Article pubs.acs.org/est
Quantile-Based Bayesian Maximum Entropy Approach for Spatiotemporal Modeling of Ambient Air Quality Levels Hwa-Lung Yu* and Chih-Hsin Wang Department of Bioenvironmental Systems Engineering, National Taiwan University, Taipei 10617, Taiwan S Supporting Information *
ABSTRACT: Understanding the daily changes in ambient air quality concentrations is important to the assessing human exposure and environmental health. However, the fine temporal scales (e.g., hourly) involved in this assessment often lead to high variability in air quality concentrations. This is because of the complex short-term physical and chemical mechanisms among the pollutants. Consequently, high heterogeneity is usually present in not only the averaged pollution levels, but also the intraday variance levels of the daily observations of ambient concentration across space and time. This characteristic decreases the estimation performance of common techniques. This study proposes a novel quantilebased Bayesian maximum entropy (QBME) method to account for the nonstationary and nonhomogeneous characteristics of ambient air pollution dynamics. The QBME method characterizes the spatiotemporal dependence among the ambient air quality levels based on their location-specific quantiles and accounts for spatiotemporal variations using a local weighted smoothing technique. The epistemic framework of the QBME method can allow researchers to further consider the uncertainty of space-time observations. This study presents the spatiotemporal modeling of daily CO and PM10 concentrations across Taiwan from 1998 to 2009 using the QBME method. Results show that the QBME method can effectively improve estimation accuracy in terms of lower mean absolute errors and standard deviations over space and time, especially for pollutants with strong nonhomogeneous variances across space. In addition, the epistemic framework can allow researchers to assimilate the site-specific secondary information where the observations are absent because of the common preferential sampling issues of environmental data. The proposed QBME method provides a practical and powerful framework for the spatiotemporal modeling of ambient pollutants.
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INTRODUCTION
understanding of the spatiotemporal distribution of ambient air quality concentrations.14,15 Researchers have extensively studies the spatiotemporal distribution of ambient pollutants using various stochastic models e.g., landuse regression method,4,16−19 generalized additive model,6,20,21 and geostatistical methods.14,15,22−25 The landuse regression (LUR) method establishes a linear relationship between air quality measures and geographical landuse information and generates maps of average air quality at a high spatial resolution.13,26−29 Because the data of temporal changes in landuse is limited, LUR generally quantifies the long-term average (e.g., yearly or multiyearly average) of air quality levels across space,17,28−30 though some attempts was proposed to integrate temporal information into the framework of the LUR method.15,16 The generalized additive model (GAM) has been widely used for time series analysis for environmental health studies (e.g., estimation of the health effect of the exposure to air pollutants).31−34 The GAM provides a flexible way to assess the
Many studies have shown that increased ambient pollutants have an adverse effect on human and environmental health.1−3 Researchers have developed spatiotemporal techniques to achieve greater accuracy and resolution in measuring ambient pollution concentration for the purpose of exposure assessment. However, the nonstationarity of spatiotemporal processes is a major challenge in the spatiotemporal modeling of air pollution, especially for processes with fine temporal scales (e.g., hour or day). This is because of the complex short-term generating and interacting dynamics among physical and chemical processes. Studies have shown that the spatial distribution of pollution depends on the geographical distribution of various landuse patterns and emission sources.4,5 The temporal heterogeneity of ambient concentrations can appear in various scales, including seasonal and short-term changes that are associated with the temporal variations of meteorological variables in the corresponding scales. The nonstationary characteristics of averaged ambient pollutant concentration across space and time has been the major focus of ambient pollution modeling6−13 (i.e., nonstationarity in the mean trend). The nonstationarity of variances across space and time is also important to the © 2012 American Chemical Society
Received: Revised: Accepted: Published: 1416
July 6, 2012 December 17, 2012 December 19, 2012 December 19, 2012 dx.doi.org/10.1021/es302539f | Environ. Sci. Technol. 2013, 47, 1416−1424
