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The framework follows a K-nearest neighbor approach. We describe the framework and the implementation algorithm for temporal BaP concentrations. K-nearest neighbor approach queries days similar to a given feature vector and identify a subset of days (K) similar to the feature day. These K days are then weighted using a bi-square weight function and randomly sampled to generate ensembles. Here we incorporate K-nearest neighbor approach to detrended logarithm of BaP concentrations, denoted by εtc. That is,

where Ptc is an index of the precipitation occurrence for site c on day t.

In a simple K-nearest neighbor approach, the nearest neighbors are obtained from the observed data by computing the distance between the unobserved BaP concentration and the observed BaP concentration, and the neighbors are assigned weights based on their distance. The weight function gives more weight to the nearest neighbors and less to the farthest neighbors. In general, the number of nearest neighbors, K is based on the heuristic scheme K=N, where N equals the sample size (Lall and Sharma, 1996), following the asymptotic arguments of Fukunaga (1990). Objective criteria, such as generalized cross validation, can also be used. The proposed approach aims to compute the distance based on the similarity of factors such as air quality variables and meteorological variables. That is, for two distinct time points, t and t′ for site c, the distance d_c(t,t′) can be defined by

where || · || is a matric or distance function. Note that air quality variables xtc and meteorological variables ztc need to be standardized before computing distances. The weights w₁ and w₂ plays a role of tuning the proximities between air quality variables and meteorological variables. For a fixed time point t, its neighbors are assigned weights based on the K smallest distances (say d₁,d₂,…,d_K) from the time point t, as follows:

for k=1,…,K. Further extensions to (4) are possible, such as extensions to include a more general class of functions of distance. The index for site c is omitted for brevity for now. The unobserved detrended logarithm of BaP concentrations can then be generated by resampling from the observed detrended logarithm of BaP concentrations based on the assigned weights, or alternatively, can be estimated based on the weighted average of the observed detrended logarithm of BaP concentrations. Based on resampled or estimated detrended logarithm of the BaP concentration, ε^tc, we get the resampled or estimated BaP concentration Y^tc by the following equation:

where μ^c,α^c, γ^1c,γ^2c and γ^3c are the estimated coefficients of the trend GLM model fitted to the observed BaP concentrations. In principle, we may extend our neighbor structure not only to a temporal scale but also to a spatial scale. That is, other sites can also be used as neighbors. In such cases, a more sophisticated and complex measure of distance is required.

In general, the approach based on resampling has unrealistic properties for extremes. That is, it is not possible to generate value greater than the highest observed. Because those air quality and meteorological factors which are more similar to BaP processes are assigned higher weights, the expression (3) accommodates a spatial dependence structure. In general, BaP concentrations are assumed to be log-normally distributed. Here, the BaP concentration Ytc is modeled as a lognormal distribution, as follows:

where μc=μ1c,μ2c,….,μTc' is a mean trend vector of the logarithm of the BaP concentration and Σ^c is a covariance matrix of the logarithm of the BaP concentration process. The mean process μtc is modeled by

The temporal dependence among the logarithms of BaP concentrations is incorporated by modeling T×T covariance matrix Σ^c whereas the temporal mean trend is represented by μtc. We can parameterize the covariance matrices by variance terms and correlation functions as Σ^c(t,t')=σ²r(t,t').

To avoid the difficulty in dealing with covariance matrices, we can consider a simple exponential correlation function as follows:

where β(>0) is a function which is used to tune the range of individual neighbors. This function provides a simple covariance matrix, and the resulting matrices can be easily manipulated. Let r_β(d_c(t,t′))=ρ^d_c(t,t′) (i.e., ρ=e^-β>0). The correlation parameter, β, can be empirically estimated from the data. In principle, we can consider a more general correlation function

which is called the Matérn correlation function (Matérn, 1986) with parameters ρ and ν, evaluated at distance d_c(t,t′).

Let U and O be the vector of logarithms of unobserved BaP concentrations and the vector logarithms of observed BaP concentrations, respectively. In our scheme, assume that U and O are jointly distributed, that is,

where Σ₁₁ is the marginal covariance matrix of U, Σ₂₂ is the marginal covariance matrix of O, and Σ₁₂ is the cross-covariance matrix of U and O. Note that μ_u, μ_o, Σ₁₁, Σ₁₂, Σ₂₁ and Σ₂₂ are obtained from the rearrangement

of μ_c and Σ^c. Then,

The predictor for logarithms of unobserved BaP concentrations, U, is then the usual best linear unbiased estimator (BLUE) for the prediction problem. Therefore, we may estimate BaP concentration by μu-Σ12Σ22-1O-μo with the uncertainty in terms of (Σ11-Σ12Σ22-1Σ21). Note that the parametric approach provides prediction with uncertainty.

