Nonparametric forecasting of low-dimensional dynamical systems

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.

Figure 1

Validation of first two moments of the diffusion forecast for a stochastic dynamical system on a torus in ${\mathbb{R}}^3$.

Figure 2

Comparing forecast methods for Lorenz-63. Top: $\Delta t=0.1$. Middle: $\Delta t=0.5$. Bottom: Eigenfunctions ${\phi }_{40},{\phi }_{500},{\phi }_{1500},$ and ${\phi }_{4000}$ of the coarse approximation of the fractal attractor by a manifold.

Figure 3

Forecasting for the El Niño 3.4 index. rms and correlation (top); 14-month lead forecast (bottom).