An Example. The question Data Analyses Conclusions

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? - ...

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 - 3068290000 Years 1949-1988 Searched an inappropriate ste for a long time (Looked at Brasil sites too, but that didn't turn up the data)

notice -9999 replace by NA file: recifecurado The web data. monthly notice -9999 replace by NA file: recifecurado 30682900001941 274 279 268 267 260 250 246 245 256 260 262 266 30682900001942 273 268 270 270 256 247 236 233 252 260 270 265 30682900001943 270 270 273 261 253 245 236 238 247 260 264 268 30682900001944 269 272 275 263 254 247 239 232 241 256 267 273 30682900001945 278 268 278 271 256 240 236 243 251 258 268 267 30682900001946 268 277 271 259 258 247 247 247 249 257 261 266 30682900001947 269 270 268-9999 253 246 244 245 249 262 268-9999 30682900001948 273 271 270 269 255 249 243 240 248-9999 264 270 30682900001949 272 274 278 266 252 246 240 236 250 261 265 271 30682900001950 273 280 269 251 248 243 235 234 248 258 264-9999 30682900001951 269 267 275 265 254 238 233 238 247 259 263-9999 30682900001952 272 278 267 262 252 245 240 238 253 256 263 268

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)/10 # 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]}

plot(xaxis,series1,type="l",xlab="year",ylab="mean temp (degrees C)",las=1) title("Mean monthly temperatures 1949-88 Recife Curado") abline(h=mean(series1))

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))

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

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

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

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

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))

Cooler in July-Aug Southern Hemisphere Uncertainty?

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

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)))

original variance 1.342 adjusted .248

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")

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)

Work remains on seasonal Residual "not" white noise

Time domain distributions

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

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: ar1 ma1 sar1 sma1 intercept 0.7604 -0.3344 0.9990 -0.9201 25.6117 s.e. 0.0610 0.0968 0.0007 0.0220 0.6100 sigma^2 estimated as 0.1771: log likelihood = -337.48, aic = 686.97

tsdiag(Junk,gof.lag=25)

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])

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

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

Southern Oscillation reflects monthly or seasonal fluctuations in the air pressure difference between Tahiti and Darwin www.cpc.ncep.noaa.gov/data/soi

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]}}

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")

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

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")

SARIMAX Yt = αYt-1 + βYt-12 + γXt + Nt ar1 ma1 sar1 sma1 intercept teries1 0.7885 -0.3717 0.9996 -0.9474 25.5572 -0.0255 s.e. 0.0610 0.1014 0.0006 0.0321 0.6403 0.0149 sigma^2 estimated as 0.1792: log likelihood = -275.68, aic = 565.35 Junk1<-arima(series1,order=c(1,0,1),seasonal=list(order=c(1,0,1),period=12),xreg=teries1)

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