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nonlinear association between a dependent variable and its timevarying covariates using nonparametric local smoothing functions.35 The GAM is also suitable for the estimation of spatiotemporal ambient pollution by integrating random effects (i.e., generalized additive mixed model, GAMM) or a Markov random field to account for the spatial effects among multiple time series.20,21,36 Despite the popularity of the GAM and LUR methods, they generally focus on the large-scale trends of ambient pollutant observations in space and time, and fail to rigorously consider the spatiotemporal dependence of the spacetime processes.16,37 Geostatistical methods can account for not only parametric or nonparametric trends, but also the dependence structure of a space-time process. Therefore, these methods have been one of the most popular methods in the spatiotemporal mapping of ambient pollutants.12,22,24,25,38−43 Although nonstationarity in the trend of averaged space-time process can be considered by common geostatistical methods (e.g., universal kriging42), the spatiotemporal dependence function is commonly assumed to be homogeneous and stationary across the entire study domain.22,39 This assumption of a stationary dependence function implies that the self-similarity pattern of air pollution levels is consistent across space and time in the study domain. This assumption can be problematic and therefore degrade the accuracy of spatiotemporal estimation. Studies have shown that the dependence structure of ambient pollutants can also be associated with seasonality or meteorological variables.14,15 Similarly, the nonhomogeous variance of ambient pollutants across space is also prevalent, particularly in studies including areas with varying emission intensities. For example, the intensity of traffic emissions changes significantly between urban and rural areas, leading to the nonhomogeneity of pollution variance across space. To account for the nonstationarity and nonhomogeneity of spatiotemporal ambient pollution processes, we propose a novel quantile-based geostatistical approach based on the Bayesian maximum entropy (BME) framework44,45 (i.e., Quantile-BME or QBME). The proposed approach considers both general and specific knowledge for spatiotemporal mapping of ambient concentration: (1) the nonstationarity of spatiotemporal variation at various temporal scales, (2) spatiotemporal dependence among the location-specific quantiles of the observations, and (3) site-specific information of ambient pollutant measurements, which can be certain or uncertain. This study applies the proposed method to the spatiotemporal modeling of the selected criteria ambient pollutants (i.e., CO and PM10) across the entire Taiwan for the period of 1998−2009. The selected pollutants show distinct spatial patterns of space-time nonstationarity resulting from the varying in urbanization levels across Taiwan.
Figure 1. Spatial distribution of monitoring stations and soft data of CO and PM10 across Taiwan.
concentration levels of ambient pollutants and atmospheric chemicals, and serves as a part of the NASA/AERONET global network.46 The ambient pollutant data used in this study (i.e., CO, and PM10) were obtained by aggregating the hourly observations at each of the TWEPA stations and the Mt. Lulin station separately. The TWEPA and Mt. Lulin data are available for the periods of 1998−2009 and 2006−2012, respectively. The data aggregation follows the “18 h” rule, which states that the daily data can be derived only if at least eighteen hours of data are available for that day.47 Because of the preferential sampling issue of ambient pollutant monitoring, soft data (i.e., uncertain data) were generated at selected locations to complement the scarcity of air quality observations in mountainous areas (i.e., areas situated higher than 1000 m). This study represents the estimated concentrations at the soft data locations in the probabilistic form, which is derived monthly from observations at the Mt. Lulin station. In other words, the probability distribution of daily data at all soft data locations for a certain month during 2006−2012 were estimated from the weighted histograms of the daily observations of Mt. Lulin station for that month (Figures 2a and b). The Mt. Lulin station data show no increasing or decreasing trends in background ambient pollutant concentrations during the period of 2006−2012. This study assumes that the mountainous air quality level follows the same observed stationary pattern during the period when the Mt. Lulin station is absent (i.e., 1998−2005). In other words, the estimated daily data at the soft data locations for a certain month from 1998 to 2005 are represented in a probabilistic form from the weighted
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MATERIALS AND METHODS Ambient Pollutant Data. Air pollution has been one of the major environmental issues in Taiwan, the densely populated island with complex terrain. More details of the study area is shown in Supporting Information. For the purposes of air quality control, the Taiwan Environmental Protection Agency (TWEPA) established an air quality monitoring network in 1994, with many stations placed in densely populated areas (Figure 1). These stations regularly record the level of various pollutants, including PM10, ozone, NOx, SO2, and CO. In 2006, an air quality background station was established at Mt. Lulin, which is at 2862 m in elevation, and lies in the central mountain ridge of Taiwan. The Mt. Lulin station records the background 1417
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Figure 2. Probabilistic form of the soft data in February for (a) CO and (b) PM10 during April 2006−2009 and (c) CO and (d) PM10 during 1998− April 2006.
