Adaptive Autoregressive (AAR) and Adaptive Autoregressive Moving Average (AARMA) Estimation

This document presents the implementation of adaptive autoregressive (AAR) and adaptive autoregressive moving average (AARMA) estimation for real-valued data series using the Kalman filter algorithm. The Kalman filter is a recursive filtering algorithm that provides optimal linear unbiased estimates of system states from observational data sequences. We detail the fundamental concepts and mathematical principles of the Kalman filter algorithm, along with practical implementation techniques for parameter estimation in time series analysis. Key implementation aspects include state-space formulation with process and measurement equations, recursive covariance matrix updates using prediction-correction cycles, and innovation-based parameter adaptation. The discussion covers model parameter selection criteria, including order determination via information-theoretic approaches (AIC/BIC) and regularization techniques for numerical stability. We further address accuracy evaluation metrics such as mean squared error analysis and stability assessment through covariance matrix eigenvalue monitoring. By studying this material, readers will acquire fundamental methodology for time series analysis using Kalman filtering, including practical implementation strategies for real-world applications through algorithmic pseudocode and MATLAB function prototypes.