Straight parameter estimations

Drawing straight lines through data points is a slightly arbitrary procedure. The slope of the straight line does not depend very much on this arbitrariness but the value of the intercept usually depends very much on it. Consequently, the value of the kinetic parameter related to the intercept will be estimated with the accuracy of the eyes capability of finding the best fit between experimental points and those lying on the line drawn. An objective method of parameter estimation consist in evaluation of the minimum of the function ... 

The data and the least squares straight line relationship are shown in Figure 11.2. It is to be remembered that the parameter estimates are those for the coded factor levels (see Section 11.2) and refer to the model... 

In general, nonlinear fitting of the data is essential for parameter estimation. Linear transformations, however, are useful for visualization of trends in data. Variances from a straight edge are more discernible to the human eye than are differences from curvilinear shapes. Therefore, linear transforms can be a useful diagnostic tool. 

The equation in this form is a useful tool to estimate parameters of reaction kinetics. Instead of performing nonlinear parameter estimation procedures, the functional dependence of XSf on X/(1 - X) can be plotted. The plot should look like a straight line whose slope is (-K m) and whose intersection with the XSf axis is given by the point of coordinate (Vmaxr). Hence, it furnishes a rapid graphic procedure to obtain rough estimates of kinetic parameters. 

The diagonal of the variance-covariance matrix consists of the variances for the parameter estimates and the off-diagonal of those of the related covariances. In the case of the straight-line model with the parameters Lq and b, the corresponding matrix... 

For the confidence intervals of the predicted y values, the approaches given for the straight-line model hold (Eqs. (6.25) - (6.27)). Instead of the parameter vector b, the matrix of the parameter estimations B is now used. For prediction of a single y value according to Eq. (6.25), we get... 

The problems of parameter estimation and model discrimination by statistical analysis are most easily understood in relation to a specific example. Let us consider a bisubstrale reaction that obeys the rate Eq. (18.50). The initial rate data from Table 1 are presented graphically in Figs. 3 and 4. The reciprocal plot of l/Uo versus 1/A is a family of straight lines which, in this case, may look either parallel or intersecting due to the fact that the crossover point is distant from the vertical axis, because If is much smaller than K, it would be... 

Finding the best fit involves calculation of appropriate values for the coefficients Aq and Aj in such a way that the net error between observed values of y and value of y predicted by straight line equation is the least. The least square parameter estimation method estimates... 

The situation is even more complex in the case of reaction related parameters. Independent of the appropriate reactor model, for complex reactions not only the reaction scheme but also the adequate type of rate equation for each reaction step has to be chosen before the parameters can be estimated. As a rule, in reaction engineering only the analytically measureable reactants (and reactions) should form the basis of the reaction network in contrast to physico-chemical research where the true" reaction mechanism (involving radical intermediates or active complexes) is sought. Certainly, the stoichiometry of the experimental product spectrum is important, but also the concentration/reactiontime dependencies like those given in Figure 6 are helpful. In contrast to parameter estimation and model discrimination, there exists no unique and straight forward analytic procedure for the built-up of even a simplified reaction scheme. The intuition of the chemical reaction engineer is therefore heavily relied upon. 

If estimated of distribution parameters are desired from data plotted on a hazard paper, then the straight line drawn through the data should be based primarily on a fit to the data points near the center of the distribution the sample is from and not be influenced overly by data points in the tails of the distribution. This is suggested because the smallest and largest times to failure in a sample tend to vary considerably from the true cumulative hazard function, and the middle times tend to lie close to it. Similar comments apply to the probability plotting. 

Because only two parameters need to be estimated, the equation of the straight line is far easier to calculate than that of most curves. 

Thus, a plot of In k versus the reciprocal temperature should yield a straight line with slope -E/Rg and In ko. These two kinetic parameters are strongly interconnected even a minor change in slope evaluation will result in a major change of the intercept. Theoretically, values of rate constants at two temperatures are sufficient to estimate the activation energy ... 

After each new observation, the estimates of the model parameters are updated (= new estimate of the parameters). In all equations below we treat the general case of a measurement model with p parameters. For the straight line model p = 2. An estimate of the parameters b based ony - 1 measurements is indicated by b(/ - 1). Let us assume that the parameters are recursively estimated and that an estimate h(j - 1) of the model parameters is available from y - 1 measurements. The next measurement y(j) is then performed at x(j), followed by the updating of the model parameters to b(/). 

By way of illustration, the regression parameters of a straight line with slope = 1 and intercept = 0 are recursively estimated. The results are presented in Table 41.1. For each step of the estimation cycle, we included the values of the innovation, variance-covariance matrix, gain vector and estimated parameters. The variance of the experimental error of all observations y is 25 10 absorbance units, which corresponds to r = 25 10 au for all j. The recursive estimation is started with a high value (10 ) on the diagonal elements of P and a low value (1) on its off-diagonal elements. 

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