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

TL;DR

This paper introduces an efficient N-fold cross-validation method for selecting the polynomial order in Savitzky-Golay smoothers, improving noise reduction while avoiding over- or under-smoothing.

Contribution

It develops a novel connection between prediction error and SG projection spaces, enabling effective polynomial order selection based on the minimum norm formulation.

Findings

The proposed method outperforms BIC in low SNR and small sample scenarios.

It reduces computational complexity of cross-validation in SG smoothing.

MATLAB codes are provided for reproducibility.

Abstract

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.