Savitzky-Golay

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. 

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.

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.

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. 

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. 

Most of our discussion so far has centered on the use of the two-point-difference method of computing an approximation to the true derivative, but since we have already brought up the Savitzky-Golay method, it is appropriate here to consider both ways of computing derivatives, when considering how they behave when used for quantitative calibration purposes. 

Therefore, if A represents the spectrum, the various a represent convolution coefficients and Var(A) represents a noise source that gives a constant noise level to the spectral values, then equation 57-36 gives the noise variance expected to be found on the computed resultant value, whether that is a smoothed spectral value, or any order derivative computed from a Savitzky-Golay convolution. For a more realistic computation, an interested (and energetic) reader may wish to compute and use the actual noise that will occur on a spectrum, from the information determined in the previous chapters [6-7] instead of using a constant-noise model. But for our current purposes we will retain the constant-noise model then equation 57-36 can be simplified slightly ... 

We turn now to the effect of using the Savitzky-Golay convolution functions. Table 57-1 presents a small subset of the convolutions from the tables. Since the tables were fairly extensive, the entries were scaled so that all of the coefficients could be presented as integers we have previously seen this. The nature of the values involved caused the entries to be difficult to compare directly, therefore we recomputed them to eliminate the normalization factors and using the actual direct coefficients, making the coefficients more easily comparable we present these in Table 57-2. For Table 57-2 we also computed the sums of the squares of the coefficients and present them in the last row. 

Table 57-1 Some of the Savitzky-Golay convolution coefficients using a quadratic fitting function...

Table 57-2 The Savitzky-Golay convolution coefficients multiplied out. All coefficients are for a quadratic fitting function. See text for meaning of SSK...

The authors thank David Hopkins for valuable discussions regarding several aspects of the behavior of Savitzky-Golay derivatives, and also for making sure we spelled Savilzky and "Steinier correctly ... 

Our new method of determining nonlinearity (or showing linearity) is also related to our discussion of derivatives, particularly when using the Savitzky-Golay method of convolution functions, as we discussed recently [6], This last is not very surprising, once you consider that the Savitzky-Golay convolution functions are also (ultimately) derived from considerations of numerical analysis. 

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