An algebraic method for constructing stable and consistent autoregressive filters

TL;DR

This paper presents an algebraic method for constructing stable and consistent autoregressive models for nonlinear turbulent signals, which does not require training data and improves short-term prediction accuracy.

Contribution

The paper introduces a novel algebraic approach to build AR models using only long-term statistics, ensuring stability and consistency without direct training data.

Findings

AR models with the new method outperform regression-based models in short-term prediction.

The method guarantees model stability and consistency across various discretization and observation parameters.

Improved forecasting of the Madden-Julian Oscillation variability.

Abstract

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.