Spatio-temporal modeling of fine particulate matter
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1 Spatio-temporal modeling of fine particulate matter Sujit Sahu a University of Southampton Lisbon: March 2004 a Joint work with Alan Gelfand and David Holland
2 Fine particulate matter Particles with diameter less than 2.5 m, PM. PM has been linked to respiratory and cardiovascular problems. Contribute to other air-pollutants, stress to vegetation and ecosystems. In 1997, EPA gave new regulations and setup a network of monitoring sites.
3 PM formation PM can be emitted directly or formed in the atmosphere: road-dust, wood burning. It contains particles from gaseous emissions such as sulfates and nitrates. Power plant, automobile. PM concentration levels are affected by emissions, topography, land cover, and other factors, e.g. air movement.
4 Our data set Mid-western United states: Illinois, Indiana and Ohio. Data on 114 sites in the year Data are 24-hr averages. Sampling frequencies are varied: daily, every three days, weekly etc. We work with weekly averages. Have 5928 (114 times 52) data points. Only 2% missing observations.
5 114 sites in Illinois, Indiana, and Ohio cements
6 Objectives in modeling Investigate rural/urban differences Measure effect of population/density Produce weekly prediction surfaces Obtain annual predictions
7 Outline of the talk Exploratory analysis Methods Modeling details Prediction details Analysis Model choice Parameter estimates Prediction Discussion
8 Scale of Data Variance increases as mean increases. Model square root of data, but report predictions on original scale. Mean Variance Mean Variance Figure 1: Top: original; bottom: square root scale.
9 Rural/Urban separation cements PM Rural Suburban and urban 10 0 Site Figure 2: First 10 for rural; last 104 for suburban and urban.
10 Rural/Urban mean variance Rural Suburban Urban Will combine suburban and urban into one category.! " # $ if # rural, =1 otherwise. Problem in prediction: don t have these for a fine grid. Will develop a spatial model for! %. '&
11 Population density Have density for the year Can pop-density separate the means and vars? No! see below. mean variance low-pop high-pop
12 * 60 () PM cements Week cements PM Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month
13 Temporal effects Higher in summer due to higher sulfate particle concentrations Higher in some winter months due to increased levels of nitrate concentrations, possible re-intrainment of road salt We shall use monthly indicators to model. '
14 Geodetic Distance Earth as a sphere of radius 6371 kilometers. Let,.- and, denote the latitudes and /0- and /1 denote the longitudes (in radian). d # :<;=;?>A@B!C (km) C I, - J@EDKFH I,4 LNM ;?>O@P I, - J;?>O@P I,4 QJ;?>O@P R/ - S /1 L.
15 Variograms Obtain variation in the residuals after fitting a linear model. There is spatial variation. Probably no sill. Points to non-stationary spatial variation. variogram All sites Urban Sites distance +
16 Hierarchical model T UWVYX[Z\] # PM ' ^ at location?v, week Z\. data = systematic process + white noise _!`VaX[Zb\] # c!`vdx[zb\] M ef!`vyx[zb\] systematic = mean + rural/background process + urban indicator * urban process c!wvyx[z\] #!`VdX[Z\P+M g!`vhx[z\]+m!`vijik!wvyx[z\]ql Mean process = pop-density + urban + urban * pop-density + month '
17 t v Spatio-temporal process Assume that g U?VYX[Zb\] and ik UWVaX[Z\P are zero mean spatio-temporal processes. Adopt separable covariance structure. Cov mnik!`vdx[zdopx ik Urq%X[Zb\]Bs # t u v w.x u!`v Syqzy{ x u v wa u }Zao S Z\~zy{ u X Covm g! V X[Z o X g! q X[Z \ s # w x!wv Syq%zy{ x v wa Zao S Zb\ zy{ l Take w s to be exponential: w! ƒzy{ # J S { 8 l Decay parameters: { x Xy{ Xy{ x u Xy{ u X are chosen by model choice.
