Parameter Estimation of ARMA Models Using Least Squares Method

Parameter Estimation of ARMA Models Using Least Squares Method and Its Application in Harmonic Retrieval

In the field of signal processing, ARMA (AutoRegressive Moving Average) models are commonly used to describe the statistical characteristics of random signals. Estimating their parameters using the least squares method is a classical and effective approach, particularly suitable for applications such as harmonic retrieval.

Fundamental Approach to Parameter Estimation The process involves constructing equations for the ARMA model from observed data and transforming them into a linear regression problem. The core principle of least squares is to find a set of parameters that minimizes the sum of squared residuals between the model output and actual observations. Key implementation steps include constructing the data matrix (typically using the Yule-Walker equations or innovation algorithm), designing the objective function (residual sum of squares), and solving for optimal parameters through optimization algorithms like gradient descent or QR decomposition.

Implementation of Harmonic Retrieval Harmonic retrieval refers to extracting specific frequency components from noise-contaminated signals. The least squares estimation based on ARMA models effectively separates signal from noise, enabling reconstruction of original harmonic amplitude and phase information through the estimated parameters. In code implementation, this involves spectral estimation using the ARMA model's transfer function and applying inverse Fourier transforms.

Key Aspects of Simulation Programs Simulation programs typically generate synthetic data conforming to ARMA models (using arima.sim function in R or signal processing toolboxes in MATLAB), then add noise to simulate real-world scenarios. The least squares algorithm iteratively optimizes parameters through numerical methods like the Levinson-Durbin recursion for AR parts and nonlinear optimization for MA components. Finally, the method's effectiveness is validated by comparing estimation errors against true parameters using metrics like MSE (Mean Squared Error) and parameter confidence intervals.