Choosing Between Additive and Multiplicative Model? I have a set of data that I am currently analysing.
I am having difficulty in deciding whether an Additive model should be used to forecast the data, or if I should use a Multiplicative model.
I know the difference between the two, and I can apply the correct model when the raw data is linear...but in this case, my data is non linear.
I have attached a time-series of my data - which of the two models should I use and why?
(My instinct is to go with the Additive Model on the basis that the magnitude of the seasonal fluctuations (or the variation around the trend-cycle) doesn't appear to vary with the level of the time series.

 A: I would go for additive too. As your apparent signal seems of low frequency, you can go a little beyond, at least empirically. You can check for instance  the homoscedasticity of  finite differences of the data (first or second order). This would act as a very crude high-pass filter, where you could expect the noise to be dominant. 
If your signal is much longer, moving windows and Fourier transforms could be of help. 
However, as for forecasting, you can perform both models in parallel, and  decide which one you apply based, for instance, on the best performance of one of them based on past statistics. This is a heuristic method that I have recently used in the prediction of outcomes for hybrid system co-simulation, where no model is known: perform different extrapolations in parallel, very fast, and decide. It is not very theoretical, but it  works well on our data.
If interested, I could develop. The reference is called: CHOPtrey: contextual online polynomial extrapolation for enhanced multi-core co-simulation of complex systems
As the data is quite short, and I am not sure we have a full seasonal period, I tried to perform some Fourier analysis on the data, its gradient and Laplacian. The fluctuation seems to be quite periodic, so on the bottom plot I have attempted to design a "filtering" moving average. The residue does not vary in amplitude a lot. It really does not seem to be random.
 
A: I took the 55 values and used AUTOBOX to automatically detect a hybrid model possibly including deterministic structure as well as ARIMA structure. The plot of the original data and the ACF plot of the  original series is here. AUTOBOX concluded that a single trend and 3 seasonal dummies were more appropriate tham SARIMA while also including AR structure of order 1 . Here is the model       AND  here   with the following statistical summaries  . 
The residual plot is here suggesting sufficiency  with the companion ACF of the residuals here .
The Actual, Fit and Forecast plot is here  and the OUTLIER adusted plot clearly suggesting the need for the 4 pulses in the model  . Finally the Forecast plot is here for the next 8 periods.
Transformations such as logarithms or multiplicative models need to be justified and suggested by the data or by the user who has certain domain knowledge. This was not so in this case. See here for when power transforms are needed When (and why) should you take the log of a distribution (of numbers)? . Note that AUTOBOX essentially converged on the HW Additive Seasonal Model with TREND and 4 anomalies and a highly significant AR(1) coefficient.
COMMENTS FOR LAURENT:
Three of the four deterministic comments were required (Trend,Seasonal(QUARTERLY) Dummies and Pulses) while also needing the AR(1) structure to deal with short-term memory. 