Because BaP concentrations are typically available on a daily basis, considering various time lag models for x_t or z_t allows for greater flexibility in exploring the lag between air quality and meteorological variables and BaP concentrations, as compared to singlelag models. Another topic of special interest lies in the hierarchical modeling of the multivariate or spatial structure of BaP concentrations. Here, this paper focuses on the temporal behavior of BaP concentrations.

The BaP and other air quality data sets used in this paper are obtained from a field study conducted in the Sihwa-Banwol national industrial complex, one of the largest industrial areas in Korea (Seo, 2010). The main purpose of the field study was to characterize the occurrence and concentrations of a wide range of HAPs in the atmosphere of the industrial complex. Temporal, spatial, and seasonal variations in HAP concentrations were determined at four sites (two industrial and two residential sites) over a two-year period, from August 2005 to July 2007. Fig. 1 shows the location of the industrial complex as well as the specific locations of air sampling sites. Seasonal HAP monitoring campaigns were carried out throughout the complex, where 12 and 10 consecutive days of monitoring are carried out for each season in the first year and the second year of operation, respectively. Detailed monitoring periods and air quality data for each site are summarized in Table 1. Samples of total suspended particles (TSPs) are collected by high-volume sampling for 24 h, and are used for PAH analysis. A total of 38 different PAH compounds are determined by GC/MS. Sampling, analyses, and quality control for PAHs are carried out in accordance with the US EPA TO-13A protocol (US EPA, 1999) and the ISO method (ISO, 2000). Interpretations of PAH determinations should be based on the objective of PAH measurement, e.g., whether the intention is to monitor for concentrations which exceed reference values, analyzing causes, or conducting epidemiological surveys. In the case of BaP, the PAH substance most frequently measured in air pollution analyses, close attention should be paid to the implications which are associated with these concentrations, as the concentrations themselves can be used to indicate the carcinogenic activity of PAH mixtures that regularly occur in ambient air (ISO, 2000). According to the WHO Air Quality Guidelines (WHO, 2000), BaP is a useful indicator of the carcinogenic potential of total PAH emissions with respect to lung cancer. For these reasons, BaP is selected for this paper’s analysis as a good representative PAH.

Fig. 1. 
Location of sampling sites for benzo(a)pyrene.

PAH sampling sites are all located within national air quality monitoring stations operated by the Korean Ministry of Environment (see Fig. 1). Hourly data on other general air quality parameters such as SO₂, CO, NO₂, PM₁₀, and O₃, were obtained from the national air quality database. In addition, hourly data on ambient temperature and wind speed were obtained from the National AWS (automatic weather station), which is located approximately 5 km from the PAH sampling sites. Data for relative humidity and solar radiation data were not available from the AWS. The daily average air quality and meteorological data was recalculated so as to match the PAH sampling time. It has been reported that atmospheric concentrations of PAHs are well related to CO and NO₂ as byproducts of combustion from many sources (Saud et al., 2011; Ravindra et al., 2008; Park et al., 2002). In addition, ambient levels of PAHs are higher in winter and lower in summer, and thus, indicate a negative correlation between PAHs and ambient temperature (Lee et al., 2011; Park et al., 2002).

Through the semi-parametric approach, we considered the weight function to be proportional to the exponentiated distance, as the augmented BaP concentrations based on the weight function proportional to the inverse of distance fail to produce enough variability of BaP concentrations. Various percentiles of daily BaP concentrations are compared in Table 2. Based on the parametrically augmented BaPs, BaP concentrations show similar variability to the observed BaP concentrations. However, semiparametrically augmented BaPs tends to be underestimated, especially during their peak season (December). The proposed approaches can provide a useful tool for exploring unobserved periods. Because of the small number of disconnected observations, the effects of autocorrelation terms in the longterm trend model are limited. However, the autocorrelation of air quality and meteorological variables still indirectly takes into account the effect of the autocorrelation term on BaP concentrations. In addition, retaining the autocorrelation term as a covariate makes the interpretation of the model more difficult and complicates the relationships between air quality and meteorological variables.

