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%{\large\textbf {Predicting traffic-related pollutant concentrations in near-road environments using a Bayesian spatio-temporal model}}\\~\\
\received{00 Month 2018}
\revised{00 Month 2018}
\accepted{00 Month 2018}

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\runninghead{O. Gilani, V.J.~Berrocal, and S. A. Batterman}{Non-stationary Spatiotemporal Bayesian Data Fusion}

\begin{document}

\title{Non-stationary Spatiotemporal Bayesian Data Fusion for Pollutants in the Near-road Environment}

\author{O. Gilani\affil{a}\corrauth, V. J. Berrocal\affil{b}, and S. A. Batterman\affil{c}}

\corraddr{Owais Gilani, Department of Mathematics, Bucknell University, Lewisburg, PA 17837, USA.
Email: owais.gilani@bucknell.edu}

\address{\affilnum{a} Bucknell University, Department of Mathematics, Lewisburg, PA, USA
\affilnum{b} University of Michigan, Department of Biostatistics, School of Public Health, Ann Arbor, MI, USA
\affilnum{c} University of Michigan, Department of Environmental Health Sciences, School of Public Health, Ann Arbor, MI, USA}


\begin{abstract}  

Concentrations of near-road air pollutants (NRAPs) have increased to very high levels in many urban centers around the world, particularly in developing countries. The adverse health effects of exposure to NRAPs are greater when the exposure occurs in the near-road environment as compared to background levels of pollutant concentration. Therefore, there is increasing interest in monitoring pollutant concentrations in the near-road environment. However, due to various practical limitations, monitoring pollutant concentrations near roadways and traffic sources is generally rather difficult and expensive. As an alternative, various deterministic computer models that provide predictions of pollutant concentrations in the near-road environment, such as the Research Line-source dispersion model (RLINE), have been developed. A common feature of these models is that their outputs typically display systematic biases and need to be calibrated in space and time using observed pollutant data. In this paper, we present a non-stationary Bayesian data fusion model that uses a novel dataset on monitored pollutant concentrations (nitrogen oxides  or \noxns, and fine particulate matter or \pmtns) in the near-road environment and, combining it with the RLINE model output, provides predictions at unsampled locations. The model can also be used to evaluate whether including the RLINE model output leads to improved pollutant concentration predictions, and whether the RLINE model output captures the spatial dependence structure of NRAP concentrations in the near-road environment. A defining characteristic of the proposed model is that we model the non-stationarity in the pollutants concentrations by using a recently developed approach that includes covariates, postulated to be the driving force behind the non-stationary behavior, in the covariance function.


\end{abstract}

\keywords{Nitrogen oxide; Fine particulate matter; Numerical model output; Data fusion; \newline  Non-stationarity; Covariates in covariance}
\maketitle

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\section{Introduction} \label{introduction}

Concentrations of near-road air pollutants (NRAPs) have increased to very high levels in many urban centers around the world, particularly in developing countries. Due to the well established adverse effects of exposure to air pollutants on various health outcomes, a number of environmental regulations have been established to monitor  concentrations of pollutants. Episodes of exceedance over compliance threshholds have occured in some urban environments, which have led to drastic measures to reduce pollutant concentration levels within cities \citep{paris2018}. Evidence suggests that the adverse health impacts of exposure to NRAPs are greater when the exposure occurs closer to major roads and highways as compared to background levels of pollutants \citep{hoek2002, lewis2005}. An important societal implication of this result is that the greatest burden of this exposure is carried by populations belonging to lower socio-economic status, since they constitute a higher proportion of residents living in close proximity to major roads \citep{gunier2003}. 

Accurate maps of various pollutants concentrations on a fine spatial and temporal resolution, particularly within the near-road environment, are important both for regulatory purposes and for assigning exposure to subjects in environmental epidemiologic studies. However, due to practical and monetary limitations, good quality data on concentrations of pollutants on a dense spatial and temporal scale are generally not available. Instead, to monitor pollutant concentrations in real time, environmental agencies often rely on a network of fixed site monitors, most commonly, those operated by states and local offices of the US Environmental Protection Agency \citep{epaAQS}. There are a number of limitations to using such monitors for both regulatory purposes and for estimating concentrations at unsampled locations. First, the network of monitors is generally spatially very sparse. Studies conducted within the near-road environments have observed that fixed/central site monitors do not capture the spatial variability exhibited by NRAPs \citep{vardoulakis2005}, and the hourly maxima recorded at these sites is often well below the concentrations measured near major highways \citep{wallace2009}. Second, due to practical limitations, these monitors are placed at considerable heights (between 3-15 meters above ground) \citep{epaProbe}, which is generally not the height of air inhaled at ground level. Third, monitors are often preferentially located, so that the concentrations measured by these monitors may not be representative of the entire population's exposure. Finally, due to maintenance downtime and mechanical failures, data are sometimes not recorded. Therefore, while these monitors contain valuable information about urban and background levels of pollutant concentrations, they do not provide very accurate information about concentrations closer to highways or the specific contribution of traffic sources to air pollution.   

In light of these limitations, there is growing interest in utilising outputs from deterministic computer models that apply numerical simulation techniques in various ways to estimate air pollution concentrations. Two examples of such models are the Community Multiscale Air Quality model  \citep{byun2006} which numerically simulates the underlying pollutant chemistry, and the Research Line-source dispersion model - RLINE \citep{snyder2013} which simulates the dispersion of air pollutants from traffic and other sources. While such models may provide estimates of pollutants concentrations on a dense temporal resolution over the geographic region of interest, they often have systematic biases associated with them due to multiple reasons. % such as simplifications of the complex atmospheric processes and uncertain model inputs, among others. 
 Therefore, these model outputs need to be calibrated in space and time. To address the joint goals of calibrating numerical model outputs (i.e., estimating their bias) and providing more accurate predictions of atmospheric pollutants at unsampled locations, various statistical modeling techniques have been developed and successfully applied to a number of pollutants including sulfur dioxide \citep{fuentes2005bdf}, ground-level ozone \citep{berrocal2010, berrocal2012biometrics, zidek2012, reich2014}, nitrogen dioxide \citep{gilani2016scarr}, and fine particulate matter \citep{choi2009, berrocal2010bivariate, mcmillan2010, crooks2014, rundel2015}. %These various statistical models can be classified into two general categories: (i) regression-based approaches, in which measured monitor data are considered as the outcome, and are related to a function of numerical model output in a spatial/spatiotemporal (Bayesian) hierarchical model, see \citet{berrocal2010, berrocal2012biometrics, zidek2012, crooks2014, reich2014, rundel2015, gilani2016scarr}, among others; and (ii) joint modeling or data fusion approaches, in which both measured monitor data and the numerical model output are treated as stochastic realizations of an underlying unobserved ``true'' pollutant concentration field, see \citet{fuentes2005bdf, choi2009, mcmillan2010, sahu2010}, to list a few. 
Although these spatial and spatiotemporal models perform well when calibrating regional-scale deterministic model outputs over an extended spatial domain, they have some limitations for modeling pollutant concentrations within a localized urban setting in a near-road environment. %Studies utilizing the regression-based approach \citep{berrocal2010, berrocal2012biometrics, crooks2014, reich2014, rundel2015, gilani2016scarr}, have typically assumed a stationary covariance function for the residual spatial dependence in pollutants concentrations after accounting for the spatial correlation that comes from the numerical model output, i.e., these models commonly express the residual correlation between pollutant concentrations at two locations as a function only of their distance. While this assumption may be valid for the residuals of some computer model outputs over certain geographical domains, it may not hold for other models and/or geographical domains. In particular, the residual correlation between pollutant concentrations at sites across major highways in an urban setting might be more dependent on local characteristics, such as wind speed and direction than simply geographical distance. 
In addition, most of these models assume a stationary covariance function for the residual spatial dependence in pollutants concentrations after accounting for the spatial correlation that comes from the numerical model output, i.e., these models commonly express the residual correlation between pollutant concentrations at two locations as a function only of their distance, which might not be appropriate in an urban setting.

Modeling spatial and spatiotemporal processes using non-stationary covariance functions has received much attention over the last few years \citep{sampson1992, higdon1998, holland1999, fuentes2001, paciorek2006, lindgren2011}. Recent work has further extended some of the non-statiovary covariance methods to include spatial covariates in the covariance function thus aiding in identifying factors that drive the non-stationarity \citep{calder2008, reich2011, schmidt2011,neto2014,risser2015, gilani2016}.

%The joint modeling approach of \citet{fuentes2005bdf} models the covariance function of a pollutant concentration using the non-stationary model based on kernel mixing of \citet{fuentes2001}. However, the Bayesian melding model of \citet{fuentes2005bdf} is purely spatial, and was not designed to handle spatio-temporal data. On the other hand, the Bayesian hierarchical model by \citet{mcmillan2010} is spatiotemporal and non-stationary. However, to simplify computations, \citet{mcmillan2010} adopt an upscaling approach, using a space-time grid rather than a continuous spatial domain, and yield predictions of pollutant concentrations at the aereal scale of the computer model grid cells rather than at point locations. Additionally, neither the non-stationary covariance function of \citet{fuentes2005bdf} nor of \citet{mcmillan2010} include covariates within the covariance function.
 
