Since my posts at Tamino’s site seem to be causing confusion, I feel I should expand upon them here. Comment as appropriate:
From my 3 June post on Open Mind:
The post above me said the linear trend doesn’t tell much – and I responded it did actually convey information.
There’s two ways to look at what the data says about acceleration – one, if the past is any indicator of the future, it says that a linear prediction is pretty darn good. And since the mathematical characteristic of a cubic is exponential increases in the future, one has to be very cautious about assuming anything from the curvefit.
To examine this, let’s only look at the data up to 2000. Excel can quickly plot the data and create a linear and cubic curve fit. Both of these can easily be extended into the future and used a predictions. The cubic fit clearly starts accelerating once we start extrapolating. But does that acceleration match up with reality? To check, add another series to the plot, this one of the same data out to 2013. The data clearly lies on the linear line, and the cubic line diverges from the readings. Therefore, the acceleration shown by the cubic is more likely a function of the nature of a cubic fit than some physical acceleration.
EDIT: I just calculated the AIC as if we had taken a time machine back to the year 2000. The blue line of the plot shows the data I used (annual data for Sewell Point). The AIC for the linear model is 720.3584. The AIC for the cubic model is 715.7580. Hence this metric would say that the cubic
fits the data better is better at capturing the nuances of the data than the linear. But is that useful in determining which model should be used for predicting the future?
Let’s jump back into our time machine and fast forward to 2013. We now have the data up to 2013 and it’s plotted as the red line above. It’s clear that the linear model tracked the data much better than the cubic model.
Therefore, AIC is not in this case a good indicator of predictive capability!
EDIT #2: In response to Bernd Palmer’s question:
“they choose a statistical model which gives a low forecast: a straight line.” The last figure above (3rd degree) doesn’t show a forecast, or does it?
And it doesn’t say anything about what’s causing the swings. Let’s say you were an observer in the year 1960, Would the 3rd-degree line have enabled you to forecast the reversal of the line in the following years?
No, if you limited your information to ending in 1960, then by definition the end of that line in a cubic would be up-turned. It would not be a predictor of the future. But it still has a better AIC than the linear model.
If I did this right (using annual data for Sewell Point and the linear and cubic fit models in R), the AIC for the linear model for the data up to 1960 is 316.7091, the AIC for the cubic fit is 311.7673, hence the cubic fit is “better”. And yet, it still had no predictive ability. If we extrapolated the cubic fit from 1960 to 2010, the predicted rise would be almost 2 meters! The actual rise is about 200 mm.