Metrics errors

Example 1 Sample Quantity for Composition Quality Control Testing An example is sampling for quality control of a 1,000 metric ton (VFg) trainload of-Ks in (9.4 mm) nominal top-size bentonite. The specification requires silica to be determined with an accuracy of plus or minus three percent for two standard errors (s.e.). With one s.e. of 1.5 percent, V is 0.000225 (one s.e. weight fraction of 0.015 squared). The problem to be solved is thus calculating weight of sample to determine sihca with the specified error variance. 

No other a priori assumptions about the form or the structure of the function will be made. For a given choice of g. Kg) in Eq. (1) provides a measure of the real approximation error with respect to the data in the entire input space X. Its minimization will produce the function g (x) that is closest to G to the real function, /(x) with respect to the, weighted by the probability P(x,y) metric p.. The usual choice for p is the Euclidean distance. Then 1(g) becomes the L -metric ... 

Given a space G, let g (x) be the closest model in G to the real function, fix). As it is shown in Appendbc 1, if /e G and the L°° error measure [Eq. (4)] is used, the real function is also the best function in G, g = f, independently of the statistics of the noise and as long as the noise is symmetrically bounded. In contrast, for the measure [Eq. (3)], the real function is not the best model in G if the noise is not zero-mean. This is a very important observation considering the fact that in many applications (e.g., process control), the data are corrupted by non-zero-mean (load) disturbances, in which cases, the error measure will fail to retrieve the real function even with infinite data. On the other hand, as it is also explained in Appendix 1, if f G (which is the most probable case), closeness of the real and best functions, fix) and g (x), respectively, is guaranteed only in the metric that is used in the definition of lig). That is, if lig) is given by Eq. (3), g ix) can be close to fix) only in the L -sense and similarly for the L definition of lig). As is clear,... 

Another noteworthy example is x-ray absorption fine structure (EXAFS). EXAFS data contain information on such parameters as coordination number, bond distances, and mean-square displacements for atoms that comprise the first few coordination spheres surrounding an absorbing element of interest. This information is extracted from the EXAFS oscillations, previously isolated from the background and atomic portion of the absorption, using nonlinear least-square fit procedures. It is important in such analyses to compare metrical parameters obtained from experiments on model or reference compounds to those for samples of unknown structure, in order to avoid ambiguity in the interpretation of results and to establish error limits. 

The t value associated with each model descriptor, defined as the descriptor coefficient divided by its standard error, is a useful statistical metric. Descriptors with large f values are important in the predictive model and, as such, can be examined in order to gain some understanding of the nature of the property or activity of interest. It should be stated, however, that the converse is not necessarily true, and thus no conclusions can be drawn with respect to descriptors with small f values. 

The knowledge and application of pharmaceutical and clinical calculations are essential for the practice of pharmacy and related health professions. Many calculations have been simplified by the shift from apothecary to metric system of measurements. However, a significant proportion of calculation errors occur because of simple mistakes in arithmetic. Further, the dosage forms prepared by pharmaceutical companies undergo several inspections and quality control tests. Such a luxury is almost impossible to find in a pharmacy or hospital setting. Therefore it is imperative that the health care professionals be extremely careful in performing pharmaceutical and clinical calculations. In the present chapter, a brief introduction is provided for the three systems of measurement and their interconversions ... 

Summation of absolute differences (I) results in an ME in which all differences have the same statistical weight. Summation of squared differences (II) is the more common practice and gives an MSE in which large deviations have higher weight than small ones. In order to make the metric independent of the number N of observations, the error sum must be related to N or an equivalent sum of the observations ... 

To leam about how a liquid junction potential, j, arises, and appreciate how it can lead to significant errors in a calculation which uses potentio-metric data. 

The % R8cR-value (repeatability and reproducibility), addresses the portion of the observed total process variation that is taken up by the measurement error. The % R R-value is an important metric for the method capability towards application in process improvement studies. 

Deuterium exchange with DjO was used by Shuravlev and Kiselev (199) in the determination of surface hydroxyl groups of silica gel. Adsorption isotherms of HgO and DjO were determined gravi-metrically they agreed with each other within the limits of experimental error. 

