ts.plot(AirPassengers)
airpass.log<-log(AirPassengers)
ts.plot(airpass.log) 
length(airpass.log) 

par(mfrow=c(1,2))
acf(airpass.log)
pacf(airpass.log)


 
# use a linear model for detrending #

airpass.fit<-lm(airpass.log ~ c(1:144))
airpass.fit.detrend<-airpass.fit$residual 
plot.ts(airpass.fit.detrend) 

# filter 


spencer.wts<- c(-3,-6,-5,3,21,46,67,74,67,46,21,3,-5,-6,-3)/320
airpass.filt<-filter(airpass.log, spencer.wts)
ts.plot(airpass.log, airpass.filt, lty=c(2,1)) 

airpass.detrend.filt<-filter(airpass.fit.detrend, spencer.wts)
ts.plot(airpass.fit.detrend, airpass.detrend.filt, lty=c(2,1)) 


# check that spencer filter leaves cubic polynomials undisturbed

x<-c(1:1000) 
for(i in 1:1000){
x[i] <- 0.003 * i^3  - 2 * i^2 + 1  
}
ts.plot(x)


x.filt<-filter(x, spencer.wts)
ts.plot(x, x.filt, lty=c(2,1)) 

# differencing

airpass.diff<-diff(airpass.log)
ts.plot(airpass.diff) 

airpass.diff2<-diff(airpass.diff, lag=12)
ts.plot(airpass.diff2) 


# stl decomposition

airpass.log.freq<-ts(as.vector(airpass.log), start=1, frequency=12)
airpass.log.stl<-stl(airpass.log.freq, s.window="periodic")
plot(airpass.log.stl) 

# fit arima model

par(mfrow=c(2,1)) 
acf(airpass.diff2)
pacf(airpass.diff2)

airpass.arima<-arima(airpass.diff2, order =c(0,0,1))
tsdiag(airpass.arima) 


airpass.arima2<-arima(airpass.diff2, order=c(0,0,1), seasonal=list(order=c(0,0,1),
period=NA) ) 
tsdiag(airpass.arima2) 



airpass.arima3<-arima(airpass.log, order=c(0,1,1), seasonal=list(order=c(0,1,1),
period=12 )  ) 
tsdiag(airpass.arima3) 


# cumulative periodogram 

fit2.res<-airpass.arima3$residuals 
fit2.res

par(mfrow=c(1,2))
acf(fit2.res)
pacf(fit2.res) 

# are consistent with white noise 

cpgram(fit2.res)