Quadratic fitting function

Another limitation is perhaps not so much a limitation as, perhaps, a strange characteristic, albeit one that can catch the unwary. To demonstrate, we consider the simplest S-G derivative function, that for the first derivative using a 5-point quadratic fitting function. The convolution coefficients (after including the normalization factor) are... 

Table 57-1 Some of the Savitzky-Golay convolution coefficients using a quadratic fitting function...

Table 57-2 The Savitzky-Golay convolution coefficients multiplied out. All coefficients are for a quadratic fitting function. See text for meaning of SSK...

Index 7-point smoothing with quartic fitting function 5-point first derivative with cubic fitting function 5-point second derivative, quadratic fitting function... 

For comparison 2, we find one case five-point first derivative versus five-point second derivative, both using a quadratic fitting function. Here again, the noise multiplier increased with increasing derivative order. 

We extrapolated the quadratic fitting function to 1200 K for this work. 

Regression for Nonlinear Data the Quadratic Fitting Function... 

This formula is exact for a quadratic function, but for real problems a line search may be desirable. This line search is performed along the vector — x. . It may not be necessary to locate the minimum in the direction of the line search very accurately, at the expense of a few more steps of the quasi-Newton algorithm. For quantum mechanics calculations the additional energy evaluations required by the line search may prove more expensive than using the more approximate approach. An effective compromise is to fit a function to the energy and gradient at the current point x/t and at the point X/ +i and determine the minimum in the fitted function. 

The calculations begin with given values for the independent variables u and exit with the (constrained) derivatives of the objective function with respec t to them. Use the routine described above for the unconstrained problem where a succession of quadratic fits is used to move toward the optimal point for an unconstrained problem. This approach is a form or the generahzed reduced gradient (GRG) approach to optimizing, one of the better ways to cany out optimization numerically. 

For comparison 1, we find two cases 7-point smooth with quadratic versus quartic fitting function, and 5-point first derivative with quadratic versus cubic fitting function. From these two comparisons we find that the noise multiplier of the derivative (of the same order and number of data points) increases as the degree of the fitting function increases. 

We note that the coefficients for the quadratic terms are the same in both cases. However, the best fitting functions have a constant intercept and varying slopes, while the functions based on the orthogonalized quadratic term has a constant slope and varying intercept. 

Where the at and the ki are values obtained by the least-squares fitting of the quadratic and linear fitting functions, respectively. The differences, then, are represented by... 

Figure 9.25 Second-harmonic intensity of LB films as function of number of layers R/S/... structures (open dots) and R/R/... stmctures (filled dots). Solid line is quadratic fit of data points for R/R/... structures.

Figure 18. The centers of mass of the induced electronic charge (xe), and induced polarization charge (x,) as a function of the amount of induced surface charge, in units of 10-3 e/(a.u.)3. The black filled circles show the calculated values of xe, and the solid black line is a quadratic fit to these values. The open circles indicate the calculated center of mass of the xs (equal to the edge position of an equivalent, classical uniform dielectric), and the line labeled Exp. dielectric edge indicates where they would need to be in order to reproduce the experimental compact capacity. The line labeled Shifted Oxygen dist. is the position of the oxygen surface layer as a function of charge, shifted downward by 2.4 a.u. From Ref. 52, by permission.

Examination of the fit of the straight line to the data points (as characterized by the residuals) shows that the points tend to be a little below the line near the ends and a little above near the middle, as if the correct fitting function should have a small amount of curvature. Let us now consider a quadratic model, which may be justified by a more refined theory or may be purely empirical. Accordingly a new fit was made with the function... 

Table 5. Optimized parameters and standard deviations for best-fit functions representing corrections to Koopmans 2pi p and 2p3/2 ionization energies and spin-orbit splitting for the investigated atoms. There are shown the results obtained from the functions giving the best fits linear, quadratic, exponential-decay, or sigmoid. The solid lines shown in Figures 4 and 5 are computed using these functions.

With any such algorithm it is necessary to specify some tolerance value below which any peaks are assumed to arise from noise in the data. The choice of window width for the quadratic differentiating function and the number of points about the observed inflection to fit the cubic model are selected by the user. These factors depend on the resolution of the recorded spectrum and the shape of the bands present. Results using a IS-point quadratic differentiating convolution function and a nine-point cubic fitting equation are illustrated in Figure 6. 

In contrast to the case of cyclobutanone, the addition of two more adjustable parameters does not seem warranted in the case of silacyclobutane in that only a small improvement in the fit results. The barrier determined is 442 cm-1, within 2 cm"1 of the barrier determined from the simpler quartic-quadratic potential function. As pointed out by Pringle, the tendency is to weight the microwave data heavily because of the precision of the rotational data compared to that of the measurement of the vibrational intervals in the far-infrared or Raman spectrum. However, in doing so, one fails to recognize the limitations of the Hamiltonian. If the potential func-... 

Fig. 5.14. Energy as a function of volumetric strain as computed using atomic-scale analysis in terms of embedded-atom potentials (courtesy of D. Pawaskar). The atomistic result is compared with the quadratic fit like that suggested in eqn (5.93).

Modeling the Experimental Data. The data collected from each experiment in a given experimental design can be mathematically modeled so that the response, such as migration time, resolution, and so on, can be correlated with the experimental conditions that produced it. This way, by using the model, the desired output can be maximized and the corresponding experimental conditions defined in a predictive manner. Frequently, data are fitted to quadratic polynomial functions similar to Equation 5.7,... 

It is always useful to look at the residuals, i.e., the differences between the data and the fitted function in the present example, the residuals are the differences y,— acalc xt. The reason for this is that use of an incorrect model (such as fitting to, say, a linear or quadratic relation rather than to a proportionality) often leads to a discernible trend in the residuals, whereas random deviations do not. Therefore plot the residuals y- ycaic = y-... 

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