Functional Time Series Forecasting: Functional Singular Spectrum Analysis Approaches

Functional data analysis (FDA) is a growing field of statistics that is showing promising results in analysis due to the fact that functional algorithms act on possibly more informative and smooth data. Often times statistical techniques that act on real-valued scalars or vectors are extended into the functional realm to handle such curved data. One example is principal component analysis (PCA) which was extended into functional PCA (FPCA) and multivariate FPCA so that dimension reduction may be performed on time-independent functional observations and many variants of these methods have been developed, see Ramsay and Silverman (2005), Jeng-Min et al. (2014), and Happ and Greven (2018) for more details. Another example of this concept can be seen in singular spectrum analysis (SSA) (Golyandina et al., 2001) which is a decomposition technique for time series. The SSA algorithm was extended into functional SSA (FSSA) in Haghbin et al. (2020a). They showed that the FSSA algorithm outperforms SSA and FPCA-based approaches in separating out sources of variation for smooth, time-dependent, functional data which is defined as a functional time series (FTS). In addition to SSA being extended to FSSA, the multivariate SSA (MSSA) approaches (Golyandina et al., 2015; Hassani and Mahmoudvand, 2013) have also been extended to the functional realm in Trinka et al. (2020) where dimension reduction was performed on a multivariate FTS of intraday temperature curves and images of vegetation in a joint analysis giving more prominent results. An important problem often confronted by researchers is prediction of stochastic processes. Golyandina et al. (2001) expanded the results of the SSA and MSSA algorithms to deliver a nonparametric forecasting method, called SSA recurrent forecasting, and Hassani