In this paper, we present a non-stationary spatiotemporal data fusion model for two NRAPs: nitrogen oxides (\noxns) and fine particulate matter (\pmtns). The goals of this model are: (a) to model contribution of traffic sources to NRAP concentrations, particularly in the near-road environment, and provide concentration predictions at unsampled locations; (b) to evaluate whether incorporating additional information on NRAPs in the near-road environment provided by the RLINE dispersion model output improves estimation of pollutant concentrations; and (c) if so, to assess whether the RLINE model output captures the spatial dependence structure of NRAP concentrations in near road environments. The proposed model follows the joint modeling approach, with a couple of innovations. First, since exposure to NRAPs in the near-road environment has potentially significantly worse adverse impact on human health as compared to exposure to background concentration levels, instead of modeling the background concentration of pollutants, we model the near-road increment in pollutants concentration due to vehicular sources. This is achieved using data obtained from a novel pollutant concentration monitoring approach that used a mobile air pollution lab to monitor concentrations across highways within an urban setting. Second, we capture the potential non-stationarity in the residual spatial correlation between sites that are upwind versus downwind of the highway using a recently developed model \citep{gilani2016} that includes covariates in the covariance function. To our knowledge, this is the first time a non-stationary data fusion model is presented in which the covariance function includes spatial covariates. 
 
A few recent studies have evaluated RLINE model output for  various pollutant concentrations. However, these studies have either just visually compared observed concentrations versus RLINE model output \citep{saha2018}, or compared the correlation between RLINE model output against various forms of observed concentrations: across regional air quality monitors \citep{milando2018, yu2018} or smoothed averages based on mobile air pollution lab measurements \citep{patton2017}. They do not attempt to quantify the potential biases in RLINE model output. While \citet{zhai2016} do quantify the calibration bias in RLINE output for \noxns, \pmtns, and carbon monoxide (CO), they use a linear regression approach with independent error terms, without accounting for the spatial/spatiotemporal structure of the residual process. 

The manuscript is organized as follows: Section \ref{data} provides details on the monitored data and the RLINE computer model output; Section \ref{model} gives details of the proposed non-stationary joint Bayesian data fusion model; Section \ref{results} presents results from fitting the model to our data as well as a comparison with various special cases of our model including  stationary models; and finally, Section \ref{discussion} provides concluding remarks and future directions.



\section{Data} \label{data}

\subsection{Monitor Data} \label{data_mapl}
A Mobile Air Pollution Lab (MAPL), a recreational vehicle equipped with air monitoring instruments, measured ambient concentrations of nitrogen oxides (\noxns) and fine particulate matter (\pmtns) along nine transects that crossed major roadways in Detroit, MI on seven consecutive days in December 2012 during morning and afternoon rush hour periods (Figure~\ref{fig:map}a). Details of the sampling are described in another report analyzing factors that affect pollutants  concentrations in the near-road environment \citep{Baldwin2015}. In short, at each of the nine ``areas", pollutants concentrations were sampled in residential neighborhoods adjacent to the major roadways. In areas 1 through 8, the sampling sites were located at  nominal distances of 50 m (two sites), 150 m (two sites) and 500 m (one site) from both edges of the roads. In these areas, the ten sampling sites formed roughly two parallel transects across the roads. On the other hand, in area 9, six sampling sites were used at distances of 50, 150 and 500 m, respectively, from both edges of the road. 

\begin{figure}[h!t]
	\centering
	\begin{tabular}{c c}
	\multicolumn{2}{c}{\subf{\includegraphics[width=100mm]{Figures/Area_Map_c.png}} {(a)}} \\
	\subf{\includegraphics[width=70mm]{Figures/Area_5_NOx_071216.png}} {(b)} & \subf{\includegraphics[width=70mm]{Figures/Area_5_PM25_071216.png}} {(c)} \\
	\end{tabular}	
	\caption{(a) Nine transect areas and monitoring sites in Detroit, MI. (b) Near-road increment (NRI) in \nox concentrations (in ppb) at MAPL and RLINE sites in area 5 on the morning of Dec. 18, 2012; \mbox{(c) NRI} in PM$_{2.5}$ concentrations (in $\mu g/m^3$) at MAPL and RLINE sites in Area 5 on the same day. }
	\label{fig:map}
\end{figure} 

Areas 1 through 8 were monitored during both morning (07:15 - 09:45) and afternoon \mbox{(15:45 - 18:15)} rush hour periods, while area 9 was monitored once or twice each day, just after the morning rush hour, or before the afternoon rush hour. Sites were monitored from the morning of Friday, December 14, 2012 to the morning of Thursday, December 20, 2012. On any given day, up to three different areas were monitored. The MAPL visited the monitoring sites and collected air samples for 5 minutes at each site. \nox concentrations were measured using conventional federal reference monitors (Model 42i, Thermo, MA USA), while particle number concentrations (PNCs) in multiple size bins were measured with a GRIMM Model 1.109 Spectrometer (Grimm Aerosol Technik, Germany). The PNCs were converted to mass concentrations using the protocol described by \citet{grimm2009} assuming particle density of 1.67 $g/cm^3$. \pmt concentrations were calculated as the sum of mass concentrations for particles with diameters less than or equal to 2.5 $\mu m$.  For each pollutant, the ``background'' concentration for a transect area was determined as the lowest  pollutant concentration observed in that area during the monitoring period.  While different methods could be used to estimate background concentration, including using observations from regulatory monitors, we selected our approach so as to account for urban local sources near the transect. Regulatory monitors are typically designed to capture regional sources of pollution, and thus might not be representative of the urban environment.

The ``near-road increment'' (NRI) in pollutant concentration at a monitoring site was defined as the difference between the observed and background pollutant concentrations. By this definition, for \noxns, across the nine areas, a total of 254 5-minute average NRI in concentrations were collected, while 235 average NRI in concentrations were sampled for \pmtns.  Figures~\ref{fig:map}(b) and \ref{fig:map}(c) show \nox and \pmt NRI in pollutants concentrations, respectively, at sites within area 5 on the morning of December 12, 2012.

Initial exploratory data analysis revealed that there was considerable variability in the NRI in concentrations of the two pollutants between the different areas. The mean NRI in concentrations of both pollutants was lowest at area 9 (9.6 ppb for \nox (SD=8.5 ppb) and 0.5 \mgm (SD=0.4 \mgmns) for \pmtns). This was expected, since area 9 sites were sampled either just after the morning rush hour or before the afternoon rush hour period, while the other sites were monitored during morning and afternoon rush hour periods. The highest mean NRI in concentrations of \nox was recorded at area 5 (21.5 ppb (SD=18.7 ppb)), and at area 1 for \pmt (2.2 \mgm (SD=2.2 \mgmns)). Histograms of the NRI in pollutants concentrations for both pollutants displayed right-skewed distributions, as shown in Figure 1 in the Supplementary Material, which suggests that modeling the NRI in pollutants concentrations on the log-scale would be more appropriate.  A time-series plot of the mean of the log of NRI in pollutant concentrations for sites monitored within the same time period (before or after 12 noon on the same day) is given in Figure~\ref{fig:mean_daily_log}(a) for \nox and Figure~\ref{fig:mean_daily_log}(b) for \pmtns.

% Daily mean time series on log scale
\begin{figure}[h!t]
	\centering
		\includegraphics[width=\textwidth]{Figures/EDA.png}
	\caption{Mean log(NRI) in (a) \nox concentration (in log ppb) and (b) \pmt concentration (in log $\mu g/m^3$) at MAPL sites at each time period.  The mean log(NRI) concentrations are stratified by downwind vs. upwind in (c) for \nox concentration (in log ppb) and in (d) for \pmt concentration (in log $\mu g/m^3$); and by morning vs. evening in (e) for \nox concentration (in log ppb) and in (f) for \pmt concentration (in log $\mu g/m^3$). The vertical bars represent $\pm$ 1 standard deviation from the mean.}
	\label{fig:mean_daily_log}
\end{figure} 

\subsection{RLINE Data} \label{data_rline}

Concentrations of \nox and \pmt in the near-road environment attributable to on-road vehicular traffic were modeled using a modified version of RLINE, a research line-source dispersion model \citep{snyder2013}. The dispersion mechanisms were unchanged, but the model was modified to incorporate Motor Vehicle Emission Simulator (MOVES) emission factors (gram/vehicle-mile) that vary by vehicle class, ambient temperature and month, and used reported traffic volumes for each modeled roadway adjusted  by hourly temporal allocation factors (Batterman et al., 2015) for commercial and non-commercial traffic. 

At each transect area, RLINE model estimates for average hourly pollutant concentrations were obtained at 96 regularly spaced point locations within a 2 km square centered around the major road, during the same time period that the MAPL data were recorded.  This selection of RLINE prediction sites allowed for some buffer around the MAPL monitoring sites, in addition to providing data at sites relatively far from the highway to better estimate the spatial range for the spatially varying additive calibration bias (detailed in \S 3.1). Since in data fusion models, the number of observations from each data source can control the relative influence of the data source on the estimation of the underlying true process, 96 RLINE sites for 10 MAPL sites provided a reasonable balance between the relative influence of each data source, the maximum distance from the highway, and the distance between neighboring points for the RLINE prediction.  Across the nine areas, a total of 3,038 1-hour average NRI in concentrations were estimated for both \nox and \pmtns. Figures~\ref{fig:map}(b) and \ref{fig:map}(c) show \nox and \pmt NRI in pollutants concentrations, respectively, at both the MAPL sites and RLINE estimation locations within area 5 on the morning of December 12, 2012.



