Regularized Singular Spectrum Analysis

Functional time series (FTS) are constituted by dependent functions and can be used to modelseveral applied processes. Several machine-learning approaches have been developed in theliterature to gain insight into the stochastic processes that generate FTS. In this work, we presentregularization techniques in the analysis of FTS. Singular spectrum analysis (SSA) is a non-parametric technique for decomposing time series into trends, periodicities, and noise components.Functional SSA (FSSA) is the functional extension of SSA applied to FTS. We begin by representingFTS as multivariate time series (MTS) data and develop a regularization technique for multivariatesingular spectrum analysis (MSSA). MSSA is a decomposition technique for MTS, and we denote theregularized version of the algorithm as reMSSA. reMSSA is formulated as a penalized lossminimization problem where we employ regularized singular value decomposition (RSVD) to findlow-rank trajectory matrix approximations of the data. Next, we develop a similar regularizationtechnique for FSSA. Regularized FSSA (reFSSA) is developed as an extension of FSSA. A penaltyfunction with a smoothing parameter is added to the loss function measuring the reconstructionerror of a low-rank trajectory operator approximation. Regularized functional SVD (RfSVD) is usedto solve the minimization problem. RfSVD allows the derivation of a closed-form generalized cross-validation (GCV) criterion for selecting smoothing parameters. Hilbert SSA (HSSA) is the applicationof SSA to FTS objects created by defining a basis system in the Hilbert space. The basis system forHSSA is different from the known basis function systems (monomial, Fourier, b-spline, etc.) used forFSSA. We develop a regularization technique for HSSA. Finally, we apply reMSSA, reFSSA, andregularization based on HSSA to call center data that contains the number of incoming calls to abank's call center in Israel. We show that the proposed regularization techniques, reMSSA, reFSSA,and regularization based on HSSA, outperform MSSA, FSSA, and HSSA, respectively, by effectivelysmoothing the rough components generated by MSSA, FSSA, and HSSA of the MTS and FTS objects.

Functional Time Series

Functional SSA

Hilbert Space

Regularization 

Mid-Level

Computational Statistics

Symposium on Data Science and Statistics (SDSS) 2023