  Many real-world time series exhibit multiple seasonality with different
lengths. The removal of seasonal components is crucial in numerous applications
of time series, including forecasting and anomaly detection. However, many
seasonal-trend decomposition algorithms suffer from high computational cost and
require a large amount of data when multiple seasonal components exist,
especially when the periodic length is long. In this paper, we propose a
general and efficient multi-scale seasonal-trend decomposition algorithm for
time series with multiple seasonality. We first down-sample the original time
series onto a lower resolution, and then convert it to a time series with
single seasonality. Thus, existing seasonal-trend decomposition algorithms can
be applied directly to obtain the rough estimates of trend and the seasonal
component corresponding to the longer periodic length. By considering the
relationship between different resolutions, we formulate the recovery of
different components on the high resolution as an optimization problem, which
is solved efficiently by our alternative direction multiplier method (ADMM)
based algorithm. Our experimental results demonstrate the accurate
decomposition results with significantly improved efficiency.
