  The diffusion forecasting is a nonparametric approach that provably solves
the Fokker-Planck PDE corresponding to It\^o diffusion without knowing the
underlying equation. The key idea of this method is to approximate the solution
of the Fokker-Planck equation with a discrete representation of the shift
(Koopman) operator on a set of basis functions generated via the diffusion maps
algorithm. While the choice of these basis functions is provably optimal under
appropriate conditions, computing these basis functions is quite expensive
since it requires the eigendecomposition of an $N\times N$ diffusion matrix,
where $N$ denotes the data size and could be very large. For large-scale
forecasting problems, only a few leading eigenvectors are computationally
achievable. To overcome this computational bottleneck, a new set of basis
functions constructed by orthonormalizing selected columns of the diffusion
matrix and its leading eigenvectors is proposed. This computation can be
carried out efficiently via the unpivoted Householder QR factorization. The
efficiency and effectiveness of the proposed algorithm will be shown in both
deterministically chaotic and stochastic dynamical systems; in the former case,
the superiority of the proposed basis functions over purely eigenvectors is
significant, while in the latter case forecasting accuracy is improved relative
to using a purely small number of eigenvectors. Supporting arguments will be
provided on three- and six-dimensional chaotic ODEs, a three-dimensional SDE
that mimics turbulent systems, and also on the two spatial modes associated
with the boreal winter Madden-Julian Oscillation obtained from applying the
Nonlinear Laplacian Spectral Analysis on the measured Outgoing Longwave
Radiation (OLR).
