SavitzkyGolayFilter

The Savitzky-Golay filter is a digital smoothing and differentiation technique used to enhance data quality while preserving features such as peak shapes. It works by fitting a low-degree polynomial to successive subsets of adjacent data points via linear least squares, and then replacing each central point with the value of the fitted polynomial (or its derivative) at that point.

Mechanism and mathematics: For a window containing 2m+1 points, a polynomial of degree p (p ≤ 2m) is fitted to the data in the window. The smoothed value at the center is a linear combination of the window values with coefficients determined by the least-squares fit. If the derivative of a given order d is desired, the corresponding derivative of the polynomial is used, yielding a different set of coefficients. The process is equivalent to convolving the data with a fixed set of symmetric coefficients for a given window length and polynomial order.

Parameters and usage: The method requires evenly spaced data. Key parameters are the window length (usually odd, equal to 2m+1) and the polynomial order p (p ≤ 2m). The derivative order d can be 0 for smoothing, or higher orders to estimate derivatives. A larger window or higher order increases smoothing accuracy but can blur sharp features; inappropriate choices can introduce bias or artifacts.

Implementation and considerations: The filter is widely implemented in scientific software (eg, sgolayfilt, savgol_filter) with options for boundary handling. Common boundary approaches include padding, padding by reflection, or extending data. The Savitzky-Golay filter is particularly popular in spectroscopy and chemometrics for noise reduction while preserving peak characteristics.