Savitsky-Golay derivatives

Fig. 7. (Top) Five equal increments of slope added to a single spectrum. Second derivative (Savitsky-Golay, 10 convolution points) applied to five equal increments of slope added to a single spectrum. (Bottom) The derivative treatment removes slope for this case.

FIGURE 3.18 Top A benzonitrile peak plotted in percent transmittance. Bottom Its first derivative (Savitsky-Golay algorithm, 2" order, 5 points). 

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... 

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

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

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. 

From top to bottom a three point moving average, a Hanning window and a five point Savitsky-Golay quadratic derivative window... 

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

Calculate the five point Savitsky-Golay quadratic first and second derivatives of A. Plot the graphs, and interpret them compare both first and second derivatives and discuss the appearance in terms of the number and positions of the peaks. 

Calculate the five point Savitsky-Golay quadratic second derivatives of all three spectra and superimpose the resultant graphs. Repeat for the seven point derivatives. Which graph is clearer, five or seven point derivatives Interpret the results for spectrum B. Do die derivatives show it is clearly a mixture Comment on the appearance of die region between 270 and 310 nm, and compare with the original spectra. 

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... 

Table 2 Savitsky-Golay filters for calculation of derivatives ...

FIGURE3.17 Top Part of the spectrum of benzonitiile plotted in absorbance. Bottom The first derivative of this spectrum (Savitsky-Golay algorithm, 2nd order, 5 points). The intersecting dotted lines indicate that the top of the ahsorhance band has a slope of zero. 

FIGURE 3.20 Bottom Part of the absorbance spectrum of a mixture of polystyrene and a polycarbonate. Top The second derivative of this spectrum (Savitsky-Golay algorithm, 2" order, 5 points). Note how the negative lobes of the 2nd derivative point at the peaks in the spectrum. 

FIGURE 3.21 Top An absorbance spectrum with slope and offset in it. Bottom The 2nd derivative of this spectrum. Note that the 2nd derivative has no slope or offset (Savitsky-Golay algorithm, 2" order, 13 points). 

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