Similar to the MAPL data, there was considerable variability observed in the concentrations of the two pollutants from the RLINE model output between the different areas. However, unlike the MAPL observations, the lowest mean NRI in concentrations for both pollutants were observed at area 7 (8.0 ppb (SD=10.1 ppb) for \noxns, and  0.23 \mgm (SD=0.30 \mgmns) for \pmtns), while the highest concentrations were observed at area 6 (42.4 ppb (SD=52.1 ppb) for \noxns, and  1.2 \mgm  (SD=1.5 \mgmns) for \pmtns). Figure~\ref{fig:scatterplot2}(a) shows a comparison of the log of NRI in monitored \nox concentrations against RLINE model predictions at the nearest monitor site (the dotted line has slope 1 and intercept 0), while Figure~\ref{fig:scatterplot2}(b) gives a similar plot for \pmt concentrations. These plots suggest that the RLINE model tends to provide higher predictions for NRI in \nox concentrations as compared to the MAPL data, while the predictions for \pmt tend to be slightly lower than monitored concentrations. 

\begin{figure}[h!t]
	\centering
	\begin{tabular}{c c}
	\subf{\includegraphics[width=70mm]{Figures/log_NOx_pairwise.png}} {(a)} & \subf{\includegraphics[width=70mm]{Figures/log_PM25_pairwise.png}} {(b)} \\
	\end{tabular}	
	\caption{Scatter-plot of  natural log of NRI in monitored (a) \nox concentrations (in log-ppb), and (b) \pmt concentrations (in log-$\mu g/m^3$) against RLINE model prediction at the nearest site. }
	\label{fig:scatterplot2}
\end{figure} 

\subsection{Covariate Data} \label{data_covariate}
In addition to the monitored and estimated pollution concentrations, we also obtained data on traffic and meteorological factors that may help explain some of the variablity in the pollutants concentrations at these areas. Wind speed and wind direction data were obtained from the National Oceanic and Atmospheric Administration (Quality Controlled Local Climatological Data; http://cdo.ncdc.noaa.gov/qclcd/QCLCD) for four airport sites in the study area: Detroit City Airport, Detroit Metro Airport, Grosse Pointe Farms, and St.Clair Shores. To obtain representative data, hourly wind direction and wind speed were averaged across the four airport sites. Additionally, we obtained annual average daily traffic (AADT) and commercial AADT (CAADT) volumes for major roads in Detroit from the Michigan Department of Transportation (MDOT) Traffic Monitoring Information System (TMIS) (2014). AADT and CAADT were adjusted to hourly values using Detroit-specific temporal allocation factors \citep{batterman2015}, with noncommercial AADT (NCAADT) defined as AADT - CAADT. Commercial traffic, mostly using heavy-duty diesel, tends to contribute a much greater proportion to atmospheric pollutants levels compared to noncommercial traffic, which operates mostly on light-duty gasoline \citep{watkins2012, batterman2015}. To more accurately account for the relative contribution of these vehicles to air pollution, a composite ``traffic volume" was calculated as \mbox{NCAADT + $c\cdot$CAADT}. In our study, following \citet{Baldwin2015}, we set the parameter $c$ to 10 for \noxns, and to 50 for \pmtns. 




\section{Model} \label{model}

In this section, we present the proposed non-stationary Bayesian data fusion model for near-road increments in pollutants concentrations of \nox and \pmt that allows us to both evaluate the RLINE model output, and provide predictions at unsampled locations along with a measure of uncertainty associated with the predictions. Our proposed model utilizes the joint modeling approach, while we capture the non-stationary behavior observed in the concentration of pollutants in the near-road environment using a recently developed approach by \citet{gilani2016}, which expresses pollutant concentrations as a mixture of two independent spatial processes, each equipped with a non-stationary covariance function with covariates driving the non-stationarity and the mixture weights. 



\subsection{Non-stationary spatiotemporal Bayesian data fusion model} \label{nonstat_bdf}

For each pollutant $k$, \mbox{$k \in \{$\noxns, \pmtns $\}$,} let $t$ represent ``time periods'', where a time period is defined by day, and time of day (before or after 12:00) of pollution monitoring. By this definition, there were $\mathcal{T}=11$ distinct time periods during the study data collection. As is customary in the joint modeling approach, we do not consider the monitored MAPL measurements or the RLINE model output as the \emph{truth} since these observations/estimates have measurement errors and/or biases associated with them. Instead, for each pollutant $k$,  we assume that there exists an underlying (unobserved) spatiotemporal field $Y_t^{(k)}(\s)$ which represents the \emph{true} NRI in concentration at location $\s \in \mathbf{\mathcal{S}}$ and time period $t=1, \ldots, \mathcal{T}$. Working on the log-scale, let $Z_t^{(k)}(\s) = \mbox{log}\left(Y_t^{(k)}(\s)\right)$ denote the natural log of the \emph{true} near-road increment in the concentration of pollutant $k$, at location $\s$ and time period $t$. For each pollutant $k$, we decompose $Z_t^{(k)}(\s)$ in the sum of two terms
\begin{equation}\label{eqn:general_decomp}
Z^{(k)}_{t}(\mathbf{s}) = \mu^{(k)}_{t}(\mathbf{s}) + \eta^{(k)}_{t}(\mathbf{s}), 
\end{equation}
\noindent where $\mu^{(k)}_{t}(\mathbf{s})$ accounts for the large-scale spatial trend in the $k$-th pollutant log NRI concentration at location $\s$ and time period $t$, and $\eta^{(k)}_{t}(\mathbf{s})$ is a spatially correlated, mean-zero error term. 

At a MAPL monitoring site $\s$ and time period $t$, we denote the measured log NRI concentration of pollutant $k$ by $\hat{Z}^{(k)}_{t}(\s)$, and assume that it is measured with some error, i.e., 
\begin{equation} \label{eqn:mapl_fn}
\hat{Z}^{(k)}_{t}(\s) = Z^{(k)}_{t}(\s) + e_t^{(k)}(\s), \quad \quad e_t^{(k)}(\s) \stackrel{iid}{\sim} N(0, \tau^{2^{(k)}}_e)
\end{equation}
where $e_t^{(k)}(\s)$ represents the measurement error at site $\s$ and time period $t$,  independent of the true underlying process $Z^{(k)}_{t}(\s)$. 

For the RLINE model output, we assume that the log NRI concentration estimate for pollutant $k$ at location $\s$ and time $t$, denoted by $\widetilde{Z}^{(k)}_{t}(\s)$, has both additive and multiplicative biases associated with it. These biases may be spatially varying or constant; while previous studies have shown evidence of spatially varying additive bias, the multiplicative bias is generally modeled as constant in space and time \citep{berrocal2010}. Exploratory data analysis (such as Figure \ref{fig:scatterplot2}) also suggests that while there does appear to be some multiplicative bias for the RLINE model output, it does not vary across space and time. We therefore model the log NRI concentration from the RLINE model output as

\begin{equation} \label{eqn:rline_fn}
\widetilde{Z}^{(k)}_{t}(\s) = a_0^{(k)} + a_t^{(k)}(\s) + b^{(k)} \cdot Z^{(k)}_{t}(\s) + \delta_t^{(k)}(\s),
\end{equation}
where $a_0^{(k)}$ is a constant (over space and time) additive bias;  $a_t^{(k)}(\s)$ is time series of serially independent Gaussian spatial processes; $b^{(k)}$ is a multiplicative bias that is constant over space and time; and $\delta_t^{(k)}(\s)$ is a mean-zero, independent across time and space, error process, i.e., $\delta_t^{(k)}(\s) \stackrel{iid}{\sim}N\left(0, \tau^{2^{(k)}}_{\delta}\right)$. At each time period $t$, the spatial covariance for the spatially correlated additive bias $a_t^{(k)}(\s)$ is modeled using an exponential covariance function so that the covariance between two sites $\s_i$ and $\s_j$ at time period $t$ is given by $\mbox{Cov}\left(a_t^{(k)}(\s_i), a_t^{(k)}(\s_j)\right) = \sigma^{2^{(k)}}_a \mbox{exp}\{-\phi^{(k)}_a |\s_i-\s_j|\}$. With this modeling choice, we assume that the additive bias of the RLINE output is similar at nearby locations, but less so at distant ones, with a correlation that decays exponentially with distance.


