Convoluting integers

For so-called equidistant data sets (where equidistance applies to the independent variable), least-squares fitting is even simpler, and takes a form tailor-made for an efficient moving polynomial fit on a spreadsheet, requiring only access to a table of so-called convoluting integers, or software (such as described in section 10.9) where these integers are automatically computed. 

Instead, we mean here the use of experimental data that can be expected to lie on a smooth curve but fail to do so as the result of measurement uncertainties. Whenever the data are equidistant (i.e., taken at constant increments of the independent variable) and the errors are random and follow a single Gaussian distribution, the least-squares method is appropriate, convenient, and readily implemented on a spreadsheet. In section 3.3 we already encountered this procedure, which is based on least-squares fitting of the data to a polynomial, and uses so-called convoluting integers. This method is, in fact, quite old, and goes back to work by Sheppard (Proc. 5th... 

In column F we will use a 15-point quadratic, for which the convoluting integers are... 

For the first derivative, using a five-point quadratic as our moving polynomial, the tables list the convoluting integers as 2, — 1, 0,1, and 2, and the normalizing factor as 10 times the increment Ax. In cell G8 therefore deposit the instruction = (— 2 F6 F7... 

Again, the above program can readily be modified. For example, a reader who wants to use the equidistant least-squares program to compute a third derivative (for which the convolution integers are not available in the usual tables) will find that this can be done with a few changes around InputBox 4, because the limitation to a second-order derivative is not inherent to the algorithm, but was inserted merely to illustrate how this is done. Similarly, in order to modify the criterion used in the F-test to, say, 1%, one only needs to change the value 0.05 in the line FValueTable (i, j) =... 

FIGURE 5-14 Least-squares polynomial smoothing convolution integers (a) quadratic five-point integers, (b) first-derivative cubic five-point integers, (c) second-derivalive quadralic five-point integers. 

Table 3-8. Mode of convolution integers, given by Savitzky and Golay after [71].

Ratzlaff, K. L., Computation of Two-Dimensional Polynomial Least-Squares Convolution Smoothing Integers, Ana/. Chem. 61, 1989, 1303-1305. 

For higher integer spins the number of allowed zero-field interaction terms further increases, and so does the convolution of comparable effects, except once more for a unique term that directly splits the highest non-Kramer s doublet. For S > 3 we have the addition, valid in cubic (and, therefore, in tetragonal, rhombic, and triclinic) symmetry ... 

We turn now to the effect of using the Savitzky-Golay convolution functions. Table 57-1 presents a small subset of the convolutions from the tables. Since the tables were fairly extensive, the entries were scaled so that all of the coefficients could be presented as integers we have previously seen this. The nature of the values involved caused the entries to be difficult to compare directly, therefore we recomputed them to eliminate the normalization factors and using the actual direct coefficients, making the coefficients more easily comparable we present these in Table 57-2. For Table 57-2 we also computed the sums of the squares of the coefficients and present them in the last row. 

A third method to simulate random variables is convolution, where the desired random variates are expressed as a sum of other random variables that can easily be simulated. For example, the Erlang distribution is a special case of the Gamma distribution when the shape parameter is an integer. In this case, an Erlang random variate with shape parameter can be generated as the sum of j3 exponential random variates each with mean a. A last method to simulate random variables is decomposition (sometimes called composition), where a distribution that can be sampled from is composed or decomposed by adding or subtracting random draws into a distribution that cannot be simulated. Few distributions are simulated in this manner, however. These last two methods are often used when the first two methods cannot be used, such as if the inverse transformation does not exist. 

For the convenience of the user, the Convolution macro has been written such as to accept a time scale that can start at any arbitrary value. Of course, the time increments must still be equidistant, and there must be 2N input data, where 2Vis a positive integer subject to the constraint 2 /V=s 10. 

Note that too wide a window function may make the convolution spill over an edge. You can avoid this by adding zeros to the beginning and end of columns B and C, with an accompanying extension of the time scale in column A. The total number of points must remain an integer power of 2, so that it is best in the present example to add 32 zeros at both the beginning and end. 

Noise in a spectrum can be diminished by smoothing. After a spectrum is smoothed it becomes similar to the result of an experiment obtained at a lower resolution. The features are blended into each other and the noise level decreases. A smoothing function is basically a convolution between the spectrum and a vector whose points are determined by the degree of smoothing you wish to apply. Generally, you will be asked to enter a degradation factor, which will be some positive integer. A low value, e.g. one, will produce only... 

The raw data are the corrected chromatogram (10 of Fig. lA, the (uniform) broadening function g(V) of Fig. 1 A, and the linear calibration log M(V) of Fig. IB. By convolution of f(y) and g(V), a noise-free measurement was first obtained. Then, this noise-free function was rounded to the last integer, this procedure is equivalent to adding a zero-mean random noise of uniform probability distribution in the range 0.5. The resulting chromatogram is u (V) of Fig. 1 A. Note that the multimodality of w (V) is lost in w(V). 

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