Polynomial order

DESCRIPTION OF PARAMETERS XCOF -VECTOR OF M-H COEFFICIENTS OF THE POLYNOMIAL ORDERED FROM SMALLEST TO LARGEST POWER COF -WORKING VECTOR OF LENGTH M-t-1 M -ORDER OF POLYNOMIAL... 

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... 

Figure 12.6 The effect of Savistky-Colay first and second derivative preprocessing on a set of NIR diffuse reflectance spectra (A) the raw (uncorrected) spectra, (B) spectra after Ist derivative preprocessing, (C) spectra after 2nd derivative preprocessing. In both cases the window width was 15 points (7.5 nm), and the polynomial order was 2.

For a given data set, the optimal window size and polynomial order depend on the nature of the data. Of primary importance is the width of the peaks relative to the window width (e.g., choosing a window width 10 times the width of a peak will most likely distort or eliminate it)- An approach to selecting a reasonable window width and polynomial order is to apply several combinations and evaluate the resulting preprocessed data and final results. 

When each peak in the reference spectrum has been matched with a corresponding peak in the spectrum acquired, the mass difference is calculated for each pair of peaks (see Section 3.1.2). These mass differences are plotted as points on a graph each data point has the mass of the acquired peak as its x coordinate, and the mass difference above as its coordinate, and a smooth curve is drawn through the points (Figure 13.10) [5]. The polynomial order parameter controls the type of curve that is drawn and can be set to any value between 1 and 5 ... 

Polynomial order = 1. A straight line is drawn through the points. This polynomial of order 1 is not used to calibrate quadrupole MS. 

Polynomial order = 4. Used for calibrations that include the lower end of the mass scale, with closely spaced reference peaks. This is suitable for calibrations with polyethylene and poly propylene glycols (see Section 13.3.3) that extend below 300 amu. 

Polynomial order = 5. Rarely has any benefit over a fourth order fit. 

Apart from optimization, a problem is often set for mathematical modeling or interpolation. The optimum does not interest us in that case but the model that adequately describes the obtained results in the experimental field. A subdomain is not chosen in that case, but the polynomial order is moved up until an adequate model is obtained. When a linear or incomplete square model (with no members with a square factors) is adequate it means that the research objective corresponds to the optimization objective. 

Dorao and Jakobsen [40, 41] did show that the QMOM is ill conditioned (see, e.g.. Press et al [149]) and not reliable when the complexity of the problem increases. In particular, it was shown that the high order moments are not well represented by QMOM, and that the higher the order of the moment, the higher the error becomes in the predictions. Besides, the nature of the kernel functions determine the number of moments that must be used by QMOM to reach a certain accuracy. The higher the polynomial order of the kernel functions, the higher the number of moments required for getting reliable predictions. This can reduce the applicability of QMOM in the simulation of fluid particle flows where the kernel functions can have quite complex functional dependences. On the other hand, QMOM can still be used in some applications where the kernel functions are given as low order polynomials like in some solid particle or crystallization problems. 

Recursive formulations with respect to both polynomial order and data position are given in Eq. 10.3. Zero-order is only used to account for the data mean if needed. For a mean-centered data vector, the effective polynomials are pi through pw-i- First five of these are plotted in Figure 10.10 for N = 50. Note that the polynomials are explicitly defined through the data length. 

Use Trendline Polynomial Order 2, using the Option to Display Equation on Chart. 

Figure 3.3-2 shows typical results. The simple first-order averaging filter, panel (c) in Fig. 3.3-2, is most effective in reducing the noise, but also introduces the largest distortion, visible even on the broadest peaks. This is always a trade-off noise reduction is gained at the cost of distortion. The same can be seen especially with the narrower peaks, where the higher-order filters distort less, but also filter out less noise. In section 10.9 we will describe a more sophisticated filter, due to Barak, which for each point determines the optimal polynomial order to be used, and thereby achieves a better compromise between noise reduction and distortion. 

The output will appear immediately to the right of the input data, in two to four columns. Make sure that this space is empty, because the output will overwrite any data in those columns. The output will show, from left to right, the smoothed data, the polynomial order used, and (if requested) the values of the 1st and/or 2nd derivative respectively. 

Polynomial Order) or its maximum value (from ELS InputBox 3 ... 

Maximum Polynomial Order). MaxOrder must be an integer, > 0, and... 

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