Parameter estimation straight line

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 ... 

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 ... 

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. 

The denominator n 2 is used here because two parameters are necessary for a fitted straight line, and this makes s2 an unbiased estimator for a2. The estimated residual variance is necessary for constructing confidence intervals and tests. Here the above model assumptions are required, and confidence intervals for intercept, b0, and slope, b, can be derived as follows ... 

Let us first consider the effect of these experimental designs on the uncertainty of estimating the parameter P,. This parameter represents the slope of a straight line relationship between y, and x,. 

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... 

It is often assumed in regression calculations that the experimental error only affects the y value and is independent for the concentration, which is typically placed on the x axis. Should this not be the case, the data points used to estimate the best parameters for a straight line do not have the same quality. In such cases, a coefficient Wj is applied to each data point and a weighted regression is used. A variety of formulae have been proposed for this method. 

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