# Additive vs Multiplicative decomposition

My question is a really simple one but those are the ones that really get me :) I don't really know how to evaluate if a specific time series is to be decomposed using an additive or a multiplicative decomposition method. I know there are visual cues as to telling them apart from one another but i don't get them.

Take for instance this time series:

How would you describe it?

Thanks in advance for your help.

• A multiplicative decomposition roughly corresponds to an additive decomposition of the logarithms, so much of the thread on deciding whether to take log (or square root) transformations at stats.stackexchange.com/questions/74537 applies here, too. (Ignore any answers there that caution against applying transformations because that's not the point.) In your example a decomposition based on the reciprocals of the data might even be called for, especially if the reciprocals have a meaningful interpretation (such as converting miles per gallon into gallons per mile). – whuber Jul 1 '14 at 16:34
• @whuber Thank you very much for your answer and for the SO post you've lunked to. I'm afraid I was hoping to learn how to tell them apart and tell when to use one over another using that time series for illustration purposes. I've never heard of decomposition based reciprocals :-/ I'll do some research on that. – 4everlearning Jul 1 '14 at 18:26
• Two answers in the thread I referenced give procedures to tell them apart: the one by "forecaster" refers to the "STL method" and illustrates it; my answer describes (and gives R code for) a simple robust exploratory method, the "spread vs. level plot." I can eyeball your graphic and see that when values are near 600 the amplitudes of their short-term variation are almost an order of magnitude greater than when they are near 200: that indicates considering a log, reciprocal, or maybe reciprocal square root. – whuber Jul 1 '14 at 18:30

## 1 Answer

In addition to what @whuber has recommended, I would refer you to https://www.otexts.org/fpp/6/1 which explains why you would choose additive vs. multiplicative decomposition.

In specifically looking at your data, because the seasonality varies, i.e., seasonality at the beginning is large and as it seasonality is almost not present in the later years, this would suggest a multiplicative decomposition. According to the text referenced above, an alternative would be to do an appropriate transformation and apply additive decomposition.

There is a level shift in the data some time around mod 1972 which also needs to be treated when decomposing.

There is another decomposition based method called unobserved components model which takes most of the guess work out of decomposition and provides you with some good statistics to make sound decision such as stochastic vs. deterministic trends/seasonality etc.

Hope this helps.

• +1 Just to clarify: I was not necessarily recommending an appropriate transformation as an alternative. My suggestion was that one indication of a multiplicative structure would be that a log transformation appears to stabilize the spread-vs-mean relationship. – whuber Jul 1 '14 at 20:21
• I agree completely @whuber. – forecaster Jul 1 '14 at 20:32