Savitsky-Golay method

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. 

Problem 3.8 First and Second Derivatives of UV/VIS Spectra Using the Savitsky-Golay method... 

Calculate the first derivative at each wavelength and each point in time. Normally the Savitsky-Golay method, described in Chapter 3, Section 3.3.2, can be employed. The simplest case is a five point quadratic first derivative (see Table 3.6), so that... 

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. ... 

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. 

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-... 

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. 

Because it is computationally intense to calculate fit polynomials to each point, Savitsky and Golay proposed an alternative and simpler computational method in which a new data point is expressed as... 

---