in D_a\}}E\left[ Cov\left( \varepsilon (\varvec{s}_i),\hat{S}_M(\varvec{s}_i;\varvec{\theta })\big |\varvec{S}\right) \right] \nonumber \\= & {} \sum _{\{\varvec{s}_i\in D_a\}}E\left[ Cov\left( Z(\varvec{s}_i),\hat{S}_M(\varvec{s}_i;\varvec{\theta })\big |\varvec{S}\right) -Cov\left( S(\varvec{s}_i),\hat{S}_M(\varvec{s}_i; \varvec{\theta })\big |\varvec{S}\right) \right] \nonumber \\= & {} \sum _{\{\varvec{s}_i\in D_a\}}E\left[ Cov\left( Z(\varvec{s}_i),\hat{S}_M(\varvec{s}_i;\varvec{\theta })\big |\varvec{S}\right) \right] , \end{aligned}$$

(35)

where the last equality follows from $Cov\left( S(\varvec{s}_i),\hat{S}_M(\varvec{s}_i;\varvec{\theta })\big |\varvec{S}\right) =0$. In addition, because $\varvec{Z}|\varvec{S}\sim N(\varvec{S},\sigma ^2_{\varepsilon }\varvec{I})$, we have the following result for all $i=1,\dots ,n$.

$$\begin{aligned} \displaystyle \frac{\partial }{\partial S(\varvec{s}_i)}E\left( \hat{S}_M(\varvec{s}_i;\varvec{\theta })\big |\varvec{S}\right)= & {} (2\pi \sigma ^2_{\varepsilon })^{-n/2}\displaystyle \frac{\partial }{\partial S(\varvec{s}_i)}\int \hat{S}_M(\varvec{s}_i;\varvec{\theta }) \exp \left\{ -\sigma _{\varepsilon }^{-2}\displaystyle \sum ^{n}_{j=1}(z_j-S(\varvec{s}_j))^2/2\right\} d\varvec{z}\nonumber \\=\, & {} (2\pi \sigma ^2_{\varepsilon })^{-n/2}\sigma _{\varepsilon }^{-2}\nonumber \\&\times \displaystyle \int \hat{S}_M(\varvec{s}_i;\varvec{\theta })(z_i-S(\varvec{s}_i)) \exp \left\{ -\sigma _{\varepsilon }^{-2}\displaystyle \sum ^{n}_{j=1}(z_j-S(\varvec{s}_j))^2/2\right\} d\varvec{z}\nonumber \\=\, & {} \sigma _{\varepsilon }^{-2}E\left( \hat{S}_M(\varvec{s}_i;\varvec{\theta })(Z(\varvec{s}_i)-S(\varvec{s}_i))\big |\varvec{S}\right) \nonumber \\=\, & {} \sigma _{\varepsilon }^{-2}Cov\left( \hat{S}_M(\varvec{s}_i;\varvec{\theta }),Z(\varvec{s}_i)\big |\varvec{S}\right) . \end{aligned}$$

(36)

It follows from (11), (12), (35), and (36) that

$$\begin{aligned} \sum _{\{\varvec{s}_i\in D_a\}}E\left[ \varepsilon (\varvec{s}_i)\hat{S}_M(\varvec{s}_i;\varvec{\theta })\right]= & {} \sigma _{\varepsilon }^2\sum _{\{\varvec{s}_i\in D_a\}}E\left[ \displaystyle \frac{\partial }{\partial S(\varvec{s}_i)}E\left( \hat{S}_M(\varvec{s}_i; \varvec{\theta })\big |\varvec{S}\right) \right] \nonumber \\=\, & {} \sigma _{\varepsilon }^2\sum _{\{\varvec{s}_i\in D_a\}}E\left[ \displaystyle \frac{\partial }{\partial S(\varvec{s}_i)}E\left( \left[ \varvec{H}_M(\varvec{\theta })\right] _i \varvec{Z}\big |\varvec{S}\right) \right] \nonumber \\=\, & {} \sigma _{\varepsilon }^2\sum _{\{\varvec{s}_i\in D_a\}}E\left[ \displaystyle \frac{\partial }{\partial S(\varvec{s}_i)}\big (E\left( \left[ \varvec{H}_M(\varvec{\theta })\right] _i \varvec{S}\big |\varvec{S}\right) + E\left( \left[ \varvec{H}_M(\varvec{\theta })\right] _i \varvec{\varepsilon }\big |\varvec{S}\right) \big )\right] \nonumber \\=\, & {} \sigma _{\varepsilon }^2\sum _{\{\varvec{s}_i\in D_a\}}E\left[ \displaystyle \frac{\partial }{\partial S(\varvec{s}_i)}E\left( \left[ \varvec{H}_M(\varvec{\theta })\right] _i \varvec{S}\big |\varvec{S}\right) \right] \nonumber \\=\, & {} \sigma _{\varepsilon }^2\sum _{\{i:\varvec{s}_i\in D_a\}}\left[ \varvec{H}_M(\varvec{\theta })\right] _{ii}\nonumber \\=\, & {} \sigma _{\varepsilon }^2k_a, \end{aligned}$$

