Linear regression uncertainty

Even thought a significant difference does not exist between the linear regression uncertainty and the overall uncertainty, this approach takes into account all sources of uncertainty and underlines the link between the field measurement results and the values of the standards used for the instrument s calibration. The ratio uncertainty between the CRMs and the photometer involved, gives the strength of the traceability link. Moreover, the evaluation of the overall uncertainty in spectrochemical measurements must take into account the steps of the spectrometric measurement process. 

Standardizations using a single standard are common, but also are subject to greater uncertainty. Whenever possible, a multiple-point standardization is preferred. The results of a multiple-point standardization are graphed as a calibration curve. A linear regression analysis can provide an equation for the standardization. 

The first two examples show that the interaction of the model parameters and database parameters can lead to inaccurate estimates of the model parameters. Any use of the model outside the operating conditions (temperature, pressures, compositions, etc.) upon which the estimates are based will lead to errors in the extrapolation. These model parameters are effec tively no more than adjustable parameters such as those obtained in linear regression analysis. More comphcated models mav have more subtle interactions. Despite the parameter ties to theoiy, tliey embody not only the uncertainties in the plant data but also the uncertainties in the database. 

A valuable inference that can be made to infer the quality of the model predictions is the (l-a)I00% confidence interval of the predicted mean response at x0. It should be noted that the predicted mean response of the linear regression model at x0 is y0 = F(x0)k or simply y0 = X0k. Although the error term e0 is not included, there is some uncertainty in the predicted mean response due to the uncertainty in k. Under the usual assumptions of normality and independence, the covariance matrix of the predicted mean response is given by... 

Each stated uncertainty in this and other tables represents one estimated standard error, propagated to parameters from uncertainties of measurements of wave numbers the uncertainties of the latter measurements were provided by authors of papers [91,93] reporting those data, and the weight of each datum in the non-linear regression was taken as the reciprocal square of those uncertainties. As the reduced standard deviation of the fit was 0.92, so less than unity, the authors... 

The linear regression is done as described in the NEA Guidelines for the Assignment of Uncertainties [24]. The following results are obtained ... 

Fig. 6.1 Plot of logio/li + 4Z) vs. 7 for Eq. (6.13). The straight hne shows the result of the weighted linear regression, and the area between the dashed lines represents the uncertainty range of logioi i and As.

Preparation of standard samples always should be as accurate as possible Therefore gravimetric procedures should be preferred compared to volumetric ones and several dilutions should be avoided since each dilution step adds to the uncertainty. For a basic calibration we need 6 to 10 standard samples. They should be distributed equidistant over the whole working range. Linear regression requires equidistant distribution. Otherwise... 

Designed Experiments Produce More Precise Models. In the context of linear regression, this is demonstrated by examining the statistical uncertainties of the regression coefficients. Equation 2.1 is the regression model where the response for the th sample (r ) of an instrument is shown as a linear function of the sample concentration (c.) with measurement error... 

Statistical Prediction Errors (Model and Sample Diag Jostic) Uncertainties in the concentrations can be estimated because the predicted concentrations are regression coefficients from a linear regression (see Equations 5.7-5.10). These are referred to as statistical prediction errors to distinguish them from simple concentration residuals (c — c). Tlie statistical prediction errors are calculated for one prediction sample as... 

For reaction 3 to replace an oxygen with a methylene group to form a primary alcohol, there are enthalpies of formation for only seven alcohols to compare with the nineteen hydroperoxides, almost all of them only for the liquid phase. The enthalpies of the formal reaction are nearly identical, —104.8 1.1 kJmol, for R= 1-hexyl, cyclohexyl and ferf-butyl, while we acknowledge the experimental uncertainties of 8.4 and 16.7 kJmol, respectively, for the enthalpies of formation of the secondary and tertiary alcohols. We accept this mean value as representative of the reaction. For R = 1- and 2-heptyl, the enthalpies of reaction are the disparate —83.5 and —86.0 kJmol, respectively. From the consensus enthalpy of reaction and the enthalpy of formation of 1-octanol, the enthalpy of formation of 1-heptyl hydroperoxide is calculated to be ca —322 kJ mol, nearly identical to that derived earlier from the linear regression equation. The similarly derived enthalpy of formation of 3-heptyl hydroperoxide is ca —328 kJmol. The enthalpy of reaction for R = i-Pr is only ca —91 kJmol, and also suggests that there might be some inaccuracy in its previously derived enthalpy of formation. Using the consensus enthalpy of reaction, a new estimate of the liquid enthalpy of formation of i-PrOOH is ca —230 kJmoU. ... 

Multiple linear regression (MLR), although a popular technique, does not meet the requirements of the experimental design describe above. MLR can only deal with one dependent variable at a time and assumes that all variables are orthogonal (uncorrelated), and tiiat they are all completely relevant to the experiment, en dealing with an experimental system for the first time, it is not always possible to predict which variables will be relevant to the experiment, and which will not. So a technique is needed that can reconcile such uncertainties. 

If a calibration function is used with coefficients obtained by fitting the response of an instrument to the model in known concentrations of calibration standards, then the uncertainty of this procedure must be taken into account. A classical least squares linear regression, the default regression... 

The linear calibration uncertainty is estimated by the interval that can be expected to encompass a large fraction of the distribution of values that could be reasonably attributed to the linear curve. This interval, indicated in Fig. 5, is due to the linear adjustement of the concentration values used to determine the regression line and obtained values of the absorbance. 

In addition to the uncertainty due to the linear regression which was 0.034 for 6.010 mg/1 copper, the overall uncertainty of the instrument calibration includes the uncertainty due to the photometic measurement and the uncertainty due to the CRMs. The overall calibration uncertainty was 0.036 for 6.010 mg/1 copper. 

Multi point calibration will be recommended if minimum uncertainty and maximum consistency are required over a wide range of pH(X) values [21, 22]. The calibration function of the electrode is then calculated by linear regression of the difference in cell voltage results from the standard pH values. This calibration procedure is also recommended for characterising the performance of electrode systems. 

If, by reason of difficulty or lack of time, problems of weighting are to be ignored, all factors Wj can be set equal to 1 in the equations that follow. In that case however, subsequent sections dealing with evaluation of uncertainties and goodness of fit lose much of their validity unless applied to a case in which all the observations just happen to be of equal weight. The assumption w,- = 1 is implicit in standard linear regression operations of spreadsheet programs. 

We consider here only the simplest case of simple linear regression, in which x is considered the accurately determinable independent variable and y the dependent variable subject to experimental uncertainty. The data are to be fitted to the straight line... 

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