Number of parameters

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. 

If there is sufficient flexibility in the choice of model and if the number of parameters is large, it is possible to fit data to within the experimental uncertainties of the measurements. If such a fit is not obtained, there is either a shortcoming of the model, greater random measurement errors than expected, or some systematic error in the measurements. 

In many process-design calculations it is not necessary to fit the data to within the experimental uncertainty. Here, economics dictates that a minimum number of adjustable parameters be fitted to scarce data with the best accuracy possible. This compromise between "goodness of fit" and number of parameters requires some method of discriminating between models. One way is to compare the uncertainties in the calculated parameters. An alternative method consists of examination of the residuals for trends and excessive errors when plotted versus other system variables (Draper and Smith, 1966). A more useful quantity for comparison is obtained from the sum of the weighted squared residuals given by Equation (1). 

The type of rig which will be selected depends upon a number of parameters, in particular ... 

A site survey will be carried out, from which a number of parameters can be established, e.g. carrying capacity of the soil at the planned location, possible access routes, surface restrictions like built-up areas, lakes, nature reserves, the general topography, possible water supplies. The survey will allow the adequate preparation of the future location. For instance, onshore in a swamp area the soil needs to be covered with support mats. 

Homogeneity of data. Homogeneous data will be uniform in structure and composition, usually possible to describe with a fixed number of parameters. Homogeneous data is encountered in simple NDT inspection, e.g. quality control in production. Inhomogeneous data will contain various combinations of indications from construction elements, defects and noise sources. An example of inhomogenous data are ultrasonic B-scan images as described in [Hopgood, 1993] or as encountered in the ultrasonic rail-inspection system described later in this paper. 

The camera model has a high number of parameters with a high correlation between several parameters. Therefore, the calibration problem is a difficult nonlinear optimization problem with the well known problems of instable behaviour and local minima. In out work, an approach to separate the calibration of the distortion parameters and the calibration of the projection parameters is used to solve this problem. 

While simulations reach into larger time spans, the inaccuracies of force fields become more apparent on the one hand properties based on free energies, which were never used for parametrization, are computed more accurately and discrepancies show up on the other hand longer simulations, particularly of proteins, show more subtle discrepancies that only appear after nanoseconds. Thus force fields are under constant revision as far as their parameters are concerned, and this process will continue. Unfortunately the form of the potentials is hardly considered and the refinement leads to an increasing number of distinct atom types with a proliferating number of parameters and a severe detoriation of transferability. The increased use of quantum mechanics to derive potentials will not really improve this situation ab initio quantum mechanics is not reliable enough on the level of kT, and on-the-fly use of quantum methods to derive forces, as in the Car-Parrinello method, is not likely to be applicable to very large systems in the foreseeable future. 

Traditionally, least-squares methods have been used to refine protein crystal structures. In this method, a set of simultaneous equations is set up whose solutions correspond to a minimum of the R factor with respect to each of the atomic coordinates. Least-squares refinement requires an N x N matrix to be inverted, where N is the number of parameters. It is usually necessary to examine an evolving model visually every few cycles of the refinement to check that the structure looks reasonable. During visual examination it may be necessary to alter a model to give a better fit to the electron density and prevent the refinement falling into an incorrect local minimum. X-ray refinement is time consuming, requires substantial human involvement and is a skill which usually takes several years to acquire. 

Sensitivity From equations 10.32 and 10.33 we can see that the sensitivity of a fluorescent or phosphorescent method is influenced by a number of parameters. The importance of quantum yield and the effect of temperature and solution composition on f and p already have been considered. Besides quantum yield, the sensitivity of an analysis can be improved by using an excitation source that has a greater... 

Rate of Mass Transfer in Bubble Plates. The Murphree vapor efficiency, much like the height of a transfer unit in packed absorbers, characterizes the rate of mass transfer in the equipment. The value of the efficiency depends on a large number of parameters not normally known, and its prediction is therefore difficult and involved. Correlations have led to widely used empirical relationships, which can be used for rough estimates (109,110). The most fundamental approach for tray efficiency estimation, however, summarizing intensive research on this topic, may be found in reference 111. 

The specific enzyme to be used in an EIA is deterrnined according to a number of parameters including enzyme activity and stabiUty (before, during, and after conjugation), cost and availabiUty of the enzyme substrate, and the desired end point of the EIA, such as color. Most EIAs utilize a colored end point which can be readily deterrnined both visually and spectrophotometricaHy. Table 1 Hsts a number of enzymes which have been used in immunoassays and their substrates. 

Radioactive isotopes are characterized by a number of parameters in addition to those attributable to chemistry. These are radioactive half-life, mode of decay, and type and quantity of radioactive emissions. The half-life, defined as the time required for one-half of a given quantity of radioactivity to decay, can range from milliseconds to biUions of years. Except for the most extreme conditions under very unusual circumstances, half-life is independent of temperature, pressure, and chemical environment. 

The choice of a particular mining method depends on a number of parameters, typically the physical properties of the host matrix, the fiber content of the ore, the amount of sterile materials, the presence of contaminants, and the extent of potential fiber degradation during the various mining operations (33). However, most of the asbestos mining operations are of the open pit type, using bench drilling techniques. 

Molecular Connectivity Indexes and Graph Theory. Perhaps the chief obstacle to developing a general theory for quantification of physical properties is not so much in the understanding of the underlying physical laws, but rather the inabiUty to solve the requisite equations. The plethora of assumptions and simplifications in the statistical mechanics and group contribution sections of this article provide examples of this. Computational procedures are simplified when the number of parameters used to describe the saUent features of a problem is reduced. Because many properties of molecules correlate well with stmctures, parameters have been developed which grossly quantify molecular stmctural characteristics. These parameters, or coimectivity indexes, are usually based on the numbers and orientations of atoms and bonds in the molecule. 

What furnace engineers most need is a closed-form solution of the problem, theoretically sound in structure and therefore containing a minimum number of parameters and no empirical constants and, preferably, physically visuaHzable. They can then (1) correlate data on existing furnaces, (2) develop a performance equation for standard design, or (3) estimate performance of a new furnace type on which no data are available. 

Design of explosion suppression systems is clearly complex, since the effectiveness of an explosion suppression system is dependent on a large number of parameters. One Hypothesis of suppression system design identifies a limiting combustion wave adiabatic flame temperature, below which combustion reactions are not sustained. Suppression is thus attained, provided that sufficient thermal quenching results in depression of the combustion wave temperature below this critical value. This hypothesis identifies the need to deliver greater than a critical mass of suppressant into the enveloping fireball to effect suppression (see Fig. 26-43). 

A large number of parameters are involved in the choice of the corrosion protection system and the provision of the proteetion eurrent these are deseribed elsewhere (see Chapters 6 and 17). In partieular, for new locations of fixed production platforms, a knowledge of, for example, water temperature, oxygen content, conductivity, flow rate, chemical composition, biological activity, and abrasion by sand is useful. Measurements must be carried out at the sea location over a long period, so that an increased margin of safety can be calculated. 

Optimization of external parameters tends to be more difficult as the quantity of the target data is decreased relative to the number of parameters to be optimized compared to the internal parameters, leaving the solution more undetermined. This increases the... 

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