Estimation criterion

The iterative parameter-estimation procedure can also be represented as in Fig. 29. An experimenter performs an experiment, obtains the data and estimates the parameters, for example, by nonlinear least squares. Then he checks the confidence region of the parameter estimates, perhaps through A as shown in Fig. 27. If it is too large, the experimenter finds those experimental settings that maximize the parameter estimation criterion of Eq. (144), that is, experimental settings that will reduce the size of confidence region as... 

While you will use the least squares method in most cases, do not forget that selecting an estimation criterion you make assumptions on the error structure, even without a real desire to be involved with this problem. Therefore, it is better to be explicit on this issue, for the sake of consistency in tlie further steps of the estimation. 

Maximum likelihood estimation Criterion under which the best estimate is the one which maximizes the likelihood of the observed event. Maximum likelihood estimation is the classical statistical criterion for estimating unknown parameter values from observed data (Sielken, Ch. 8). 

The estimated ultimate load of pile foundation is an important project and involves a number of challenges from the geotechnical and structural safety viewpoint. The strength reduction method of pile foundation and estimation criterion of ultimate load is studied based on nonlinear finite element and cusp catastrophe theory. The finite element limit analysis of pile is performed using the single reduction factor and two reduction factors of strength reduction method and the criterion of the cusp catastrophe curve method, and has been shown to be a reliable and objective method for estimating the ultimate load of pile. 

The molecular weight reduction, which is a first consequence of the mechanochemical act, is macroscopically reflected in the change of some mechanical properties, among them being the tensile strength [701]. Accordingly, the variation of with the parameters of ultrasonic treatment, especially with the time, can constitute an estimation criterion for the intensity of the mechanochemical processes which develop in the mechanically stressed material. 

The most reliable estimates of the parameters are obtained from multiple measurements, usually a series of vapor-liquid equilibrium data (T, P, x and y). Because the number of data points exceeds the number of parameters to be estimated, the equilibrium equations are not exactly satisfied for all experimental measurements. Exact agreement between the model and experiment is not achieved due to random and systematic errors in the data and due to inadequacies of the model. The optimum parameters should, therefore, be found by satisfaction of some selected statistical criterion, as discussed in Chapter 6. However, regardless of statistical sophistication, there is no substitute for reliable experimental data. 

To illustrate the criterion for parameter estimation, let 1, 2, and 3 represent the three components in a mixture. Components 1 and 2 are only partially miscible components 1 and 3, as well as components 2 and 3 are totally miscible. The two binary parameters for the 1-2 binary are determined from mutual-solubility data and remain fixed. Initial estimates of the four binary parameters for the two completely miscible binaries, 1-3 and 2-3, are determined from sets of binary vapor-liquid equilibrium (VLE) data. The final values of these parameters are then obtained by fitting both sets of binary vapor-liquid equilibrium data simultaneously with the limited ternary tie-line data. 

If this criterion is based on the maximum-likelihood principle, it leads to those parameter values that make the experimental observations appear most likely when taken as a whole. The likelihood function is defined as the joint probability of the observed values of the variables for any set of true values of the variables, model parameters, and error variances. The best estimates of the model parameters and of the true values of the measured variables are those which maximize this likelihood function with a normal distribution assumed for the experimental errors. 

Equations (7-8) and (7-9) are then used to calculate the compositions, which are normalized and used in the thermodynamic subroutines to find new equilibrium ratios,. These values are then used in the next Newton-Raphson iteration. The iterative process continues until the magnitude of the objective function 1g is less than a convergence criterion, e. If initial estimates of x, y, and a are not provided externally (for instance from previous calculations of the same separation under slightly different conditions), they are taken to be... 

Convergence of the iteration requires the norm of the objective vector 1g to be less than the convergence criterion, e. The initial estimates used, if not provided externally, are, in addition to Equation (7-28)... 

The type of development, type and number of development wells, recovery factor and production profile are all inter-linked. Their dependency may be estimated using the above approach, but lends itself to the techniques of reservoir simulation introduced in Section 8.4. There is never an obvious single development plan for a field, and the optimum plan also involves the cost of the surface facilities required. The decision as to which development plan is the best is usually based on the economic criterion of profitability. Figure 9.1 represents a series of calculations, aimed at determining the optimum development plan (the one with the highest net present value, as defined in Section 13). 

Various partitions, resulted from the different combinations of clustering parameters. The estimation of the number of classes and the selection of optimum clustering is based on separability criteria such as the one defined by the ratio of the minimum between clusters distance to the maximum of the average within-class distances. In that case the higher the criterion value the more separable the clustering. By plotting the criterion value vs. the number of classes and/or the algorithm parameters, the partitions which maximise the criterion value is identified and the number of classes is estimated. 

To estimate the quality of restoration having both, simulated and reconstructed images, using some kind of quality criterion For this last purpose the following mean square measure was applied ... 

The results of the estimation of the distortion criterion s are presented in fig. 4. It is seen that the distortion increases with the noise. But the main conclusion is that the Gibbs functional (12) provides a satisfactory reconstruction having quite large noise up to 0.15, what can not be provided by other known reconstruction techniques. 