In either interpretation of the Langevin equation, the form of the required pseudoforce depends on the values of the mixed components of Zpy, and thus on the statistical properties of the hard components of the random forces. The definition of a pseudoforce given here is a generalization of the metric force found by both Fixman [9] and Hinch [10]. An apparent discrepancy between the results of Fixman, who considered the case of unprojected random forces, and those of Hinch, who was able to reproducd Fixman s expression for the pseudoforce only in the case of projected random forces, is traced here to an error in Fixman s use of differential geometry. 

Root Mean Square Error of Prediction (RMSEP) Plot (Model Diagnostic) Prediction error is a useful metric for selecting the optimum number of factors to include in the model. This is because the models are most often used to predict the concentrations in future unknown samples. There are two approaches for generating a validation set for estimating the prediction error internal validation (i.e., cross-validation with the calibration data), or external validation (i.e., perform prediction on a separate validation set). Samples are usually at a premium, and so we most often use a cross- validation approach. 

PEG-200 was a commercial product (Chemische Werke Hiils, A. G.) with mean molecular weight 182 as determined osmometrically in the vapor phase (with an error of 5%) and with water content of 4.42% as determined by K. Fischer s method. (All weights were corrected with respect to this moisture.) CaCl2 was of analytical reagent grade. For the standard solutions, the stock saturated solution was prepared and maintained at the constant temperature of 25°C. The concentration of CaCl2 was tested by the chloride content argento-metrically. 

Complexes with (R)-lactate have been studied by ESR631 and CD.580,632 Lactate is near water in the spectrochemical series and its low pAa helps reduce error due to hydrolysis and oxidation. Table 32 summarizes the data. The pAa of the R—OH is quite high (especially for the aliphatic acids). Many have considered coordination in V02+ hydroxycarboxylate to involve the C02 and OH groups and assumed the dissociation of the OH. As structures (55) and (56) or (57) and (58) are not pH-metrically distinguishable, if the pA°H is not known, other techniques must establish the species present. 

There are many possible sources of error in the calculation of equilibrium from dipole moments. It is difficult to calculate accurately the moments of model compounds. As has been pointed out,124 the carbon skeleton itself contributes to the moment, as shown by the different dipole moments of ethyl bromide (2.069 D) and n-hexyl bromide (2.156 D) and of axial and equatorial fluorocyclohexane (1.81 D and 2.11 D, respectively). For nonsym-metrical molecules in general, the estimation of the precise direction of a dipole moment is subject to considerable error. 

In retrospect, one can see that Boltzmann s inspired conjecture (13.69) served, through (13.77), to anticipate an essential feature of the probabilistic quantum description that was to supplant classical determinism in the 20th century. Boltzmann and his followers specifically captured the key probabilistic feature (13.69) that could bring proper metric geometrical character (13.77) to macroscopic-level thermodynamic description, despite gross errors of then-current microscopic dynamical theory. 

The QCE model also allows numerical evaluation of the heat capacities, thermal coefficients, and compressibilities needed to construct the thermodynamic metric geometry. Unfortunately, the higher derivatives of Q that are needed to evaluate the QCE thermodynamic metric are subject to considerable errors, both from underlying theoretical approximations and from increasingly severe numerical errors in finite-difference evaluations. Significant improvements, including extension to multicomponent chemical mixtures and more accurate description of cluster-cluster interactions, are needed before QCE-like models can provide additional ab initio insights into the mysteries of nonideality in phase equilibria. 

Write the units In 1999, the 125 million Mars Climate Orbiter spacecraft was lost when it entered the Martian atmosphere 100 km lower than planned. The navigation error would have been avoided if people had labeled their units of measurement. Engineers who built the spacecraft calculated thrust in the English unit, pounds of force. Jet Propulsion Laboratory engineers thought they were receiving the information in the metric unit, newtons. Nobody caught the error. 

There are certain sources of error which are inherent to potentio-metric measurements. The first is the relative error (in %) in a potent-iometric measurement of the ith analyte Act/ct — 100 zi F/RTAE (A E is... 

Figure 3.2 The effect of prolonged subcutaneous implantation on biosensor function. Blood glucose values shown in solid circles and glucose sensor values in the continuous lines. The early study (top panel), but not the late study (bottom), shows excellent sensor accuracy and minimal lag between blood glucose and sensed glucose values. MARD (mean absolute relative difference) refers to a sensor accuracy metric. EGA refers to the Clarke error grid analysis accuracy metric.

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