To forecast a spatially-large nonlinear system X, we seek a low-dimensional subspace XW of the system that contains all of the predictable information about that system (where W is a projection matrix). For linear systems, there are currently two methods that can find such a subspace: Average Predictability Time Decomposition and Forecasteable Components Analysis. For nonlinear systems, there are currently no ideal methods, but there are two possible prediction models: i) X_τ = Φ(X)WP, and ii) X_τ = Φ(XW)P, where Φ(x) is a nonlinear mapping of x. One solution for the first model is Nonlinear Laplacian Spectral Analysis. NLSA is "nonlinear principal components of the phase space'', and thus like principal components regression, NLSA does not find the smallest subspace which is most predictable. Here I introduce a new solution for the second prediction model which equates to finding the eigenvectors of the average Jacobian matrix product integrated over multiple lead times (∫∇_τ∇_τ∂τ), where the Jacobians are estimated using Reproducing Kernel Hilbert Spaces. The eigenvectors of the integrated average gradient product matrices define the smallest subspace which contains a system's predictable information in a linear or nonlinear sense. The resulting predictable component time series can be forecast using simple nonlinear autoregression models. The method will be illustrated using examples from simple chaotic models to observational climate data.

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Fischer, M. (2017) Investigating nonlinear dependence between climate fields. Journal of Climate, 30, 5547--5562.