You observe a time series plot where the seasonal variation seems to grow as the level of the series increases. Which decomposition and transformation combination is most appropriate to start with?

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We start by restating the scenario to ensure clarity: a time series plot shows seasonal variation increasing as the series level rises. This pattern suggests that seasonality is proportional to the level, not constant.
Option 1: 'An additive decomposition on the log-transformed data.' Using a log transformation can stabilize multiplicative effects, but performing an additive decomposition on the log-transformed data corresponds to assuming additive seasonality in the log scale, which translates to multiplicative seasonality in the original scale. This is a viable path, but it changes the form of the decomposition and may not be the most direct starting point when the goal is to address multiplicative seasonality in the original data.
Option 2: 'A multiplicative decomposition on the original data.' This directly models seasonality that scales with the level, which matches the observed pattern. Multiplicative decomposition assumes components multiply together (Trend × Seasonal × Error), capturing increasing seasonal amplitude with the level.
Option 3: 'A multiplicative decomposition on the differenced data.' Differencing removes trend and can alter the interpretation of seasonality; applying multiplicative decomposition after differencing is less standard and may not correctly reflect the original multiplicative relationship, making this choice awkward as a starting approach.
Option 4: 'An additive decomposition on the original data.' Additive decompositions assume constant seasonal effects regardless of level. Since the seasonal amplitude grows with the level in this scenario, this option would fail to capture the observed pattern, making it unsuitable as a starting point.
In summary, the core issue is whether seasonality scales with the level. An operation that directly encodes this behavior—multiplicative seasonality on the original data—best aligns with the observed pattern right from the outset, while the other options either assume constant seasonality, alter the data in ways that complicate the interpretation, or rely on transformations that are not as direct for this particular characteristic.

You observe a time series plot where the seasonal variation seems to grow as the level of the series increases. Which decomposition and transformation combination is most appropriate to start with?

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