Error and residuals

Table 3.7 Results of bias, errors and residuals for best fit line through the centroid...

H. Error and Residual Vectors. Vector and Matrix Norms. Condition Number... 

In the first generation of HRA methods, human failure was seen and investigated as random phenomenon, with some distribution in time formed by performance shaping factors influence. In HRA second generation method/framework ATHEANA, treatment of human failure is different, as it is seen as cause based consequence of error forcing context actuation. Still, the plant specific experience can lead to the conclusion that some residual randomness should be kept in hiunan failure model, similarly to the case of (equipment) dependent errors and residual common cause failures. 

Table 15.6 Recommended limits of voltage and phase displacement errors, applicable for all types of protection VTs (electromagnetic, capacitor and residual VTs)...

In general, the R factor is between 0.15 and 0.20 for a well-determined protein structure. The residual difference rarely is due to large errors in the model of the protein molecule, but rather it is an inevitable consequence of errors and imperfections in the data. These derive from various sources, including slight variations in conformation of the protein molecules and inaccurate corrections both for the presence of solvent and for differences in the orientation of the microcrystals from which the crystal is built. This means that the final model represents an average of molecules that are slightly different both in conformation and orientation, and not surprisingly the model never corresponds precisely to the actual crystal. 

The weighting model with which the goodness-of-fit or figure-of-merit (GOF = E(m,)) is arrived at can take any of a number of forms. These continuous functions can be further modified to restrict the individual contributions M, to a certain range, for instance r, is minimally equal to the expected experimental error, and all residuals larger than a given number r ax are set equal to rmax- The transformed residuals are then weighted and summed over all points to obtain the GOF. (See Table 3.5.)... 

Van der Voet [21] advocates the use of a randomization test (cf. Section 12.3) to choose among different models. Under the hypothesis of equivalent prediction performance of two models, A and B, the errors obtained with these two models come from one and the same distribution. It is then allowed to exchange the observed errors, and c,b, for the ith sample that are associated with the two models. In the randomization test this is actually done in half of the cases. For each object i the two residuals are swapped or not, each with a probability 0.5. Thus, for all objects in the calibration set about half will retain the original residuals, for the other half they are exchanged. One now computes the error sum of squares for each of the two sets of residuals, and from that the ratio F = SSE/JSSE. Repeating the process some 100-2(K) times yields a distribution of such F-ratios, which serves as a reference distribution for the actually observed F-ratio. When for instance the observed ratio lies in the extreme higher tail of the simulated distribution one may... 

Equation (7.28) can also be used for a different situation. Consider that initially c, specified measurements are suspected to possess gross errors, and let be the corresponding covariance matrix of the residuals in the balances. If a different set Q+i of suspect measurements is obtained by adding measurements to the set c,-, the... 

A drawback of Gran plots is the fact that all deviations from the theoretical slope value cause an error and that side reactions are not considered. The method was modified by Ingman and Still [63], who considered side reactions to a certain degree, but the equilibrium constants and the concentrations of the components involved must be known. The Gran method is, however, advantageous for determinations in the vicinity of the determination limit The extrapolation of the linear dependence yields the sum + c, where c, is the residual concentration of the test component produced by impurities, dissolution of the ISE membrane, etc. 

The concentration residuals are plotted as a function of the predicted concentration (Draper and Smith. 1981). One side note is that with replicate samples it can appear that there is structure in the residuals when in fact the model is adequate. This is demonstrated in Figure 5.15 where there are 10 replicates at three concentration levels. The residuals form slanted lines, but this is only because some samples arc predicted low (and so have positive errors), and ojher samples are predicted high (and so have negative errors). 

Sometimes crystallographers consider that measuring a crystal at very low temperature is a kind of panacea, able to solve all defects of the sample, all kinds of experimental errors, and enhance the response indefinitely. Young students might be disappointed to leam that these miracles do not take place. A bad crystal sample remains as such even at 10 K, and sometimes it becomes even worse because the cooling process and the residual stress induced by a temperature gradient may produce further damage to the sample. Many other kinds of experimental problems and sources of error (for example absorption, extinctions, disorder, etc.) are not attenuated by the low temperature. 