Earlier work on modeling pollutant concentrations in the near-road environment in Detroit \citep{Baldwin2015, gilani2016} has revealed that observed pollutant concentrations are dependent on day of the week, time of the day, being upwind versus downwind of major highways, meteorology, traffic activity and composition, and distance to the highway. Figure ~\ref{fig:mean_daily_log} show a comparison of log NRI in MAPL concentrations between upwind versus downwind sites, and between morning versus evening for both pollutants. Boxplots comparing the distribution of log NRI in MAPL concentrations between upwind and downwind sites, between morning and evening, and between weekday and weekend for both pollutants are given in Figure 2 in the Supplementary Material. Even though these plots do not reveal much difference in the log NRI pollutant concentrations between downwind and upwind sites, between morning and evening, and between weekdays and weekends, for consistency with earlier work \citep{gilani2016}, we model the mean spatial trend $\mu_t^{(k)}(\s)$ at location $\s$ and time period $t$ for each pollutant $k$ as 

\begin{align} 
\mu_t^{(k)}(\s) &= \beta_0^{(k)} + \beta_1^{(k)} \mbox{Down}_t(\s) + \beta_2^{(k)} \mbox{Weekday}_t + \beta_3^{(k)} \mbox{Morning}_t + \beta_4^{(k)} \mbox{Weekday}_t * \mbox{Morning}_t \nonumber \\
&+ \beta_5^{(k)} \left(\frac{\mbox{Traffic}_t(\s)}{\mbox{Wind Speed}_t(\s)} \right) + \beta_6^{(k)} \left(\frac{\mbox{Traffic}_t(\s)}{\mbox{Wind Speed}_t(\s)} \right) * \mbox{Down}_t(\s) + \beta_7^{(k)} \mbox{Distance}(\s)  \nonumber \\
& + \beta_8^{(k)} \mbox{Distance}(\s)* \mbox{Down}_t(\s), \label{eqn:mean_fn}
\end{align}
where Down$_t(\s)$ is an indicator for site $\s$ at time $t$ being downwind (versus upwind) of the highway; Weekday$_t$ an indicator for time period $t$ falling on a weekday (versus weekend); Morning$_t$ an indicator for morning (before 12:00) versus evening at time period $t$; $\frac{\mbox{Traffic}_t(\s)}{\mbox{Wind Speed}_t(\s)}$ a scalar that quantifies the amount of traffic on the nearest highway around $\s$ during time period $t$ normalized by the wind speed at site $\s$ during time period $t$; and Distance$(\s)$ records the distance of $\s$ from the nearest highway. This formulation allows different intercepts at downwind and upwind sites, as well as different gradients at downwind versus upwind sites for the covariates related to traffic normalized by wind speed $\left( \frac{\mbox{Traffic}_t(\s)}{\mbox{Wind Speed}_t(\s)}\right)$, and distance from nearest highway (Distance$(\s)$). We also assume that the temporal variability of log NRI concentration $Z^{(k)}_{t}(\s)$ is adequately captured by the time varying covariates in $\mu_t^{(k)}(\s)$, so that the residual error term $\eta_t^{(k)}(\s)$ is purely spatial in nature, and can be modeled  as independent realizations over time periods of a mean-zero spatial Gaussian process.

Previous studies modeling concentrations of atmospheric pollutants have observed differing patterns of spatial dependence depending on the pair of locations considered \citep[among others]{higdon1998, fuentes2001, paciorek2006, calder2007}, which indicates that we should model the residual process $\eta_t^{(k)}(\s)$ as a non-stationary spatial process. Recent work in modeling concentrations of various pollutants in the near-road environment \citep{gilani2016} has further indicated that certain spatio-temporally varying covariates, such as wind speed, might be important covariates that help account for the different spatial dependency patterns observed in a localized spatial domain that includes sites that are both upwind and downwind of major highways. Figure 3 in the Supplementary Material shows the empirical semivariogram of the residuals obtained from a linear regression model fitted to log NRI concentrations for the two pollutants separately using the mean function given in Equation \ref{eqn:mean_fn}, stratified by upwind and downwind sites. The figure provides evidence of a non-stationary residual process.  Therefore, in this study, for each pollutant $k$, we model the term $\eta_t^{(k)}(\s)$ as a non-stationary spatial process using a recently developed approach by \citet{gilani2016}, which builds upon the kernel mixing approach of \citet{fuentes2001} and a method proposed by \citet{schmidt2011} that allows inclusion of covariates in a covariance function. The new approach expresses pollutant concentrations as a mixture of two independent spatial processes, each equipped with a non-stationary covariance function with covariates driving the nonstationarity and the mixture weights.

Following the specification of \citet{gilani2016}, for each pollutant $k$, we model $\eta_t^{(k)}(\s), ~t=1, \ldots, \mathcal{T}$, as independent realizations in time of a weighted mixture of two mutually independent mean-zero Gaussian spatial processes $\eta_{1,t}^{(k)}(\s)$ and $\eta_{2,t}^{(k)}(\s)$, independent over time, with non-stationary covariance functions $C_{\boldsymbol{\theta}_1^{(k)}}(\cdot, \cdot)$ and $C_{\boldsymbol{\theta}_2^{(k)}}(\cdot, \cdot)$, respectively. That is, for each time period $t$ and site $\s \in \mathcal{S}$,
\begin{equation} \label{eqn:eta_mixture}
\eta_t^{(k)}(\s) = w_{1,t}(\s) \cdot \eta_{1,t}^{(k)}(\s) + w_{2,t}(\s) \cdot \eta_{2,t}^{(k)}(\s),
\end{equation}
where the spatially and temporally-varying weights $w_{1,t}(\s)$ and $w_{2,t}(\s)$ add up to 1, which is achieved by setting
\begin{equation*}
w_{i,t}(\s) = \frac{\widetilde{w}_{i,t}(\s)}{\widetilde{w}_{1,t}(\s)+\widetilde{w}_{2,t}(\s)}, \quad \quad i=1,2.
\end{equation*}
The un-normalized weights $\widetilde{w}_{1,t}(\s)$ and $\widetilde{w}_{2,t}(\s)$ are specified in such a way that when site $\s$ is downwind of the highway at time $t$, the spatial process $\eta_{1,t}(\s)$ receives a larger weight as compared to $\eta_{2,t}(\s)$, and vice versa. One weighting scheme that achieves this is given by
\begin{equation} 
\label{eqn:weights}
\widetilde{w}_{i,t}(\mathbf{s}) = \begin{cases} \mbox{exp} \big(-\psi \cdot (1-\mbox{Down}_t(\mathbf{s})) \cdot |\mathbf{s} - \mathbf{s^*}|\big), & i=1 \\
                                                                          \mbox{exp} \big(-\psi \cdot (\mbox{Down}_t(\mathbf{s})) \cdot |\mathbf{s}-\mathbf{s^*}|\big), & i=2, \end{cases}
\end{equation}
where $\mathbf{s^*}$ is defined as the projection of the site $\mathbf{s}$ onto the further edge of the highway.

To model the two mutually independent spatial processes $\eta_{1,t}^{(k)}(\s)$ and $\eta_{2,t}^{(k)}(\s)$ for each pollutant $k$ as non-stationary processes, where their non-stationarity is driven by a spatial covariate (e.g. wind speed), we define a non-linear 1-1 smooth transformation $\widetilde{\mathbf{f}}_t(\cdot)$ from the \emph{geographical} space $\mathcal{S} \subset \mathbf{R}^2$ to the \emph{deformation} space $\mathcal{D} \subset \mathbf{R}^3$ that, for each time period $t$, maps a site $\s = (\s_x, \s_y) \in \mathcal{S}$ onto $\widetilde{\mathbf{f}}_t(\s) = (\s_x, \s_y, h_t(\s))$, where $h_t(\s)$ is a spatial covariate. In this formulation, for any pair of sites $\s, \s' \in \mathcal{S}$, 

$$ \mbox{Cov}\left(\eta_{i,t}^{(k)}(\s), \eta_{i,t}^{(k)}(\s') \right)= C_{\mathcal{D},i}^{(k)}\left(\widetilde{\mathbf{f}}_t(\s), \widetilde{\mathbf{f}}_t(\s')\right) = C_{\mathcal{D},i}^{(k)}\left(|\widetilde{\mathbf{f}}_t(\s) - \widetilde{\mathbf{f}}_t(\s')|\right),  \quad i=1,2,$$
where $C_{\mathcal{D},i}^{(k)}(\cdot, \cdot)$ is a stationary, isotropic covariance function (e.g. a Mat\'{e}rn covariance function) on $\mathcal{D}$, and $|\widetilde{\mathbf{f}}_t(\s) - \widetilde{\mathbf{f}}_t(\s')|$ is the Mahalanobis distance between $\widetilde{\mathbf{f}}_t(\s)$ and $\widetilde{\mathbf{f}}_t(\s')$. \citet{schmidt2011} show that while the spatial processes $\eta_{1,t}^{(k)}(\widetilde{\mathbf{f}}(\s))$ and $\eta_{2,t}^{(k)}(\widetilde{\mathbf{f}}(\s))$ in the deformation space $\mathcal{D} \subset \mathbf{R}^3$ are stationary and isotropic (since the correlation between any two vectors $\widetilde{\mathbf{f}}(\s)$ and $\widetilde{\mathbf{f}}(\s')$ is dependent only on the distance $|\widetilde{\mathbf{f}}(\s) - \widetilde{\mathbf{f}}(\s')|$), the spatial processes $\eta_{1,t}^{(k)}(\s)$ and $\eta_{2,t}^{(k)}(\s)$ in the geographic space $\mathcal{S} \subset \mathbf{R}^2$ are non-stationary. 

Since we believe that the different spatial dependence behavior in the log NRI concentrations exhibited between sites that are upwind and downwind of major highways is driven by local meteorological factors, we define $h_t(\s)$ as signed wind speed at $\s$ at time period $t$, i.e., wind speed with a positive sign if $\s$ is downwind at time $t$ and with a negative sign otherwise. Finally, for each pollutant $k$ and time period $t$, we model the covariance function of the underlying process $\eta_{i,t}^{(k)}(\s)$ using an exponential covariance function with distance given by the Mahalanobis distance, so that for any pair of sites $\s, \s' \in \mathcal{S}$,
\begin{equation} 
\label{eqn:covf_eta}
\mbox{FigureCov} \left( \eta^{(k)}_{i,t}(\mathbf{s}), \eta^{(k)}_{i,t}(\mathbf{s^\prime}) \right) = \sigma^{2^{(k)}}_i \cdot \mbox{exp} 
   \left\{ -\sqrt{ \left( \widetilde{\mathbf{f}}_{t}(\mathbf{s})- \widetilde{\mathbf{f}}_{t}(\mathbf{s}^\prime) \right)^T \cdot \boldsymbol{\Phi}^{(k)}_i  \cdot  
   \left( \widetilde{\mathbf{f}}_{t}(\mathbf{s})- \widetilde{\mathbf{f}}_{t}(\mathbf{s}^\prime) \right)} \right\},  \quad i=1,2 .
\end{equation}
For each $i=1,2$, and each pollutant $k$, the matrix $\boldsymbol{\Phi}^{(k)}_i$ is a $3\times 3$ diagonal matrix with diagonal elements $(\phi^{(k)}_{i}, \phi^{(k)}_{i}, \varphi^{(k)}_{i})$ and off-diagonal elements equal to 0, where $\phi^{(k)}_{i}$ controls the range or smoothness of the covariance function along the x-y direction and $\varphi^{(k)}_i$ controls it along the signed wind-speed direction.

