Derivation and General Form of an Optimal Recursive Algorithm for Moving Polynomial Regression

This study introduces a general recursive formulation for moving polynomial regression, applicable to polynomials of arbitrary degree mm. The proposed model, deterministic in nature, enables iterative updates of estimates with constant computational complexity O(1), and is analytically compact, exhibiting formal analogies with ARMA models. Each increase in the polynomial degree entails the addition of a single autoregressive term and two convolutive terms, preserving the simplicity and efficiency of the recursive structure. Moving linear regression, previously analyzed in the earlier work Optimal Recursive Algorithm for Moving Linear Regression, is here revisited as a special case within the general formulation. While the proposed method is structurally suited for large-scale implementation, particularly in high-frequency contexts and environments with limited computational resources, issues related to numerical stability have not been addressed in the present study, and its practical deployment remains subject to further investigation.