Savitzky Golay method

Gorry, P. A., General Least-Squares Smoothing and Differentiation by the Convolution (Savitzky-Golay) Method, Anal. Chem. 62, 1990, 570-573. 

Most of our discussion so far has centered on the use of the two-point-difference method of computing an approximation to the true derivative, but since we have already brought up the Savitzky-Golay method, it is appropriate here to consider both ways of computing derivatives, when considering how they behave when used for quantitative calibration purposes. 

Our new method of determining nonlinearity (or showing linearity) is also related to our discussion of derivatives, particularly when using the Savitzky-Golay method of convolution functions, as we discussed recently [6], This last is not very surprising, once you consider that the Savitzky-Golay convolution functions are also (ultimately) derived from considerations of numerical analysis. 

FIGURE 7.2 First derivatives of the seven NIR spectra from Figure 7.1. The Savitzky-Golay method was applied with a second-order polynomial for seven points. 

FIGURE 8 Change in reaction rate during racemization of (-)-adrenaline as obtained using Savitzky-Golay method [58] by derivative of VTK profile reported in Figure 7. 

Compute the second derivative by divided finite difference approximation and compare the result with that of the Savitzky — Golay method. 

Smoothing by Sliding Polynomials (Savitzky-Golay Method)... 

Numerically the convolution of a step scan is merely the application of a sliding weighted mean (e.g. like the Savitzky-Golay method). The Fourier transform of the rectangular function has the shape of sin(nv)/(nv) (whereby n is inversely proportional to the width of the rectangle) and unfortunately approaches 0 only very slowly. To make do with a small number of points for a convolution, one must tolerate a compromise and renounce the ideal rectangular shape of the low pass filter (in the frequency domain). 

Savitzky-Golay method, and then find the peak maximum from the coefficients as Xpeak = a /2a-2. 

When we have too few points to justify linearizing the function between adjacent points (as the trapezoidal integration does) we can use an algorithm based on a higher-order polynomial, which thereby can more faithfully represent the curvature of the function between adjacent measurement points. The Newton-Cotes method does just that for equidistant points, and is a moving polynomial method with fixed coefficients, just as the Savitzky-Golay method used for smoothing and differentation discussed in sections 8.5 and 8.8. For example, the formula for the area under the curve between x, and xn, is... 

The Savitzky-Golay method combines filtering with single or multiple differentiation in one operation. Moreover, as we have already seen in section 8.5, it is very convenient for spreadsheet use. In the spreadsheet exercise we will differentiate a noisy sine wave and compare the result with its analytical derivative, a cosine. Then we will compute the second derivative, and again compare the result with the theoretical second derivative, an inverted sine wave. We could also compute that second derivative stepwise, as the derivative of the derivative, but the present route is simpler and loses fewer points at the edges. 

When the experimental data are not equidistant, a moving polynomial fit can still be used, but the convenience of the Savitzky-Golay method is lost. In that case you may have to write a macro to fit a given data set to a polynomial, and then change the cell addresses to make the polynomial move along the curve. Consult chapter 10 in case you want to write your own macros. 

In chapter 7 we saw that Fourier transformation can also be used to differentiate data. Just as the Savitzky-Golay method, Fourier transformation... 

As an example of writing a macro we will here illustrate its progress, including its validation at various stages. As our example we will use the Savitzky-Golay method to interpolate in a set of equidistant data points. In other words, given a set of data x,y for evenly spaced x-values, we want to... 

The macro must of course be based on an algorithm, for which we will use the Savitzky-Golay method, which we briefly encountered in section 3.3, described in somewhat more detail in section 8.5, and further apply in section 10.9. The macro should do the following ... 

Savitzky Golay method, using the first derivative (left and right point = 1 polynomial order = 1) of FTIR selected spectra. The spectra used were from three regions pins 250 - 400 pins 445 - 589 pins 631 - 770 a total of 436 X-variables. The PLS analysis used every fourth wavelength from pins 250 to 490 (total of 48 channels) with mean centering, full-cross validation with 7 PCs, an outlier warning limit of 4.0 and the bilinear PLSR model. 

Figure 1 Synthetic example demonstrating numerical derivative spectra up to third order derived from a synthetic absorbance spectrum - the lower left graph compares the Savitzky-Golay method to the noise sensitive difference approach [3].

Savitzky-Golay method This method determines a derivative spectrum by moving a spectral window comprising 2 +1 measurement points over an absorbance spectrum. Then a polynomial of order m... 

Of the two parameters the user has to select for the Savitzky-Golay method, the polynomial order m [4] is the less important one. Much more important is the window width 2n- -l. In Figure 6, second-derivative spectra of a synthetic absorption spectrum (Figure 1, upper left graph) are compared, which have been calculated using different window widths. A too narrow window does not incorporate sufficient smoothing (upper left graph) and suppresses relevant broad features - a too broad window causes artifacts... 

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