Under the above specification, the covariance function of the residual spatial process $\eta^{(k)}_{t}(\mathbf{s})$ is a mixture of two non-stationary covariance functions with non-stationarity driven by a covariate (i.e., signed wind speed), and with spatio-temporally varying weights that also depend on a covariate (i.e., being upwind or downwind). Thus, for each pollutant $k$, at time period $t$ and sites $\mathbf{s},\mathbf{s}^\prime \in \mathcal{S}$,

\begin{align}\label{eq:covfinal}
\mbox{Cov}&\left(\eta^{(k)}_t(\mathbf{s}),\eta^{(k)}_t(\mathbf{s}^\prime)\right) = \nonumber \\ 
& \sum_{i=1}^2 w_{i,t}(\mathbf{s}) w_{i,t}(\mathbf{s}^\prime) \cdot \sigma^{2^{(k)}}_i \cdot \mbox{exp} 
   \left\{ -\sqrt{ \left( \widetilde{\mathbf{f}}_{t}(\mathbf{s})- \widetilde{\mathbf{f}}_{t}(\mathbf{s}^\prime) \right)^T \cdot \boldsymbol{\Phi}^{(k)}_i  \cdot  
   \left( \widetilde{\mathbf{f}}_{t}(\mathbf{s})- \widetilde{\mathbf{f}}_{t}(\mathbf{s}^\prime) \right)} \right\}
\end{align}

The hierarchical Bayesian data fusion model for each pollutant is completed by assigning prior distributions to the various parameters in the process model (Equation \ref{eqn:general_decomp}) and the data models (Equations \ref{eqn:mapl_fn} and \ref{eqn:rline_fn}), as detailed below. Denoting the vector of process model parameters for pollutant $k$ by $\boldsymbol{\theta}_p^{(k)} = \{\boldsymbol{\beta}^{(k)}, \boldsymbol{\theta}_1^{(k)}, \boldsymbol{\theta}_2^{(k)} \}$, the vector of data models parameters by $\boldsymbol{\theta}_{\hat{Z}}^{(k)}$ and $\boldsymbol{\theta}_{\widetilde{Z}}^{(k)}$, and the vector of realizations of the true process, observed data, and model output collected over all time periods by $\mathbf{Z}^{(k)}, \hat{\mathbf{Z}}^{(k)}, \mbox{and } \widetilde{\mathbf{Z}}^{(k)}$, respectively, and noting that given $\mathbf{Z}^{(k)}, \hat{\mathbf{Z}}^{(k)}$ and $\widetilde{\mathbf{Z}}^{(k)}$ are independent, the joint distribution of all of the unknowns and available data (MAPL measurements and RLINE model output) can be decomposed as

\begin{align} \label{eqn:joint_distribution}
 p(\hat{\mathbf{Z}}^{(k)}, \widetilde{\mathbf{Z}}^{(k)},&~\mathbf{Z}^{(k)}, \boldsymbol{\theta}_p^{(k)}, \boldsymbol{\theta}_{\hat{Z}}^{(k)},  \boldsymbol{\theta}_{\widetilde{Z}}^{(k)}) = \nonumber \\
& p(\hat{\mathbf{Z}}^{(k)}|\mathbf{Z}^{(k)},\boldsymbol{\theta}_{\hat{Z}}^{(k)}).p(\widetilde{\mathbf{Z}}^{(k)}|\mathbf{Z}^{(k)},\boldsymbol{\theta}_{\widetilde{Z}}^{(k)}).p(\mathbf{Z}^{(k)}|\boldsymbol{\theta}_p^{(k)}).p(\boldsymbol{\theta}_{\hat{Z}}^{(k)}).p(\boldsymbol{\theta}_{\widetilde{Z}}^{(k)}).p(\boldsymbol{\theta}_p^{(k)})
\end{align}

Posterior estimates of the model parameters $\boldsymbol{\theta}_p^{(k)}, \boldsymbol{\theta}_{\hat{Z}}^{(k)}, \mbox{and } \boldsymbol{\theta}_{\widetilde{Z}}^{(k)}$, as well as prediction of the true underlying process $Z^{(k)}_t(\s_0)$ at an unsampled location $\s_0$ at time period $t$ can be obtained using Markov chain Monte Carlo (MCMC) approaches. 


\subsection{Prior specification} \label{priors}

For each pollutant $k$, the full model contains 21 parameters: nine regression coefficients $\beta^{(k)}_0, \ldots, \beta^{(k)}_8$ as given in Equation (\ref{eqn:mean_fn}); constant additive and multiplicative biases $a^{(k)}_0$ and $b^{(k)}$, respectively, as given in Equation (\ref{eqn:rline_fn}); a nugget effect $\tau^{2^{(k)}}_e$ for the MAPL data model given in Equation (\ref{eqn:mapl_fn}); a nugget effect $\tau^{2^{(k)}}_{\delta}$ for the RLINE data model given in Equation (\ref{eqn:rline_fn}); two spatial covariance parameters $\sigma^{2^{(k)}}_a$ and $\phi^{(k)}_a$ for the spatially correlated additive bias $a^{(k)}_t(\s)$ given in Equation (\ref{eqn:rline_fn}); and six spatial covariance parameters $\sigma^{2^{(k)}}_1, \phi^{(k)}_1, \varphi^{(k)}_1, \sigma^{2^{(k)}}_2,  \phi^{(k)}_2,  \varphi^{(k)}_2$. On all of these parameters except for $\tau^{2^{(k)}}_{e}$, we specify non-informative prior distributions. Following \citet{gilani2016}, the parameter $\psi$ in the definition of the mixture weights $w_{1, t}(\s)$ and $w_{2, t}(\s)$ given in (\ref{eqn:weights}) was kept constant and determined through empirical considerations. Specifically, we chose a value of $\psi$ such that if a site $\s$ is the furthest downwind site during time period $t$, the mixture weight $w_{2, t}(\s)$ is equal to about 0.05. 

The nugget effect $\tau^{2^{(k)}}_e$ provides the variability observed in the MAPL data due to measurement error. The precision of the monitoring equipment used to record concentrations of \nox and \pmt were therefore used to provide informative priors on these parameters. We place Inverse Gamma priors such that the prior mean of $\tau^{2^{(k)}}_e$ is equal to 10\% of the observed mean in NRI concentration, and the prior variance is very small. To specify the priors on the remaining 20 pollutant-specific model parameters, we use an empirical Bayes approach. Specifically, on each of the $\beta^{(k)}_l, l=0, \ldots, 8$, we place independent normal priors with large variances and means set equal to the estimates of the regression coefficients obtained by fitting a linear regression model on the MAPL data for each pollutant with the nine variables in Equation (\ref{eqn:mean_fn}) as the covariates. On the variance component parameters  $\sigma^{2^{(k)}}_1$ and $\sigma^{2^{(k)}}_2$, we place vague Inverse Gamma priors with shape and scale parameters chosen so that the prior variance is infinite, and the prior mean is equal to 40\% of the estimated residual variance from the linear regression model fit to get the prior means for the regression coefficients. On the variance component parameters $\sigma^{2^{(k)}}_a$ and $\tau^{2^{(k)}}_{\delta}$ for the RLINE data model given in Equation  (\ref{eqn:rline_fn}), we place vague Inverse Gamma priors with shape and scale parameters chosen such that the prior variance is infinite, and the prior mean is equal to, respectively, 40\% and 60\% of the estimated residual variance from a linear regression model on the RLINE data for each pollutant with the same nine covariates. On the parameters $\phi^{(k)}_1, \phi^{(k)}_2, \varphi^{(k)}_1, \varphi^{(k)}_2$ controlling the roughness/smoothness of the covariance function of the underlying spatial processes $\eta^{(k)}_{1,t}$ and $\eta^{(k)}_{2,t}, t=1, \ldots, \mathcal{T}$ in the $x-y$ and signed wind-speed direction, we place Gamma priors with shape and scale chosen so that the variance of the Gamma distribution is large while the prior mean is such that the correlation is equal to 0.05 at a distance equal to 3 divided by the prior mean. Specifically, we set the prior means to be approximately 3 divided by the maximum inter-monitoring sites distance and 3 divided by the maximum signed wind-speed ``distance". Finally, for the overall additive and multiplicative bias terms of the log RLINE output, $a_0^{(k)}$ and $b^{(k)}$, we used two vague independent normal priors with means 0 and 1, respectively, assuming a priori no bias in the log RLINE output.

