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Data-driven kernel methods for dynamical systems

Eniko Szekely

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Date and time:Ìý

Tuesday, September 29, 2015 - 2:00pm

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ECCR 257

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Datasets generated by dynamical systems are often high-dimensional, but they only display a small number of patterns of interest. The underlying low-dimensional structure governing such systems is generally modeled as a manifold with observations lying on (or near) the manifold. Its intrinsic geometry is well described by local measures that vary smoothly on the manifold, such as kernels, rather than by global measures, such as covariances. I use a kernel-based nonlinear dimension reduction method, namely nonlinear Laplacian spectral analysis (NLSA), to extract a reduced set of basis functions that describe the large-scale behavior of the dynamical system. These basis functions are the leading Laplace-Beltrami eigenfunctions of a discrete Laplacian operator.Ìý Specifically, I use NLSA to extract coherent signals in organized tropical convection. The patterns of interest exist on a wide range of timescales, from interannual to annual, semiannual, intraseasonal and diurnal scales.

In the second part of the talk, the NLSA-based modes are used as predictors to quantify regime predictability of a signal of interest. Dynamical regimes can be associated with coarse-grained partitions of the feature space identified through clustering and the information carried by each regime is assessed using information-theoretic measures. The framework is adapted to handle variables with cyclostationary statistics that are frequent in atmosphere-ocean science. As dynamical systems are inherently nonlinear, we use a kernel-based clustering method, namely kernel k-means, to partition the predictor space. The regimes associated with these partitions are found to carry significant information and have physically meaningful interpretation, e.g. El-Niño.

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