Fuzzy-valued and complex-valued time series analysis using multivariate and complex extensions to singular spectrum analysis

This paper provides evidence for the effectiveness of two extensions of Singular Spectrum Analysis, Complex SSA (CSSA) and Multivariate SSA (MSSA), when performing tasks such as smoothing, change point detection and forecasting of time series. CSSA is well suited for bivariate time series (usually displaying co-movements) and interval-valued time series. Functionally quasi-equivalent with CSSA in the bivariate case, MSSA comes, however, with its extra-potential for multivariate objects, such as fuzzy-valued time series (expressed in terms of α-levels). Our extension of the univariate SSA based change point detection algorithm to complex and multivariate cases is a novel approach. CSSA and MSSA are formally compared with each other and intensively tested in numerical experiments for smoothing, change point detection and forecasting with real-world data (a couple of foreign exchange rates with strong co-movements and a triangular-shaped fuzzy daily temperature time series).