Alternative Approaches to Estimating the Linear Propagator and Finite-Time Growth Rates from Data

Abstract

The propagator of a linear model plays a central role in empirical no mal mode and finite-time instability problems. Its estimation will affect whether the linear stability character is tics of the corresponding dynamic system can be properly extracted. In this study, we introduce two alternative methods for estimating the linear propagator and finite-time growth rates from data. The first is the generalized singular value de composition (GSVD) and the second is the singular value de composition combined with the cosine-sine decomposition (SVD-CSD). Both methods linearize the relation between the predictor and the predict and by de composing them to have a common evolution structure and then make the estimations. Thus, the linear propagator and the associated singular vectors can be simultaneously derived. The GSVD clearly reveals the connection between the finite-time amplitude growth rates and the singular values of the propagator. However, it can only be applied in situations when given data have more state variables than observations. Further more, it generally encounters an over-fitting problem when data are contaminated by noise. To fix these two draw backs, the GSVD is generalized to the SVD-CSD to include data filtering capability. Therefore, it has more flexibility in dealing with general data situations. These two methods as well as the Yule-Walker equation were applied to two synthetic datasets and the Kaplan's sea surface temperature anomalies (SSTA) to evaluate their performance. The results show that, because of linearization and flexible filtering capabilities, the propagator and its associated properties could be more accurately estimated with the SVD-CSD than other methods.

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