ARMA Spectral Estimation - Total Least Squares Method for AR Parameter Estimation

In this article, we explore ARMA spectral estimation using the Total Least Squares (TLS) method, with specific focus on AR parameter estimation. The Autoregressive (AR) model represents a classical time series approach commonly employed to characterize the autocorrelation properties within time series data. To implement AR models effectively, accurate parameter estimation becomes crucial. Here, we utilize the TLS method for estimating AR parameters, which constitutes a statistical approach for parameter estimation in regression models. The TLS method simultaneously accounts for errors in both dependent and independent variables, unlike ordinary least squares that only considers errors in the dependent variable. From an implementation perspective, the TLS algorithm typically involves singular value decomposition (SVD) of an augmented data matrix. The key steps include: 1. Constructing the augmented matrix containing both input and output data 2. Performing SVD to identify the minimal perturbation direction 3. Extracting the parameter estimates from the right singular vector Through TLS implementation, we obtain optimal parameter estimates for AR models that enhance prediction accuracy and robustness against noise. The method proves particularly valuable when dealing with noisy measurements or when both variables contain observational errors. Practical implementation in programming environments like MATLAB would involve functions such as svd() for matrix decomposition and appropriate matrix operations to extract the parameter vector corresponding to the smallest singular value.