Weighted Least Square

Least squares weighting, 237, 2381 Lew and Angus method, 113—114 Ligand(s)... 

Dunitz and Seiler (1973) have used the equivalence to modify least-squares weighting, such as to emphasize the fit near the density peak positions, in order to obtain parameters less biased by bonding effects. The resulting weights emphasize high-order reflections, similar to the higher-order refinement method, but with a smoothly varying cut-off rather than a sharp sin 6//. limit. 

LARGE (worksheet function), use with array formulas 103 LBound Function 280 least squares weighted 212... 

The behavior of B(p) and C(p) for propane is shown in Figure 7. The number of PpT data used here for adjusting the equation of state is 843, with different least-squares weightings than in Refs. 3 and 5. Overall deviations, with equal weighting for all points, are 2.07 bar for the mean of absolute pressure deviations and 0.34% for the rms of relative density deviations. 

A.1 Ordinary least-squares weights for linear contrasts... 

Discrete least square Ordinary least square Regular least square Weighted least square... 

Nealen, A. (2004.). An as-short-as-possible introduction to the least squares, weighted least squares and moving least squares methods for scattered data approximation and interpolation. Technical Report, TU Darmstadt. 

No. of Measured Reflections Function Minimized Least-squares Weights p-factor... 

Thus, the least-squares weight functions are Wp = dR/d(pp- These may be difficult to compute and thus simpler methods are often favored in practice. 

The standard least-squares approach provides an alternative to the Galerkin method in the development of finite element solution schemes for differential equations. However, it can also be shown to belong to the class of weighted residual techniques (Zienkiewicz and Morgan, 1983). In the least-squares finite element method the sum of the squares of the residuals, generated via the substitution of the unknown functions by finite element approximations, is formed and subsequently minimized to obtain the working equations of the scheme. The procedure can be illustrated by the following example, consider... 

Potentiometric titration curves are used to determine the molecular weight and fQ or for weak acid or weak base analytes. The analysis is accomplished using a nonlinear least squares fit to the potentiometric curve. The appropriate master equation can be provided, or its derivation can be left as a challenge. 

A weighted least-squares analysis is used for a better estimate of rate law parameters where the variance is not constant throughout the range of measured variables. If the error in measurement is corrected, then the relative error in the dependent variable will increase as the independent variable increases or decreases. 

The weighted least-squares analysis is important for estimating parameter involving exponents. Examples are the eoneentration time data... 

Assumption 3 The variance of the random error term is constant over the ranges of the operating variables used to collect the data. When the variance of the random error term varies over the operating range, then either weighted least squares must be used or a transformation of the data must be made. However, this may be violated by certain transformations of the model. 

The simplest procedure is merely to assume reasonable values for A and to make plots according to Eq. (2-52). That value of A yielding the best straight line is taken as the correct value. (Notice how essential it is that the reaction be accurately first-order for this method to be reliable.) Williams and Taylor have shown that the standard deviation about the line shows a sharp minimum at the correct A . Holt and Norris describe an efficient search strategy in this procedure, using as their criterion minimization of the weighted sum of squares of residuals. (Least-squares regression is treated later in this section.)... 

Carrying through the treatment as before yields Eqs. (2-78) as the normal equations for weighted linear univariate least-squares regression. 

Therefore in applying weighted least-squares analysis to Eq. (2-83), each c = In c should be weighted inversely to o /c rather than to cr. ... 

Correct weighting procedures for least-squares analysis have been discussed in... 

Obtain the weighting function required to carry out weighted least-squares regression analysis of Eq. (2-15). 

We wish to apply weighted linear least-squares regression to Eq. (6-2), the linearized form of the Arrhenius equation. Let us suppose that our kinetic studies have provided us with data consisting of Tj, and for at least three temperatures, where o, is the experimental standard deviation of fc,. We will assume that the error in T is negligible relative to that in k. For convenience we write Eq. (6-2) as... 

Table 6-1 lists the experimental quantities, k, T, ct, the transformed variables x, y, and the weights w. (It is necessary, in least-squares calculations, to carry many more digits than are justified by the significant figures in the data at the conclusion, rounding may be carried out as appropriate.) The sums required for the solution of the normal equations are... 

Examination of Table 6-1 reveals how the weighting treatment takes into account the reliability of the data. The intermediate point, which has the poorest precision, is severely discounted in the least-squares fit. The most interesting features of Table 6-2 are the large uncertainties in the estimates of A and E. These would be reduced if more data points covering a wider temperature range were available nevertheless it is common to find the uncertainty in to be comparable to RT. The uncertainty of A is a consequence of the extrapolation to 1/7" = 0, which, in effect, is how In A is determined. In this example, the data cover the range 0.003 23 to 0.003 41 in 1/r, and the extrapolation is from 0.003 23 to zero thus about 95% of the line constitutes an extrapolation over unstudied tempertures. Estimates of A and E are correlated, as are their uncertainties. ... 

Table 6-1. Data for Weighted Linear Least-Squares Arrhenius Analysis...

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