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I am working on a research problem related to time series analysis. Now, I have STL decomposition and FB Prophet to decompose my data into trend, seasonality and residual. I struggle with measuring the strength of my seasonality component. In the book Forecasting: Principles and Practice, there is a formula which is the ratio of var(residual) to var(residual + seasonality) (takes 0 if negative). Yet, it's very difficult to build an intuition behind it, on repeated trials it shows that daily seasonality is pretty much zero (though my algorithms suggest that there is one.) I would appreciate any hints and suggestions as to measuring the strength of seasonality. Thank you!

Dina Kunets
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on repeated trials it shows that daily seasonality is pretty much zero (though my algorithms suggest that there is one.)

I am not sure about Prophet, but STL will fit a seasonal component whether or not one is present. This may account for your observation.

I personally would fit seasonal and non-seasonal models to your data and assess forecast accuracy on a holdout sample for both models. Then you can quantify by how much including seasonality reduces your MSE. (And whether it does so at all.)

Stephan Kolassa
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Are you dealing with simple or complex seasonality in your data? For example, monthly data may show monthly seasonality whereas hourly data may show daily and weekly seasonality.

I think your decomposition would need to reflect whether the seasonality in your data is simple or complex. Simple seasonality would produce a single seasonal component when decomposing the time series into its component parts, whereas multiple seasonality would produce multiple seasonal components. For an example of the latter, see https://otexts.com/fpp2/complexseasonality.html.

One way to retrieve the long term trend and seasonal component(s) in your time series would be using a GAM model. For this type of model, you can report an adjusted R squared and keep track of improvements in adjusted R squared associated with introducing additional systematic components to the model.

Isabella Ghement
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    thank you for your response! I am dealing with complex seasonality and am using a gam model. in the book that you've referenced, there is a formula called seasonality strength (https://otexts.com/fpp2/seasonal-strength.html), do you happen to know if there are alternative ways to measure this 'strength'? thank you! – Dina Kunets Oct 24 '19 at 16:17
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    That's what I thought - you would have to worry about complex seasonality. Rob Hyndman shows some detailed R code for cases where simple seasonality would be at play here: https://robjhyndman.com/hyndsight/tscharacteristics/. But the code won't work for complex seasonality. Maybe you could reach out to Rob and ask him if he has worked on extending that approach to complex seasonality? I am not familiar with this area so I can't really help. – Isabella Ghement Oct 25 '19 at 23:12