Data-Driven Methods for Dynamical Systems: Quantifying Predictability and Extracting Spatiotemporal Patterns

This chapter reviews two examples of applied mathematics techniques for data analysis in dynamical systems. The two examples are: (1) Methods for quantifying predictability and model error based on data clustering and information theory and (2) nonlinear Laplacian spectral analysis (NLSA) algorithms for extracting spatiotemporal patterns from high-dimensional data. The chapter highlights these techniques with applications to climate atmosphere ocean science (CAOS), in particular, predictability assessment and Markov modeling of circulation regimes in a simple ocean model and extraction of modes of organized convection in the tropics from infrared brightness temperature satellite data. A common theme in these methods has been the coarse-grained geometry of the data. The machinery of discrete exterior calculus and spectral graph theory was combined with delay-coordinate mappings of dynamical systems to extract spatiotemporal modes of variability which are describable in terms of low-dimensional sets of diffusion eigenfunctions.