Savitzky

Figure 3.11. Smoothing a noisy signal. The synthetic, noise-free signal is given at the top. After the addition of noise by means of the Monte Carlo technique, the panels in the second row are obtained (little noise, left, five times as much noise, right). A seven-point Savitzky-Golay filter of order 2 (third row) and a seven-point moving average (bottom row) filter are...

SMOOTH.dat A 26-point table of values interpolated from a figure in Ref. 162, to demonstrate the capability of the discussed extended Savitzky-Golay filter to provide a smoothed trace from the first to the last point in the time series. 

Bromba, M. U. A., and Ziegler, H., Application Hints for Savitzky-Golay Digital Smoothing Filters, Anal. Chem. 53, 1981, 1583-1586. 

Madden, H. H., Comments on the Savitzky-Golay Convolution Method for Least-Squares Fit Smoothing and Differentiation of Digital Data, Anal. Chem. 50, 1978, 1383-1386. 

Gorry, P. A., General Least-Squares Smoothing and Differentiation by the Convolution (Savitzky-Golay) Method, Anal. Chem. 62, 1990, 570-573. 

Savitzky, A. and Golay, M. J. E., Smoothing and Differentiation of Data by Simplified Least-Squares Procedures, Anal. Chem. 36, July 1964, 1627-... 

Fig. 40.29. Fourier spectrum of second-order Savitzky-Golay convolutes. (a) 5-point, (b) 9-point, (c) 17-point, (d) 25-point (arrows indicate cut-off frequencies).

Fig. 40.30. Smoothed second-derivative (window 7 data points, second-order) according to Savitzky-Golay.

A. Savitzky and M.J.E. Golay, Smoothing and differentiating of data by simplified least-squares procedures. Anal. Chem., 36 (1964) 1627-1639. 

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. ... 

George Boris Savitzky, American physical chemist (1925-) Marcel J, E. Golay, Swiss American physicist (19G2-). 

Figure 3. Plot of the loss tangent with temperature for Hytrel in the temperature range of 0 to 140 Deg. The x s represent the experimental values, while the solid line represents the results of smoothing using the Savitzky-Golay technique.

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

Evaluate the expression for the derivative of that polynomial at the point for which the derivative is to be computed. In the Savitzky-Golay paper, this is the central point of the set used to fit the data. As we shall see, in general this need not be the case, although doing so simplifies the formulas and computations. 

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. 

The publication of the Savitzky-Golay paper (augmented by the Steinier paper) was a major breakthrough in data analysis of chemical and spectroscopic data. Nevertheless, it does have some limitations, and some more caveats that need to be considered when using this approach. 

Another limitation is that, also because of the computation being applicable to the central data point, there is an end effect to using the Savitzky-Golay approach it does not provide for the computation of derivatives that are too close to the end of the spectrum. The reason is that at the end of the spectrum there is no spectral data to match up to the coefficients on one side or the other of the central point of the set of coefficients, therefore the computation at or near the ends of the spectrum cannot be performed. 

Of course, an inherent limitation is the fact that only those combinations of parameters (derivative order, polynomial degree and number of data points) that are listed in the Savitzky-Golay/Steinier tables are available for use. While those cover what are likely to be the most common needs, anyone wanting to use a set of parameters beyond those supplied is out of luck. 

Through the use of these formulas, Savitzky-Golay convolution coefficients could be computed for a convolution function using any odd number of data points for the convolution. 

The paper contains formulas for only those derivative orders and degrees of polynomials that are contained in the original Savitzky-Golay paper, therefore we are still limited to those derivative orders and polynomial degrees. 

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. 

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