Solved – Getting best fitted model using Auto ARIMA but prediction result is very bad

arimaforecastingrtime series

I saw this: time series – Poor prediction using ARIMA model
But the answers aren't clear and isn't directing to me for solving the problem I have. Using only AR is giving me better prediction whereas Auto arima told me to use ARIMA.

acf(diff(sunspots)) #check if there's any seasonal pattern
pacf(diff(sunspots))

ACF plot suggesting to use MA(2) as ACF is cutting off and PACF is decreasing slowly.

enter image description here
enter image description here

auto.arima(sunspots, start.p=0, max.p=3, start.q=0, max.q=3)

Auto arima gave me (2,1,2)X(2,0,1):

fit <- arima(sunspots, c(2, 1, 2), seasonal = list(order = c(2, 0, 1), period = 12))
AIC(fit)
tsdisplay(residuals(fit), main="fit2residual")
pred <- predict(fit, n.ahead = 240)

#ts.plot(sunspots,pred$pred, log = "x", lty = c(1:3))

years20_pred<-pred$pred
years20_se<-pred$se

plot(sunspots,xlim=c(1700,2015),col="grey",lwd=1.5,ylab="sunspots")
lines(years20_pred, col="green",lwd=1.5)

enter image description here
This is evident how wrong the prediction is as it's not matching the previous patterns.

Using only AR is giving better prediction graph see below:

#++++++++++++++ ANOTHER WAY FOr AR=====
y<- ar(sunspots)
years20<-predict(y,n.ahead=240)
years20_pred<-predict(y,n.ahead=240)$pred
years20_se<-predict(y,n.ahead=240)$se

plot(sunspots,xlim=c(1700,2015),col="grey",lwd=1.5,ylab="sunspots")
lines(years20$pred, col="green",lwd=1.5)

enter image description here

I hvae tried lots of combination for ARIMA nothing is working and stuck on this for seven days. Can someone please advise where am I going wrong?

After doing BoxCoxplot as @stephan said I am getting the positive bounds of the prediction intervals very high which shouldn't be considering previous patterns. Also, I replaced the x-axis values with my own to show years using these lines:

forcastvar<- forecast(model_seasonal,h=240)
plot(forcastvar, xaxt='n')
axis(1, at=1:43, labels=seq(1790, 2003,5))

enter image description here

If I force change X-axis values it's not coming correctly according to data points. Is there a way for that?

Best Answer

ARIMA has well-known problems with seasonal time series if the seasonal cycle is "too long". Monthly data, with a seasonal length of 12 months, is fine. Weekly data, with a season of length 52 (disregarding fractional week numbers) are already a problem for ARIMA.

In the present case, sunspots have a cycle of length 11 years. The sunspots data are a monthly time series. Thus, the implicit seasonality of sunspots is 12 (months), not 11$\times$12=132 (months).

ARIMA and auto.arima() were never built to automatically detect a seasonal cycle whose length is not pre-specified. It is not overly surprising it does not see that it should do 11 seasonal differences to model a seasonality that repeats every 11 cycles of its prespecified frequency.

So, the first order of business would be to specify that seasonal cycles are indeed of length 132:

library(forecast)
sunspots_seasonal <- ts(sunspots,frequency=11*12)

Unfortunately, auto.arima() does not pick up on this. This is the problem with long seasons I allude to above.

model <- auto.arima(sunspots_seasonal)
plot(forecast(model,h=11*12))

auto.arima

So in this case, we need to help auto.arima(), by forcing a seasonal difference:

model_seasonal <- auto.arima(sunspots_seasonal,D=1)
plot(forecast(model_seasonal,h=11*12))

auto.arima seasonal

This is still not perfect; in particular, the negative bounds to the prediction intervals are nonsensical, so a Box-Cox transformation may be called for. But I would expect it to be better than a nonseasonal AR-only model.

auto.arima() does not aim at being a magic wand. Its aim is to be a robust method that works reliably on a large number of time series, and it is very good at this. If you have subject matter knowledge that it does not model, then by all means, help it along.

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