title: Nonparametric forecasting of low-dimensional dynamical systems.
abstract: This paper presents a nonparametric modeling approach for forecasting stochastic dynamical systems on low-dimensional manifolds. The key idea is to represent the discrete shift maps on a smooth basis which can be obtained by the diffusion maps algorithm. In the limit of large data, this approach converges to a Galerkin projection of the semigroup solution to the underlying dynamics on a basis adapted to the invariant measure. This approach allows one to quantify uncertainties (in fact, evolve the probability distribution) for nontrivial dynamical systems with equation-free modeling. We verify our approach on various examples, ranging from an inhomogeneous anisotropic stochastic differential equation on a torus, the chaotic Lorenz three-dimensional model, and the Niño-3.4 data set which is used as a proxy of the El Niño Southern Oscillation.
authors: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802-6400, USA.
authors: Tyrus
authors: T
authors: Berry
authors: Courant Institute of Mathematical Sciences, New York University, New York, New York 10012, USA.
authors: Dimitrios
authors: D
authors: Giannakis
authors: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802-6400, USA.|Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania 16802-5013, USA.
authors: John
authors: J
authors: Harlim