Weighted, Least-Squares Regression

In weighted, least-squares analysis, it is assumed that the variance of the individual data points may be variable. In order to reduce this problem to the standard linear regression framework, a weight, w is introduced for each observation that reflects how good the data point is. Thus, the regression model for weighted, least-squares is 

This implies that the objective function to be optimised in weighted, least squares is given as 

Following a similar procedure as for the ordinary, least-squares case, Eq. (3.80) can be solved to obtain the unknown, estimated coefficients,  

The weighted residual is defined as the difference between the predicted and measured values, that is, 

The 100(1—a)% predictive confidence intervals, that is, the region within which the actual value will lie 100(1—a)% of the time, for the point Xd is given by 

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

Robust and resistant isochrons can have very different characteristics than traditional least-squares or error-weighted least squares regressions. Some methods ignore analytical errors entirely, others infer them from the observed scatter of the data, and still others make use of analytical errors only to the extent that they are validated by their observed scatter. 

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... 

The three saturated long-chain tert-butyl peresters are members of a homologons series, and as such, the weighted least-squares regression analysis of the enthalpies of formation V5. number of carbons yields a methylene increment of —26.7 kJmol , a typical valne for liquids. The methylene increment for the terf-butyl esters of the Cg, Cjo, Cn and C14 acids is —28.0 kJmol. The closeness of these two values ensures that the enthalpies of formal reaction 16 will be nearly constant. For the three pairs from Table 3, the value is —70.3 8.1 kJmol. The standard deviation from the mean is quite large because the arithmetic difference for the C12 ester and perester, —79.5 kJmol, is quite a bit more negative than the differences for the Cio and C14 pairs, —64.4 and —66.9 kJmol, respectively. Unfortunately, the acids and esters are in different phases and so we are reluctant to attempt any comparison between them, such as a formal hydrolysis reaction or disproportionation with hydrogen peroxide. 

Weighted least squares regressions based on the second and third sets of weights (l/ a,2) produces the following estimates (standard errors are shown in parentheses). In each case, the weights are the reciprocals of the estimated standard deviations as described above. 

The condition of variance homogeneity can be proven with the help of statistical tests (/- -test for quotient of variances at the lower and upper end of the calibration range, or better, in the case of rij>5 by using the Bartlett-test, which includes all of the variances in the calibration range). If variance homogeneity is violated, a weighted least squares regression (WLS) should be used. 

In this paper we present the results of an evaluation of selected thermodynamic data of arsenic species. The results are of two types with one that consists of data that have been fitted with a weighted least-squares regression, and a second that is derived from the first least-squares determined group by standard thermodynamic relationships. The results (in Tables la, lb, and 2) and their implications for geochemistry and geochemical modelling are then discussed with known occurrences, observed mineral transformations in the environment and calculated pe-pH relationships. The objective is to provide a consistent and coherent framework of thermodynamic calculations and field relationships for mineral stabilities among arsenic species. 

The results of simultaneous weighted least-squares regression of the data and some of the unfitted but derived quantities are shown in Tables la, lb, and 2. Table la displays elemental arsenic, its simple oxides, and the reactions for arsenic oxidizing to arsenic trioxides. Table lb introduces the hydrolysis species for As(III) and As(V) in solution, the hydrolysis reactions, and the solubility reactions for the simple oxides. Single species are shown at the top of each table with the reactions underneath. The following discussion describes some of the mineral occurrences for these substances, describes their relative stabilities from field observations, and considers the implications of the evaluated thermodynamic data in terms of these occurrences. 

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

Refined calculations using weighted least squares regression... 

A simple transformation procedure can often remove the unequal scatter in e. But this is not the only procedure available weighted least-squares regression can also be useful. 

REM THIS PROGRAM ESTABLISHES THE CALIBRATION CURVE FOR X,Y PAIRS OF DATA USING NON-WEIGHTED LEAST SQUARES REGRESSION STATISTICS. 

It may be necessary to use weighted least squares regression calculations in the method evaluation to obtain the Oji value, corresponding to the standard deviation at the conventional true value pti having unit weight. A valid estimate of aj,- at the concentration p-x,- of the control sample can be computed using the equation [12] ... 

Example 3.2 Determining the Weights for Weighted, Least-Squares Regression... 

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