1. An Example.
The question
Data
Analyses
Conclusions

2. Monthly mean air temperature at Recife 1953-1962
The question:
What is the relationship between temperatures in Recife and El Nino?
Objectives
- to layout analyses
- to explore the data for surprises
- predicted values
- signal + noise?
- ...

3. Finding the data.
Google with various key words: temperature, Recife, ...
"Eventually lead" to:
cdiac.ornl.gov/ftp/ndp041
Carbon dioxide information analysis center!
Had to discover Recife Curado station id
Years
Searched an inappropriate ste for a long time
(Looked at Brasil sites too, but that didn't turn up the data)

4. notice -9999 replace by NA file: recifecurado
The web data. monthly
notice replace by NA file: recifecurado

5. How to handle missing values? Interpolate? Model? ...?
junk<-scan("recifecurado")
junk1<-matrix(junk,ncol=48)
junk2<-junk1[2:13,] # years in first row
series<-c(junk2)/ # for degrees centigrade
length(series[is.na(series)]) #17 - need to understand missingness
Interpolation
series1<-series
for(i in 2:(length(series)-1)){if(is.na(series[i]))series1[i]<-.5*series[i-1] +.5*series[i+1]}

6. plot(xaxis,series1,type="l",xlab="year",ylab="mean temp (degrees C)",las=1)
title("Mean monthly temperatures Recife Curado")
abline(h=mean(series1))

7. There is seasonality and variability
Restricted range in mid-sixties - nonconstant mean level?
ylim<-range(series1)
par(mfrow=c(2,1))
plot(lowess(xaxis,series1),type="l",ylim=ylim,xlab="year",ylab="degrees C",main="Smoothed Recife series")
abline(h=mean(series1))
junk20<-lowess(xaxis,series1)
plot(xaxis,series1-junk20$y,type="l",xlab="year",ylab="degrees C",main="Residuals")
abline(h=mean(series1-junk20$y))

9. par(mfrow=c(1,1))
acf(series1,las=1,xlab="lag(mo)",ylab="",main="autocorrelation recife temperatures",lag.max=50,ylim=c(-1,1))

10. More confirmation of period 12
Remember the interpretation of the error lines
Note that nearby values are highly correlated

11. spectrum(series1,xlab="frequency (cycles/month)",las=1)

12. Note peaks at frequency 1/12 and harmonics
Further confirmation of period 12
Note log scale for y-axis
Note vertical line in upper right
Gives uncertainty

13. What is the shape of the seasonal?
junk4<-matrix(series1,nrow=12)
junk5<-apply(junk4,1,mean)
plot(junk5,type="l",las=1)
abline(h=mean(junk5))

14. Cooler in July-Aug
Southern Hemisphere
Uncertainty?

15. Cooler in July-August. Southern hemisphere
Part of a longer cycle? El Nino explanatory?
After "removing" trend middle has been pulled up
Need uncertainties
Back to original data

16. Remove seasonal
series2<-series1
for(i in 1:48){
for(j in 1:12){
series2[(i-1)*12+j]<-series1[(i-1)*12+j]-junk5[j] }
par(mfrow=c(2,1))
plot(xaxis,series2,type="l",xlab="year",ylab="residual",main="Series after removing seasonal",las=1)
abline(h=0)
ylim<-range(series2)
plot(xaxis,series1-mean(series1),type="l",xlab="year",ylab="degreesC",main="Mean removed series",las=1,ylim=ylim)
abline(h=mean(series1-mean(series1)))

17. original variance 1.342 adjusted .248

18. par(mfrow=c(2,1))
acf(series2,lag.max=50,las=1,xlab="lag (mo)",main="Ajusted by removing monthly means",las=1)
acf(diff(series1,lag=12),lag.max=50,xlab="lag (mo)",main="Order 12 differenced series")

19. Frequency domain analysis.
par(mfrow=c(2,1))
junk9<-spec.pgram(series1,taper=0,detrend=F,demean=F,spans=5,plot=F)
ylim<-range(junk9$spec)
junk9<-spec.pgram(series1,taper=0,detrend=F,demean=F,spans=5,xlab="frequency (cycles/mo)",las=1,main="Original series")
junk10<-spec.pgram(series2,taper=0,detrend=F,demean=F,spans=5,ylim=ylim,main="Monthly means removed",las=1)

20. Work remains on seasonal
Residual "not" white noise

21. Time domain
distributions

22. Parametric model. SARIMA ?
Thinking about prediction, consider
Yt = αYt-1 + βYt-12 + Nt
with some ARMA for Nt
Check seasonal residuals for normality
Hope to end up with white noise

