# Additive and multiplicative seasonality – can you identify them correctly?

By | November 9, 2014

Seasonality is a common characteristic of time series. It can appear in two forms: additive and multiplicative. In the former case the amplitude of the seasonal variation is independent of the level, whereas in the latter it is connected. The following figure highlights this:

Note that in the example of multiplicative seasonality the season is becoming “wider”. Obviously if the level was decreasing the seasonal amplitude of the multiplicative case would decrease as well. For selecting the appropriate model to produce our forecasts we need to know the type of seasonality we are dealing with. How do you compare against statistical identification? Select additive or multiplicative in the demonstration below and submit your choice to see if you can do better than statistics and the average accuracy of participants so far.

## 7 thoughts on “Additive and multiplicative seasonality – can you identify them correctly?”

1. Martin

Hi Nikolaos,
Thanks for this interactive tool !

What does it stand for?

Thanks

1. Nikos Post author

That is an unpublished test that I have been working on with a colleague of mine, Carlos Diaz. Sometimes it takes quite a bit of time to get an academic publication!
You can find a prototype of it in TStools for R. The function is called mseastest< \code>.

2. Benjamin

How did you obtain the plot comparison (additive vs multiplicative) graphs in R that are shown above?

1. Nikos Post author

It has been a while since I built this plot, so I do not have code any readily, but I think what I did is construct the series, fit a centred moving average through them (red line) and then add or multiply a factor to it to get the dotted blue lines for the two types of seasonality. Hope this helps!

3. Nitesh Jindal

Thanks.
But I still have some confusions:
1. What does level stands for here and why are we using level here?
2. I see that data points are closer to each other within a year( time frame) in additive case whereas data points are far from each other within a year in multiplicative case. Is this additive/multiplicative seasonality refers to?

1. Nikos Post author

1. As level, in these cases, I am using a local level estimate as defined in exponential smoothing state space models. In many cases, you get a similar view by using a centred moving average.
2. The general definition of additive or multiplicative seasonality is: level + seasonal indices, or level x seasonal indices. Effectively, with multiplicative seasonality the width of the seasonal pattern is proportional to the level. For additive seasonality it is independent.

4. Maria

Excellent demonstration. I finally understand. Many thanks!