Uncertainty least squares parameters

The value of the second free variable will be refined freely and converges at the mean Cl—0-distance. The advantage of the second way of restraining the CIO4 ion to be tetrahedral is that the average Cl—O distance will be calculated with a standard uncertainty (in addition to the individual Cl—O distances with their standard uncertainties). The disadvantage is that an additional least squares parameter has to be rehned. 

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... 

It can be argued that the main advantage of least-squares analysis is not that it provides the best fit to the data, but rather that it provides estimates of the uncertainties of the parameters. Here we sketch the basis of the method by which variances of the parameters are obtained. This is an abbreviated treatment following Bennett and Franklin.We use the normal equations (2-73) as an example. Equation (2-73a) is solved for <2o-... 

The expression x (J)P(j - l)x(j) in eq. (41.4) represents the variance of the predictions, y(j), at the value x(j) of the independent variable, given the uncertainty in the regression parameters P(/). This expression is equivalent to eq. (10.9) for ordinary least squares regression. The term r(j) is the variance of the experimental error in the response y(J). How to select the value of r(j) and its influence on the final result are discussed later. The expression between parentheses is a scalar. Therefore, the recursive least squares method does not require the inversion of a matrix. When inspecting eqs. (41.3) and (41.4), we can see that the variance-covariance matrix only depends on the design of the experiments given by x and on the variance of the experimental error given by r, which is in accordance with the ordinary least-squares procedure. 

The maximum search function is designed to locate intensity maxima within a limited area of x apace. Such information is important in order to ensure that the specimen is correctly aligned. The user must supply an initial estimate of the peak location and the boundary of the region of interest. Points surrounding this estimate are sampled in a systematic pattern to form a new estimate of the peak position. Several iterations are performed until the statistical uncertainties in the peak location parameters, as determined by a linearized least squares fit to the intensity data, are within bounds that are consistent with their estimated errors. 

In Hitchell s work unequal variance of the response data was compensated for by weighting the data by the variance at each level. The regression parameters and the confidence band around the regression line were estimated by least squares ( ) The overall level of uncertainty, OL, was divided between the variation in response values and the variance in the regression estimation. His overall a was 0.05. The prediction interval was estimated around a single response determination. 

Figure 6.4 shows the relationship between (the least squares estimate of Pq) nd zero for the data in Figure 6.1. In this example, the parameter po has been estimated on the basis of only two experimental results if another independent set of two experiments were carried out on the same system, we would very likely obtain different values for and y,4 and thus obtain a different estimate for Pq from the second set of experiments. It is for this reason that the estimation of Po is usually subject to uncertainty.

In case, the network is overdetermined a least square approach is possible. The information obtained by MS from tracer studies can be combined with metabolite balancing for the estimation of flux parameters [25, 27]. Measurement errors can be included to predict uncertainties for the obtained flux parameters [23]. 

Cecchi, G.C., 1991. Error analysis of the parameters of a least-squares determined curve when both variables have uncertainties. Meas. Sci. Technol., 2 1127-1128. Chong, D.P., 1991. Comment on linear leat-squares fits with errors in both variables. Am. J. Phys., 59 472. 

With small molecules, it is usually possible to obtain anisotropic temperature factors during refinement, giving a picture of the preferred directions of vibration for each atom. But a description of anisotropic vibration requires six parameters per atom, vastly increasing the computational task. In many cases, the total number of parameters sought, including three atomic coordinates, one occupancy, and six thermal parameters per atom, approaches or exceeds the number of measured reflections. As mentioned earlier, for refinement to succeed, observations (measured reflections and constraints such as bond lengths) must outnumber the desired parameters, so that least-squares solutions are adequately overdetermined. For this reason, anisotropic temperature factors for proteins have not usually been obtained. The increased resolution possible with synchrotron sources and cryocrystallography will make their determination more common. With this development, it will become possible to obtain better estimates of uncertainties in atom positions than those provided by the Luzzati method. 

In order to estimate the uncertainty in the value obtained for the rate parameter K, one should also consider random errors. There are two sources of random error in this case the least-squares fitting which gives an error (dK)ls due to random noise in the experimental spectrum [equation (164)] and the measurement of w0 the line-width of the standard employed. The total random error in K is approximated by ... 

According to the rules of combined uncertainty evaluation [8, 9], ox can be considered as negligible, if it leads to increase of SXo for less then one-third of its initial value (calculated by ordinary least squares technique). An example of the ox influence on the calibration parameters bh b0 and SXo, and corresponding lifetime of the traceability chain are analysed below. 

Normally, the uncertainties in the concentrations of the calibration solutions (variable x) are small in relation to the uncertainties of the response of the measurement system (variable y), so that the regression parameter can be estimated using ordinary least squares (OLS). In exceptional cases, the test quantity SeJSx (Sex2 means the variance in the concentration of the calibration solutions for a particular calibration level, and Sx2 indicates the total variance in concentration) can be calculated, and if Sex/Sx>0.2 the regression parameters should be estimated by orthogonal distance regression (ODR) [10],... 

Where the number of points is sufficiently large, the limits of error of the position of plotted points can be inferred from their scatter. Thus an upper bound and a lower bound can be drawn, and the lines of lintiting slope drawn so as to lie within these bounds. Since the theory of least squares can be applied not only to yield the equation for the best straight line but also to estimate the uncertainties in the parameters entering into the equation (see Chapter XXI), such graphical methods are justifiable only for rough estimates. In either case, the possibility of systematic error should be kept in ntind. 

It should be stressed that, in any experiment for which there is to be a least-squares refinement of parameters, intended to yield not only the best possible parameter values but also respectable estimates of their uncertainties and a test of the validity of the model, it is vital to take the trouble to analyze the methods and the circumstances of the experiment carefully in order to get the best possible values of the a priori weights. 

We now assume that the least-squares refinement has converged satisfactorily, that any necessary rejection of discordant data has taken place before the final cycles were carried out, and that statistical tests on the weighted residuals have given reassuring results. It is now appropriate to estimate the uncertainties in the determined values of the adjustable parameters a,.f... 

M = Estimated uncertainties, in parentheses, represent one standard deviation in the least squares fit to the experimental data. Table IV summarizes several parameters of interest for... 

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