In many cases, the methods used to solve identification problems are based on an iterative minimization of some performance criterion measuring the dissimilarity between the experimental and the synthetic data (generated by the current estimate of the direct model). In our case, direct quantitative comparison of two Bscan images at the pixels level is a very difficult task and involves the solution of a very difficult optimization problem, which can be also ill-behaved. Moreover, it would lead to a tremendous amount of computational burden. Segmented Bscan images may be used as concentrated representations of the useful... 

Note The segmentation operation yields a near-optimal estimate x that may be used as initialization point for an optimization algoritlim that has to find out the global minimum of the criterion /(.). Because of its nonlinear nature, we prefer to minimize it by using a stochastic optimization algorithm (a version of the Simulated Annealing algorithm [3]). 

The state of an adsorbate is often described as mobile or localized, usually in connection with adsorption models and analyses of adsorption entropies (see Section XVII-3C). A more direct criterion is, in analogy to that of the fluidity of a bulk phase, the degree of mobility as reflected by the surface diffusion coefficient. This may be estimated from the dielectric relaxation time Resing [115] gives values of the diffusion coefficient for adsorbed water ranging from near bulk liquids values (lO cm /sec) to as low as 10 cm /sec. 

In [13], an efficient residual error estimation scheme has been introduced for controlling the quality of the approximation. This gives us a stopping criterion for the iteration guaranteeing that the quality of the approximation fits to the accuracy requirements of the stepsize control. 

Having a numeric criterion for tolerable risk would be everyone s choice when making decisions using absolute risk estimates. Unfortunately, no universally accepted or mandated criterion exists. Nevertheless, when attempting to establish risk guidelines satisfying the requirements described, an organization has a number of resources avail-able. 3 Some particularly valuable sources of information include ... 

However, because of the diversity of short-term and long-term hazards to the public and workers in the CPI, no single criterion will ever meet everyone s needs. Even if such a criterion could be developed, there would still be controversy over its use. Absolute risk estimates are not... 

In the case of exothermic reactions, underestimating the transfer coefficients makes the real gradients less than the estimated ones. As such, this makes our estimates conservative, in the sense that if a criterion calls gradients negligible then they surely are. The intent here is to do most of the kinetic study and catalyst testing at gradientless conditions and this book will make use of the Colburn-type correlations as developed by Hougen (1951) and his associates. 

Finally, there is an interesting article" that shows how to use Taylor s series to generate shortcut methods from established theory. Examples are given for developing a criterion for replacing log mean temperature differences with average differences and for estimating the effect of temperature on reaction rate. 

The decreasing value of K with decreasing flaw size for fracture of IG-11 graphite suggests that the near crack tip failure criterion would be better expressed by combining the near crack tip stress intensification estimated by K with the far field applied stress, Sfraa(Eq. 28). Using simple superposition of the stress perpendicular... 

As piping drawings are developed, re-evaluate those lines where estimated pressure drop was a criterion in size selection, taking into account the actual piping configuration and effects of control and piping components. 

The elution of such gels is an example not of size exclusion but rather of hydrodynamic fractionation (HDF). However, it must be remembered that merely being able to physically fit an insoluble material through the column interstices is not the only criterion for whether the GPC/HDF analysis of an insoluble material will be successful. A well-designed HDF packing and eluant combination will often elute up to the estimated radius in Eq. (5), but adsorption can drastically limit this upper analysis radius. For example, work in our laboratory using an 8-mm-bead-diameter Polymer Laboratories aqueous GPC column for HDF found that that column could not elute 204 nM pSty particles, even though Eq. (5) estimates a critical radius of —1.5 jam. 

In these circumstances a decision must be made which of two (or more) kinet-ically equivalent rate terms should be included in the rate equation and the kinetic scheme (It will seldom be justified to include both terms, certainly not on kinetic grounds.) A useful procedure is to evaluate the rate constant using both of the kinetically equivalent forms. Now if one of these constants (for a second-order reaction) is greater than about 10 ° M s-, the corresponding rate term can be rejected. This criterion is based on the theoretical estimate of a diffusion-controlled reaction rate (this is described in Chapter 4). It is not physically reasonable that a chemical rate constant can be larger than the diffusion rate limit. 

A weaker but more widely applicable criterion is that the rate constant estimate should be consistent with the body of experimental work on closely related reactions. A third factor is that of style, which is essentially equivalent to the contemporary state of mechanistic chemistry it may seem more reasonable to write a mechanism for one of the forms than for the alternative. Styles change, however. 

The final estimation of the value of ay may appear tedious and several assumptions are made in its derivation, but experimental evidence suggests that it may be used with reasonable accuracy to assess the levels of potentially damaging cavitation erosion. In small valves with nominal bores up to 65 mm cavitation inception occurs in intermittent bursts when the value oy is approximately unity. The cavitation becomes continuous and audible as Oy is reduced to about 0.6, but the risk of damage does not become significant until the value falls below 0.4. As a design criterion the condition of light, steady noise has been described by Tullis as the critical level and is sug-... 

---