  The problem of estimating trend and seasonal variation in time-series data
has been studied over several decades, although mostly using single time
series. This paper studies the problem of estimating these components from
functional data, i.e. multiple time series, in situations where seasonal
effects exhibit arbitrary time warpings or phase variability across different
observations. Rather than ignoring the phase variability, or using an
off-the-shelf alignment method to remove phase, we take a model-based approach
and seek MLEs of the trend and the seasonal effects, while performing
alignments over the seasonal effects at the same time. The MLEs of trend,
seasonality, and phase are computed using a coordinate-descent based
optimization method. We use bootstrap replication for computing confidence
bands and for testing hypothesis about the estimated components. We also
utilize log-likelihood for selecting the trend subspace, and for comparisons
with other candidate models. This framework is demonstrated using experiments
involving synthetic data and three real data (Berkeley Growth Velocity, U.S.
electricity price, and USD exchange fluctuation).
