Nonparametric forecasting of low-dimensional dynamical systems

Tyrus Berry, Dimitrios Giannakis, John Harlim

Research output: Contribution to journalArticle

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 languageEnglish (US)
Article number032915
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Volume91
Issue number3
DOIs
StatePublished - Mar 19 2015

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forecasting
dynamical systems
Forecasting
Dynamical system
Shift Map
Stochastic Dynamical Systems
Southern Oscillation
Large Data
three dimensional models
Modeling
Invariant Measure
Galerkin
Stochastic Equations
Torus
Quantify
Probability Distribution
differential equations
Semigroup
projection
Projection

ASJC Scopus subject areas

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

Cite this

Nonparametric forecasting of low-dimensional dynamical systems. / Berry, Tyrus; Giannakis, Dimitrios; Harlim, John.

In: Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, Vol. 91, No. 3, 032915, 19.03.2015.

Research output: Contribution to journalArticle

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