Savitzky-Golay smoothing

Figure 4.4 Similar to the sliding polynomial smoothing (Savitzky Golay filter, the coefficients for 2nd order fit to a parabola) is the effect of Bromba Ziegler filters [Bromba and Ziegler, (1983c), coefficients fit to a triangle upper figure]. Both have bad low pass filter characteristics, as shown in the lower figure with the Fourier transforms of filters through 21 points each.

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. 40.30. 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.

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 ... 

First the signal was smoothed through a Savitzky-Golay algorithm, whose parameters were optimized considering the mean fringe number for a fixed temperature variation. 

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. 

The function SavGo 1, m performs a Savitzky-Golay smoothing. The parameters are the x- and y-vectors, the number (n) of neighbouring left or right data points that are used for one polynomial fit (i.e. if n= 5, 2n+l=ll data points are fitted) and the degree (ndf of the polynomial to be fitted. 

The Savitzky-Golay algorithm could readily be adapted for polynomial interpolation. The computations are virtually identical to smoothing. In smoothing, a polynomial is fitted to a range of (x,y)-data pairs arranged around the x-value that needs to be smoothed. For polynomial smoothing, the polynomial is evaluated for a set number of data points around the desired x-value and the computed y-value at that x is the interpolated value. 

Savitzky-Golay smoothing profile. The response function used was a gaussian with a FWHM of seven points. The weighting scheme used was of the form... 

The next module is based on the five point Savitzky - Golay formulas listed in Tables 4.1 and 4.2. It returns both the smoothed function values and the estimates of the derivative. The formulas are extended also to the four outermost points of the sample, where (4.10) does not directly apply. 

Figure 4.10 Graphical representation of (a) baseline correction, (b) smoothing and (c) simplified view of a Savitzky Golay filter (moving window, five points per window, linear interpolation).

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