Least-square polynomials

Let us lump together the m observations y (i = 1. m) into a vector y, the polynomial coefficients otj (/=0.n — 1) into a vector jc of unknowns, and define the (/—l)th power of the ith observable (uj 1 as the current term atJ of the matrix Am, . We now apply the usual method. Polynomials of high degrees tend to generate nearly singular matrices A which result in excessive fluctuations. 

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Kragten, J., Least-Squares Polynomial Curve-Fitting for Calibration Purposes (STATCAL-CALIBRA), Ana/yrica Chimica Acta 241, 1990, 1-13. 

Enke, C. G., and Nieman, T. A., Signal-to-Noise Ratio Enhancement by Least-Squares Polynomial Smoothing, Ana/. Chem. 48, 1976, 705A-712A. 

Leach, R. A., Carter, C. A., and Harris, J. M., Least-Square Polynomial Filters for Initial Point and Slope Estimation, Anal. Chem. 56, 1984, 2304-2307. 

Kahn, A., Procedure for Increasing the Accuracy of the Initial Data Point Slope Estimation by Least-Squares Polynomial Filters, Anal. Chem. 60, 1988,... 

C.G. Enke and T.A. Nieman, Signal-to-noise ratio enhancement by least-squares polynomial smoothing. Anal. Chem., 48 (1976) 705A-712A. 

With the aid of Eq. (48), we can show that 6ik (o) = (k + l)N(co) for t(co) = 0. The object estimate consists of noise at frequencies that t does not pass. The noise grows with each iteration. This problem can be alleviated if we bandpass-filter the data to the known extent of z to reject frequencies that t is incapable of transmitting. Practical applications of relaxation methods typically employ such filtering. Least-squares polynomial filters, applied by finite discrete convolution, approximate the desired characteristics (Section III.C.5). For k finite and t 0, but nevertheless small,... 

Prior to deconvolution, the background was subtracted and the data were smoothed with a 15-point quadratic least-squares polynomial followed by a 19-point quartic least-squares polynomial. The data were then scaled from 0 to 1. The S3 profile was deconvolved using a weight constraint of the form... 

We attempt to fit least squares polynomials of degree 0 through 11, describing the vapor pressure as a function of temperature. 

Figure 2 shows the mean cascading velocity versus distance down the granular cascade for experiments run at the same tangential velocity. Despite a nearly fourfold difference in diameter, the velocity data all fall on nearly the same curve over the first 3 cm down the flowing layer. This agreement indicates that initial particle accelerations may be nearly equivalent, regardless of vessel size. Scatter in the experimental data shown in Figure 2 precludes direct calculation of accelerations, so least-square polynomials were fit to the experimental data. By differentiating the polynomial fit, we obtain an estimate of the downstream acceleration, shown in Figure 3. Over the initial upper third [0 to ( )F of the flowing layer, the acceleration profiles for all cylinders are nearly identical, with only mi-...

Marchand, P. and Marmet, L., Binomial smoothing filter a way to avoid some pitfalls of least squares polynomial smoothing, Rev. Sci. Instrum., 54, 1034—1041, 1983. 

Consider five data points forming a part of a spectrum described by the data set X recorded at equal wavelength intervals. Polynomial smoothing seeks to replace the value of the point Xj by a value calculated from the least-squares polynomial fitted to Xj-2, Xj-u Xj, Xj+i, and Xj+2 recorded at wavelengths denoted by Xj-2, k/ i, j, Xj+1, and j+2-... 

Intensity/wavelength/time cross-sectional diagrams (or time-resolved fluorescence "contour" diagrams) are generated using a weighted nonlinear least squares polynomial surface procedure (20). Area-normalized TRE spectra can be used for convenient pictorial representation, since the absolute emission intensity of individual time-resolved spectra vary substantially with time after excitation. 

Consider the concentration of a chemical species reacting according to a first-order irreversible reaction A—>B with reaction rate constant k. In this case the concentration of A is described by [A] = [A]0 exp(— kt) where [A] is the concentration of species A, t is the time elapsed since the beginning of the experiment (at t= 0), and [A] 0 is the corresponding initial concentration, [A]0 = [A] t = 0. Imagine that we follow the concentration of A spectrometri-cally, and want to extract from the resulting data the initial concentration [A]0 and/or the rate constant k. In that case we must fit the exponential to a polynomial. Using a least-squares polynomial fit, this is no simple task. 

Methods least squares, polynomial regression, non-linear regression PLS, PCR, MARS, PPR, ANN GRAM, TLD... 

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. 

Ilecause least-squares polynomial smoothing is so widely used to enhance the quality of analytical data, it is important to note the advaniages and limiiations of... 

Several different methods have been used to obtain derivative spectra. For modern com iuter-controlled digital spectrophotometers, the differcjuiation can be performed numerically using procedures such as derivative least-squares polynomial smoothing, which is discussed in Section. SC 2. With older analog instruments, derivatives of spectral data could be obtained electronically with a suitable operational amplifier circuit (see... 

A plot of these results is shown in Figure 4.10. The first-order plot is given in Figure 4.10a, and one can see that there is an approximate correlation to the values obtained from the experiment. However, look at Figure 4.10b. This is a least-squares polynomial fit to the data, and it obviously shows a trend with the logarithmic function of equation (iii). The question is whether we go back and reformulate the rate equation to test another form, or do we leave the interpretation as is, admitting... 

The same computational scheme was followed for calcite. The values obtained for E, c/a and x(0) as functions of V/Vo are reported and compared to the corresponding quantities ofmagnesite in Table 3. By substituting for Fin least-squares polynomial interpolations ofq(V) and x(V), and... 

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