site-specific KB (i.e., the operational Bayesian updating in the second part of 1).52 The terms g and ξS are the inputs of 1, whereas the unknowns include μ and f K across space-time. Spatiotemporal estimation of air pollution. In this study, we estimate the spatiotemporal distribution of ambient pollutants using the BME framework with S-KB, as discussed in the previous section. This estimation includes the hard and probabilistic data of two pollutants (Figure 2). The G-KB includes the spatiotemporal trends and location-specific quantile-based covariance of the ambient pollutants. This study decomposes the time series observations at every station into the additive components of multiyear trend, seasonal trend, and residuals using the STL algorithm,53 (i.e., the seasonal decomposition of time series by a local weighted regression method (LOESS)54,55). The local temporal trend at each station was obtained by the addition of the multiyear and seasonal trends. The geostatistical framework commonly assumes a spacetime process to be the combination of its deterministic trend and the stochastic residuals.10,56,57 Following this assumption, the spatiotemporal trend across Taiwan can be obtained from the spatial interpolation of the temporal trend components without considering their stochastic dependence. The modeling of spatiotemporal covariance function among residuals typically involves the stationary and homogeneous assumption. However, residuals usually present various degrees of variability across space because of the distinct emission patterns among the station locations. To remove variance nonstationarity, we apply the normal score (NS) transform to the residuals at each station to produce a new set of values that have the same variance at all locations. The NS transform converts an arbitrary data distribution into a Gaussian distribution with zero mean and unit variance by establishing a NS table that contains one-to-one relationships between the quantiles of two random variables.49,58 This approach generates a number of NS values and their corresponding NS tables with the same predetermined quantiles at each observed location. In other words, the NS
histogram of the data observed at Mt. Lulin for the corresponding month from 2006 to 2010 (Figures 2c and d). BME Method. Geostatistical methods consider air pollution attributes (e.g., CO and PM10) in terms of spatiotemporal random fields (S/TRF48). Let Xp = Xs,t denote the S/TRF of an air pollution attribute; the vector p = (s,t) denotes a spatiotemporal point, where s is the geographical location and t is the time. The S/TRF model represents the collection of all physically possible realizations of the attribute we seek to represent mathematically. The BME method is based on the epistemic framework that distinguishes the knowledge bases (KBs) of ambient pollutants into (a) the general KB, denoted by G-KB, which includes physical and biological laws, primitive equations, and space-time moments, and (b) the site-specific KB (S-KB), which includes observations across space-time with exact numerical values (hard data) and uncertain formats (in which the data are not represented by a unique data value available, but an interval or probabilistic distribution of possible values).49 The BME method integrates both knowledge bases (i.e., K = G ∪ S) for the spatiotemporal estimation using the equation (for further details of the BME theory, refer to refs 45, 50, and 51) ⎫ ⎪ ⎬ − AfK (χ ) = 0 ⎪ ⎪ ⎭ T
∫ d χ (g − g ̅ )e μ g = 0 T
∫ dχξSe μ g
(1)
where g is a vector of the gα-functions of χ that are α stochastic constraints of the G-KB under consideration (the bar denotes statistical expectation), μ is a vector of μα-coefficients that depends on the space-time coordinates and is associated with g (i.e., μα expresses the relative significance of each gα-function in the composite solution sought), ξS represents the S-KB available, A is a normalization parameter, and f K is the pollutant probability density function (pdf) at each space-time point. Here, the subscript K means that f K is based on the blending of the core and 1418
