A flexible spatio-temporal model for air pollution with spatial and spatio-temporal covariates

Abstract

The development of models that provide accurate spatio-temporal predictions of ambient air pollution at small spatial scales is of great importance for the assessment of potential health effects of air pollution. Here we present a spatio-temporal framework that predicts ambient air pollution by combining data from several different monitoring networks and deterministic air pollution model(s) with geographic information system covariates. The model presented in this paper has been implemented in an R package, SpatioTemporal, available on CRAN. The model is used by the EPA funded Multi-Ethnic Study of Atherosclerosis and Air Pollution (MESA Air) to produce estimates of ambient air pollution; MESA Air uses the estimates to investigate the relationship between chronic exposure to air pollution and cardiovascular disease. In this paper we use the model to predict long-term average concentrations of $\text {NO}_{x}$ in the Los Angeles area during a 10 year period. Predictions are based on measurements from the EPA Air Quality System, MESA Air specific monitoring, and output from a source dispersion model for traffic related air pollution (Caline3QHCR). Accuracy in predicting long-term average concentrations is evaluated using an elaborate cross-validation setup that accounts for a sparse spatio-temporal sampling pattern in the data, and adjusts for temporal effects. The predictive ability of the model is good with cross-validated $R^2$ of approximately $0.7$ at subject sites. Replacing four geographic covariate indicators of traffic density with the Caline3QHCR dispersion model output resulted in very similar prediction accuracy from a more parsimonious and more interpretable model. Adding traffic-related geographic covariates to the model that included Caline3QHCR did not further improve the prediction accuracy.

Appendix: Proof of equivalence for the simplified likelihood

To prove the equivalence of the two likelihood forms (9) and (10) we need the following:

Lemma 1

If $\varSigma _1$ and $\varSigma _2$ are two nonsingular matrices of size $n_1$-by-$n_1$ and $n_2$-by-$n_2$ respectively, and $A$ is a $n_2$-by-$n_1$ matrix, then:

$$\begin{aligned} \left|A\varSigma _1 A^\top + \varSigma _2\right| = \left|\varSigma _1\right|\left|\varSigma _2\right|\left|\varSigma _1^{-1} + A^\top \varSigma _2^{-1} A\right|. \end{aligned}$$

(Harville 1997, Thm. 18.1.1)

Lemma 2

The Woodbury identity (Harville 1997, Thm. 18.2.8): If $A$ and $B$ are two invertable matrices of size $n$-by-$n$ and $p$-by-$p$ respectively, and $C$ is an arbitrary $n$-by-$p$ matrix, then

$$\begin{aligned} \left( A+CBC^\top \right) ^{-1} = A^{-1} - A^{-1} C\left( B^{-1}+C^\top A^{-1}C\right) ^{-1}C^\top A^{-1}. \end{aligned}$$

Rearranging the terms and multiplying with $A$ from both sides, Lemma 2 becomes

$$\begin{aligned} C\left( B^{-1}+C^\top A^{-1}C\right) ^{-1}C^\top = A - A \left( A+CBC^\top \right) ^{-1} A. \end{aligned}$$

(13)

Lemma 3

The Searle identity (Harville 1997, Thm. 18.2.3): If $A, B$ are matrices of size $p$-by-$n$ and $n$-by-$p$ respectively, $\mathbf I$ denotes identity matrices of appropriate size, and $(\mathbf I + AB)$ is nonsingular, then

$$\begin{aligned} \left( \mathbf I + AB\right) ^{-1}A = A \left( \mathbf I + BA\right) ^{-1}. \end{aligned}$$

Lemma 4

Blockwise inversion (Harville 1997, Thm. 8.5.11): Let $A, B, C$, and $D$ be block matrices, with $A$ and $(D - C A^{-1} B)$ being nonsingular, then

$$\begin{aligned} \left[ \begin{array}{ll} A &{} B \\ C &{} D \end{array}\right] ^{-1} = \left[ \begin{array}{ll} A^{-1} + A^{-1} B \bigl (D - C A^{-1} B \bigr )^{-1} C A^{-1} &{} -A^{-1} B \bigl (D - C A^{-1} B \bigr )^{-1} \\ -\bigl (D - C A^{-1} B \bigr )^{-1} C A^{-1} &{} \bigl (D - C A^{-1} B \bigr )^{-1} \end{array}\right] . \end{aligned}$$

To make the notation clearer we suppress the dependence on $\varPsi$. Superscripts above equality signs denote the identities used in each step.

For the determinant in (9) we have

$$ \begin{aligned} \left|\widetilde{\varSigma }\right| \overset{\text {(7)}}{=} \left|\varSigma _\nu + F \varSigma _B F^\top \right| \overset{\text {Lem. 1} \& \text {(11a)}}{=} \left|\varSigma _\nu \right|\left|\varSigma _B\right| \left|\varSigma _{B\vert Y}^{-1}\right|, \end{aligned}$$

proving equality with the determinants in (10).