18 Posterior Assume vague prior for t mean parameters. Have Š! ŒXEt and Š ŒXEt ˆ XEt u XEt x Ž, u x u Ž u. and the The joint posterior is a long expression but provides conjugate full conditional distributions for Gibbs sampling. '
19 Predicting urban indicators Want to predict at 3210 new locations. Haven t got the urban indicator h % for those. Use a spatial probit model. Set h % # 7 if U $ and 0 otherwise.! % is a spatial process with mean 6 M -! and variance 1, correlation function w! S = pzy{ š4. Select {ƒš using model choice. Run a Gibbs sampler for this first and save the predictions for!. '
20 Weekly predictions Model is: _!` X[Zd K Š!` X[ZY GHM g!` œx[zd G M UW žik!` X[Zd KX t ˆ Ÿl Bayesian predictive distributions: T < I T d rª # T y G I «9: 9: T ä yª 9: l Need to simulate from: g! X[Z and ik U X[Z l Distributions are normal but parameters are expensive to calculate. Q&
21 _ t t Annual predictions On the sqrt scale. By averaging over 52 weeks: U Š U HM g! M! ik U X ˆ Ō l To obtain the predictive, we need the distributions: g!` K and ik UW K. By sufficiency g!? g!` K <± g U - X=l=l=lfX g!w²6y³. Can show that: g!` K Š µ x U SW x - ' º¹A»½¼¾ d a is equivalent to X ± 7 S x! S` K - x x! S` Ky³PÀ l P
22 _ # È - Ä- Ä - _ M l Original scale orig UW # _ a d a \<Á - m d _ \<Á - UW px[z\] UW px[z\] S!` Â s _!` K is the square of the predictions on the square-root scale. UW Âl For the first, use: - d d \<Á - m q[á - m!` X[Z\P S!` X[Zb\hÅ S Take Æ # Ç weeks (equi-spaced) in four Ã!W Â s _ Ä UW s quarters. Do these predictive calculations for each of É# 24$O$4$6 MCMC iterations. The results for Æ # 2 look similar.
23 3 Ã We obtain annual mean surface, standard deviation and an estimate of the probability that annual PM ^ greater than 15. Thus we store, read and manipulate 7f$ v v 24$O$4$ 96 million numbers. Were we to average over all 52 weeks we had to work with 1000 million or a billion numbers! Q
24 Ê l Analysis: Model Choice Define the predictive model choice criterion Ë Ë (PMCC): # Var T < G I 'M T ä yª S T < I PMCC = P(enalty) + G(oodness of fit) Find the optimal decay parameters. Assume { x # { x u # { x and { # { u # {. Search for { x among (0.015, 0.03, 0.06, 0.12 and 0.30). Search for { among (3, 1.5, 1, 0.75 and 0.6). Optimal { x # $lì$43 (range = 100 km = about 1/10th of the maximal distance). Optimal { # 74l (range = 2 weeks).
25 Analysis: Model comparison Compare the proposed model with the basic model which is not spatio-temporal. PMCC values: Model P G P + G Base model Our model There are other intermediate models. We are not interested in those.
26 Analysis: Parameter estimates mean sd 95% interval Grand mean (4.150, 4.401) Pop-density (0.010, 1.023) Urban (0.244, 0.391) interaction ( 1.124, 0.123) Feb ( 0.788, 0.477) Mar ( 0.864, 0.560) Apr ( 1.365, 1.033) May ( 1.023, 0.706) Jun ( 0.288, 0.046) Jul ( 0.875, 0.526) Aug ( 0.800, 0.496) Sep ( 1.318, 0.977) Oct ( 1.357, 1.018) Nov ( 1.093, 0.784) Í Î Dec ( 1.244, 0.905) Ï (0.057, 0.070) Rural Var (0.363, 0.405) Urban Var (0.001, 0.003) Q
27 Discussion Current work Proposed and fitted non-stationary spatio-temporal model. Positive shift in mean level for urban areas. Also larger variability in urban areas. Obtained both weekly and annual predictions. Future work Produce annual map for the entire US. Simultaneous modeling with other pollutants. See if rural/urban differences are there for other pollutants.
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