Bias, model

For example, in Figure 5.11-left-down a bias method is graphically illustrated the real stmcture factor is F, the one calculated from the model is (the sub-index C indicates the calculated nature)- which, despite indexed, provides the calculated phase exp(/a ) which by combining with the observed (measured) amplitude F generates the bias model (calculation + experiment) of the bias structure factor F exp(/a ) this should be closer to the real one than the one F provided by applying of theoretical model alone. Thus, properties of the stmcture will be identified, as close to the real ones, namely if inside the calculated model there were not certain atoms, they will appear in the bias electronic map, built on the bias stmcture factor. 

Ri/2 — R1/3R3/2 and rxii = rxiiriji Substituting this into the linear mass bias model, we obtain ... 

Inherent to all mass bias models stemming from Eq. (5.7) is the built-in variable (discrimination exponent) that distinguishes the various mass discrimination phenomena. Hence we have linear, exponential, equilibrium, power, and other discrimination models. This variable, in turn, is often used to identify the presence of a particular discrimination model [32, 41], Consequently, considerable effort has been spent in extracting the numerical value of the mass bias discrimination... 

Like the traditional mass bias correction approaches, the double-spike method also relies on the choice of the mass bias model. The original formalism of the double spikes employed the linear mass bias law and, although double-spike calibration equations adapted for the exponential mass bias discrimination are available, linear models are still often used owing to their simplicity (see, for example, [50-52]). The caveat here is that erroneous results can be obtained when a linear correction is applied to data that do not follow such behavior. This is illustrated below. 

Two main types of reaction occur when alkylcobaloximes react with cobalox-ime(ii) complexes in methanol first, homolytic displacement of cobaloxime(ii) from the alkylcobaloxime is brought about by the attack of the cobaloxime(ii) reagent on the alkyl group, and this is followed by the rapid additional exchange of equatorial ligands between reagent cobaloxime(ii) and displaced cobaloxime(ii). The rate constants for the reduction of a series of Bia models (25) by [Fe(edta)] " are very sensitive to the nature of the axial ligands L. 

Mathematical Consistency Requirements. Theoretical equations provide a method by which a data set s internal consistency can be tested or missing data can be derived from known values of related properties. The abiUty of data to fit a proven model may also provide insight into whether that data behaves correctiy and follows expected trends. For example, poor fit of vapor pressure versus temperature data to a generally accepted correlating equation could indicate systematic data error or bias. A simple sermlogarithmic form, (eg, the Antoine equation, eq. 8), has been shown to apply to most organic Hquids, so substantial deviation from this model might indicate a problem. Many other simple thermodynamics relations can provide useful data tests (1—5,18,21). 

Systematic Operating Errors Fifth, systematic operating errors may be unknown at the time of measurements. Wriile not intended as part of daily operations, leaky or open valves frequently result in bypasses, leaks, and alternative feeds that will add hidden bias. Consequently, constraints assumed to hold and used to reconcile the data, identify systematic errors, estimate parameters, and build models are in error. The constraint bias propagates to the resultant models. 

Because the technical barriers previously outhned increase uncertainty in the data, plant-performance analysts must approach the data analysis with an unprejudiced eye. Significant technical judgment is required to evaluate each measurement and its uncertainty with respec t to the intended purpose, the model development, and the conclusions. If there is any bias on the analysts part, it is likely that this bias will be built into the subsequent model and parameter estimates. Since engineers rely upon the model to extrapolate from current operation, the bias can be amplified and lead to decisions that are inaccurate, unwarranted, and potentially dangerous. 

Analysts The above is a formidable barrier. Analysts must use limited and uncertain measurements to operate and control the plant and understand the internal process. Multiple interpretations can result from analyzing hmited, sparse, suboptimal data. Both intuitive and complex algorithmic analysis methods add bias. Expert and artificial iutefligence systems may ultimately be developed to recognize and handle all of these hmitations during the model development. However, the current state-of-the-art requires the intervention of skilled analysts to draw accurate conclusions about plant operation. 

The critical role of analysts introduces a potential for bias that overrides all others—the an ysts evaluation of the plant information. Analysts must recognize that the operators methods, designers models, and control engineers models have merit but must so beware they can be misleading. If the analysts are not familiar with the unit, the explanations are seductive, particiilarly since there is the motivation to avoid antagonizing the operators and other engineers. 

The third interaction compromising the parameter estimate is due to bias in the model. If noncondensables blanket a section of the exchanger such that no heat transfer occurs in that section, the estimated heat-transfer coefficient based on a model assuming all of the area is available will be erroneous. 

The above assumes that the measurement statistics are known. This is rarely the case. Typically a normal distribution is assumed for the plant and the measurements. Since these distributions are used in the analysis of the data, an incorrect assumption will lead to further bias in the resultant troubleshooting, model, and parameter estimation conclusions. 

An example adapted from Verneuil, et al. (Verneuil, V.S., P. Yan, and F. Madron, Banish Bad Plant Data, Chemical Engineeiing Progress, October 1992, 45-51) shows the impact of flow measurement error on misinterpretation of the unit operation. The success in interpreting and ultimately improving unit performance depends upon the uncertainty in the measurements. In Fig. 30-14, the materi balance constraint would indicate that S3 = —7, which is unrealistic. However, accounting for the uncertainties in both Si and S9 shows that the value for S3 is —7 28. Without considering uncertainties in the measurements, analysts might conclude that the flows or model contain bias (systematic) error. 

Overview Reconciliation adjusts the measurements to close constraints subject to their uncertainty. The numerical methods for reconciliation are based on the restriction that the measurements are only subject to random errors. Since all measurements have some unknown bias, this restriction is violated. The resultant adjusted measurements propagate these biases. Since troubleshooting, model development, ana parameter estimation will ultimately be based on these adjusted measurements, the biases will be incorporated into the conclusions, models, and parameter estimates. This potentially leads to errors in operation, control, and design. 

Representativeness can be examined from two aspects statistical and deterministic. Any statistical test of representativeness is lacking becau.se many histories are needed for statistical significance. In the absence of this, PSAs use statistical methods to synthesize data to represent the equipment, operation, and maintenance. How well this represents the plant being modeled is not known. Deterministic representativeness can be answered by full-scale tests on like equipment. Such is the responsibility of the NSSS vendor, but for economic reasons, recourse to simplillcd and scaled models is often necessary. System success criteria for a PSA may be taken from the FSAR which may have a conservative bias for licensing. Realism is more expensive than conservatism. 

The ANN model had four neurones in the input layer one for each operating variable and one for the bias. The output was selected to be cumulative mass distribution thirteen neurones were used to represent it. A sigmoid functional... 

The schematic model is depicted in Fig. 8. As the bias voltage increases, the number of the molecular orbitals available for conduction also increases (Fig. 8) and it results in the step-wise increase in the current. It was also found that the conductance peak plotted vs. the bias voltage decreases and broadens with increasing temperature to ca. 1 K. This fact supports the idea that transport of carriers from one electrode to another can take place through one molecular orbital delocalising over whole length of the CNT, or at least the distance between two electrodes (140 nm). In other words, individual CNTs work as coherent quantum wires. 

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