A Spatio-Temporal Model for Mexico City Ozone Levels
Gabriel Huerta
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131 USA. Email: [email protected], URL: www.stat.unm.edu/ ghuerta
Bruno Sansó
Centro de Estadı́stica y Software Matem ático, Universidad Sim ón Bolı́var, Caracas, Venezuela and Department of Applied Mathematics and Statistics, University of California Santa Cruz, USA.
E-mail: [email protected], URL: www.ams.ucsc.edu/ bruno
Jonathan R. Stroud
Department of Statistics, The Wharton School, University of Pennsylvania, 465 Jon M. Huntsman Hall,
3730 Locust Walk, Philadelphia, PA 19104 USA. E-mail: [email protected],
URL: stat.wharton.upenn.edu/ stroud
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1. Introduction
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G. Huerta, B. Sans ó and J. Stroud
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2. Data description
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A Spatio-Temporal Model for Mexico City Ozone Levels
3
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Ozone only
20
•
Northing (km)
10
•
TLA
•
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AZC
EAC
CHA
•
TAC
0
•
SAG
LAG
• •
MER HAN
CUA
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•
•
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UIZ
TAX
•
CES
PED
TAH
-20
TPN
-30
-30
-20
-10
0
10
20
30
Easting (km)
Fig. 1. Locations of 19 monitoring stations. Circles represent stations where ozone and temperature are measured. Squares represent stations where only ozone is measured. The dotted lines represent topographical
contours; the solid line represents the political boundaries of Mexico, D.F.; and the inner box indicates our
modelling region.
G. Huerta, B. Sans ó and J. Stroud
200
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100
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0
Sept. 8
Ozone (ppb)
Sept. 9
Sept. 10
Sept. 11
Sept. 12
Sept. 13
Sept. 14
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200
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Sept. 11
Sept. 12
Sept. 13
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Temp (C)
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200
21.6
100
13.3
Temp (C)
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Sept. 9
Sept. 10
Sept. 11
Sept. 12
Sept. 13
Sept. 14
Fig. 2. Ozone and temperature time series for September 8-September 14.
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3. Univariate Analysis
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A Spatio-Temporal Model for Mexico City Ozone Levels
5
100
log likelihood
80
60
40
20
0
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wavelength (hours)
Fig. 3. Bayesian periodogram of ozone concentrations at ten monitoring stations in Mexico City, during the first
two weeks of September 1997.
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4. Spatio-Temporal Model
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A Spatio-Temporal Model for Mexico City Ozone Levels
7
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d e d e A Spatio-Temporal Model for Mexico City Ozone Levels
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10
G. Huerta, B. Sans ó and J. Stroud
Table 1. Posterior medians and 95% intervals for nondynamic parameters.
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A Spatio-Temporal Model for Mexico City Ozone Levels
11
Xalostoc (NE)
Ozone (ppb)
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Benito Juarez (CE)
Ozone (ppb)
300
200
100
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300
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Fig. 4. Ozone data, retrospective medians and predictive values with 95% probability bands for 3 monitoring
stations.
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11:00
50
44
26 28
67
0
Fig. 5. Retrospective median ozone maps for September 8, 1997 based on 5000 MCMC samples, using a
1.4km (25 25 pixel) mapping grid. The values written on the maps represent the observed data at that time
and station.
G. Huerta, B. Sans ó and J. Stroud
11:00
12:00
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Fig. 6. Retrospective 95% range maps of ozone for September 8, 1997.
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6.1. Discussion
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A Spatio-Temporal Model for Mexico City Ozone Levels
13
Xalostoc (NE)
Temp (C)
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Sept. 14
Benito Juarez (CE)
Temp (C)
30
25
20
15
10
Sept. 8
Sept. 9
Sept. 10
Sept. 11
Pedregal (SW)
Temp (C)
30
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Sept. 9
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Sept. 14
Fig. 7. Temperature data, retrospective means and predictive values with 95% probability bands for 3 monitoring
stations.
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0.4
L
M
H
B
0.3
0.3
P
L
C
T
0.30
Southwest
H LB
10
Sqrt Ozone
12
14
2
4
CX
S
6
8
10
12
14
Sqrt Ozone
Fig. 8. Posterior predictive densities at 2 pm on September 10 for the “leave-one-out” analysis. Each panel
corresponds to a different geographical region. Stations are identified with the first letter of their names. Letters
on the axis correspond to actual observations (not used in the predictions).
G. Huerta, B. Sans ó and J. Stroud
Central
Northeast
0.15
L
M
H
B
C
0
P
LT
5
10
15
Sqrt Ozone
S
X
C
0.10
0.05
0.0
0.0
0.05
0.05
0.10
0.10
0.15
P
L
C
T
0.15
Southwest
M
0
5
0.0
14
LH
B
10
15
XCS
0
2
4
6
8
10
14
Sqrt Ozone
Sqrt Ozone
Fig. 9. Posterior forecast densities at 2pm on September 14. Each panel corresponds to a different geographical
region. Stations are identified with the first letter of their names. Letters on the axis correspond to actual future
observations.