For each pollutant $k$, we fit the full model by running an MCMC algorithm with Gibbs sampling for all the regression coefficients and variance component parameters, and with Metropolis-Hastings steps to update the spatial decay parameters. 

\subsection{Model Comparison} \label{modComp}

For each pollutant separately, we compared the predictive performance of six joint Bayesian data fusion models that differed in (a) the approach used to model the additive bias of the log RLINE output, and (b) the type of spatial dependence structure hypothesized for the two latent processes, $\eta^{(k)}_{1,t}(\s)$ and $\eta^{(k)}_{2,t}(\s)$. Table \ref{tab:modspec} provides a summary of the models considered. For the additive bias of the log RLINE output, three different approaches were used. The simplest model, called ``Model 1", assumes that the additive error of the RLINE output in representing the true, unobserved field is constant in space, thus $a^{(k)}_{t}(\s) \equiv 0$ in Equation \ref{eqn:rline_fn}. ``Model 2" postulates that even though the RLINE output has a spatially varying additive bias on the local scale (where local means a small region on the order of a couple of km around the urban highway), overall, the RLINE output does not have a global additive bias (where global means over the entire study region), i.e., $a^{(k)}_0 \equiv 0$. ``Model 3" is the full model as described in \S \ref{nonstat_bdf}. For each of these three models, we contrast two cases: the first assumes that $\eta^{(k)}_t(\s)$ is a stationary Gaussian process independent in time with mean 0 and with an exponential covariance function (with marginal variance $\sigma^{2^{(k)}}_Z$ and spatial decay parameter $\phi^{(k)}_Z$); while the second assumes that $\eta^{(k)}_t(\s)$ is equipped with the non-stationary covariance function described in Equation \ref{eqn:eta_mixture}. These cases are distinguished by appending ``S" or ``NS" to the model labels, respectively.

We evaluated the predictive performance of each of the six models for both pollutants through comparisons between observations and predictions at out-of-sample sites. Out of the 254 observed NRI in \nox concentrations from the MAPL data, we randomly selected 221 observations for model fitting, and held out 33 observations for model validation. Similarly, for \pmt we randomly selected 202 observations in NRI concentrations from the MAPL data for model fitting, and held out 33 observations for model validation. In each case, the held out observations were sampled so that we had 3 observations held out per time period. Since the modeling was conducted on the log-scale, we assessed the predictive performance by back transforming predictions to the original scale. Using the median of the posterior predictive distribution as the predicted value at each site, we evaluated the predictive performance of each model in terms of \emph{mean absolute prediction error} (MAPE), \emph{average length of the 90\% prediction interval} (PI), \emph{empirical coverage} of the 90\% PI, \emph{and the average bias}. The empirical coverage of the 90\% PI is used to assess whether the uncertainty in the prediction is correctly quantified: if empirical coverage is below the nominal level, assuming no bias in the predictions, the model is underestimating the variability/uncertainty in the predictions. Conversely, empirical coverage of the prediction intervals above the nominal level indicates that the model overestimates the variability/uncertainty, assuming no bias in the predictions. 


\begin{table}
\caption{\label{tab:modspec} Summary of the joint Bayesian data fusion models for Log NRI of \nox and \pmtns.}
\centering
\begin{tabular}{c | c c}
\toprule \toprule
  & \bf Form of additive & \\
\bf Model name & \bf bias for RLINE & \bf Covariance structure of $\eta^{(k)}_t(\s)$ \\
\midrule
Model 1-S & $a^{(k)}_{t}(\s) \equiv 0$ & Stationary: exponential \\
Model 2-S & $a^{(k)}_0 \equiv 0$ & Stationary: exponential\\
Model 3-S & Full & Stationary: exponential \\
Model 1-NS & $a^{(k)}_{t}(\s) \equiv 0$ & Non-stationary, as in Eq. \ref{eqn:eta_mixture}\\
Model 2-NS & $a^{(k)}_0 \equiv 0$ & Non-stationary, as in Eq. \ref{eqn:eta_mixture}\\
Model 3-NS & Full & Non-stationary, as in Eq. \ref{eqn:eta_mixture} \\
\bottomrule
\end{tabular}
\end{table}

For each fitted model for both pollutants, we examined trace plots of the posterior samples for the model parameters to verify that the MCMC algorithm had converged. For all six models for both pollutants, trace plots indicated that the posterior samples for all model parameters had converged within the first 10,000 iterations. We therefore treated the first 10,000 iterations as burn-in. We also examined the autocorrelation within posterior samples for each parameter. Posterior samples for the stationary models (with fewer model parameters) exhibited weaker autocorrelation than the non-stationary models. For each model, a thinning factor was used to ensure that the posterior samples used to determine the posterior distribution had very low autocorrelation. For the stationary models for both pollutants, a thinning factor of 12 was sufficient to ensure near independence of the thinned posterior samples. For the non-stationary models for \noxns, thinning factors of 64, 29, and 31 were used for Models 1, 2, and 3, respectively, while for \pmtns, thinning factors of 60, 62, and 53 were used for Models 1, 2, and 3. Each MCMC algorithm was run for a number of iterations large enough so that the posterior inference was based on posterior samples of size 1000 after thinning for each model. While some models needed considerably long chains, we did not observe any identifiability issue with any model.

The second goal of the study, i.e., evaluating whether incorporating RLINE output in our model improves estimation of pollutants concentration, can be assessed by examining the posterior distribution of $b^{(k)}$ in Equation \ref{eqn:rline_fn}. When $b^{(k)} \equiv 0$, the conditional distribution of $\mathbf{Z}^{(k)}$, given the MAPL and RLINE data and model parameters is equivalent to the conditional distribution of $\mathbf{Z}^{(k)}$, given the MAPL data and model parameters only. That is,  when $b^{(k)} \equiv 0$, $p(\mathbf{Z}^{(k)}| \hat{\mathbf{Z}}^{(k)}, \widetilde{\mathbf{Z}}^{(k)}, \boldsymbol{\theta}_p^{(k)}, \boldsymbol{\theta}_{\hat{Z}}^{(k)},  \boldsymbol{\theta}_{\widetilde{Z}}^{(k)}) \equiv p(\mathbf{Z}^{(k)}| \hat{\mathbf{Z}}^{(k)}, \boldsymbol{\theta}_p^{(k)}, \boldsymbol{\theta}_{\hat{Z}}^{(k)})$. Therefore, if the posterior distribution of $b^{(k)}$ is centered at 0 and has a small standard deviation, there isn't sufficient evidence that including RLINE output in the model helps in improving estimates of the true concentration of pollutant $k$.  In fact, when $b^{(k)} = 0$, Equation 3 is simply an ordinary kriging model fitted to the RLINE predictions, while Equations 1 and 2 together specify a universal kriging model for the MAPL pollution concentration process.  Similarly, the third goal of the study, i.e., evaluating whether RLINE output captures the spatial dependence structure of NRAP concentrations in the near road environment, can be assessed by comparing Model 1 to Models 2 and 3. If Model 1 outperforms Models 2 and 3, then the additive bias does not have any spatial pattern, so that RLINE model output would appear to capture the spatial variability exhibited by the true underlying pollutant concentration process. However, if Models 2 or 3 outperform Model 1, then there is evidence of some residual spatial structure that is not captured by the RLINE model output. 



\section{Results} \label{results}

Summaries of the predictive performance statistics of the joint Bayesian data fusion models for \nox and \pmt are shown in Table \ref{tab:modComparison}. No single model was clearly dominant with respect to all criteria for either pollutant, i.e., no single model had the lowest MAPE and the shortest PI with empirical coverage close to the nominal level. 

\begin{table}
%\centering
\caption{\label{tab:modComparison} \footnotesize Predictive performance of the joint Bayesian data fusion models averaged across validation sites and time periods. Table shows the mean absolute prediction error (MAPE), average length and empirical coverage of the 90\% prediction intervals (PI), and the average bias. Models are defined in Table \ref{tab:modspec}.} 
\centering
\scriptsize
\begin{tabular}{l l c c c c c c}
\toprule \toprule
 &\bf Criteria & \bf Mod 1-S &\bf Mod 2-S & \bf Mod 3-S & \bf Mod 1-NS &\bf Mod 2-NS & \bf Mod 3-NS \\
\midrule				
\multirow{4}{*}{\rotatebox[origin=c]{90}{\bf \nox }} & MAPE  &6.59  	&6.35	&7.00	&6.95	&6.69	&8.13 \\
 & Avg. length of 90\% PI	&23.81	&28.31	&32.49	&108.25	&33.98	&300.41\\
 & Emp. coverage of 90\% PI 	&0.84	&0.77	&0.77	&0.84	&0.74	&0.74 \\
& Avg. bias &  4.98 & 4.34 & 4.17 & 3.08 & 4.68 & 2.94 \\
\midrule
\multirow{3}{*}{\rotatebox[origin=c]{90}{\bf \pmt }} & MAPE  &1.11		&0.72	&0.78	&0.48	&0.42	&0.44\\
& Avg. length of 90\% PI	&4.72	&7.10	&7.66	&3.30	&4.87	&4.22\\
& Emp. coverage of 90\% PI	&0.55	&0.90	&0.86	&0.69	&0.66	&0.69\\
 & Avg. bias & -0.77	& -0.48 & -0.53 & -0.10 & -0.07 & -0.09 \\

