Confidence representation

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 potential fiinctions for the mteractions between pairs of rare-gas atoms are known to a high degree of accuracy [125]. Flowever, many of them use ad hoc fiinctional fonns parametrized to give the best possible fit to a wide range of experimental data. They will not be considered because it is more instmctive to consider representations that are more finnly rooted in theory and could be used for a wide range of interactions with confidence. 

Extended Plant-Performance Triangle The historical representation of plant-performance analysis in Fig. 30-1 misses one of the principal a ects identification. Identification establishes troubleshooting hypotheses and measurements that will support the level of confidence required in the resultant model (i.e., which measurements will be most beneficial). Unfortunately, the relative impact of the measurements on the desired end use of the analysis is frequently overlooked. The most important technical step in the analysis procedures is to identify which measurements should be made. This is one of the roles of the plant-performance engineer. Figure 30-3 includes identification in the plant-performance triangle. 

Required Sensitivity This is difficult to establish a priori. It is important to recognize that no matter the sophistication, the model will not be an absolute representation of the unit. The confidence in the model is compromised by the parameter estimates that, in theoiy, represent a limitation in the equipment performance but actually embody a host of limitations. Three principal limitations affecting the accuracy of model parameters are ... 

The way that the teacher conducted the lessons contributed to making some details of the systems explicit and helped students in interpreting (i) the empirical evidence, (ii) the questions to be answered by the models, and (iii) the symbolic representations she presented for each system. Moreover, the teacher s questions supported the students as they tried to remember previous ideas and/or models, to identify the limitations of their models, to propose new models or new explanations for the use of their models in new contexts. Finally, the teachers questions were very helpful for increasing students confidence in their models. 

The statistical fundamentals of the definition of CV and LD are illustrated by Fig. 7.8 showing a quasi-three-dimensional representation of the relationship between measured values and analytical values which is characterized by a calibration straight line y = a + bx and their two-sided confidence limits and, in addition (in z-direction) the probability density function the measured values. 

Fig. 7.8. Schematic three-dimensional representation of a calibration straight line of the form y = a + bx with the limits of its two-sided confidence interval and three probability density function (pdf) p(y) of measured values y belonging to the analytical values (contents, concentrations) X(A) = 0 (A), x = x(B) (B) and X(q = ld (C) yc is the critical value of the measurement quantity a the intercept of the calibration function yBL the blank x(B) the analytical value belonging to the critical value yc (which corresponds approximately to Kaiser s a3cr-limit ) xLD limit of detection... 

Fig. 8.10. Schematic representation of three classes in a two-dimensional discriminant space dfi vs df2 R, R2> R3 are the confidence radii of the respective classes... 

Y as a function of a change in X. These include, but are not limited to correlation (r), the coefficient of determination (R2), the slope (, ), intercept (K0), the z-statistic, and of course the respective confidence limits for these statistical parameters. The use of graphical representation is also a powerful tool for discerning the relationships between X and Y paired data sets. 

Given the same underlying spread of data (standard deviation, s), as more data are gathered, we become more confident of the mean value, x, being an accurate representation of the population mean, x. 

Any inferences about the difference between the effects of the two treatments that may be made upon such data are the observed rates, or proportions of deteriorations by the intrathecal route. In this example, amongst those treated by the intrathecal route 22/58 = 0.379 of patients deteriorated, and the corresponding control rate is 37/60 = 0.617. The observed rates are estimates of the population incidence rates, jtt for the test treatment and Jtc for the controls. Any representation of differences between the treatments will be based upon these population rates and the estimated measure of the treatment effect will be reported with an associated 95% confidence interval and/or p-value. 

The workshop favored the use of graphical representations that combine the key elements of the assessment outcome the magnitude and frequency of effects, together with appropriate confidence bounds. This should always be accompanied... 

A control chart is a visual representation of confidence intervals for a Gaussian distribution. The chart warns us when a monitored property strays dangerously far from an intended target value. 

Source discrimination was accomplished by examining a series of two- and three-dimensional plots of the obsidian source data. Discovery of graphical representations which show the clearest picture of inter-source versus intra-source variation, makes possible source discrimination with a high degree of confidence. The greater the number of elements that one can use to reinforce the observed discrimination the smaller becomes the chances for misassignment of artifacts when compared to the obsidian source database. 

Sanchis and coworker [64] in order to insight something about this fact and in order to get confidence about this phenomenon, have used the electric modulus formalism [146], (M = l/e ) to represent the experimental data. The advantages of this kind of representation are evident due to the better resolution observed for dipolar and conductive processes. The imaginary part of M as a function of frequency at 423 K, for both polymers, is shown in Fig. 2.45. The curve corresponding to PTHFM shows a complex behavior at low frequencies, which presumably is the result of the superposition of the two conductive processes. 

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