23. Junk<-arima(series1,order=c(1,0,1),seasonal=list(order=c(1,0,1),period=12))
Call:
arima(x = series1, order = c(1, 0, 1), seasonal = list(order = c(1, 0, 1), period = 12))
Coefficients:
ar ma1 sar1 sma1 intercept
s.e
sigma^2 estimated as : log likelihood = , aic =

25. tsdiag(Junk,gof.lag=25)

27. Junk<-arima(series1,order=c(1,0,1),seasonal=list(order=c(1,0,1),period=12))
postscript(file="recifeplots1a.ps",paper="letter",hor=T)
Junk2<-predict(Junk,n.ahead=24)
Junk3<-c(series1,Junk2$pred)
Junk3a<-c(rep(0,576),2*Junk2$se)
Junk3b<-c(rep(0,576),-2*Junk2$se)
Junk4a<-Junk3+Junk3a;Junk4b<-Junk3+Junk3b
ylim<-range(Junk4a,Junk4b)
par(mfrow=c(1,1))
xaxis1<-1941+(1:length(Junk3)/12)
plot(xaxis1[xaxis1>1983],Junk4a[xaxis1>1983],type="l",las=1,ylim=ylim,col="red",xlab="year",ylab="degrees C",main="Data + predictions")
lines(xaxis1[xaxis1>1983],Junk4b[xaxis1>1983],col="red")
lines(xaxis1[xaxis1>1983],Junk3[xaxis1>1983],col="blue")
lines(xaxis[xaxis>1983],series1[xaxis>1983])

28. Two series
Bivariate case {Xt, Yt} - jointly distributed
Linear time invariant / transfer function model
nonparametric/parametric approaches

29. Southern Oscillation Index
El Niño: global coupled ocean-atmosphere phenomenon.
The Pacific ocean signatures, El Niño and La Niña are important temperature fluctuations in surface waters of the tropical Eastern Pacific Ocean

30. Southern Oscillation reflects monthly or seasonal fluctuations in the air pressure difference between Tahiti and Darwin

31. junk<-scan("recifecurado")
junk1<-matrix(junk,ncol=48)
junk6<-junk1[1,]
junk1<-junk1[,junk6>1950]
junk2<-junk1[2:13,]
series<-c(junk2)/10
length(series[is.na(series)]) #13
xaxis<-1951+(1:length(series)/12)
series1<-series
junk4<-matrix(series1,nrow=12)
junk5<-apply(junk4,1,mean)
for(i in 2:(length(series)-1)){if(is.na(series[i]))series1[i]<-.5*series[i-1]+.5*series[i+1]}
series2<-series1
for(i in 1:38){
for(j in 1:12){
series2[(i-1)*12+j]<-series1[(i-1)*12+j]-junk5[j]}}

32. kunk<-scan("SOIa.dat")
kunk1<-matrix(kunk,ncol=58); kunk6<-kunk1[1,]
kunk1<-kunk1[,kunk6<1989]
kunk2<-kunk1[2:13,]
teries<-c(kunk2)
length(teries[is.na(teries)]) #0
teries1<-teries; teries2<-teries1
postscript(file="recifeplots3.ps",paper="letter",hor=T)
par(mfrow=c(2,1))
plot(xaxis,series2,type="l",las=1,xlab="year",ylab="",main="Seasonally adjusted Recife temps")
plot(xaxis,teries2,type="l",las=1,xlab="year",ylab="",main="Southern Oscillation Index")

33. postscript(file="recifeplots2.ps",paper="letter",hor=F)
par(mfrow=c(1,1))
acf(cbind(series2,teries2))

35. junk10<-cbind(series2,teries2)
junk11<-spec.pgram(junk10,plot=F,taper=0,detrend=F,demean=F,spans=11)
par(mfcol=c(2,2))
plot(junk11$freq,10**(.1*junk11$spec[,2]),log="y",main="SOIspectrum", xlab="frequency", ylab="", las=1,type="l")
plot(junk11$freq,junk11$coh,main="Coherence",xlab="frequency",ylab="",las=1,ylim=c(0,1),type="l")
junkh<-1-(1-.95)**(1/(.5*junk11$df-1))
abline(h=junkh)
plot(junk11$freq,10**(.1*junk11$spec[,1]),log="y",main="Seasonally corrected Recife spectrum",xlab="frequency", ylab="",las=1, type="l")

36. SARIMAX
Yt = αYt-1 + βYt-12 + γXt + Nt
ar ma1 sar1 sma1 intercept teries1
s.e
sigma^2 estimated as : log likelihood = , aic =
Junk1<-arima(series1,order=c(1,0,1),seasonal=list(order=c(1,0,1),period=12),xreg=teries1)

37. The answer to the question:
There is a hint of a linear time invariant relationship.