\bottomrule 
\end{tabular}
\end{table}


\begin{table}
\caption{\label{tab:finMods} Posterior medians and 95\% credible intervals of regression coefficients $\beta_0, \ldots, \beta_8$ and covariance parameters estimated by the best fitting models (Model 2-S for \nox and Model 2-NS for \pmtns). Traffic is in adjusted number of vehicles per hour; wind speed is in m/s; and distance is in km.} 
\centering
%\footnotesize
\begin{tabular}{l rc rc}
\toprule \toprule
 &\multicolumn{2}{c}{\bf \nox}&\multicolumn{2}{c}{\bf \pmt}  \\
\cmidrule(r){2-3} \cmidrule(r){4-5} 
\bf Parameter &   \bf Median & \bf 95\% CI &    \bf Median & \bf 95\%CI  \\
\midrule
$\beta_0$ \quad Intercept &   1.65 & (1.26, 2.10) & -0.89 & (-2.06, 0.51) \\
$\beta_1$ \quad Down &   0.96 & (0.83, 1.11) & 1.37 & (-0.65, 3.36)\\
$\beta_2$ \quad Weekday &  0.57 & (0.03, 1.15) &  -0.25 & (-1.57, 1.10) \\
$\beta_3$ \quad Morning &  0.32 & (-0.43, 1.07) & 1.06 & (-0.88, 3.05) \\
$\beta_4$ \quad Weekday*Morning & -0.77 & (-1.67, 0.06) & -0.63 & (-2.79, 1.59) \\
$\beta_5$ \quad Traffic/WS ($10^{-5}$) &   5.78 & (2.71, 8.63) & 14.1 & (4.79, 24.3) \\
$\beta_6$ \quad Traffic/WS*Down ($10^{-5}$) &  -4.54 & (-6.04, -3.13) &   -15.7 & (-27.0, -5.22) \\
$\beta_7$ \quad Distance ($10^{-4}$) &  -15.1 & (-17.8, -12.6) &   -12.9 & (-19.2, -6.88)\\
$\beta_8$ \quad Distance*Down ($10^{-4}$) &  -3.01 & (-6.28, 0.15) &  -1.49 & (-8.44, 5.40)\\
\midrule
$a_0$ \quad Additive bias (RLINE)& - & - &- & -  \\
$b$ \quad  ~ Multiplicative bias (RLINE)& 1.25 & (1.11, 1.35) & 0.66 & (0.59, 0.75) \\
$\tau^2_e$ \quad Nugget (MAPL)&   0.42 & (0.42, 0.42) &0.06 & (0.06, 0.06)  \\
$\tau^2_{\delta}$ \quad Nugget (RLINE) & 0.07 & (0.05, 0.09) & 0.07 & (0.06, 0.09)   \\

$\sigma^2_{1}$ \quad Sill - Downwind & - & - & 6.34 & (2.70, 13.0) \\
$\phi_{1}$ \quad Downwind & - & - & 0.005$^a$ & (0.001, 0.030)  \\
$\varphi_{1}$ \quad Downwind, Wind speed & - & - & 65.8$^b$ & (6.18, 353)   \\
$\sigma^2_{2}$ \quad Sill - Upwind & - & - & 8.90 & (5.06, 14.3) \\
$\phi_{2}$ \quad Upwind & -  & - &0.07$^c$ & (0.02, 0.19)  \\
$\varphi_{2}$ \quad Upwind, Wind speed & - & - & 256$^d$ & (64.5, 765)   \\
$\sigma^2_Z$ \quad Sill &  0.49 & (0.35, 0.69) & - & - \\
$\phi_Z$ \quad Decay & 1.01$^e$ & (0.65, 1.52) & - & - \\
$\sigma^2_a$ \quad Sill (Additive bias - RLINE) & 0.60 & (0.43, 0.92) & 1.01 & (0.68, 1.69) \\
$\phi_a$  \quad Decay (Additive bias - RLINE) & 0.78$^f$ & (0.44, 1.26) & 0.15$^g$ & (0.07, 0.26)\\
\bottomrule 
\multicolumn{5}{l}{Effective range for spatial correlation: $^a$: 42.4 km; $^b$: 0.37 m/s; $^c$: 11.3 km; $^d$: 0.19 m/s;} \\
\multicolumn{5}{l}{ \mbox{$^e$: 2.97 km;} $^f$: 3.85 km; $^g$: 7.75 km.} \\
\end{tabular} 
\end{table}


\subsection{Nitrogen Oxides}

For \noxns, the stationary models provided better predictive performance than the non-stationary counterparts, and Model 2-S yielded the lowest MAPE and 90\% PIs that have an empirical coverage that was the second closest to the nominal coverage, even though on average they were wider than the 90\% PIs from Model 1-S  (estimated parameters for Model 2-S are shown in Table \ref{tab:finMods}.)  While earlier studies have suggested non-stationarity in the concentrations of pollutants such as \nox  across major highways \citep{gilani2016}, these results suggest that the residual process for \emph{near-road increment} in \nox concentrations for sites across major highways is stationary.

The estimated coefficient for the multiplicative bias $\hat{b}$ is 1.25 (95\% CI: 1.11, 1.35), which is statistically significantly greater than 1. This estimate suggests that including RLINE output in our model does help improve our estimate of the true NRI concentrations, and also confirms our initial belief that the RLINE model tends to provide higher predictions for NRI in \nox concentrations than the true NRI concentrations. 

The superior performance of Model 2 (with a mean-zero spatially varying additive bias term for the RLINE model) suggests that while the additive bias on average is negligible, there is some spatial structure to the bias term, so that RLINE output does not fully capture the spatial dependence structure of NRI in \nox concentrations. The estimate of the marginal variance of the spatially-varying error $a_t(\s)$ of the log RLINE output $\hat{\sigma}_a^2$ is 0.60 (0.43, 0.92), while the spatial correlation of this bias term decays fairly slowly, with an effective range of 3.85 km.   

The estimated nugget effect $\hat{\tau}^2_e$ is 0.42 (0.42, 0.42), which provides a measure of the variability in the measurement error of the MAPL measurements. This is greater than the estimated nugget effect for the RLINE model output, ($\hat{\tau}^2_{\delta}$ is 0.07 (0.05, 0.09)), suggesting smaller residual variability in the RLINE model output after accounting for multiplicative bias as compared to MAPL measurements.

The effective range of the stationary model error $\eta_t(\s)$ is 2.97 km, so that the errors for sites within the same area (2 km x 2 km) are fairly correlated. This provides evidence that while the model covariates, MAPL measurements, and RLINE output when combined together account for the large scale spatial variation in log NRI concentrations for \noxns, there is still some small scale residual spatial variability that they do not account for. 

As Table \ref{tab:finMods} shows, the estimated coefficient for downwind indicator ($\hat{\beta}_1$) is positive; controlling for other variables in the model, downwind NRI \nox concentrations are on average exp(0.96) = 2.61 ppb greater than at upwind sites. Similarly, the estimated coefficient for weekday indicator ($\hat{\beta}_2$) is positive, while the morning indicator ($\hat{\beta}_3$) and the interaction term between morning and weekday ($\hat{\beta}_4$) are not significantly different from 0. This means that controlling for other variables, there is no difference in the NRI concentrations of \nox between mornings and evenings, while NRI concentrations on weekdays tend to be 1.77 ppb higher on average than on weekends. As expected, the estimated coefficient for distance to highway ($\hat{\beta}_7$) is negative, and the effect of distance is not different between upwind and downwind sites ($\hat{\beta}_8$ is not significantly different from 0).

% Observed/Model output from MAPL and RLINE at Area 8 on 12/20/12
\begin{figure}[h!t]
	\centering
		\includegraphics[width=110mm]{Figures/NOxPlots/area_8_transect_rline_map_NOx_ab.png}\\
		\includegraphics[width=\textwidth]{Figures/NOxPlots/Mod3_area_8_pred_map_median_CI_NOx_b.png}
	\caption{NRI in \nox concentration (in ppb) (a) monitored  at MAPL sites and (b) output from RLINE model at area 8 near I-94 in Detroit, MI at 8:00 am on Thursday, December 20, 2012. (c) Predictions of hourly NRI in \nox concentration (in ppb) at area 8 at 8:00 am on Thursday, December 20, 2012, as yielded by Model 2-S. (d) Lower and (e) upper bounds of the 90\% prediction interval.}
	\label{fig:observed_area8_nox}
\end{figure} 


Figure~\ref{fig:observed_area8_nox}(a) shows NRI in \nox concentrations as monitored at MAPL sites at area 8 near I-94 on the morning of December 20, 2012, while Figure~\ref{fig:observed_area8_nox}(b) shows output from the RLINE model for the same area. The relative size of the bubbles in Figure~\ref{fig:observed_area8_nox}(a) are proportional to the magnitude of the observed concentration. As noted earlier, RLINE model output provides much higher estimates of the NRI in \nox concentrations, particularly at downwind sites close to the highway as compared to the monitored MAPL observations. Figure~\ref{fig:observed_area8_nox}(c) shows predictions of hourly NRI in \nox concentrations obtained using Model 2-S for the same region and time period, while Figure~\ref{fig:observed_area8_nox}(d) and (e) show the lower and upper bounds, respectively, of the 90\% prediction intervals. Predictions from the best fitting Bayesian data fusion model are much closer to the monitored MAPL concentrations, while they also capture the influence of secondary roads provided by the RLINE model output that was not captured by MAPL observations. 