For the quadratic form in (9) we first note that

$$\begin{aligned}&\widetilde{\varSigma }^{-1} \overset{\text {Lem. 2}}{=} \varSigma _\nu ^{-1} - \varSigma _\nu ^{-1} F \varSigma _{B\vert Y} F^\top \varSigma _\nu ^{-1}, \end{aligned}$$

(14a)

$$\begin{aligned}&F^\top \widetilde{\varSigma }^{-1} F \overset{\text {(13)}}{=} \varSigma _B^{-1} - \varSigma _B^{-1} \varSigma _{B\vert Y} \varSigma _B^{-1}, \end{aligned}$$

(14b)

$$\begin{aligned}&\widetilde{\varSigma }^{-1} F \overset{\text {(7)}}{=} \left( {\varvec{I}} + \varSigma _\nu ^{-1} F \varSigma _B F^\top \right) ^{-1} \varSigma _\nu ^{-1} F&\overset{\text {Lem. 3}}{=} \varSigma _\nu ^{-1} F \varSigma _{B\vert Y} \varSigma _B^{-1}. \end{aligned}$$

(14c)

Using (14) we have that

$$ \begin{aligned} \varSigma _{\alpha \vert Y}^{-1}&\overset{\text {(11b)} \& \text { (14b)}}{=}&X^\top F^\top \widetilde{\varSigma }^{-1} F X \end{aligned}$$

(15a)

$$ \begin{aligned} \widehat{\varSigma }&\overset{\text {(11c)} \& \text {(14a)} \& \text {(14c)}}{=}&- \widetilde{\varSigma }^{-1} F X \varSigma _{\alpha \vert Y} X^\top F^\top \widetilde{\varSigma }^{-1} + \widetilde{\varSigma }^{-1}. \end{aligned}$$

(15b)

For the quadratic form in (9) we have

$$ \begin{aligned}&Y^\top \widetilde{\varSigma }^{-1} \widetilde{X} \Bigl (\widetilde{X}^\top \widetilde{\varSigma }^{-1} \widetilde{X} \Bigr )^{-1} \widetilde{X}^\top \widetilde{\varSigma }^{-1} Y - Y^\top \widetilde{\varSigma }^{-1} Y\\&\quad \overset{ (7) \& (15\text {a})}{=} Y^\top \widetilde{\varSigma }^{-1} \widetilde{X} \left[ \begin{array}{cc} \varSigma _{\alpha \vert Y}^{-1} &{} \mathcal M ^\top \widetilde{\varSigma }_\nu ^{-1} F X\\ X^\top F^\top \widetilde{\varSigma }_\nu ^{-1} \mathcal M &{} \mathcal M ^\top \widetilde{\varSigma }_\nu ^{-1} \mathcal M \end{array}\right] ^{-1} \widetilde{X}^\top \widetilde{\varSigma }^{-1} Y- Y^\top \widetilde{\varSigma }^{-1} Y\\&\quad \overset{ \text {Lem. 4} \& \text {(15b)}}{=} Y^\top \widetilde{\varSigma }^{-1} \biggl [ F X \varSigma _{\alpha \vert Y} X^\top F^\top + \Bigl ( \mathbf I - F X \varSigma _{\alpha \vert Y} X^\top F^\top \widetilde{\varSigma }^{-1}\Bigr )\\&\mathcal M \bigl ( \mathcal M ^\top \widehat{\varSigma } \mathcal M \bigr )^{-1} \mathcal M ^\top \Bigl ( \mathbf I - \widetilde{\varSigma }^{-1} F X \varSigma _{\alpha \vert Y} X^\top F^\top \Bigr ) \biggr ]\widetilde{\varSigma }^{-1} Y- Y^\top \widetilde{\varSigma }^{-1} Y\\&\quad =Y^\top \biggl [\Bigl ( \widetilde{\varSigma }^{-1} -\widetilde{\varSigma }^{-1} F X \varSigma _{\alpha \vert Y} X^\top F^\top \widetilde{\varSigma }^{-1} \Bigr )\mathcal M \bigl ( \mathcal M ^\top \widehat{\varSigma } \mathcal M \bigr )^{-1} \mathcal M ^\top \\&\quad \Bigl ( \widetilde{\varSigma }^{-1} - \widetilde{\varSigma }^{-1} F X \varSigma _{\alpha \vert Y} X^\top F^\top \widetilde{\varSigma }^{-1} \Bigr )\\&\quad - \Bigl ( \widetilde{\varSigma }^{-1} -\widetilde{\varSigma }^{-1} F X \varSigma _{\alpha \vert Y} X^\top F^\top \widetilde{\varSigma }^{-1} \Bigr )\biggr ] Y\\&\quad \overset{\text {(15b)}}{=} Y^\top \widehat{\varSigma } \mathcal M \Bigl ( \mathcal M ^\top \widehat{\varSigma } \mathcal M \Bigr )^{-1} \mathcal M ^\top \widehat{\varSigma } Y- Y^\top \widehat{\varSigma } Y, \end{aligned}$$

showing that the quadratic forms in (9) and (10) are equal.