Optimal Choice of AR and MA Parts in Autoregressive Moving Average Models

Area of Science:

• Statistics
• Signal Processing
• Machine Learning

Background:

• Model selection is crucial for analyzing time series data in fields like speech and image processing.
• Existing methods for comparing dynamical models often have limitations.

Purpose of the Study:

• To develop an optimal Bayesian decision rule for selecting the best model from a finite set of candidates (AR, MA, ARMA).
• To minimize average probability of error and average loss function value in model selection.
• To simplify decision rules for Gaussian Autoregressive Moving Average (ARMA) models.

Main Methods:

• Derivation of an optimum decision rule based on Bayesian principles.
• Application to one-dimensional series, including time series and 2D image pixel intensities.
• Consideration of non-Gaussian observation sets.

Main Results:

• An optimal decision rule is derived to minimize average error probability.
• A second optimal rule is derived to minimize average loss function value.
• Simplified decision rules are presented for Gaussian ARMA models of varying orders.

Conclusions:

• The proposed Bayesian method offers an optimal approach to model selection for diverse data types.
• The derived decision rules are consistent and provide a benchmark for comparing dynamical models.