Least-squares analysis weighted

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. 

Consider a first order reaction with the final concentration expressed 

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

It is also possible to determine A and B that minimize the weighted sum of squares. The weighting funetion is the square of the independent variable, and the funetion to be minimized is 

The validity of least squares model fitting is dependent on four prineipal assumptions eoneerning the random error term , whieh is inherent in the use of least squares. The assumptions as illustrated by Baeon and Downie [6] are as follows  

Consider a first order reaction with the final concentration expressed by CA = CAOe kt. If the error in concentration measurement is 0.01CAO, 

The weighted least-squares analysis is important for estimating parameter involving exponents. Examples are the concentration time data for an irreversible first order reaction expressed by CA = CAOe kt, and the reaction rate-temperature data expressed by (-rA) = k0CAe Ea/RT. These equations are of the form 

---

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. ... 

These plots can also provide information about the assumption of constant error variance (Section III) made in the unweighted linear or nonlinear least-squares analyses. If the residuals continually increase or continually decrease in such plots, a nonconstant error variance would be evident. Here, either a weighted least-squares analysis should be conducted (Section III,A,2) or a transformation should be found to stabilize the error variance (Section VI). 

There remain three Bronsted acids that have no liquid phase enthalpy of formation data that we know of dodecanethiol, cyclohexyl methyl amine and A-methyl dodecanamide. Although the enthalpy of formation of 1-dodecanethiol has not been measured, there are experimental values available for other members of its homologous series, C2-C7, Cjo. From a weighted least-squares analysis of the data from which a slope, —25.4, and an intercept, —23.3, are derived, the enthalpy of formation of dodecanethiol is —328.1 kJ moH The enthalpy of formation of dodecanethiolate magnesium bromide is thus estimated as —744.6 kJmoH. We can estimate the enthalpy of formation of cyclohexyl methyl amine by assuming equation 12 is thermoneutral. 

The rates of acid-catalysed medioxy exchange between methanol and the three diastereomers of 2-medioxy-4,6-dimethyl-l,3-dioxalane (37a-c) were measured in benzene and methanol-chloroform. Rate constants were evaluated in a novel way using 1D-EXSY NMR pulse sequence and a weighted least-squares analysis. The critical intermediate is (38) and rates of methanol attack on it in benzene show a 24-fold axial selectivity whereas in methanol-chloroform the selectivity difference is ninefold.34... 

In such an analysis, one selects a suitable objective function and then varies the parameters so as to maximize or minimize the function. Theoretically, the objective function should be derived using the statistical principles of maximum-likelihood estimation. In practice, however, it is satisfactory to use a weighted-least-squares analysis, as follows ... 

Because both AH°f and IE are linear with respect to n , so also is ST. The results from the weighted least squares analysis [15] are for alkanes (nc = 4-6)... 

Both the linear and nonlinear least-squares analyses presented above assume that the variance is constant throughout the range of the measured variables. ff this is not the case, a weighted least-squares analysis must be used to obtain better estimates of the rate law parameters. If the error in measurement is at a fixed level, the relative error in the dependent variable will increase as the independent variable increases (decreases). For example, in a first-order decay reaction (C = Cao< 0. if the error in concentration measurement is O.OIC ao. the relative error in the concentration measurement [0.01Cao/Ca(0] will increase with time. When this error condition occurs, the sum to be nuni-mized for N measurements is... 

For parameter estimation involving exponents, it has been shown that a weighted least-,squares analysis is usually necessaty. Further discussion on weighted least squares as applied to afirst-orderreaction is given on the CD-ROM. 

The average component numbers from these unweighted fits are 1n close agreement with the results from the weighted least squares analysis as expected. The standard deviation, and hence the unreal lability of the results from a single series, 1s generally higher In the estimation of component number from the slope than from the Intercept. This observation Is attributed to the variances In slope and Intercept which. In turn, are functions of the peak capacities and the number of counted peaks (6). The variance In the estimation from the Intercept Increases with the value m, and a reversal of the trend Is observed In Set E. 

Table II contains the optimal empirical resolutions calculated from the weighted least squares analysis of counted peak maxima. The empirical r from each set was used to calculate, from an unweighted east squares fit, a component number from each series in that set. The results for each set were averaged, and the mean and standard deviation for each set are reported In Table II.

Weighted least-squares analysis is also called for when we must average data of different precision. In section 2.5 we already encountered the need for weighting of the experimental data when their individual standard deviations are known. In that case the individual weights are simply the recipro -cals of the variances of the individual measurements,... 

Also call the Regression analysis, and apply it to the same data set. (You could also use the Weighted Least Squares analysis for this, after copying the values in columns H and... 

Weighted least-squares analysis provides the variance of experimental errors on the basis of which the linearity of the Arrhenius plots can be assessed. 

Figure 38. Linear Arrhenius regression by weighted least-squares analysis (a) and least-squares analysis (b). y -axis (1) logarithmic scale, (2) arithmetical scale. (Reproduced from Ref. 317 with permission.)...

H. Seki, T. Hayashi, and N. Okusa, An application of weighted least-squares analysis to Weibull probability paper for prediction of stability of drug products [in Japanese], Yakuzaigaku 40,201-207 (1980). 

The selected log K° determined from the weighted least squares analysis is ... 

The stability constant for the formation of ZrF from ZrF" (g = 2 in Eq.(V.23)), was determined using a weighted (least squares) analysis technique (see Figure V-16). The selected log Q AT° determined from the regression is ... 

For the formation of ZrF " from Zr i.e. <7 = 1 in Eq.(V.23)), the SIT formulation has been used to determine the stability constant at zero ionic strength using the data given in Table V-14, as is shown in Figure V-17. The selected value of logio K° determined from the weighted least squares analysis is ... 

---