Functional Singular Spectrum Analysis: Nonparametric Decomposition and Forecasting Approaches for Functional Time Series

In this dissertation, we develop nonparametric decomposition methods and the subsequent forecasting techniques for functional, time-dependent data known as functional time series (FTS). We use ideas from functional data analysis (FDA) and singular spectrum analysis (SSA) to introduce the nonparametric decomposition method known as functional SSA (FSSA) and its associated forecasting techniques. We also extend these developed methodologies into multivariate FSSA (MFSSA) over different dimensional domains and its subsequent forecasting routines so that we may perform nonparametric decomposition and prediction of multivariate FTS (MFTS). The FSSA algorithm may be viewed as a signal extraction technique and we find that the method outperforms other competing approaches in estimating the underlying deterministic nature of an FTS. We then develop the FSSA recurrent forecasting (FSSA R-forecasting) and FSSA vector forecasting (FSSA V-forecasting) algorithms to predict future observations and we find that these methods outperform the current gold standard for nonparametric forecasting of periodic FTS. Finally, we finish with the implementation of MFSSA and the respective forecasting algorithms (MFSSA R-forecasting and MFSSA V-forecasting), which are used to decompose and forecast MFTS. We find that the MFSSA methods outperform their univariate FSSA counterparts in signal extraction and forecasting of MFTS data