Savitzky and Golay

The interpolation method outlined above can be applied as well to the "smoothing of experimental data. In this case a given experimental point is replaced by a point whose position is calculated from the values of m points on each side. The matrix X then contains an odd number of columns, namely 2m + 1. The matrices A have also been tabulated for this application. This smoothing method has been used for a number of years by molecular spectroscopists, who generally refer to it as the method of Savitzky and Golay. ... 

So the steps that Savitzky and Golay took to create their classic paper was as follows ... 

And finally, while this work was all of very important theoretical interest, Savitzky and Golay took one more step that turned the theory into a form that could be easily put to practical use. 

We start by creating a matrix. This matrix is based on the index of coefficients that are to be ultimately produced. Savitzky and Golay labeled the coefficients in relation to the central data point of the convolution, therefore a three-term set of coefficients are labeled -1, 0, 1. A five-term set is labeled -2, -1, 0, 1, 2 and so forth. 

Polynomials do not play an important role in real chemical applications. Very few chemical data behave like polynomials. However, as a general data treatment tool, they are invaluable. Polynomials are used for empirical approximations of complex relationships, smoothing, differentiation and interpolation of data. Most of these applications have been introduced into chemistry by Savitzky and Golay and are known as Savitzky-Golay filters. Polynomial fitting is a linear, fast and explicit calculation, which, of course, explains the popularity. 

A popular method of smoothing analytical measurement data is the least squares technique presented by Savitzky and Golay (1964) [see also Steinier, Termonia, and Deltour (1972)]. The technique is useful for data consisting of a single response as a function of a single factor with equally spaced factor levels. This type of data is... 

Show that if the x, values are coded -2, -1, 0, +1, and +2, calculation of (X X) X gives the five-point convolutes found in the Savitzky and Golay paper. 

We have carried out simulations using polynomial least-squares filters of the type described by Savitzky and Golay (1964) to determine the impact of such smoothing on apparent resolution. For quadratic filters, a filter length of one-fourth of the linewidth (at FWHM) does not seriously degrade the apparent resolution of two Gaussian lines in very close proximity. 

Coefficients for local quadratic or cubic smoothing by Savitzky and Golay... 

Treat the raw voltammograms by using the Savitzky and Golay filter (level 2) included in the General Purpose Electrochemical... 

The data were smoothed using a 15 point cubic-quartic Savitzky and Golay (1A) algorithm, the x-ray satellites and a Shirley background were subtracted using computer routines available in the Vacuum Generators data analysis software. Only the treated data are presented here. 

Savitzky and Golay published the coefficients for a range of least-squares fit curves with up to 25-point wide smoothing windows for each. Corrections to the original tables have been published by Steinier et al. ... 

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. 

Table 4.2 Coefficients for a sliding polynomial fit after Savitzky and Golay (1964, corrected). The weighted sum (weights c,) must be divided by the norm.

The refinement of a preliminary peak position can easily be obtained by fitting a parabola a + bv + cx into 5 (or more) points around the local maximum, using the solutions for a, b, and c after Savitzky and Golay. The maximum of this parabola is the wanted refinement of the peak position and is calculated as the root of the 1st derivative y ( max) = 0 = b + 2c.Xinax- So far the 5-point appoximation one gets ... 

Various other methods have been proposed to compute derivatives, including the use of suitable polynomial functions as suggested by Savitzky and Golay. The use of a suitable array of weighting coefficients as a smoothing function with which to convolute spectral data was described in... 

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