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!
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.)
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.