Andrew J. Majda

Darin Comeau, Zhizhen Zhao, Dimitrios Giannakis et al.

The North Pacific exhibits patterns of low-frequency variability on the intra-annual to decadal time scales, which manifest themselves in both model data and the observational record, and prediction of such low-frequency modes of variability is of great interest to the community. While parametric models, such as stationary and non-stationary autoregressive models, possibly including external factors, may perform well in a data-fitting setting, they may perform poorly in a prediction setting. Ensemble analog forecasting, which relies on the historical record to provide estimates of the future based on past trajectories of those states similar to the initial state of interest, provides a promising, nonparametric approach to forecasting that makes no assumptions on the underlying dynamics or its statistics. We apply such forecasting to low-frequency modes of variability for the North Pacific sea surface temperature and sea ice concentration fields extracted through Nonlinear Laplacian Spectral Analysis. We find such methods may outperform parametric methods and simple persistence with increased predictive skill.

Dimitrios Giannakis, Andrew J. Majda

We present a technique for spatiotemporal data analysis called nonlinear Laplacian spectral analysis (NLSA), which generalizes singular spectrum analysis (SSA) to take into account the nonlinear manifold structure of complex data sets. The key principle underlying NLSA is that the functions used to represent temporal patterns should exhibit a degree of smoothness on the nonlinear data manifold M; a constraint absent from classical SSA. NLSA enforces such a notion of smoothness by requiring that temporal patterns belong in low-dimensional Hilbert spaces V_l spanned by the leading l Laplace-Beltrami eigenfunctions on M. These eigenfunctions can be evaluated efficiently in high ambient-space dimensions using sparse graph-theoretic algorithms. Moreover, they provide orthonormal bases to expand a family of linear maps, whose singular value decomposition leads to sets of spatiotemporal patterns at progressively finer resolution on the data manifold. The Riemannian measure of M and an adaptive graph kernel width enhances the capability of NLSA to detect important nonlinear processes, including intermittency and rare events. The minimum dimension of V_l required to capture these features while avoiding overfitting is estimated here using spectral entropy criteria.