Parameter accuracy

What is the accuracy parameter in the calculation of the tunneling electron matrix element ... 

Fig. 3 Comparison of precision and accuracy parameters of an anal3ftical method.

To perform a simulation, we solve Eq. (3.44) repeatedly, updating the value of the potential, 0, at each subsequent timestep. In order to do this, five parameters must be specified the initial potential, vertex potential, Oy, scan rate, a, and the space and time increments, AX and AT respectively. We may describe the first three as being experimental parameters, as changing them corresponds to a change in the corresponding experimental system. The last two parameters may be described as accuracy parameters. 

Quantitative estimation of /q (and of the following ones) is performed using the Romberg algorithm in Mathcad software with accuracy parameter Tol=10. Iq values are listed in Table 8.1. 

The accuracy of our calculations is strongly dependent on the accuracy of the experimental data used to obtain the necessary parameters. While we cannot make any general quantitative statement about the accuracy of our calculations for multicomponent vapor-liquid equilibria, our experience leads us to believe that the calculated results for ternary or quarternary mixtures have an accuracy only slightly less than that of the binary data upon which the calculations are based. For multicomponent liquid-liquid equilibria, the accuracy of prediction is dependent not only upon the accuracy of the binary data, but also on the method used to obtain binary parameters. While there are always exceptions, in typical cases the technique used for binary-data reduction is of some, but not major, importance for vapor-liquid equilibria. However, for liquid-liquid equilibria, the method of data reduction plays a crucial role, as discussed in Chapters 4 and 6. 

Since the accuracy of experimental data is frequently not high, and since experimental data are hardly ever plentiful, it is important to reduce the available data with care using a suitable statistical method and using a model for the excess Gibbs energy which contains only a minimum of binary parameters. Rarely are experimental data of sufficient quality and quantity to justify more than three binary parameters and, all too often, the data justify no more than two such parameters. When data sources (5) or (6) or (7) are used alone, it is not possible to use a three- (or more)-parameter model without making additional arbitrary assumptions. For typical engineering calculations, therefore, it is desirable to use a two-parameter model such as UNIQUAC. 

Figure 4-16. Representation of ternary liquid-liquid equilibria using the UNIQUAC equation is improved by incorporating ternary tie-line data into binary-parameter estimation. Representation of binary VLB shows small loss of accuracy. ---- Binary...

While many methods for parameter estimation have been proposed, experience has shown some to be more effective than others. Since most phenomenological models are nonlinear in their adjustable parameters, the best estimates of these parameters can be obtained from a formalized method which properly treats the statistical behavior of the errors associated with all experimental observations. For reliable process-design calculations, we require not only estimates of the parameters but also a measure of the errors in the parameters and an indication of the accuracy of the data. 

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). 

Appendix C-6 gives parameters for all the condensable binary systems we have here investigated literature references are also given for experimental data. Parameters given are for each set of data analyzed they often reflect in temperature (or pressure) range, number of data points, and experimental accuracy. Best calculated results are usually obtained when the parameters are obtained from experimental data at conditions of temperature, pressure, and composition close to those where the calculations are performed. However, sometimes, if the experimental data at these conditions are of low quality, better calculated results may be obtained with parameters obtained from good experimental data measured at other conditions. 

The model is predictive and uses a method of contributing groups to determine the parameters of interaction with water. It is generally used by simulation programs such as HYSIM or PR02. Nevertheless the accuracy of the model is limited and the average error is about 40%. Use the results with caution. 

As will be shown in the next section, the methods discussed so far do not take account of the uncertainties and lateral variations in reservoir parameters. Hence the accuracy of the results is not adequate for decision making. The next section introduces a more comprehensive approach to volumetric estimation. 

Once the distortion parameters are calibrated, undistorted images are used to estimate the projection parameters. The. sample under investigation is recorded together with a calibration body. Although a 3D calibration body in general provides higher accuracy, a planar calibration body is used, because it is much easier to handle. To overcome... 

The data from Table 2 show that the algorithm developed in allows sizing of different cracks with complex cross-sections and unknown shapes for orientation angles not exceeding 45°. It is seen that the width 2a and the parameter c (or the surface density of charge m=4 r // e at the crack walls) are determined with 100% accuracy for all of the Case Symbols studied. The errors in the computation of the depths dj and di are less than 4% while the errors in the computation of d, dj, d, and d are less than 20% independent of the shape of the investigated crack and its orientation angle O <45°. 

Shear Horizontal (SH) waves generated by Electromagnetic Acoustic Transducer (EMAT) have been used for sizing fatigue cracks and machined notches in steels by Time-of-Flight Diffraction (TOED) method. The used EMATs have been Phased Array-Probes and have been operated by State-of-the-art PC based phased array systems. Test and system parameters have been optimised to maximise defect detection and signal processing methods have been applied to improve accuracy in the transit time measurements. 

Measurements have been made in a static laboratory set-up. A simulation model for generating supplementary data has been developed and verified. A statistical data treatment method has been applied to estimate tracer concentration from detector measurements. Accuracy in parameter estimation in the range of 5-10% has been obtained. 

After having proved the principles a dynamic test facility has been constructed. In this facility it is possible to inject 3 tracers in a flownng liquid consisting of air, oil and water. By changing the relative amounts of the different components it is possible to explore the phase diagram and asses the limits for the measurement principle. Experiments have confirmed the accuracy in parameter estimation to be below 10%, which is considered quite satisfactorily for practical applications. The method will be tested on site at an offshore installation this summer. 

To improve the accuracy of the solution, the size of the time step may be decreased. The smaller is the time step, the smaller are the assumed errors in the trajectory. Hence, in contrast (for example) to the Langevin equation that includes the friction as a phenomenological parameter, we have here a systematic way of approaching a microscopic solution. Nevertheless, some problems remain. For a very large time step, it is not clear how relevant is the optimal trajectory to the reality, since the path variance also becomes large. Further-... 

The salient comparisons are between the bars marked P3-Dk, our initial parallel PME implementation, and DP-4, the macroscopic multipole method with four levels of macroscopic boxes. Though it is difficult to create a completely fair comparison in terms of the relative accuracy of the potentials and forces as computed by the two methods, the parameters for these simulations were tuned to give comparable overall accuracy. PME is clearly... 

For every type of angle including three atoms, two parameters (force constant fe and reference value 0q) are needed. Also, as in the bond deformation case, higher-order contributions such as that given by Eq. (23) are necessary to increase accuracy or to account for larger deformations, which no longer follow a simple harmonic potential. 

A number of other software packages are available to predict NMR spectra. The use of large NMR spectral databases is the most popular approach it utilizes assigned chemical structures. In an advanced approach, parameters such as solvent information can be used to refine the accuracy of the prediction. A typical application works with tables of experimental chemical shifts from experimental NMR spectra. Each shift value is assigned to a specific structural fragment. The query structure is dissected into fragments that are compared with the fragments in the database. For each coincidence, the experimental chemical shift from the database is used to compose the final set of chemical shifts for the... 

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