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projections can vary across space. In practice, the quantiles of the original space of the NS table at the unmonitored location can be estimated using the BME method with spatiotemporal covariance of the original space. Following the BME estimation of the transformed space, the stochastic part of spatiotemporal process can be obtained by the back-transform of the estimated NS values with respect to the NS tables based upon estimated quantiles at the locations of interest. To assess the estimated NS table at unmonitored locations, Kullback−Leibler (K−L) divergence can be used to measure the “distance” between the pdfs with the true and estimated quantiles. The K−L divergence can be expressed using the equation I(p,q) = ∑kn = 1pk log(pk/qk), where pk and qk represent the pdfs in terms of the quantiles of the BME estimates and the observations, respectively. The goodness-of-fit test is usually applied to determine if two pdfs come from the same random variable. The K−L divergence test can use the chi-square distribution with n − 1 degrees of freedom.59 In summary, this study presents a spatiotemporal approach to account for the nonstationary and nonhomogeneous properties of both deterministic trends and stochastic residuals under the BME framework. More specifically, we consider the multiscale temporal trends and the spatial variation of the quantiles of random variables (i.e., QBME) in this study. The stochastic estimation of the QBME method is performed in a NS transformed space in which the projection between the quantiles of the original space-time random variable and the Gaussian variable can vary with respect to geographical locations.
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RESULTS The STL algorithm can decompose the complex temporal variations of CO and PM10 at a TWEPA station (i.e., Chungshan station) into trends in different temporal scales (i.e., multiyear, season, and the residuals) (Figure S1 in the Supporting Information). The higher concentration levels of both PM10 and CO during spring and winter compared to summer and autumn indicate the strong seasonality of these pollutants. The distinction between the trend and residuals can result in the symmetric and Gaussian-like distributions of the residuals of both CO and PM10, as in many popular geostatistical approaches (Figure S2). Examples include the kriging method, and the BME method with the G-KB of only the first two statistical moments of the data set.11,60 However, the spatial distribution of the standard deviations of the residuals of both PM10 and CO exhibit nonhomogeneity in Figure 3, which shows that the standard deviations of CO are larger in highly urbanized areas (e.g., Taipei, Kaohsiung, and Taichung). To account for this nonhomogeneity of the spatial variances among the stochastic residuals, the NS transform converts the residuals at each station into standard Gaussian distributions. The K−L divergence tests shown in Supporting Information Table S1 cross-validate the NS table estimations, showing that the NS tables at unmonitored locations across the entire study domain can be reasonably approximated in this study. To characterize the spatiotemporal dependence among the CO and PM10 residuals in both original and transformed space, the G-KB of BME framework uses stationary nested covariance (details in Supporting Information). Figure S3 shows the spatiotemporal covariances of CO and PM10 in the original, NS transform, and location-specific NS transform space, respectively. This performance assessment was performed by applying leave-one-out cross-validation in both CO and PM10 estimations using 1000 randomly sampled days at all of the TWEPA stations from 1998 to 2009. Table 1 shows a
Figure 3. Spatial distribution of the data variances of (a) CO and (b) PM10.
Table 1. Comparison among the Cross-Validation Results of the Three Spatiotemporal Approaches pollutant CO (ppm)
PM10 (μg/ m3)
methods BME with original data BME with NS-transformed data QBME BME with original data BME with NS-transformed data QBME
mean absolute error
standard deviation
0.0765 0.0715
0.0879 0.0785
0.0604 6.8537 5.8416
0.0641 7.4272 6.3213
5.8107
6.2882
comparison of the cross-validation results, including the three different approaches: (1) the BME method with the data in the original space, (2) the BME method with the data in the NStransformed space, and without considering the spatial variation 1419
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of NS projection (i.e., a single NS table is applied for the entire study domain), and (3) the BME method with the data in the NS-transformed space, which considers the nonhomogeneity of NS projection across space (i.e., QBME method that NS tables can vary with respect to the locations of interest). Previous studies have presented the details of the first two approaches.11,49 Figure 4 shows the spatial distribution of the mean absolute error (MAE) using the three approaches for CO estimation. Figure 5 shows the corresponding MAE distribution figures for PM10. Figures 6 and 7 show the spatiotemporal distribution of PM10
Figure 5. Spatial distribution of the mean absolute error of PM10 estimation in (a) original space, (b) NS-transformed space, and (c) location-specific NS-transformed space.
and CO on four selected days in 2002 (i.e., the first day of January, April, July, and October, respectively).
