  In this paper, we apply a recently developed nonparametric modeling approach,
the "diffusion forecast", to predict the time-evolution of Fourier modes of
turbulent dynamical systems. While the diffusion forecasting method assumes the
availability of a noise-free training data set observing the full state space
of the dynamics, in real applications we often have only partial observations
which are corrupted by noise. To alleviate these practical issues, following
the theory of embedology, the diffusion model is built using the
delay-embedding coordinates of the data. We show that this delay embedding
biases the geometry of the data in a way which extracts the most stable
component of the dynamics and reduces the influence of independent additive
observation noise. The resulting diffusion forecast model approximates the
semigroup solutions of the generator of the underlying dynamics in the limit of
large data and when the observation noise vanishes. As in any standard
forecasting problem, the forecasting skill depends crucially on the accuracy of
the initial conditions. We introduce a novel Bayesian method for filtering the
discrete-time noisy observations which works with the diffusion forecast to
determine the forecast initial densities.
  Numerically, we compare this nonparametric approach with standard stochastic
parametric models on a wide-range of well-studied turbulent modes, including
the Lorenz-96 model in weakly chaotic to fully turbulent regimes and the
barotropic modes of a quasi-geostrophic model with baroclinic instabilities. We
show that when the only available data is the low-dimensional set of noisy
modes that are being modeled, the diffusion forecast is indeed competitive to
the perfect model.
