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There are some threads about deterministic and stochastic trends here. But I have not found a thread regarding to this question:

Let's assume I have a given time series plot with a seasonality (and without trend). How can I detect if this present seasonality is determinstic or stochastic ?

My intuition is that if a deterministic seasonality is present we observe a peak/trough at the same frequency and with the same magnitude, e.g. every 12 months, we have a peak of roughly the same size.

If a stochastic seasonality is present the peak occurs only roughly at the same frequency but it can have a completely different magnitude.

It would be great if someone could explain this a bit further since I am not really sure about my intuition so far.

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You intuition seems to be about right. Give the theoretical assumptions of a stationary time series and a 'legible' plot, one would expect identical magnitudes for deterministic seasonality and varying magnitudes for stochastic seasonality.

However, you should keep in mind that different types of seasonality can "coexist in a single framework" (Carpolare et al., 2009, p. 350)—the three types of seasonality being 1) deterministic seasonality, 2) stochastic seasonality in a stationary process, and 3) non-stationary stochastic seasonality. Thus, identifying the type of seasonality in your time series by simply looking at the plot isn't always possible, depending your ts and your plot. In the paper mentioned above, the authors discuss a mathematical procedure to identify the type of seasonality present in a given ts.

I also found this blog post to be quite helpful—some ts-plots are provided there.

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