A study of singular spectrum analysis with global optimization techniques

Singular spectrum analysis has recently become an attractive tool in a broad range of applications. Its main mechanism of alternating between rank reduction and Hankel projection to produce an approximation to a particular component of the original time series, however, deserves further mathematical justification. One paramount question to ask is how good an approximation that such a straightforward apparatus can provide when comparing to the absolute optimal solution. This paper reexamines this issue by exploiting a natural parametrization of a general Hankel matrix via its Vandermonde factorization. Such a formulation makes it possible to recast the notion of singular spectrum analysis as a semi-linear least squares problem over a compact feasible set, whence global optimization techniques can be employed to find the absolute best approximation. This framework might not be immediately suitable for practical application because global optimization is expectedly more expensive, but it does provide a theoretical baseline for comparison. As such, our empirical results indicate that the simpler SSA algorithm usually is amazingly sufficient as a handy tool for constructing exploratory model. The more complicated global methods could be used as an alternative of rigorous affirmative procedure for verifying or assessing the quality of approximation. Moody T. Chu’s research was supported in part by the National Science Foundation under grant DMS-1014666. Matthew M. Lin’s research was supported in part by the National Science Council of Taiwan under grant NSC 101-2115-M-194-007-MY3. M. T. Chu Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA e-mail: [email protected] M. M. Lin (B) Department of Mathematics, National Chung Cheng University, Min-Hsiung, Chia-Yi 621, Taiwan e-mail: [email protected] L. Wang Department of Mathematics, Dalian University of Technology, Dalian, China e-mail: [email protected]