# Trend and Variable-Phase Seasonality Estimation from Functional Data

## TL;DR

This paper introduces a model-based method for estimating trend and seasonal components from functional data with phase variability, using maximum likelihood and coordinate descent, validated on synthetic and real datasets.

## Contribution

It proposes a novel joint estimation approach for trend, seasonality, and phase in functional data, addressing phase variability explicitly rather than ignoring or pre-aligning.

## Key findings

- Effective estimation of trend and seasonal effects demonstrated on real datasets.
- Bootstrap confidence bands and hypothesis tests validate the method's reliability.
- Model selection guided by log-likelihood improves component identification.

## Abstract

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).