On the Approximation of Stationary Processes using the ARMA Model
This work addresses theoretical foundations for ARMA model approximation in time series analysis, offering improved mathematical tools for statisticians and signal processing researchers, though it appears incremental rather than paradigm-shifting.
The paper quantifies approximation errors between true stationary processes and ARMA models by establishing that the L∞ norm of transfer functions controls Wold coefficients and forms a Banach algebra with better structural properties than cepstral norms. It provides explicit approximation bounds for continuous transfer functions and critiques heuristic approaches like Padé approximations.
We look at a problem related to Autoregressive Moving Average (ARMA) models, on quantifying the approximation error between a true stationary process $X_t$ and an ARMA model $Y_t$. We take the transfer function representation $x(L)$ of a stationary process $X_t$ and show that the $L^{\infty}$ norm of $x$ acts as a valid norm on $X_t$ that controls the $\ell^2$ norm of its Wold coefficients. We then show that a certain subspace of stationary processes, which includes ARMA models, forms a Banach algebra under the $L^{\infty}$ norm that respects the multiplicative structure of $H^{\infty}$ transfer functions and thus improves on the structural properties of the cepstral norm for ARMA models. The natural definition of invertibility in this algebra is consistent with the original definition of ARMA invertibility, and generalizes better to non-ARMA processes than Wiener's $\ell^1$ condition. Finally, we calculate some explicit approximation bounds in the simpler context of continuous transfer functions, and critique some heuristic ideas on Padé approximations and parsimonious models.