# 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.

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.

• Thank you very much for your answer! Very helpful and informative! I will be using some holdback data for the forecast, so in your opinion, what would be the best and simplest statistical test that I can use on the 'out of sample (holdback) data' to test the forecasts accuracy? – Jonas Blaps Nov 28 '16 at 10:07
• @JonasBlaps Do you have the possibility to share the data? – Laurent Duval Nov 28 '16 at 11:17
• Using holdback data from one origin can be flawed when there are anomalies in the holdout data.Optimally predicting bad data can lead to bad model selection. This is often called "the tail wagging the dog syndrome" – IrishStat Nov 28 '16 at 12:14
• @IrishStat Indeed, I was about to suggest an exponentially weighted criterion (in the spirit of EWMA) that allows to progressively forget the past past – Laurent Duval Nov 28 '16 at 12:47
• Rather than assume any form of a weighted average it is far better to determine the optimal form via ARIMA while taking into account any idebtifiable deterministic structure such as level shifts/trends/seasonal pulses and of course pulses. – IrishStat Nov 28 '16 at 13:17

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.