Polynomial Order Selection for Savitzky-Golay Smoothers via N-fold Cross-Validation (extended version)

Savitzky-Golay (SG) smoothers are noise suppressing filters operating on the principle of projecting noisy input onto the subspace of polynomials. A poorly selected polynomial order results in over- or under-smoothing which shows as either bias or excessive noise at the output. In this study, we apply the N-fold cross-validation technique (also known as leave-one-out cross-validation) for model order selection and show that the inherent analytical structure of the SG filtering problem, mainly its minimum norm formulation, enables an efficient and effective order selection solution. More specifically, a novel connection between the total prediction error and SG-projection spaces is developed to reduce the implementation complexity of cross-validation method. The suggested solution compares favorably with the state-of-the-art Bayesian Information Criterion (BIC) rule in non-asymptotic signal-to-noise ratio (SNR) and sample size regimes. MATLAB codes reproducing the numerical results are provided.

💡 Research Summary

The paper addresses a long‑standing practical problem in the use of Savitzky‑Golay (SG) smoothers: how to choose the polynomial order P that balances bias (over‑smoothing) against variance (insufficient noise reduction). While the SG filter is traditionally designed by hand or by heuristic criteria such as the Bayesian Information Criterion (BIC), the authors propose a systematic, data‑driven approach based on N‑fold (leave‑one‑out) cross‑validation (CV).

The key insight is that SG filtering can be expressed as a minimum‑norm solution of a linear system Ah = c, where A contains the sampled values of the monomials 1, t, t²,…, tᴾ over the window of length N, and c is the column that extracts the central sample. The smoothing coefficients hₛₖ are given by Aᵀ(AAᵀ)⁻¹cₖ. By zero‑ing the k‑th column of A (one sample is left out) the same framework yields a “prediction” filter hₚₖ that predicts the omitted sample from the remaining N‑1 observations.

Through a series of algebraic manipulations (Woodbury matrix identity, projection matrices) the authors derive a compact relationship between the prediction filter and the smoothing filter:

 hₚₖ = hₛₖ – γₖ eₖ, γₖ = 1 / (1 –