Holt-Winters seasonal smoothing, additive and multiplicative
Today let's understand Holt-Winters seasonal smoothing, the method that adds a repeating yearly pattern on top of a rising or falling trend.
Take Western Australia's domestic holiday trips, counted every quarter from 1998 to 2017, in thousands of overnight trips. That's 80 quarters of real tourism numbers, and they do two things at once. They climb overall, from around 750 thousand trips a quarter in the late 1990s to over 1,000 thousand two decades later. And within every single year they repeat the same shape: high in the first quarter, low in the third. Here is all 80 quarters of it, plotted in order.
Look at how the line saws up and down on its way up. Every year opens high and dips by the third quarter, while the whole pattern climbs together across the two decades. That repeating shape, riding on top of a trend, is exactly what Holt-Winters seasonal smoothing is built to forecast.
Why a trend-only forecast misses the repeating swings
See what a plain trend does with this series before adding anything new to it. Fit ETS(A,A,N), additive error, additive trend, no season, on the WA holiday trips, and forecast two years ahead.
The forecast barely moves. It creeps from 999 thousand trips in 2018 Q1 up to 1017 thousand by 2019 Q4, a rise of only 18 thousand trips over two full years. But the real WA series never sits still like that within a year. In 2017 alone it ran from 1134 thousand trips in the first quarter down to 880 thousand in the third, a swing of 254 thousand trips inside twelve months, more than ten times what this model expects to move in two full years. The model has a level and a slope, and nothing else. Nothing in that combination marks the third quarter as usually quiet.
Run report() to see the fitted model and its AICc, the score used to compare how well different models fit the same data. Lower is better.
AICc lands at 1120.46. Hold on to that number, because once a season enters the model, that score has somewhere to fall.