Smoothing, Savitsky-Golay

Figure 4.9 Columns - spectrum in Fig. 4.8(a) with high frequency noise levels of 10, 5 and 1. Rows -spectra after Savitsky-Golay smoothing with cubic having 5,10 and 15 points.

Perform Savitsky-Golay smoothing with derivatives on individual spectra or entire image cubes Spatially interpolate... 

Figure 5-1 Examples of Savitsky Golay smoothing. A second-degree polynomial was used for the left column and a third-degree polynomial was used for the right column with a 5-, 15-, and 25-point smooth.

Figure 8.34. Cosmic removal from a single spectrum of bacteriorhodopsin by smoothing routines. A is original spectrum, with asterisks indicating cosmics. B is after a conventional quadratic, 9 point Savitsky Golay smooth. C is after a similar smooth, but including the missing point routine. Adapted from Reference 30 with permission.

Sampling theory, 27 Savitsky-Golay coefficients, 41 Savitsky-Golay differentiation, 57 Savitsky-Golay smoothing, 38 Scatter plot, 24 Scores, factor, 74 Scree plot, 75... 

Figure 4.9b and 4.9c show a two-step procedure. Step 1 shows the subtraction of the dark current from both the raw Raman spectrum and the dark current spectrum. The former is then divided by the latter to give in the first instance a Raman spectrum corrected for dark noise and white light. Step 2 considers the latter spectrum and includes a Savitsky-Golay smoothing, followed by subtraction of the substrate (here quartz) and the background contributions (here a fourth order polynomial). The final Raman spectrum is thus corrected for instrument response and substrate contribution. 

FIGURE 3.14 Bottom A noisy spectrum of benzonitrile. Top The same spectrum after smoothing (17-point Savitsky-Golay smooth, polynomial order 2). Note that the peak at 1599 cm is much easier to see after the smooth. 

Another advantage of the Savitsky-Golay method is that derivatives of these functions can also be determined from the method of least squares. This method can be used to determine alpha-peak temperatures automatically since the first derivative changes sign at the peak temperature. The advantage of smoothing is that the number of extraneous peaks due to noise has been minimized. 

Regarding the three adjustable parameters for Savitsky-Golay derivatives, the window width essentially determines the amount of smoothing that accompanies the derivative. For rather noisy data, it can be advantageous to use higher window widths, although this also deteriorates the resolution improvement of the derivative. The polynomial order is typically set to two, meaning that the derivative is calcnlated based on the best fits of the local data windows to a second-order polynomial. The derivative order, of conrse, dictates... 

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

Figure 4.4 Top - original spectrum of a protein followed by the same spectrum smoothed by various Savitsky—Golay models.

Madden, H.H., Comments on the Savitsky-Golay convolution method for least-squares fit smoothing and differentiation of digital data, Anal. Chem., 50, 1383, 1978. 

Bromba, M.U.A., Application hints for Savitsky-Golay digital smoothing filters, Anal. Chem., 53, 1583, 1981. 

Figure 5-2 Original spectrum, second derivative using point difference and no smoothing and second derivative with a Savitsky-Golay 15-point smooth.

The most common applications of methods for handling sequential series in chemistry arise in chromatography and spectroscopy and will be emphasized in this chapter. An important aim is to smooth a chromatogram. A number of methods have been developed here such as the Savitsky-Golay filter (Section 3.3.1.2). A problem is that if a chromatogram is smoothed too much the peaks become blurred and lose resolution, negating the benefits, so optimal filters have been developed that remove noise without broadening peaks excessively. 

There are, however, two disadvantages of derivatives. First, they are computationally intense, as a fresh calculation is required for each datapoint in a spectrum or chromatogram. Second, and most importantly, they amplify noise substantially, and, therefore, require low signal to noise ratios. These limitations can be overcome by using Savitsky-Golay coefficients similar to those described in Section 3.3.1.2, which involve rapid calculation of smoothed higher derivatives. The coefficients for a number of window sizes and approximations are presented in Table 3.6. This is a common method for the determination of derivatives and is implemented in many software packages. 

Problem 3.1 Savitsky-Golay and Moving Average Smoothing Functions... 

Smooth the data in the following five ways (a) five point moving average (b) seven point moving average (c) five point quadratic Savitsky-Golay filter (d) seven point... 

This procedure is equivalent to the Savitsky-Golay method, algorithms for which have been included in computer software for scientific instruments such as the Fourier-transform infrared (FTIR) spectrometer. Alternatives to smoothing are weighted least-squares fitting or optimal (Weiner) filtering techniques. ... 

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