NO2 case study (5 stations)
HW
March 12, 2019
Data set from the paper by Cooley, Davis, Naveau (The Annals of Applied Statistics 2012, Vol. 6, No. 4, 1406–1429). Data available at http://www.stat.colostate.edu/~cooleyd/DataAndCode/PredExtremes/.
Description
Variable: daily maximum hourly NO2 measurement.
Five stations located in Washington DC and nearby Virginia: Alexandria (alx), McMillan (mc), River Terrace (rt), Takoma School (ts) and Arlington (arl).
Period: 5000 daily NO2 measurements recorded between 1995 and 2010, keeping only those records for which all five stations have measurements, resulting in 4497 daily measurements.
Rounding: because the data were truncated to the nearest ppb, a uniform random variable on the interval [−0.5, 0.5] was added to the data so that they behave more like the underlying continuous variable.
## Load data and display first and last 5 lines
load("nitrogenDioxideMsmts.RData")
head(no2Msmts); tail(no2Msmts)## date alx arl mc rt ts
## 1 9162 48.94594 40.02866 44.394349 41.51378 41.44504
## 2 9163 36.89465 36.48012 32.839171 35.87481 35.16941
## 3 9164 38.98373 44.07211 39.801299 40.08453 38.58505
## 4 9165 34.41888 31.91026 26.643805 31.29779 29.14976
## 5 9166 17.34388 14.71787 5.918686 12.71006 11.78087
## 6 9167 34.01735 32.12474 33.940509 31.13621 21.81810## date alx arl mc rt ts
## 4631 14635 26.86159 24.84220 26.10330 28.25237 26.56204
## 4632 14636 38.20120 32.46180 37.88082 38.51873 36.11601
## 4633 14637 31.54337 28.05154 42.41072 34.67496 41.32923
## 4634 14638 23.26186 10.96069 14.56890 21.55876 16.49128
## 4635 14639 25.54794 19.02271 20.27997 21.12096 19.73806
## 4636 14640 51.28440 44.88402 43.02091 49.27270 45.42105## Load graphical package lattice
library(lattice)
## Times series at each of the 5 stations
dates=as.Date(no2Msmts[,1], origin = "1970-01-01")
xyplot(no2Msmts[,2]~ dates,pch=19,ylab="NO2 at Alexandria")
xyplot(no2Msmts[,3]~ dates,pch=19,ylab="NO2 at Arlington")
xyplot(no2Msmts[,4]~ dates,pch=19,ylab="NO2 at McMillan ")
xyplot(no2Msmts[,5]~ dates,pch=19,ylab="NO2 at River Terrace")
xyplot(no2Msmts[,6]~ dates,pch=19,ylab="NO2 at Takoma School")
Examine dependence on weekdays
On Saturday and Sunday there is less NO2 for all series.
## data object is a matrix, create a data frame
class(no2Msmts)## [1] "matrix" "array"NO2=as.data.frame(no2Msmts)
names(NO2)## [1] "date" "alx" "arl" "mc" "rt" "ts"## in case the locale is not in english
Sys.setlocale("LC_TIME", "en_US.UTF-8")## [1] "en_US.UTF-8"## dependence on weekdays: on Sat and Sun there is decrease of NO2
day= factor( weekdays(dates,abbreviate=TRUE), levels=c("Mon","Tue","Wed","Thu", "Fri", "Sat", "Sun"),ordered=TRUE)
bwplot(day~NO2$ts, main="Tacoma school station",xlab="NO2")
Examine dependence on month
Strangely the median is highest for Feb, Mar, Apr for all series
month =factor( months(dates,abbreviate=TRUE),levels=c("Jan","Feb","Mar","Apr","May","Jun","Jul","Aug","Sep","Oct","Nov","Dec"),ordered=TRUE)
bwplot(month~NO2$alx);bwplot(month~NO2$arl);bwplot(month~NO2$mc);bwplot(month~NO2$rt);bwplot(month~NO2$ts)
QQ plots
Empirical quantiles vs normal quantiles
qqmath(NO2$alx[month=="Jul"])
Empirical quantiles vs exponential quantiles
qqmath(NO2$alx[month=="Jul"],distri=qexp)
Extreme value distribution fit
We use the library ismev that goes with the book by Stuart Coles
# the library ismev needs to be downloaded from CRAN to your machine
# install.packages("ismev")
## Load library ismev
library(ismev)## Loading required package: mgcv## Loading required package: nlme## This is mgcv 1.8-38. For overview type 'help("mgcv-package")'.GEV modelling for each station
Modelling all data without distinguishing days and months, the tail is not fitted well at all for each of the series.
## all days and months together
gev.diag(gev.fit(NO2$alx));gev.diag(gev.fit(NO2$arl));gev.diag(gev.fit(NO2$mc));gev.diag(gev.fit(NO2$rt));gev.diag(gev.fit(NO2$ts))GEV modelling performed for each month
Better fits are obtained for individual months
## for a specific month
gev.diag(gev.fit(NO2$alx[month=="Jul"]));gev.diag(gev.fit(NO2$arl[month=="Jul"]));gev.diag(gev.fit(NO2$mc[month=="Jul"]));gev.diag(gev.fit(NO2$rt[month=="Jul"]));gev.diag(gev.fit(NO2$ts[month=="Jul"]))## $conv
## [1] 0
##
## $nllh
## [1] 1648.973
##
## $mle
## [1] 32.8942383 14.1591176 -0.2119891
##
## $se
## [1] 0.79464224 0.57170539 0.03715304
## $conv
## [1] 0
##
## $nllh
## [1] 1647.064
##
## $mle
## [1] 30.7457773 14.3098402 -0.2366442
##
## $se
## [1] 0.79541361 0.56987139 0.03438239
## $conv
## [1] 0
##
## $nllh
## [1] 1616.905
##
## $mle
## [1] 28.52068010 11.86502236 -0.03510485
##
## $se
## [1] 0.66513952 0.48023036 0.03576291
## $conv
## [1] 0
##
## $nllh
## [1] 1608.132
##
## $mle
## [1] 32.7382090 12.6154456 -0.1859158
##
## $se
## [1] 0.70277439 0.50049085 0.03468554
## $conv
## [1] 0
##
## $nllh
## [1] 1593.454
##
## $mle
## [1] 30.090874 12.255358 -0.197299
##
## $se
## [1] 0.67986659 0.47858079 0.03377075
Mean excess plot
mrl.plot(NO2$alx[month=="Jul"])
Scatter plot matrix for the 5 stations
No distinction of months.
Observe the extremes: are they common for a pair of stations ?
splom(NO2[,2:6],pch=19)
Scatter plot matrix for the 5 stations
Using a different colour for each month.
splom(NO2[,2:6],group=as.factor(month),auto.key=TRUE)