Quadratic least squares

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... 

Precision can be increased if several injections of the sample and the reference solutions are made, always using equal volumes. In a multilevel calibration several different amounts of the standard are prepared and analyzed. A regression method is used (e.g. linear least-square, or quadratic least square) and this leads to a more precise value for Cunk- This quantitative method is the only one adapted to gaseous samples. 

Figure 1. Computer plot of the light-scattering analysis of HEC before enaocellulase hydrolysis. (O) Experimental Kc/R values, (M) extrapolated values (by quadratic least squares analysis). The inverse of the Kc/Re intercept, Mw, equals 1,285,000.

The interpolated values of Kc/Re at constant time are calculated or read from the graph, and then the extrapolation of the isochronous Kc/Re values to zero angle is made in the same manner as mentioned for the Zimm plot, by using quadratic least squares regression analysis. 

Figure 3. Computer plot obtained by isochronous interpolation of the experimental light-scattering data (O) of HEC during endocellulase attack. The Langrangian interpolation functions are given by the horizontal curves ana the isochronous interpolated Kc/Re values by A. The quadratic least squares extrapolations to zero angle ( ) are given by the...

Figure 3.10. Unit cell edge ratio c/a (A) and volume (B) versus composition for biogenic magnesian calcites. Solid line is quadratic least-squares fit to data for synthetic calcites, and dashed line represents straight line between calcite and disordered dolomite. (After Bischoff et al., 1985.)...

The matrix M is determined by integrals of the products of the second derivatives of the B-spline basis functions used in the representation of the unknown distribution. This is a quadratic least-squares minimization problem. 

A built-in microcomputer system performs rapid quadratic least squares fit to the data, yielding D, R, a (normalized standard deviation of the intensity weighted distribution of diffusion constants) and x squares goodness of fit. The greater the value of a the larger the degree of polydispersity present in the particle sizes -a values less than 0.2 are generally considered to correspond to pure monodisperse systems. A typical result obtained for 4 x 10 3M CdS-SDS is 5= 7.25 x 10"8 cm2/s, R = 300 A, a = 0.60 and y2... 

Some researchers have reported techniques involving least square fitting with experimental data and extrapolation method in order to calculate permittivity from the experimental plots of effective permittivity vs. solid volume fraction. Nelson [80] has mentioned some of these approaches involving complex permittivity in case of granular materials. These include a quadratic least square fit used by Kent [81] and linear extrapolation... 

The peak area of each oxygenate in the gasoline is measured relative to the peak area of the internal standard. A quadratic least-squares fit of the calibrated data of each oxygenate is applied and the concentration of each oxygenate calculated. 

For each oxygenate, s, calibration data set, obtain the quadratic least-squares fit equation in the following form ... 

Figure 3 gives an example of a quadratic least-squares fit for MTBE and the resulting equation in the form of Eq 6 ... 

The least squares derivation for quadratics is the same as it was for linear equations except that one more term is canied through the derivation and, of course, there are three normal equations rather than two. Random deviations from a quadratic are ... 

Our results indicate that dispersion coefficients obtained from fits of pointwise given frequency-dependent hyperpolarizabilities to low order polynomials can be strongly affected by the inclusion of high-order terms. A and B coefficients derived from a least square fit of experimental frequency-dependent hyperpolarizibility data to a quadratic function in ijf are therefore not strictly comparable to dispersion coefficients calculated by analytical differentiation or from fits to higher-order polynomials. Ab initio calculated dispersion curves should therefore be compared with the original frequency-dependent experimental data. 

The experiments were carried out in random order and the responses analyzed with the program X-STAT(11) which runs on an IBM PC computer. The model was the standard quadratic polynomial, and the coefficients were determined by a linear least-squares regression. 

The calibration curve is generated by plotting the peak area of each analyte in a calibration standard against its concentration. Least-squares estimates of the data points are used to define the calibration curve. Linear, exponential, or quadratic calibration curves may be used, but the analyte levels for all the samples from the same protocol must be analyzed with the same curve fit. In the event that analyte responses exceed the upper range of the standard calibration curve by more than 20%, the samples must be reanalyzed with extended standards or diluted into the existing calibration range. 

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... 

A quadratic equation will be fitted to N sets of data, (xt, yl), by least squares. 

This section describes the basic idea of least squares estimation, which is used to calculate the values of the coefficients in a model from experimental data. In estimating the values of coefficients for either an empirical or theoretically based model, keep in mind that the number of data sets must be equal to or greater than the number of coefficients in the model. For example, with three data points of y versus x, you can estimate at most the values of three coefficients. Examine Figure 2.7. A straight line might represent the three points adequately, but the data can be fitted exactly using a quadratic model... 

The adjustment of measurements to compensate for random errors involves the resolution of a constrained minimization problem, usually one of constrained least squares. Balance equations are included in the constraints these may be linear but are generally nonlinear. The objective function is usually quadratic with respect to the adjustment of measurements, and it has the covariance matrix of measurements errors as weights. Thus, this matrix is essential in the obtaining of reliable process knowledge. Some efforts have been made to estimate it from measurements (Almasy and Mah, 1984 Darouach et al., 1989 Keller et al., 1992 Chen et al., 1997). The difficulty in the estimation of this matrix is associated with the analysis of the serial and cross correlation of the data. 

The methods used were those of Mitchell ( 1 ), Kurtz, Rosenberger, and Tamayo ( 2 ), and Wegscheider T ) Mitchell accounted for heteroscedastic error variance by using weighted least squares regression. Mitchell fitted a curve either to all or part of the calibration range, using either a linear or a quadratic model. Kurtz, et al., achieved constant variance by a... 

To find L 2 it is necessary to calculate (dAH/dn2) T,p at various molalities of HCl. The data in Figure 18.5 have been fitted to a quadratic polynomial by the method of least squares. The equation obtained is... 

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