Weighted regression methods

Comparison of Anthony-Howard and Coats-Redfern Equations. Because the CR equation and the CN equation yielded similar results, the CR equation was chosen as a basis for comparison with the AH equation. The correlated results based on the AH multiple reaction model and the CR model are presented in Table VIII. The kinetic parameters were determined using an equally weighted regression method. From Table VIII, it can be seen that the AH model yields... 

Estimation is easier and less time-consuming because use is made of empirical relationships between the BCF and physicochemical properties of the compound, such as water solubility (S) [42-48], Km, (solid organic carbon/water partition coefficient) [48], Kmw (membrane water partition coefficient), iipw (liposome water partition coefficient) [49], critical micelle concentration (CMC) [45], steric factors, molecular weight [47,48], and others. The most common regression method is the estimation of BCF from the octanol-water partition coefficient (Kovl) [18,42,44-48,50,51],... 

Outliers or inhomogeneous data can affect traditional regression methods, hereby leading to models with poor prediction quality. Robust methods, like robust regression (Section 4.4) or robust PLS (Section 4.7.7), internally downweight outliers but give full weight to objects that support the (linear) model. Note that to all methods discussed in this chapter robust versions have been proposed in the literature. 

It is often assumed in regression calculations that the experimental error only affects the y value and is independent for the concentration, which is typically placed on the x axis. Should this not be the case, the data points used to estimate the best parameters for a straight line do not have the same quality. In such cases, a coefficient Wj is applied to each data point and a weighted regression is used. A variety of formulae have been proposed for this method. 

The artificial neural network (ANN) is a relatively new technique and possibly the preferred one for current and future (Q)SAR development. Basically, ANNs can be regarded as multinonlinear regression methods. Thus, the neural network software simply multiplies the input by a set of weights that in a nonlinear way transforms the... 

Residue Analysis Methods. Among the methods routinely used for residue analysis are determination of weight loss, extraction of the residue and quantitation of the extract by either gas chromatography or liquid scintillation counting, and measurement of the increase in void space within a fiber with time, or meniscus regression method. 

Meniscus Regression. The meniscus regression method is comparable to the weight loss method in its simplicity and rapidity. The increase in the length of the void space between the open end of the hollow fiber and the meniscus of the column of liquid is measured periodically using a Wilder Varibeam optical comparator which has been described by Weatherston et. al. (3). Since this instrument can also be used to measure tFe internal diameter of the fiber, the volume and therefore the weight of material lost can be calculated as a function of time. 

In Equation 9 the yi values are the fitted f-values, i.e. the values of y on the best line at the x-value of the standard. More complex equations are necessary if weighted regression has to be used, i.e. when the j -direction error is not the same at all values of x. Moreover, many analytical methods give curved calibration plots, fitted for example by polynomial graphs ... 

LR, least-squares regression method GM, geometric mean fimctional regression method. BCFw, bio concentration factor on a wet weight basis. 

A number of variable selection techniques were also suggested for the Partial Least Squares (PLS) regression method [Lindgren et al, 1994]. The different strategies for PLS-based variable selection are usually based on a rotation of the standard solution by a manipulation of the PLS weight vector w or of the regression coefficient vector b of the PLS closed form. 

The six curve-fitting parameters in Equations (23) and (24) (Q, fii, ai, Cu, ixii, and crn) can be determined through use of a nonlinear regression method that minimizes a weighted residual sum of squares (RSS),... 

More standard points are needed to adequately define a curvilinear function than to define a linear function. Weighted least-square regression methods are... 

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