I have data that was collected at a daily interval. The data has a 7-day weekly periodicity and then annual seasonality which is really a 365.25 day period. To handle complex, multiple seasonality you can follow the method described by De Livera, Hyndman, and Snyder in this paper: http://robjhyndman.com/papers/ComplexSeasonality.pdf. The seasonal components are reduced to their Fourier components, and I would describe the update procedure as looking like a "rolling Fourier transform." Using the Fourier representation of annual seasonality is an effective way to reduce the number of parameters that you have to maintain. So instead of keeping 365 values for a 365 day cycle, you reduce it to say 10 Fourier frequencies which require 20 parameters (2 times the number of frequencies). When I divide all of the values in my time series by the long term mean value (there is no trend, but one could de-trend the data if there was), the data varies from 0.5 to 1.5. After removing the day-of-week component, the data varies over a much smaller range 0.85 to 1.15. After removing the second level of seasonality (annual), the level varies between 0.96 and 1.04. That says that about 70% of the variability of the data was determined by day-of-week seasonality, another 22% of the variability was due to day-of-year seasonality. Leaving only 8% noise!
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