(37)

where $\left[ \varvec{H}_M(\varvec{\theta })\right] _i$ denotes the ith row of matrix $\varvec{H}_M(\varvec{\theta })$, the third equality is based on $\varvec{Z}=\varvec{S}+\varvec{\varepsilon }$, and the fourth equality follows from $E\left( \left[ \varvec{H}_M(\varvec{\theta })\right] _i \varvec{\varepsilon }\big |\varvec{S}\right) =0$. Thus, we obtain the desired result based on (34) and (37). This completes the proof.

Proof of Corollary 1 From the definitions of $CAIC(M;\varvec{\theta })$ and $LCAIC(M;D_a;\varvec{\theta })$ in (13) and (14), we know that

$$\begin{aligned} \displaystyle \sum _{a=1}^A LCAIC(M;D_a;\varvec{\theta })= & {} \displaystyle \sum _{a=1}^A \sum _{\left\{ \varvec{s}_i\in D_a\right\} }\left( Z(\varvec{s}_i)-\hat{S}_M(\varvec{s}_i;\varvec{\theta })\right) ^2 +2\sigma ^2_{\varepsilon }\displaystyle \sum _{a=1}^A\sum _{\left\{ i:\varvec{s}_i\in D_a\right\} }\left[ \varvec{H}_M(\varvec{\theta })\right] _{ii}\\= & {} \displaystyle \sum _{i=1}^n\left( Z(\varvec{s}_i)-\hat{S}_M(\varvec{s}_i;\varvec{\theta })\right) ^2 +2\sigma ^2_{\varepsilon }\displaystyle \sum _{i=1}^n\left[ \varvec{H}_M(\varvec{\theta })\right] _{ii}\\=\, & {} CAIC(M;\varvec{\theta }). \end{aligned}$$

Taking the expectation on both sides of the above equality, we have

$$\begin{aligned} E\left[ CAIC(M;\varvec{\theta })\right]= & {} \displaystyle \sum _{a=1}^A E\left[ LCAIC(M;D_a;\varvec{\theta })\right] \\= & {} \displaystyle \sum _{a=1}^A\left\{ E\left[ \sum _{\{\varvec{s}_i\in D_a\}}\left( \hat{S}_M(\varvec{s}_i;\varvec{\theta })-S(\varvec{s}_i)\right) ^2\right] +n_a\sigma _{\varepsilon }^2\right\} \\=\, & {} E\left[ \displaystyle \sum _{a=1}^A\sum _{\{\varvec{s}_i\in D_a\}}\left( \hat{S}_M(\varvec{s}_i;\varvec{\theta })-S(\varvec{s}_i)\right) ^2\right] +\sigma _{\varepsilon }^2\displaystyle \sum _{a=1}^A n_a\\=\, & {} E\left[ \displaystyle \sum _{i=1}^n\left( \hat{S}_M(\varvec{s}_i;\varvec{\theta })-S(\varvec{s}_i)\right) ^2\right] +n\sigma _{\varepsilon }^2, \end{aligned}$$

where the second equality is based on the result of (15) in Theorem 1, $n_a>0$ is the number of observations in the subarea $D_a$ for $a=1,\dots ,A$, and $\sum\nolimits _{a=1}^A n_a=n$ is the number of all observations in the study area D. Thus, we obtain the desired result, which completes the proof.