Recursive Moving Polynomial Regression: A Unified Constant‑Time Approach

While previously proposed recursive formulations were restricted to causal evaluation, the proposed approach extends to an arbitrary evaluation point, using the same unified expression. We present a recursive formulation of the moving polynomial regression (RPM), valid for any degree m, with O(1) constant‑complexity per‑sample updates. For each fixed degree m, the same expression provides full flexibility in the choice of the window length k and of the evaluation point. Compared with polynomial FIR filters such as Savitzky–Golay, the proposed recursion maintains a fixed and extremely small number of operations, independent of k and of the evaluation point, while performing least‑squares estimation over sliding windows with asymptotically predictable and controllable numerical drift. The structure consists of an autoregressive part of order m+1 and a convolutional part of order 2(m+1). The same architecture extends to the estimation of the derivatives of the local model, preserving the IIR+FIR structure. This yields a family of filters for the signal value and for derivatives of arbitrary order, with precomputable deterministic coefficients and full operational flexibility.

Editorial note: This is the first English release of the manuscript, based on earlier Italian drafts (v0.1– v0.3) with minor editorial refinements.