How do you decide between an additive and a multiplicative time-series decomposition?

## The two structures Additive: ``` y_t = T_t + S_t + R_t ``` Multiplicative: ``` y_t = T_t * S_t * R_t ``` The difference is what the seasonal component *means*. In the additive form, `S` is measured in the units of the series: "December runs 4,000 units above the trend". In the multiplicative form, `S` is a dimensionless factor centred on 1: "December runs 1.20, that is 20 percent above the trend". The multiplicative seasonal factors over one full period average to about 1 rather than to 0. ## How to choose **Look at the picture first.** Draw the series and mentally trace the envelope of the seasonal peaks and the seasonal troughs. If those two envelopes stay roughly parallel as the level moves, the seasonal effect is a constant number of units and additive is right. If the envelopes fan out as the level rises and pinch in as it falls, the seasonal effect is a constant proportion and multiplicative is right. The monthly international airline-passenger record is the textbook fan-out. Passenger numbers roughly triple across the record, and the summer bulge grows right along with them. Treating that series additively forces one compromise seasonal shape onto both the small early years and the large late years: the fitted seasonal is too big at the start and too small at the end, and the mistake is not random, so it shows up as a systematic, seasonal-looking pattern in the remainder whose amplitude grows over time. That growing-amplitude remainder is the diagnostic that you picked the wrong structure. **Think about the mechanism, not just the plot.** Ask what generates the seasonality. A retail effect where "December is a third bigger than a normal month" is inherently proportional and therefore multiplicative. A fixed physical or contractual quantity, such as a batch that is always processed in the same month, is a constant number of units and therefore additive. Mechanism reasoning is what keeps your answer stable when the plot is ambiguous over a short history. ## The log transform Because `log(a*b*c) = log(a) + log(b) + log(c)`, taking logarithms of a strictly positive series turns a multiplicative decomposition into an additive decomposition of the transformed series: ``` log(y) = log(T) + log(S) + log(R) ``` This is why so much of the classical toolkit only needs to support the additive case. Log first, decompose additively, and exponentiate the components back if you want factors again. Three consequences are worth stating: - **Interpretation changes.** On the log scale, an additive seasonal effect of 0.18 is roughly an 18 percent effect on the original scale (exp(0.18) is about 1.20). Differences of logs are approximately relative changes for small values, and exactly ratios after exponentiating. - **Back-transforming is not neutral.** Exponentiating a mean on the log scale gives back a median-like quantity on the original scale, not the mean. If someone downstream is going to add these numbers up into a total, that gap matters. - **Variance is stabilised as a side effect.** A log transform also shrinks the larger fluctuations at high levels, which is usually helpful and occasionally overdone: a series whose swing genuinely is constant in units becomes *under*-dispersed at high levels after logging. ## When neither form is clean - **Zeros and negatives.** `log(0)` is undefined and logs of negatives do not exist, so a series with genuine zeros, or one that can go negative such as net change or profit, cannot be logged directly. Adding a constant before logging is possible but it changes the implied relationship between level and swing, so the choice of constant quietly becomes a modelling assumption. A pseudo-additive form, where the seasonal effect is proportional to the trend but the remainder is not, exists precisely for series that hit zero. - **The structure changes partway.** A business that switched from a flat to a proportional seasonal pattern after a pricing change fits neither form over the whole history. Split the history or use a method that lets the seasonal shape evolve. - **The history is short.** With two or three seasonal periods you often cannot tell the two structures apart from the plot; lean on the mechanism argument and say out loud that the evidence is thin. ## What a strong answer sounds like "Additive if the seasonal swing is a constant number of units, multiplicative if it is a constant percentage. I check by looking at whether the peak-to-trough envelope fans out with the level, and I sanity-check against the mechanism. If it is multiplicative I usually log the series and work additively, remembering that logs need strictly positive data and that back-transformed components are ratios, not differences."