Savitzky-Golay polynomial

The most common algorithm is that of Savitzky and Golay [22, 23]. This polynomial incorporates a shortened least-square computation for data smoothing (see also Section 3.6.4.2). Some points must be considered when using this differentiation mode  

Another differentiation method is point-point differentiation (see Sec. S.6.4.2), also called the difference quotient method. We obtained very good results with this technique by foUowing simple mathematical manipulations  

5-point sliding window twice than it is to take a 6- or 10-point set of data only once. It must be kept in mind that all smoothing operations involve a compromise between a reduction of the noise with an alteration of the shape of the signal and the loss of information. 

For computing the difference quotient Aj /Ax, the difference of two amplitude values is divided by Ax, the width of differentiation and the differentiated value mostly positioned in the middle of Ax, say at Ax/2. For this reason, the data of the derivatives and, therefore, also the positions of the extrema are shifted for Ax/2 in the direction of differentiation. This alteration of the beginning of the curve can be exactly predetermined and therefore easily compensated for. 

In Table 3-9 some more complex algorithms are listed. One of them, the Fourier series, has been used more frequently in the past few years. The reason for this may be the great progress in the hard- and software for digital computation. Nevertheless, one should first evaluate in each case whether it is really necessary to generate derivatives by more complicated algorithms, and whether the quality and resolution of the spectra can be improved in this way. 

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It is well known that features of signals can be enhanced by examining their derivative. Consequently, derivative cyclic voltammetry (DCV) was developed [57]. The derivative is usually calculated numerically by simple differencing (A/ /At) if the time step increments are small enough [58], by a Savitzky-Golay polynomial least-squares procedure [59], or by Fourier transformation [60]. Also, hardware based differentiation is possible [60]. 

Note that the broader the width of intervals is, the smaller the number of data points will be. It is not a sliding data computation (e.g., the Savitzky-Golay polynomial), but a computation, dependent on the width of intervals. In the sliding data manipulation, the interval width can be varied while the number of data points remains constant. To contrast, broader data intervals correspond to a higher influence on the signal shape (deformation), and a reduced amount of information in the spectra. 

Savitzky-Golay polynomials with variable A A steps and filtering modes (number of points for smoothing)... 

In Table 4-9 there are two possibilities given for differentiation, namely. Ax for the difference quotient method, and the number of points for Savitzky-Golay polynomials. We prefer the first of these two methods because it is simpler and, nevertheless, very effective. Do not forget to correct, if necessary, all AA shifts, in order to allow for correct estimation of A positions of all maxima and minima. 

Evaluate the expression for the derivative of that polynomial at the point for which the derivative is to be computed. In the Savitzky-Golay paper, this is the central point of the set used to fit the data. As we shall see, in general this need not be the case, although doing so simplifies the formulas and computations. 

Of course, an inherent limitation is the fact that only those combinations of parameters (derivative order, polynomial degree and number of data points) that are listed in the Savitzky-Golay/Steinier tables are available for use. While those cover what are likely to be the most common needs, anyone wanting to use a set of parameters beyond those supplied is out of luck. 

The paper contains formulas for only those derivative orders and degrees of polynomials that are contained in the original Savitzky-Golay paper, therefore we are still limited to those derivative orders and polynomial degrees. 

Polynomials do not play an important role in real chemical applications. Very few chemical data behave like polynomials. However, as a general data treatment tool, they are invaluable. Polynomials are used for empirical approximations of complex relationships, smoothing, differentiation and interpolation of data. Most of these applications have been introduced into chemistry by Savitzky and Golay and are known as Savitzky-Golay filters. Polynomial fitting is a linear, fast and explicit calculation, which, of course, explains the popularity. 

Figure 4-23. Savitzky-Golay filtering. A polynomial is fitted to a range of data points and the original point (x) is replaced by the value on the polynomial (o).

The function SavGo 1, m performs a Savitzky-Golay smoothing. The parameters are the x- and y-vectors, the number (n) of neighbouring left or right data points that are used for one polynomial fit (i.e. if n= 5, 2n+l=ll data points are fitted) and the degree (ndf of the polynomial to be fitted. 

It is tempting to write a routine such as SavGo l bad. m, to perform the Savitzky-Golay filtering, but we will show its numerical weakness. F is built up by the appropriate range of x-values and used to calculate the polynomial coefficients as a=F y( i-n i+n), see e.g. equation (4.31). 

The Savitzky-Golay algorithm could readily be adapted for polynomial interpolation. The computations are virtually identical to smoothing. In smoothing, a polynomial is fitted to a range of (x,y)-data pairs arranged around the x-value that needs to be smoothed. For polynomial smoothing, the polynomial is evaluated for a set number of data points around the desired x-value and the computed y-value at that x is the interpolated value. 

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. 

Very popular is the Savitzky-Golay filter As the method is used in almost any chromatographic data processing software package, the basic principles will be outlined hereafter. A least squares fit with a polynomial of the required order is performed over a window length. This is achieved by using a fixed convolution function. The shape of this function depends on the order of the chosen polynomial and the window length. The coefficients b of the convolution function are calculated from ... 

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