  In this paper, we introduce an algebraic method to construct stable and
consistent univariate autoregressive (AR) models of low order for filtering and
predicting nonlinear turbulent signals with memory depth. By stable, we refer
to the classical stability condition for the AR model. By consistent, we refer
to the classical consistency constraints of Adams-Bashforth methods of
order-two. One attractive feature of this algebraic method is that the model
parameters can be obtained without directly knowing any training data set as
opposed to many standard, regression-based parameterization methods. It takes
only long-time average statistics as inputs. The proposed method provides a
discretization time step interval which guarantees the existence of stable and
consistent AR model and simultaneously produces the parameters for the AR
models. In our numerical examples with two chaotic time series with different
characteristics of decaying time scales, we find that the proposed AR models
produce significantly more accurate short-term predictive skill and comparable
filtering skill relative to the linear regression-based AR models. These
encouraging results are robust across wide ranges of discretization times,
observation times, and observation noise variances. Finally, we also find that
the proposed model produces an improved short-time prediction relative to the
linear regression-based AR-models in forecasting a data set that characterizes
the variability of the Madden-Julian Oscillation, a dominant tropical
atmospheric wave pattern.