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DISCUSSION The stationary and homogeneous assumption has been widely used in geostatistical modeling to reduce the degree of freedom of complex and stochastic environmental systems. However, the complexity of natural systems seldom exhibits the expected stationary property. In the case of ambient pollutants, the spatiotemporal variation of the concentration level is typically the result of emissions from various sources along with many physical and chemical processes in various spatial and temporal scales interacting internally and externally. In this study, we
Figure 4. Spatial distribution of the mean absolute error of CO estimation in (a) original space, (b) NS-transformed space, and (c) location-specific NS-transformed space. 1420
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Figure 6. Spatiotemporal distribution of CO concentraion on (a) January 1, (b) April 1, (c) July 1, and (d) October 1, in 2002.
Figure 7. Spatiotemporal distribution of PM10 concentraion on (a) January 1, (b) April 1, (c) July 1, and (d) October 1, in 2002.
investigate the daily spatiotemporal distribution of ambient pollutant concentrations across Taiwan, which is characterized by significant changes in terrain and a high discrepancy in urbanization among the plain and mountain areas. Taiwan also has a high variability of meteorological conditions because of its
complex air-ocean-land interactions. The distinct emission characteristics across the island determine the average concentration levels and the spatiotemporal dependence among ambient pollutants. This implies that the nonstationarity of the ambient pollution process can be present in both trends 1421
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functions used by the other two approaches when the spatial heterogeneity of variance is observed. Figure 3 shows the heterogeneity of the spatial variances of CO, whereas PM10 can be viewed as a process with homogeneous variances in space. Consequently, the improvement of estimation performance for CO by QBME is better than that in PM10, as Table 1 shows. This is because QBME can provide substantially improved estimation accuracy in terms of lower mean absolute errors. This is particularly true of locations with higher data variability, as shown in Figures 4 and 5 for CO and PM10, respectively. Conversely, because PM10 has a more homogeneous variance across space, it produces higher similarity in the transformed data and the covariance between the NS-transformed space and location-wise NS transformed space. Consequently, the QBME approach does not significantly improve the estimation accuracy compared to the estimations performed in the location-wise NS-transformed space. This suggests that QBME is especially suitable for spatiotemporal analysis with strong heterogeneity in the spatial dependence. The preferential sampling of ambient pollutants is an important factor in the modeling of ambient concentrations.70 In this case, monitoring stations are not uniformly distributed across the entire study area. In addition, the monitoring stations are primarily located at areas of higher environmental concern (e.g., higher concentration levels). Previous studies have shown that preferential sampling can lead to biased estimations in geostatistical modeling.70−72 Figure 1 shows that the TWEPA stations are mostly located in the western plain, and particularly in metropolitan areas (e.g., Taipei and Kaohsiung). The only station located in a high altitude mountain area did not commence operation until 2006. The absence of observations in mountain areas can result in the overestimation of ambient pollutant concentrations in mountain areas. This study derives the additional soft data of ambient concentration from the data of Mt. Lulin station based on the assumption that the areas over 1000 m in elevation should have similar concentration level with Mt. Lulin station (Figure 2). This is because these areas are relatively inaccessible to human activities and therefore are affected little by local emissions. The use of soft data is a commonly used and effective way to incorporate secondary knowledge in spatiotemporal modeling in the BME framework.14,24,39,73,74 This study uses soft data to reduce the overestimation in the mountain areas resulting from the effects of preferential sampling. Consequently, the spatial distributions of CO and PM10 show a clear terrain effect and an association with population levels. A high CO concentration is particularly associated with the spatial distribution of populated areas (Figure 6). The areas of high PM10 concentrations can vary over time because of changes in the patterns of local emissions and largescale Asian dust storm transport (Figure 7).75,76
