Savitsky-Golay

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... 

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

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... 

Polynomial Moving-Average (Savitsky-Golay) Filters.403... 

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. 

Instead of using a triangular weight function, the Savitsky-Golay (2) method uses a selectable nonlinear function on each side of the central point. The use of an odd number of points remains the same, but each side of the center weight is fitted with a polynomial curve, which can be a quad-... 

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 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. 

Savitsky-Golay Filters, Hanning and Hamming Windows... 

The calculation of moving average and Savitsky-Golay filters is illustrated in Table 3.4. 

The first point of the seven point Savitsky-Golay quadratic/cubic filtered data can be calculated as follows ... 

Raw data Moving average Quadratic/cubic Savitsky- -Golay... 

Figure 3.8(a) is a representation of the raw data. The result of using MA filters is shown in Figure 3.8(b). A three point MA preserves the resolution (just), but a five point MA loses this and the cluster appears to be composed of only one peak. In contrast, the five and seven point quadratic/cubic Savitsky-Golay filters [Figure 3.8(c)] preserve resolution whilst reducing noise and only starts to lose resolution when using a nine point function. 

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