Smoothing by Sliding Polynomials Savitzky-Golay Method

1 Smoothing by Sliding Polynomials (Savitzky-Golay Method) 

This method was successfully introduced into spectroscopy and popularized by Savitzky and Golay. Therefore, the best sources for papers on this topic are not mathematical journals and textbooks, but the chemical journal Analytical Chemistry. However, the method of sliding polynomials itself has long been known. As an example, formulas for calculating the necessary coefficients (see Table 4.2) can already be found in the textbook of Whittaker and Robinson (1924). Only because of the missing computational possibilities at this time were these formulas not used in practice. 

A prerequisite for the application of this method is a set of equally separated base points, as they are obtained in step scan measurements with a constant step width, and a signal shape that can be approximated by a polynomial of nth order. This way, X-ray reflections in the region of their half-width can be 

In m = 2n + 1 adjacent base points Xk n, +u , i one tries to approximate the measured values y .. y + by a polynomial of th order (e.g. y = a + hx + cx ) by means of the method of least squares. Because the wanted parameters a, b, c appear as linear factors, one deals with a linear system, which immediately (in one step) delivers the correct solution, which will be independent of the used step width and of any starting values (assumed to be 0). The solution for a parabola through m = 5 points (n = 2) 

As during the least-squares refinement, every other coelficient of the normal equations sums up to 0, the values for the 0th (=smoothing) and 2nd derivative hold for polynomials of 2nd and of 3rd order as well. The coelficients for 1st derivatives are different for 2nd and 3rd order (but equal for 1st and 2nd order, and for 3rd and 4th order, respectively). 

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