### 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.

Original language | English (US) |
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Article number | 032915 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 91 |

Issue number | 3 |

DOIs | |

State | Published - Mar 19 2015 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics

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## Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*91*(3), [032915]. https://doi.org/10.1103/PhysRevE.91.032915