Statistical measures of errors

Table 1.12 Statistical measures of errors for extrapolated CCSD(T) AEs relative to experiment (kJ/mol). A is the mean error, Astd is the standard deviation around the mean error, Aabs is the mean absolute error, and Amax is the maximum absolute error.

In this section, we investigate the accuracy of ab initio electronic-structure predictions of bond distances [21. In Section 15.3.1, we review the experimental data on which the investigation is based. Next, in Sections 15.3.2-15.3.5, we consider the statistical measures of errors, piecing together a picture of the performance of the standard models with respect to the calculation of equilibrium bond distances. After a discussion of higher-order effects at the CCSDT level in Section 15.3.6 and core correlation in Section 15.3.7, we attempt to rationalize the behaviour of the different models in Section 15.3.8. We conclude our discussion in Section 15.3.9, which... 

In the present subsection, we examine the statistical measures of error in the calculated atomization energies. For the sample molecules in Table 15.1, the atomization eneigies have been calculated... 

The equilibrium reaction enthalpies have been calculated as described in Section 15.6.1. The statistical measures of errors in the calculated reaction enthalpies have been plotted in Figure 15.15 and listed in Table 15.30. 

Nevertheless, as for atomization eneigies, we expect the basis-set convergence of the enthalpies to he dominated hy the errors in the correlation treatment. For this reason, we shall here attempt to carry out extrapolations of the correlation part of the enthalpy, applying the procedure described in Section 15.6.1. In Table 15.32, we have listed the usual statistical measures of errors for the ealculated and extrapolated all-electron CCSD(T) reaction enthalpies. In Figure 15.17, we have plotted the normal distributions of the errors in the calculated reaction enthalpies, with and without extrapolation. 

The standard deviation of the distribution of means equals cr/Nm. Since cris not usually known, its approximation for a finite number of measurements is overcome by the Student t test. It is a measure of error between p and x. The Student t takes into account both the possible variation of the value of x from p on the basis of the expected variance cr2/Nm and the reliability of using s in place of cr. The distribution of the statistic is ... 

The kinetics of these pyrolysis reactions were followed by several complementary methods under conditions as close to the product studies as possible. The most frequently-used ampule technique14 17) with gc analysis of 5 and the scavenger technique, with chloranil or Koelsch radical as scavenger 18), for very labile compounds 5 were complemented by the DSC method, in which the heat flow under conditions of linear temperature increase is analysed. It proved to be a particularly convenient and reliable technique 18- 21). Rates were followed over a temperature span of at least 40 °C with temperature control of 0.1-0.2 °C. All rate data and activation parameters were subjected to a thorough statistical analysis including statistical weights of errors. The maximum statistical errors in k were 3%, in AH 1 kcal mol-1 in AS 513 e.u. and in AG (at the temperature of measurement) g0.5 kcal mol-1. 

The statistical measure of the quality of the regression is used to determine whether the model provides a meaningful representation of the data. The parameter estimates are reliable only if the model provides a statistically adequate representation of the data. The evaluation of the quality of the regression requires an independent assessment of the stochastic errors in the data, information that may not be available. In such cases, visual inspection of the fitting results may be useful. Issues associated with assessment of regression quality are discussed further in Section 19.7.2 and Chapter 20. 

Precision may be defined as the closeness of agreement between independent results of measurements obtained under stipulated conditions.The degree of precision is usually expressed on the basis of statistical measures of imprecision, such as the SD or CV (CV = SD/x, where x is the measurement concentration), which thus is inversely related to precision. Imprecision of measurements is solely related to the random error of measurements and has. no relation to the trueness, of measurements. 

No information presented in the reference document suggests typical values for errors associated with the concentration measurements, or the type of statistical distribution that might represent these errors. The document does suggest that the preferred measurement of error is a 95-percent confidence limit based on the t-statistic, implying that the error distribution is normal in appearance. The equation presented to compute the 95-percent confidence limit was incorrect in the reference. The following equation is the correct form ... 

This text does not give a complete theory of error. Only the fundamentals needed for a basic understanding of the statistical analysis of errors are presented. The objective is to present methods that provide an estimate of the error of a certain measurement or a series of measurements and procedures that minimize the error. 

In order to figure out how many trees are suitable for the random forest, we further refer to the statistical parameters. From Figure 2, when there is only 1 tree, the Kapa statistics is 0.65, while it increases to 0.77 when the tree number is 7 or above. The measurement of error also becomes tolerable when the number of trees increases, and as is shown in Table 2, the MAE is 0.21 and the RMSE is 0.31 when the number of trees is 10. Although the RMSE is 0.13 when the number of trees is 1, the MAE has a value of 0.38 and this is higher than the value of other number of trees. These results indicate, if we only refer to the RMSE, we still cannot evaluate the error precisely. While it is similar for the RAE, it is only 33% when the... 

Note that Eqs. (4.557) and (4.558) parallel the statistical behavior of error in measurements that being vanishing in the first case as mean deviation, is manifested in the second as squared deviation (dispersion),... 

Accuracy The degree of conformance between the estimated or measured position and/or velocity of a platform at a given time and its true position or velocity. Radionavigation system accuracy is usually presented as a statistical measure of system error. 

Among the mathematical models of industrial chemical processes, the system of balance equations is basic. If some measured data do not satisfy the balance equations, this fact is attributed to measurement errors, and not perhaps to an inadequate description of the process by the model. In practice, measurement errors are always present. Hence before using the measured data, they are adjusted to obey the balance constraints. The adjustment by methods using statistical theory of errors is called reconciliation. 

In 2011, the OPERA experiment at CERN mistakenly observed neutrinos appearing to travel faster than light. Even before the mistake was discovered, the result was considered anomalous because speeds higher than that of light in a vacuum violate Einstein s theory of special relativity. Later, the team reported two flaws in their equipment set-up that had caused errors far outside of their original confidence interval (a statistical measure of reliability) a fibre optic cable not attached properly, which caused the apparently faster-than-light measurements, and a clock oscillator ticking too fast. 

To define numerically the quality of an HTS the metric Z is often used [10]. This metric is a statistical measure of the performance of an assay. Each plate contains a positive and a negative control sample that will serve not only to evaluate the assay signal window, namely, the dynamic range, but also to check for procedural errors and data variability over time. The formula of Z" is highlighted below wherein tj represents the standard deviation, p the mean value of the control and the positive control/the negative control The investigators should... 

Before we consider the different properties analysed in the present chapter, it is appropriate to discuss in general terms the errors that arise in quantum-chemical calculations. Thus, in Section 15.2.1, we discuss the sources of errors in quantum-chemical calculations, introducing the concepts of apparent and intrinsic errors. Next, in Section 15.2.2, we review the measures of errors that we shall employ in our statistical investigations in this chapter. 

Table 15 Statistical measures of the basis-set errors in the atomization energies (kJ/moI) relative to the cc-pV6Z energy for the Hartree-Fock model and relative to the cc-pcV(56)Z correlation energies for the correlated models...

Table 1533 Statistical measures of the basis-set errors (kj/mol) relative to the cc-pV6Z enthalpy (for the Haitree-Fock contribution) and to the extrapolated enthalpy (for the correlation contributions)...

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