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Volume 9 (2020-2021)**

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1.
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Regularized Kernel Machine Learning for Data Driven Forecasting of Chaos****Full text (PDF)**

Erik Bollt

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Pages:
1-2****6**

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Abstract:
** Forecasting outcomes from
initial states continues to be an essential task in dynamical systems
theory as applied across the sciences and engineering. The data-driven
philosophy has become prevalent across the community. While geometric
methods founded in time series to rebuild the underlying geometry based
on Taken's embedding theorem have been popular and successful in
previous decades, they are complex, computationally expensive, and
parametrically intensive. The wave of machine learning methods have come
to reveal that a black box oriented approach has a great deal to offer
the fundamental problem of forecasting the future. Modelling the flow
operator as a linear combination of nonlinear basis functions in terms
of regression least squares fitting in a data-driven manner is straight
forward to pose. However there are two major obstacles to overcome,
which are first, the problem of model complexity may lead to either
underfitting or overfitting, but these can be mitigated by Tikohonov
regularization. Another serious issue regards computational complexity,
which can be overcome by the kernel trick, where in all necessary inner
products to be computed in a high dimensional feature (basis function)
space occur implicitly within low-dimensional kernel operations. In
particular kernel methods from the broader theory of support vector
machines is founded in the functional analytic theory of Mercer's
theorem and also reproducing kernel Hilbert spaces (RKHS), but
practically this fundamental concept in machine learning has become
central to many efficient algorithms. Putting these concepts together,
the efficiency of kernel methods, and the robustness of regularized
regression are both possible within an approach called kernelized ridge
regression, that we show here makes for an especially useful way to
carry forward time-series forecasting problems, as a simple to use and
computationally efficient methodology. We demonstrate the utility of
these concepts in terms of a progression of examples from low
dimensional where direct analysis is possible, to high-dimensional and
spatiotemporally chaotic, and then an experimental data set from
physiology of heart rate and breathing interactions.