Uncertainties in the Parameters

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 

A propagation-of-error treatment of Eq. (13) yields for the estimated standard deviation in parameter a, the expression 

For the trivial one-parameter case in which the parameter to be determined is the arithmetic mean, 

Eor the two-parameter case of the linear relationship we may apply Eqs. (28) and (40) to the estimation of the standard deviations in the intercept uq and the slope of the corresponding straight line. Erom Eq. (22) we have 

As in the case of Eqs. (20) to (22), when unit weights are employed, the Wyare deleted and 2 Wj is replaced by m. 

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The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

The slope of the expectation curve indicates the range of uncertainty In the parameter presented a broad expectation curve represents a large range of uncertainty, and a steep expectation curve represents a field with little uncertainty (typical of fields which have much appraisal data, or production history). 

In frequentist statistics, probability is instead a long-run relative occurrence of some event out of an infinite number of repeated trials, where the event is a possible outcome of the trial. A hypothesis or parameter that expresses a state of nature cannot have a probability in frequentist statistics, because after an infinite number of experiments there can be no uncertainty in the parameter left. A hypothesis or parameter value is either... 

For a time-invariant system, the expected standard deviation of the innovation consists of two parts the measurement variance (r(/)), and the variance due to the uncertainty in the parameters (P(y)), given by [4] ... 

If we have very little information about the parameters, direct search methods, like the LJ optimization technique presented in Chapter 5, present an excellent way to generate very good initial estimates for the Gauss-Newton method. Actually, for algebraic equation models, direct search methods can be used to determine the optimum parameter estimates quite efficiently. However, if estimates of the uncertainty in the parameters are required, use of the Gauss-Newton method is strongly recommended, even if it is only for a couple of iterations. 

A simple procedure to overcome the problem of the small region of convergence is to use a two-step procedure whereby direct search optimization is used to initially to bring the parameters in the vicinity of the optimum, followed by the Gauss-Newton method to obtain the best parameter values and estimates of the uncertainty in the parameters (Kalogerakis and Luus, 1982). 

Having determined the uncertainty in the parameter estimates, we can proceed and obtain confidence intervals for the expected mean response. Let us first consider models described by a set of nonlinear algebraic equations, y=f(x,k). The 100(1 -a)% confidence interval of the expected mean response of the variable y at x0 is given by... 

To illustrate the usefulness of the Information Index in determining the best time interval, let us consider the grid point (l, 0.20, 35). From Figure 12.7 we deduce that the best time interval is 25 to 75 h. In Table 12.4 the standard deviation of each parameter is shown for 7 different time intervals. From cases 1 to 4 it is seen that that measurements taken before 25 h do not contribute significantly in the reduction of the uncertainty in the parameter estimates. From case 4 to 7 it is seen that it is preferable to obtain data points within [25, 75] rather than after the steady state has been reached and the Information Indices have leveled off. Measurements taken after 75 h provide information only about the steady state behavior of the system. 

The uncertainty in the estimate of Pi is smaller (s, = s x5.04), and the uncertainty in the estimate of Pn is smaller still (Sj, = s x0.04). The geometric interpretation of the parameters Pi and Ph in this model is not straightforward, but Pi essentially moves the apex of the parabola away from Xi = 0, and pn is a measure of the steepness of curvature. The geometric interpretation of the associated uncertainties in the parameter estimates is also not straightforward (for example, P, Pi, and p,i are expressed in different units). We will simply note that such uncertainties do exist, and note also that there is covariance between bg and hi, between and b,i, and between hi and hn. 

Uncertainty analysis Determination of the sources of uncertainty in the measurement or prediction of environmental parameters. The analysis can be both quantitative (computation of variances) or qualitative (lists of uncertain methods and procedures). The total uncertainty in the parameters of interest is typically a function of all of the individual sources of uncertainty. 

Limitations of the Junge-Pankow model include uncertainties in the parameters c and 0. Pankow (1987) suggested that optimal values of c might be chosen for different classes of... 

If a parametric distribution (e.g. normal, lognormal, loglogistic) is fit to empirical data, then additional uncertainty can be introduced in the parameters of the fitted distribution. If the selected parametric distribution model is an appropriate representation of the data, then the uncertainty in the parameters of the fitted distribution will be based mainly, if not solely, on random sampling error associated primarily with the sample size and variance of the empirical data. Each parameter of the fitted distribution will have its own sampling distribution. Furthermore, any other statistical parameter of the fitted distribution, such as a particular percentile, will also have a sampling distribution. However, if the selected model is an inappropriate choice for representing the data set, then substantial biases in estimates of some statistics of the distribution, such as upper percentiles, must be considered. 

Let us first note that the quality of a fit of correct models (comprehensive models can be assumed to be correct) to experiments generally increases when the number of parameters increases. But this is compensated for by increasing uncertainties in the parameter estimates. In other words, for a given amount of experimental results, only a limited number of parameters (or combinations of parameters) can be estimated with reasonable accuracy. A tentative rule of thumb could be stated for a given type of experiment, the number of rate coefficients that can be estimated from experimental results is nearly equal to the number of independent stoichiometries. (This rule is clearly true for molecular reaction schemes.) In general, except for very simple experiments where elementary processes have been quasi-isolated, the number of kinetic parameters far exceeds the amount of experimental information. Thus, only a few model parameters can be estimated. 

The obtained age for the deep water in the artesian Milo Holdings 3 well is semiquantitative because of uncertainties in the parameters applied in the calculation. Yet the order of magnitude—millions of years—is of utmost importance the water resource is not renewed, but it is entirely shielded from the surface and, hence, is immune to anthropogenic pollution. 

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. 

Evaluate the statistical uncertainties in the adjustable parameters obtained from the best fit (see Uncertainties in the Parameters). 

The model or set of models to be used in the exposure assessment to relate the presence of a substance to human exposure/absorbed dose should be stated. The model s general description should provide enough detail so that the user or reviewer understands the input variables, underlying mathematical algorithms and data transformations and output/results, such that the model can be easily compared to other alternatives. The basis for each model, whether deterministic, empirical or statistical, should be described. The statement of the model should include which variables are measured and which are assumed. A description should be provided of how uncertainties in the parameters and the model itself are to be evaluated and treated. 

Sensitivity analysis is a method used to evaluate the impact of a single variable or a group of variables on the results from a model calculation. Sensitivity analysis may be used to determine which parameters in a calculation have the greatest influence on the results such that greater emphasis is placed on characterizing these parameters. Moreover, the results from these analyses may be used to identify ways to improve the overall predictive capability of the model by reducing the uncertainty in the parameters that have the greatest influence on the outcome. Sensitivity analysis may be applied to the risk assessment process in order to identify those variables that dominate risk estimates as well as those that are relatively unimportant. 

As shown with both global and local SA, the probability of a successful trial was less than 80% across a large range of uncertainty in the parameters (Figures 35.2 and 35.3 and Table 35.3). Given prespecified criteria of >80% power, the... 

Several publications have reported on studies of the water balance of the Aral Sea. Water balance estimation for the Aral Sea is rather difficult because of several sources of uncertainties in the parameters entering the water balance equation. 

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