A general goal is to minimize the sums of the squares of the N error weighted residuals (Titterington and Halliday, 1979 Ludwig, 1998)... 

When an unreplicated experiment is run, the error or residual sum of squares is composed of both experimental error and lack-of-fit of the model. Thus, formal statistical significance testing of the factor effects can lead to erroneous conclusions if there is lack-of-fit of the model. Therefore, it is recommended that the experiment be replicated so that an independent estimate of the experimental error can be calculated and both lack-of-fit and the statistical significance of the factor effects can be formally tested. 

The purpose of the Model I ANOVA is to decompose each result as yij=iu+aj+eij where // is the population mean, aj the effect of group j and eij the randomly distributed error or residual. Significance in a fixed effect... 

Methods intended for regulatory residue condol should be designed with as much simplicity as possible to limit the variety, size, and type of glassware and equipment needed to minimize the potential for analytical error and to reduce costs. Reagents and standards must be readily available while specific instrumentation should be based on performance characteristics rather than a particular manufacturer. 

A different method of interpretation is frequently observed between inspection services and analytical laboratories. This is because inspection services are interested mainly in a yes/no answer to questions, such as Has the animal been treated with anabolics or Does the food commodity contain residues above their MRL , in order to proceed to such action as rejection of the food commodity or removal of the test-positive animals from the farm. On the other hand, laboratories mainly use quality criteria to convert analytical results into yes/no answers. This conversion, however, is often obscured by inherent analytical difficulties including estimation of the impact of systematic and random errors and the way of sampling. 

The list of error sources continues, just to mention a few the ionic strength of the sample, the liquid-junction and residual liquid-junction potentials, temperature effects, instabilities in the galvanic cell, carryover effects, improper use of available corrections (e.g., for pH-adjusted ionized calcium or magnesium). An error analysis goes beyond the limited scope of this paper more details are presented elsewhere [10]. 

The point (Xj, Yj) denotes the i-th observation. The true error or residual is Yj-(Po+PiXi), the difference between the observed Y and the true unknown value Po+PiXj. The observed residual ej is Yj-fbo+l Xj Y — Y(, which is the difference between the observed Yj and the estimated Yi = b0 + blXi. The problem is now to obtain estimates b0 and bi from the sample for the unknown parameters p0 and Pi. 

Analysis of variance with no setting apart of the residual variance into experimental error and lack of fit variance is given in the table ... 

Here, we want to emphasize that one is able to calculate the fraction of the experimental error only if replicate measurements (at least at one point x ) have been taken. It is then possible to compare model and experimental errors and to test the sources of residual errors. Then, in addition to the GOF test one can perform the test of lack of fit, LOF, and the test of adequacy, ADE, (commonly used in experimental design). In the lack of fit test the model error is tested against the experimental error and in the adequacy test the residual error is compared with the experimental error. 

The common feature variance originates from correlating features. Specific feature variance and residuals or error are now expressed by the matrix E ... 

Step 8 Measuring Results and Monitoring Performance The evaluation of MPC system performance is not easy, and widely accepted metrics and monitoring strategies are not available, ffow-ever, useful diagnostic information is provided by basic statistics such as the means and standard deviations for both measured variables and calculated quantities, such as control errors and model residuals. Another useful statistic is the relative amount of time that an input is saturated or a constraint is violated, expressed as a percentage of the total time the MPC system is in service. 

The statistical submodel characterizes the pharmacokinetic variability of the mAb and includes the influence of random - that is, not quantifiable or uncontrollable factors. If multiple doses of the antibody are administered, then three hierarchical components of random variability can be defined inter-individual variability inter-occasional variability and residual variability. Inter-individual variability quantifies the unexplained difference of the pharmacokinetic parameters between individuals. If data are available from different administrations to one patient, inter-occasional variability can be estimated as random variation of a pharmacokinetic parameter (for example, CL) between the different administration periods. For mAbs, this was first introduced in sibrotuzumab data analysis. In order to individualize therapy based on concentration measurements, it is a prerequisite that inter-occasional variability (variability within one patient at multiple administrations) is lower than inter-individual variability (variability between patients). Residual variability accounts for model misspecification, errors in documentation of the dosage regimen or blood sampling time points, assay variability, and other sources of error. 

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