Weighting schemes

T = Tolerance aggregation method A, G Modules for alternate aggregation methods E == Equal weighting scheme... 

P, U, S = Codes available for alternate weighting scheme models... 

Homoscedacity must prevail, that is the reproducibilities of y measured for small, medium, and large x-values must be approximately the same, so the uniform weighting scheme can be implemented. 

Curve-fitting need not be abandoned in this case, but some modifications are necessary so that precisely measured points influence to a greater degree the form of the curve, more so than a similar number of less precisely measured ones. Thus, a weighting scheme is introduced. There are different ways of doing this the most accepted model makes use of the experimental standard deviation,namely ... 

It is important to realize that for the typical analytical application (with relatively few measurements well characterized by a straight line) a weighting scheme has little influence on slope and intercept, but appreciable influence on the confidence limits of interpolated X(y) resp. Y(x). 

Conclusions the residual standard deviation is somewhat improved by the weighting scheme note that the coefficient of determination gives no clue as to the improvements discussed in the following. In this specific case, weighting improves the relative confidence interval associated with the slope b. However, because the smallest absolute standard deviations. v(v) are found near the origin, the center of mass Xmean/ymean moves toward the origin and the estimated limits of detection resp. quantitation, LOD resp. 

Figure 2.18. The data from the preceding example are analyzed according to Eqs. (2.27)-(2.39), (2.10)-(2.17) the effect of the weighting scheme on the center of mass and the confidence limits is clearly visible because the noise model was changed.

WLR.dat Section 2.2.10 A set of peak area vs. concentration results of a gas chromatography calibration. Use with LINREG to test the effect of a weighting scheme. The originally estimated dependence of the standard deviation of determination vs. concentration is described by the equation SD = 100 + 5 X. 

Upon substituting these expressions into (9) we write the weighted scheme in the canonical form (8), where... 

This condition is necessary and sufficient for the stability of the weighted scheme, which interests us. 

Stability of the weighted scheme. As an example of applying the theorems just established the weighted scheme comes first... 

The first analysis is connected with the case when A is a constant self-adjoint positive operator A = A > 0. As we have shown in Section 2, a necessary and sufficient condition for the stability of the weighted scheme (47) with respect to the initial data is... 

Theorem 8 Let A be a self-adjoint positive operator independent oft = nr A = A > 0. Then for the weighted scheme (47) estimate (37) is valid for... 

To avoid generality, for which we have no real need, we restrict ourselves here to the case when A = A is a skew-symmetric operator involved in the weighted scheme... 

Weighted schemes. In practice the reader frequently encountered the weighted schemes... 

Afterwards when the sweep formulae became customary, one began to analyze in full details two-layer implicit schemes (weighted schemes) for which R = crA. These schemes obviously represent a particular case of the scheme with R = (tAq. 

Stability and convergence. The general stability theory for two-layer schemes applies equally well to the stability analysis of the weighted scheme (7). With this aim, the appropriate difference scheme with the homogeneous boundary conditions comes first ... 

Summarizing, the weighted scheme (9) is stable in the space C, provided condition (17) holds. For the purely implicit scheme with [Pg.467]

Homogeneous difference schemes. In preparation for designing a homogeneous weighted scheme associated with problem (l)-(3), let... 

For the weighted scheme (20) these estimates are ensured hy a > and laj < Cjfl. 

Economical Difference Schemes for Multidimensional Problems In what follows the weighted scheme... 

We contrived to do the necessary factorization in a number of different ways. First, we initiated the construction of a primary weighted scheme on an equidistant rectangular grid... 

Through the approximation of every heat conduction equation with number a on the half-interval ij+(a-i)/p < t < ij+ajp by the standard two-layer weighted scheme we arrive at the chain of p one-dimensional schemes... 

D., Balahan, A. T. Comparison of weighting schemes for molecular graph descriptors application in quantitative structure-retention relationship models for alkylphenols in gas-liquid chromatography. J. Chem. Inf. Comput. Sci. 2000, 40, ITl-lM,. 

Avdeef, A., Weighting scheme for regression analysis using pH data from acid-base titrations, Anal. Chim. Acta 148, 237-244 (1983). 

Uneven distributions of residuals. The MaxEnt calculations in presence of an overall chi-square constraint suffer from highly non-uniform distributions of residuals, first reported and discussed by Jauch and Palmer [29, 30] the error accumulates on a few strong reflexions at low-resolution. The phenomenon is only partially cured by devising an ad hoc weighting scheme [20,31, 32]. Carvalho et al. have discussed this topic, and suggested that the recourse to as many constraints as degrees of freedom would cure the problem [33]. 

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