Error component

Table 13-8. Self-interaction error components for Coulomb and exchange energies (Ej + Ex) as well as for the correlation energy (Ec), and the resulting sum for the H atom, the H2 molecule, and the H3 transition structure [kcal/mol]. Data taken from Csonka and Johnson, 1998.

One common characteristic of many advanced scientific techniques, as indicated in Table 2, is that they are applied at the measurement frontier, where the net signal (S) is comparable to the residual background or blank (B) effect. The problem is compounded because (a) one or a few measurements are generally relied upon to estimate the blank—especially when samples are costly or difficult to obtain, and (b) the uncertainty associated with the observed blank is assumed normal and random and calculated either from counting statistics or replication with just a few degrees of freedom. (The disastrous consequences which may follow such naive faith in the stability of the blank are nowhere better illustrated than in trace chemical analysis, where S B is often the rule [10].) For radioactivity (or mass spectrometric) counting techniques it can be shown that the smallest detectable non-Poisson random error component is approximately 6, where ... 

For example, a) in (radioactivity) counting experiments a non-Poisson random error component, equal in magnitude (variance) to the Poisson component, will not be detected until there are 46 degrees of freedom ( ), and b) it was necessary for a minor component in a mixed Y-ray spectrum to exceed its detection limit by -50 , before its absence was detected by lack-of-fit (x, model error) (7). 

When the difference D between the results A and B is computed the systematic errors, which have the same magnitude and sign, will cancel. This leaves the difference of the two random error components, which do not necessarily cancel for a particular pair. 

Figure 5) Systematic error is present, but it does not follow a simple functional relationship with time. This case is simulated by the occurence of a large systematic error component (greater than three standard deviations from the mean) which appears without warning. 

Applying a mask function in real space is equivalent to combining many structure factors through a convolution in reciprocal space. This results in an improvement because the random error component of the structure factors will average out, whereas the true values of the structure factors will add up systematically. Fig. 10.4 gives a graphical example showing this phenomenon. 

Unknown Component A Statistical Prediction Error Component B Statistical Prediction Error... 

Regression analysis assumes that all error components are independent, have a mean of zero and have the same variance throughout the range of POM values. Through an examination of residuals, serious violations in these assumptions can usually be detected. The standardized residuals for each of the fitted models were plotted against the sequence of cases in the file and this scatterplot was examined visually for any abnormalities (10, 14). 

Each of these error components adds its own uncertainty to the total uncertainty budget of the analytical procedure. Therefore, the different error components are referred to as sources of uncertainty. Depending on the sources of uncertainty taken into account and thus on the conditions of the measurement, the overall MU will be different and another definition of MU will apply. This means that there is no single, straighforward definition of MU. It is rather a concept the interpretation of... 

As illustrated in Figure 5, the error of an analytical result for a specified analyte concentration is composed of different error components, forming together the ladder of errors ... 

The more traditional distinction of error components is between random errors and systematic errors. In this classical approach, random errors are generally referred to as precision (repeatability, intermediate precision, and reproducibility), while systematic errors are typically attributed to the uncertainty on the bias estimate and... 

As mentioned above, each of these error components is a potential source of uncertainty. Depending on the conditions under which the analysis is performed, different sources of uncertainty contribute to the overall uncertainty. Hund et al. 

The relevance of systematic errors in the total error budget depends on whether optimistic or conservative error estimates are used. Further studies are needed to attain realistic error estimates and to reduce those error components which may have a contribution larger than noise error. 

Bias difference between the population mean of the test results and an accepted reference value a systematic error as contrasted to random error, and there may be one or more systematic error components contributing to the bias (ASTM E-456). 

Table 1.1 Total error components and PARCC parameters...

The intricate rules for sampling, laboratory analysis, and data quality evaluation, all designed to minimize total error components, apply to every environmental data collection project. The quality of collected data is defined by this system of multidisciplinary rules, and a practical interpretation of these rules lies at the core of this Guide. 

Accuracy means the closeness of agreement between a test result and the accepted reference value. It is determined by determining trueness and precision [57], In a strict sense, the accuracy of a method is affected by systematic (bias) as well as random (precision) error components, but the term is often used to describe only the systematic error component, i.e., in the sense of bias. In this sense sometimes... 

This expression goes to zero with increasing magnitude of any error component Eiu, since the argument is nonpositive whenever 27 exists see Problem 4.C. 

Another factor that can determine the quality of the stochastic correlation of results is the standard deviation s of the random error component of the measurement results. This is calculated as follows ... 

The values of the standard deviation that are close to 0 indicate that the chosen model has a small random error component and that the regression was done properly. In case st 224+240, these factors are k = 0,995124 and... 

According to ISO, bias is the difference between the expectation of the test results and an accepted reference value [17]. It may consist of more than one systematic error component. Bias can be measured as a percent deviation from the accepted reference value. The term trueness expresses the deviation of the mean value of a large series of measurements from the accepted reference value. It can be expressed in terms of bias. 

The random error components may be expressed as SDs, and generally we can assume that random bias and analytical components are independent for each analyte yielding the relations... 

Another methodological problem concerns the question whether the dispersion of the random error components is constant or changes with the analyte concentration as considered previously in the difference plot sections. For most clinical chemical compounds, the analytical SDs vary with the measured concentration, and this relationship may also apply to the random-bias components. In cases with a considerable range (i.e., a decade or more), this phenomenon should also be taken into account when applying a regression analysis. Figure 14-20 schematically shows how the dispersions may increase proportionally with concentration. 

Having outlined the random error components related to regression analysis, some comments on the correlation coefficient may be appropriate. The ordinary correlation coefficient p, also called the Pearson product moment correlation coefficient, is estimated as r from sums of squared deviations for xl and x2 values as follows using the same notation as above ... 

Looking at the theoretical model, p is related to the ratio between the standard deviations of the distributions of target values (associated independent total random error components and 0, 2) ... 

The total random error components comprise both imprecision error and sample-related random interferences (i.e., - <5ai + [Pg.383]

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