and residuals under the general geostatistical framework. The BME method can account for various types of core and sitespecific knowledge bases, and does not make any restrictive or unrealistic assumptions (linearity, normality, and independency, etc.) under the geostatistical context.61 Under the BME framework, we propose the use of nonparametric spatiotemporal trends and a quantile-based spatiotemporal dependence model to account for the nonstationary properties of ambient pollutants in this study. The epistemic framework also allow us to consider uncertain information in mountainous areas derived from data collected at high-altitude stations. This provides a way to reduce the possibility of unrealistic estimation at the unmonitored locations. Seasonal variation is one of the major temporal patterns of ambient concentrations. Because the emission and meteorological characteristics can change across space and time, the magnitude and the phase of the temporal variation of ambient concentrations vary among the monitoring stations. To account for the geographical disparity of temporal trends among stations, this study uses the STL algorithm to decompose the time series at each station into distinct temporal components with different scales and assumes independence between the trend and the residual components (Figure 3). Thus, the spatiotemporal trend of ambient concentration can be derived from the locationspecific trends of the ambient pollutant, as shown in previous studies.14,62 Figure 3 and Supporting Information Figure S2 show that the residuals at each station are Gaussian-like distributed with distinct variances among the locations, especially for pollutants highly associated with traffic emissions (e.g., CO and NOx). The variances of these pollutants in metropolitan areas are greater than those in other areas. As for PM10, the spatial distribution of residual variances is also highly associated with that of its averaged levels. To account for the covariance heterogeneity, researchers have developed several geostatistical approaches to characterize the original spatiotemporal processes from different perspectives (i.e., to make the process stationary).63−66 The moving window approach localizes the estimation procedure within a number of relatively small “windows” in which the spatial process is assumed to be stationary.7,63,66 The deformation approach transforms the spatial data from the geographical space into a new space in which the transformed data are considered stationary and isotropic.64,67 Some approaches decompose the original heterogeneous process into several homogeneous processes.65,68,69 This study proposes a quantile-based approach (i.e., QBME) that assures the unit variance at every location across space. This allows the location-specific quantiles to better satisfy the stationary assumption of the spatial dependence function. The basic idea is to focus on the spatiotemporal relationship among the relative changes (i.e., the percentage of changes) rather than the absolute change (i.e., the magnitude of changes) of the concentration levels. This is because the varying spatial variances across the study domain often cause the spatially varying pairwise magnitude changes of ambient concentrations between a specified distance across space (i.e., heterogeneity). Supporting Information Figure S3 shows the increase in the spatial range of the spatiotemporal covariances among the quantiles of the location-wise NS-transformed space compared to those in the original and NS-transformed spaces, especially for CO (Supporting Information S.1−S.4). This implies that the homogeneous spatiotemporal dependence of the locationspecific quantiles can better characterize the inherent variability across space and time among the pollutants than the dependence
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ASSOCIATED CONTENT
S Supporting Information *
Figures showing the results of spatiotemporal trend modeling by STL algorithm and spatiotemporal dependence modeling for both CO and PM10, a table showing the cross-validation result of the quantile estimation for the NS tables of both CO and PM10 is also available, and details of spatiotemporal covariance functions for the modeling of CO and PM10 in the three estimation scenarios. This information is available free of charge via the Internet at http://pubs.acs.org. 1422
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AUTHOR INFORMATION
Corresponding Author
*Address: No. 1, Section 4, Roosevelt Road, Taipei, Taiwan. Phone: +886-3366-3454. Fax: +886-2363-5854. E-mail: hlyu@ ntu.edu.tw. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This research was supported by funds from the National Science Council of Taiwan (NSC 101-2628-E-002-003- and NSC 1012628-E-002-017-MY3) and National Taiwan University (NTU 101R7844).
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