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6.2. Measurement error
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A Spatio-Temporal Model for Mexico City Ozone Levels
15
λy
12
10
8
6
4
LAG TAC EAC SAG AZC TLA XAL MER PED CES PLA HAN UIZ BJU TAX CUA TPN CHA TAH ALL
a1
3.0
2.5
2.0
LAG TAC EAC SAG AZC TLA XAL MER PED CES PLA HAN UIZ BJU TAX CUA TPN CHA TAH ALL
Fig. 10. Boxplots of samples from the the marginal posterior distributions of (upper panel) and (lower
panel) obtained by leaving one station out. The horizontal axis denotes the omitted station. The far right boxplot
represents the posterior with all stations included, and the horizontal line represents the posterior mean when
all stations are included.
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@C:D <9: ED:;:I< KEED@KG9 P
7. Conclusions and Extensions
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FI<:DED:<K<F@I P
16
G. Huerta, B. Sans ó and J. Stroud
Acknowledgements
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Appendix A
A1. Marginal likelihood for range and variance parameters
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A Spatio-Temporal Model for Mexico City Ozone Levels
FIKBB?S ;FIG:
17
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J @CFIL OKGaXKD>; AD@J <@ P
References
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WI d:>;P P :DIKD>@ S P :DL:D S T P KXF> KI> T P J F<9eS _f U]] P RA@D>
RA@D> IFC:D;F<? D:;;P
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:RE@;=D: KI> E@E=BK<F@I >:I;F<? FI KDDF; Z@=I<?S 8:RK;P S S f^\ UgP
ZKD<:D S Z P KI>
@9I S b P dU^^ e FOO; ;KJ EBFIL A@D ;<K<: ;EKG: J @>:B;P S S g U gf P
ZD:;;F: S P KI> FaB: S Z P dU^^_e <DK<:LF:; A@D >?IKJ FG ;EKG: <FJ : ;<K<F;<FGKB J @>:BFIL F;G=;;F@I
@A 89: DFL:> KBJ KI FB<:D O? Y KD>FK S @@>KBBS b:>A:DI KI> TB@I;@ P 8:;< S S S \g ` P
D =9X FD<9 G9IK<<:DS P dU^^ e K<K K=LJ :I<K<F@I KI> >?IKJ FG BFI:KD J @>:B;P
S S U_f U]\P
=<<@DE S PS Y :FDFIL S P KI> KJ E;@I S P dU^^ e T ;EKG: <FJ : KIKB?;F; @A LD@=I> B:C:B @H@I: >K<K P
S S \ U \g P
dU^^_e EKG: <FJ : :;<FJ K<F@I @A LDF> G:BB 9@=DB? @H@I: B:C:B; A@D K;;:;;J :I< @A K >:<:DJ FIF;<FG
J @ >:BP FJ S PS 9:E9KD> S P KI> Z9FO S P dU^^_e <@G9K;<FG C@BK<FBF<? BFa:BF9@@> FIA:D:IG: KI> G@J EKDF;@I
X F<9 KDG9 J @>:B;P S S f` U f^f P
I@DD :B> S [ P KI> b=: S P d\]]\e I OB@Ga =E >K<FIL FI Y KDa@C DKI>@J :B> J @>:B; A@D >F;:K;:
J KEEFIL d<@ KEE:KDeP Y KD>FK S PS @@>KBBS Z PS b:>A:DI S Q P KI> TB@I;@ S P dU^^_e 89: aDFL:> KBJ KI B<:D P S S \ U \_g P
Y FBKIG9=;S Y PS b K@ S 8 P KI> =DO:Ia@ S WP dU^^_e QCKB=K<FIL <9: : :G<FC:I:;; @A @H@I: J KIKL:J :I<
: @D<; FI <9: ED:;:IG: @A J :<:@D@B@LFGKB CKDFKOFBF<?P S S \] US\UgP
b K@ S PS =DO:Ia@ S WPS :KL= S b PS @D<:DS PS = S P KI> :ID?S b P dU^^ e EKG: KI> <FJ : ;GKB:; FI
KJOF:I< @H@I: >K<K P S S \Ugf \U`` P
b=: S P d\]] Ue K;< ;KJ EBFIL @A K=;;FKI Y KDa@C DKI>@J :B>;P
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KJ E;@I S P KI> =<<@DE S P dU^^\e @IEKDKJ :<DFG :;<FJ K<F@I @A I@I;<K<F@IKD? ;EK<FKB G@CKDFKIG:
;<D=G<=D: P S S U]_ UU^ P
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d<@ KEE :KDeP P
18
G. Huerta, B. Sans ó and J. Stroud
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