Reconstruction of Gaps in Flow Series Using Singular Spectrum Analysis (SSA) and Multi-channel SSA (M-SSA)
Singular Spectrum Analysis (SSA) (Broomhead and King, 1986a; Fraedrich, 1986, Ghill et al., 2002) is a state-of-the-art spectral, data-adaptive and nonparametric method. SSA decomposes an original time series to trend (if exists), oscillatory and noise components by way of a singular value decomposition. The term singular spectrum comes from the spectral (eigenvalue) decomposition of a matrix into its spectrum of eigenvalues (Elsner and Tsonis, 1996). The basic SSA algorithm has stages of decomposition and reconstruction. The decomposition stage requires embedding and singular value decomposition. In this stage, there are two basic steps. The former is embedding the sampled time series in a vector space of dimension M, the latter is computing the MxM lag-covariance matrix of the data (Vautard and Ghill, 1989). The reconstruction stage demands the grouping to make subgroups of the decomposed trajectory matrices and diagonal averaging to reconstruct the new time series from the subgroups (Myung, 2009). Multi-Channel SSA (M-SSA) (Broomhead and King, 1986b) is an extension of the standard SSA to the case of multivariate time series. It is a generalization of this approach to systems of partial differential equations and the study of the spatiotemporal structures that characterize the behavior of solutions on their attractor (Constantin et al., 1989; Temam, 1997) (Ghil et al., 2001).
The accuracy and reliability of the methods depend on the pattern of missing data, the relative length of the gaps with respect to the total length of the data set, and the fraction of variance captured by robust, oscillatory modes (Kondrashov and Ghil, 2006). In this study, SSA and M-SSA is applied to flow series by using SSA-MTM Toolkit (Singular Spectrum Analysis-MultiTaper Method Group, 2007). Different percentages of the data have been removed randomly and then filled with using SSA and M-SSA. It is easily seen that missing data are filled perfectly for either univariate or multivariate time series.