Standard deviation least-squares parameters

Fuoss and Accascina 36) present graphical methods for treating conductance data according to either equation. Kay 8 43> describes a computer program for least squares analysis in which standard deviations for the parameters are calculated as well. A similar program is described in Ref. H>. 

By application of least-squares methods, most probable values and standard deviations for the parameters Em and a of this distribution have been calculated, and from the former a value for the temperature-independent rate constant 0 has been derived ... 

The standard uncertainties (or standard deviations) for each parameter determined according to the least squares method are calculated from... 

Table 2.6-1 Comparison of the various least-squares methods available in Excel. The first three methods come with Excel, as does Solver. WLS is a weighted least-squares macro provided with this book, as is the macro SolverAid which yields the standard deviations for the parameter values provided by Solver.1...

How reliable are these least squares parameters To answer this question, we first need to find the standard deviation about the least squares best-fit line by summation over N calibration points of the square of the difference between experimental and calculated detector responses according to... 

The structural parameters (r, u, sometimes k, and possibly other parameters) are then obtained by least-squares fitting, usually on the intensity curves, but occasionally on the RD curves (c/. p. 45). If the intensity data are used, the data from all the nozzle-to-plate distances are not necessarily combined into one composite curve an individual scale factor may be refined for each set of data. Since the observed data are considerably correlated, i.e. each point cannot be regarded as an independent observation, a number of problems are encountered in the application of the least-squares method. This problem is most important for the estimation of the standard deviations, since the parameters may be obtained fairly satis-factorally by conventional least-squares refinement. (A more detailed discussion is set out on p. 45.)... 

Standard Deviations from Least-squares Refinements. The standard deviations for the parameters are obtained by taking the square root of the diagonal elemenents of the moment matrix for the parametors ... 

When the final least-squares parameters Xi with standard deviations af) have been obtained with some fixed parameters, additional refinements are carried out changing one of the assumed parameters [X at a time by a small amount. In this way, dXildXl may be obtained. A standard deviation for the assumed parameter (crj) is estimated, and the standard deviation for Xi calculated from... 

When experimental data is to be fit with a mathematical model, it is necessary to allow for the facd that the data has errors. The engineer is interested in finding the parameters in the model as well as the uncertainty in their determination. In the simplest case, the model is a hn-ear equation with only two parameters, and they are found by a least-squares minimization of the errors in fitting the data. Multiple regression is just hnear least squares applied with more terms. Nonlinear regression allows the parameters of the model to enter in a nonlinear fashion. The following description of maximum likehhood apphes to both linear and nonlinear least squares (Ref. 231). If each measurement point Uj has a measurement error Ayi that is independently random and distributed with a normal distribution about the true model y x) with standard deviation <7, then the probability of a data set is... 

The parameter values found by the two methods differ slightly owing to the different criteria used which were the least squares method for ESL and the maximum-likelihood method for SIMUSOLV and because the T=10 data point was included with the ESL run. The output curve is very similar and the parameters agree within the expected standard deviation. The quality of parameter estimation can also be judged from a contour plot as given in Fig. 2.41. 

Copp and Everet (1953) have presented 33 experimental VLE data points at three temperatures. The diethylamine-water system demonstrates the problem that may arise when using the simplified constrained least squares estimation due to inadequate number of data. In such case there is a need to interpolate the data points and to perform the minimization subject to constraint of Equation 14.28 instead of Equation 14.26 (Englezos and Kalogerakis, 1993). First, unconstrained LS estimation was performed by using the objective function defined by Equation 14.23. The parameter values together with their standard deviations that were obtained are shown in Table 14.5. The covariances are also given in the table. The other parameter values are zero. 

White noise signifies that the experimental standard deviation, ay, is normally distributed and the same for all individual measurements, yij. The traditional least-squares fit delivers the most likely parameters only under the condition of white noise. 

Expression (4.31) is widely applied to calculate the error in properties derived from the least-squares variables. We will use it in chapter 7 for the calculation of the standard deviations of the electrostatic moments derived from the parameters of the multipole formalism. 

The response of many instruments is linear as a function of the measured variable, if variations due to experimental conditions or the instrument are taken into account. The objective is to determine the parameters of the linear equation that best represents the observations. The primary hypothesis in using the method of least squares is that one of the two variables should be without error while the second one is subject to random errors. This is the most frequently applied method. The coefficients a and b of the linear equation y = ax + b, as well as the standard deviation on a and on the estimation of y have been obtained in the past using a variety of similar equations. The choice of which formula to use depended on whether calculations were carried out manually, with calculator or using a spreadsheet. However, appropriate computer software is now widely used. 

Estimated standard deviations in the last place(s) are given in parentheses following the respective parameter values. These are adapted from the values given by the program orfls by taking into account the fact that the number of observations— powder-line intensities—is smaller than the number of reflections used explicitly in the least-squares treatment. 

Substitution parameters A, calculated from a least-squares regression analysis, are collected in Table 4.17. By applying eqs. (4.3)-(4.6), the 13C chemical shifts of halomethanes can be predicted with a standard deviation of about 4 ppm. 

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