\subsection{Fine Particulate Matter}
For \pmtns, the situation was different: the joint Bayesian data fusion models with non-stationary covariance functions provided strikingly better predictive performance than the stationary models.  However, just as for \noxns, no model was better in terms of all three criteria (MAPE, average length and empirical coverage of 90\% PI).  Weighting the MAPE criterion the most, Model 2-NS yielded the best predictions. The estimated coefficient for the multiplicative bias of the RLINE output $\hat{b}$ is 0.66 (95\% CI: 0.59, 0.75), which suggests that for \pmtns, just as for \noxns, including RLINE output in the model helps in estimating the log NRI in true pollutant concentrations, while again confirming our initial belief that the RLINE model tends to provide lower predictions for NRI in \pmt concentrations than the true NRI concentrations. 

Similar to \noxns, Model 2-NS hypothesized a null overall additive bias, but localized additive error at each location.  The spatially varying additive bias of RLINE had a small marginal variance ($\hat{\sigma}_a^2$ is 1.01 (0.68, 1.69)) and, similar to \noxns, a fairly large effective range of 7.75 km (Table \ref{tab:finMods}). RLINE output for \pmt does not appear to fully capture the spatial dependence structure of NRI concentrations, and the residual spatial correlation seems to persist for fairly large distances.

The estimated nugget effects for the MAPL measurements and the RLINE output are very similar ($\hat{\tau}^2_e$ is 0.06 (0.06, 0.06) and $\hat{\tau}^2_{\delta}$ is 0.07 (0.06, 0.09)), suggesting similar residual variability after the multiplicative error has been controlled for. The decay parameters $\phi_1$ and $\phi_2$ of the underlying spatial processes $\eta^{(k)}_{1,t}(\s)$ and $\eta^{(k)}_{2,t}(\s)$ relative to the $x - y$ direction were fairly different, with effective ranges of 42.4 km for sites that are downwind, and \mbox{11.3 km} for upwind sites.  Spatial correlation decays extremely slowly with increasing distance for sites that are upwind, and even slower for downwind sites. Similarly, the decay parameters in the signed wind speed direction, $\varphi_1$ and $\varphi_2$, are also fairly different for the two underlying spatial processes, with effective ranges of 0.37 m/s ($\eta^{(k)}_{1,t}(\s)$) and 0.19 m/s ($\eta^{(k)}_{2,t}(\s)$). Correlation between upwind sites decays rather rapidly with increasing signed wind speed, while more slowly for downwind sites. 

Examining the estimated regression coefficients in Table \ref{tab:finMods}, we observe that indicators for downwind, weekday, morning, and interaction for morning and weekday are all not significantly different from 0. As expected, NRI in \pmt concentrations decreases with increasing distance from highway; and the effect is similar between upwind and downwind sites. It is interesting to note that while NRI concentrations do not appear to differ between upwind and downwind sites (as also seen in Figure~\ref{fig:mean_daily_log}(d)), the spatial dependence structure is non-stationary, driven by wind speed and direction. 

\begin{figure}[h!t]
	\centering
		\includegraphics[width=110mm]{Figures/PM25Plots/area_8_transect_rline_map_pm25.png}\\
		\includegraphics[width=\textwidth]{Figures/PM25Plots/Mod3_NS_area_8_pred_map_median_CI_PM25_sameWS.png}
	\caption{NRI in \pmt concentration (in $\mu$g/m$^3$) (a) monitored  at MAPL sites and (b) output from RLINE model at area 8 near I-94 in Detroit, MI at 8:00 am on Thursday, December 20, 2012. (c) Predictions of hourly NRI in \pmt concentration (in $\mu$g/m$^3$) at area 8 at 8:00 am on Thursday, December 20, 2012, as yielded by Model 2-NS. (d) Lower and (e) upper bounds of the 90\% prediction interval.}
	\label{fig:observed_area8_pm25}
\end{figure} 

Figure~\ref{fig:observed_area8_pm25}(a) shows NRI in \pmt concentrations as monitored at MAPL sites at area 8 near I-94 on the morning of December 20, 2012, while Figure~\ref{fig:observed_area8_pm25}(b) shows output from the RLINE model for the same area. The relative size of the bubbles in Figure~\ref{fig:observed_area8_pm25}(a) are proportional to the magnitude of the observed concentration. As noted earlier, RLINE model output provides lower estimates of the NRI in \pmt concentrations, particularly at upwind sites further away from the highway as compared to the monitored MAPL observations. Figure~\ref{fig:observed_area8_pm25}(c) shows predictions of hourly NRI in \pmt concentrations obtained using Model 2-NS for the same region and time period, while Figure~\ref{fig:observed_area8_pm25}(d) and (e) show the lower and upper bounds, respectively, of the 90\% prediction intervals. Predictions from the best fitting Bayesian data fusion model are much closer to the monitored MAPL concentrations, while, just as for the predictions for \noxns, they also capture the influence of secondary roads provided by the RLINE model output that was not captured by MAPL observations. 

\section{Discussion} \label{discussion}

In this paper, we have presented a non-stationary spatio-temporal Bayesian data fusion model for near-road increments in concentrations of two NRAPs (\nox and \pmtns) with the goals of: (i) estimating the true underlying NRI pollutant concentration process and predicting concentrations at unsampled sites; (ii) evaluating whether the RLINE dispersion model output provides additional useful information in estimating the true process; and (iii) if so, whether the RLINE model adequately captures the spatial dependence structure of NRI in these pollutants' concentrations.  The data fusion model is specified using the joint modeling framework, with non-stationarity modeled using an approach that allows incorporation and identification of covariates that might drive the non-stationarity.  Results from fitting the Bayesian data fusion models suggest that while the RLINE output provides additional information that is helpful in estimating the true unobserved NRI field for both \nox and \pmtns, it needs to be calibrated to be considered as a good representation of the true unobserved NRI field for these pollutants. However, the bias adjustment mostly consists of multiplicative correction (or rescaling) of the RLINE output for both pollutants since the best performing model for both pollutants (in terms of MAPE) were models that did not include a spatially constant additive bias.  For both pollutants, RLINE displayed a spatial error that has a small to moderate marginal variance and a fairly slow-decaying spatial correlation. 

Predictions from the final model can provide improved estimates of NRAP concentrations for use in health studies. As noted earlier, adverse health effects due to exposure to NRAPs close to traffic sources such as major roads and highways differ significantly from exposure at background levels. Therefore, there is increasing interest in monitoring near-road increments in pollutant concentrations. However, monitored data on the near-road increment in pollutant concentrations are difficult to obtain. In our study, we model near-road increments in pollutant concentrations using a novel dataset on observed concentrations of pollutants in the near-road environment collected via a mobile air pollution lab. Other studies have also attempted to obtain data on pollutants in the near-road environment. Similar spatio-temporal models might be applied to data collected in such mobile monitoring campaigns, especially if repeated measurements for the same season and time of day (e.g., rush hour) are obtained. Recently, an initiative was launched to use a Google Street View car equipped with air pollution monitors to collect such data in the Oakland, California area \citep{apte2017}. While such programs can provide useful data, our study was specifically designed to investigate the near-road increment during the critical rush hour period.  Additionally, a key advantage of our modeling approach over using the RLINE model directly (which readily provides estimates of NRI in pollutants concentration) is that in addition to accounting for the multiplicative bias in RLINE output, predictions from our model are accompanied by appropriate quantification of uncertainty in the prediction, which is highly relevant when improving estimates of NRAP exposures in epidemiological studies given the nature of exposure measurement errors.  

Our results are based on short-term measurements on transects across a variety of major roads in Detroit collected during wintertime morning and afternoon rush hour periods.  Transect locations were selected to represent predominantly residential areas across the city, and the selected roads differ significantly with respect to traffic volume and fleet mix.  In these ways, the collected data are representative and relevant to health studies using cohorts based on residence location.  Areas with large industrial emissions sources, which are common in the Detroit area, were avoided.  Still, \nox and \pmt are emitted from numerous sources in addition to the local roads studied, including point sources, and non-road mobile, area and regional sources.  Results of our analysis should, therefore, be interpreted and generalized cautiously given the limited spatio-temporal nature of the observed data.  

Fitting our model using additional datasets on observed pollutant concentrations would be useful for extending our findings.  In particular, applications using long-term (e.g., seasonal to annual average) observations might be particularly revealing, especially since it is difficult for dispersion models to attain good performance at short averaging periods \citep{milando2018}.  However, spatio-temporal models require a sufficient number of sampling locations to estimate parameters.  Unfortunately, few datasets provide the sufficient spatial coverage needed to develop and assess long-term urban-scale spatio-temporal models, and current monitoring networks do not capture the small-scale gradients of NRAP concentrations.  There may be applications using satellite data, other mobile monitoring platforms, or possibly low-cost monitoring sensors if sufficient spatial resolution, accuracy, sensitivity and selectivity for NRAPs can be achieved.  In particular, new tools have been developed to cheaply monitor atmospheric pollutants, but with relatively low accuracy \citep{kumar2015}. If the range of measurement errors for these new gauges are known, our approach can be readily used to monitor NRI in pollutants concentrations across transects more frequently, so that RLINE model output can be calibrated over longer time durations and wider spatial domains.


\section*{Acknowledgements}

All three authors were supported by a grant from the Health Effects Institute (HEI - Assistance Award No. R-82811201), an organization jointly funded by the United States Environmental Protection Agency (EPA) and certain motor vehicle and engine manufacturers. The contents of this article do not necessarily reflect the views of HEI, or its sponsors, nor do they necessarily reflect the views and policies of the EPA or motor vehicle and engine manufacturers.



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