Figs. 2 and 3 show the performance of the proposed approaches in reproducing BaP concentrations for four sites. Time series of daily BaP concentrations are simulated over a one-year period. Augmented BaP concentrations show much less variability during summer months than winter months, and the observation period may miss the peak season (December) of BaP concentrations. Estimated cumulative density functions of seasonally observed BaP concentrations were compared with those of semiparametrically and parametrically augmented BaP concentrations (see Fig. 4). Figs. 5 and 6 clearly show the variability of observed and augmented BaP concentrations, both regionally and seasonally, particularly for high BaP concentrations. Their inconsistency is partly due to the lack of alignment of observation times, and is potentially due to adverse meteorological conditions (e.g., severe episodes). Fig. 7 shows that the proposed model well preserves the relationship between BaP concentrations near sites. The forecasting behavior of the proposed model and the approach taken in the previous section is assessed through a standard “hold-out” experiment (see West and Harrison, 1999). More specifically, the model parameters are estimated using observed BaP concentrations after the removal of a subset (called the validation set) of data from the 2005-2006 season, and forecasting of the removed BaP concentrations. The forecast is then compared with actual BaP concentrations during the validation period. During the forecast period, variables for the observed air quality data and meteorological conditions are used for each region. Fig. 8 shows the actual BaP concentrations for the 2005-2006 season, as well as their predicted intervals for the Shiwha residential area (site A).

Fig. 2. 
Observed BaP concentrations (black) for the period 2005-2006 and (semi-parametrically) augmented BaP concentrations (grey) for four sites (A, B, D, and C; clockwise from top left).

Fig. 3. 
Observed BaP concentrations (black) for the period 2005-2006 and (parametrically) augmented BaP concentrations (grey) with uncertainty (dotted) for four sites (A, B, D, and C; clockwise from top left).

Fig. 4. 
Esitmated cumulative density functions based on semi-parametrically augmented BaP concentrations (grey) and parametrically augmented BaP concentrations (black) for four sites (A, B, D, and C; clockwise from top left; the dashed line corresponds to observed BaP concentrations).

Fig. 5. 
Boxplots of observed and modeled BaP concentrations for four sites (A, B, D, and C; clockwise from top left).

Fig. 6. 
Boxplots of observed and modeled BaP concentrations for site A for spring (top left), summer (top right), autumn (bottom left), and winter (bottom right).

Fig. 7. 
Plots of observed BaP concentrations (black), semi-parametrically (left) and parametrically (right) augmented BaP concentrations 9 gray) near sites.

Fig. 8. 
Observed BaP concentrations (gray x) for the winter 2005-2006 period and corresponding predicted intervals of BaP concentrations (gray) for site A.

For the assessment of our approaches, as well as to determine the distribution of this variable over the population residing in the risk area, cancer risk was examined for different HAP sources. First, an exposure model was first constructed based on the lifetime average daily dose (LADD) involving the active surface area concentration of the HAP (C, μg/m³), the inhalation rate (IR, m³/day), exposure duration (ED, years), exposure frequency (EF, days/year), absorption efficiency (AB, %), body weight (BW, kg), and the averaging time (AT, days):

Finally, the cancer risk for a specific HAP can be obtained using inhalation slope factor (SF, (μg/kg/day)^-1) as follows:

where SF=UR×BWIR and UR is inhalation unit risk. The level of chemical pollution was modeled as a function of other variables, some of which are random variables, and the distribution of the variable of interest was generated through a computer simulation. This then allowed for the determination of whether the probability of the variable of interest exceeds acceptable levels.

The Monte Carlo simulation has been used for risk assessment purposes because of its increased computing power. Here, it includes LADD and Cancer Risk as functions of the other variables. Each recalculation produces new random values for IR, EF, and BW, and consequently, for LADD and Cancer Risk to simulate the situation for a random individual from the population at risk. Fig. 9 compares the distribution of the cancer risk for the observed BaP concentrations with that for augmented BaP concentrations, for both industrial and residential areas in Korea. Note that augmented BaP concentrations provide more variations in the cancer risk (ng/m³) of residential area but there is not “worst-case” scenario with extremely high cancer risk values.

Fig. 9. 
Esitimated density (left) and boxplots (right) of cancer risks based on observed BaP concentrations (gray), semi-parametrically augmented BaP concentrations (black; dash) and parametrically augmented BaP concentrations (black; line) for site A (top) and site D (bottom).