Root mean square error of approximation

The results of calculations of the time dependencies of the constants are presented in Fig. 3.5. In this case the root-mean-square error of approximation also has a maximum at a specific time, although its magnitude is substantially lower than for the three-constant model. This is to be expected with the four-contant model, because it is known that at t the relaxation spectrum of a curing polymeric material changes radically it widens abruptly, and new relaxation modes appear.130 The four-constant model is insufficient to describe a rapid change in relaxation properties furthermore, the behavior of a real material near the gel-point (at the transition of the system to the heterophase state) is a new phenomenon that is not described by a simple model. 

Root Mean Square Error of Approximation <. 07 (with CFI >. 90)... 

The next step in the analysis involved assessment of the model. Given that the multivariate kurtosis in the data was elevated, as indicated by the Mardia coefficient, the robust method was used in this analysis (Bentler, 2006). A specified model is generally indieated as a good fit with the data when the /df ratio is less than 3, the RCFI (Robust Comparative Fit Index) and NNFl (Bentler-Bonnet Non-Normed Fit Index) are above. 90, and when RMSEA (Root Mean-Square Error of Approximation) is below. 07 (Byrne, 2006). Two multi-sample analyses (Byrne, 2006) were carried out contrasting Canadian and Swedish students and, female and male students. 

RMSEA Root mean square error of approximation (<0.08) NNFI = Non-normed fit index (>0.90)... 

Root Mean Square Error of Calibration (RMSEC) Plot (Model Diagnostic) The RMSEC as a function of the number of variables included in the model is shown in Figure 5-77. It decreases as variables are added to the model and the largest decrease is observed between a one- and two-variable model. The reported error in the reference caustic concentration is approximately 0.033 vrt.% (la). The tentative conclusion is that four variables are appropriate because the RMSEC is less than the reference concentration error after five variables are included in the model. 

Fig. 3-6. Conductivity/salinity calibration for a CTD (METEOR cruise M39/2). Upper row shows the corrections (CREFC-CTD) which is needed to match the conductivity measurements CCTD to the in situ reference value CREF as a function of the cast number (left) and pressure (right), the second row the residuals (CREF-C) after calibration of conductivity C. The third and fourth rows are corrections in salinity (SREFS-CTD) and residuals (SREF-S), respectively. Accuracy in C (95% confidence level) for all 74 casts is twice the root mean square error of the linear least square approximation, 0.0004 S/m (Siemens/m), corresponding to 0.004 in salinity. The result would be improved if the obvious cast-dependancies (see lower left part of panel) could be removed.

The root mean squared error of calibration (RMSEC) has been defined above. The leverage, ha, quantifies the distance of the predicted sample (at zero concentration level) to the mean of the calibration set in the -dimensional space Hq can be estimated as an average value of the leverages of a set of validation samples having zero concentration of the analyte. For a model calculated from mean-centred spectra its calculation was presented in Section 5.3 in matrix notation /zo=l//+to (T T) 4o, where to is the (.4x1) score vector of the predicted sample and T is the (7x4) matrix of scores for the calibration set. Finally, A(a,p,v) is a statistical parameter that takes into account the a and (3 probabilities of falsely stating the presence/absence of analyte, respectively, as recommended elsewhere. When the number of degrees of freedom i.e. the number of calibration samples) is high (v>25), as is usually the case in multivariate calibration models, and a =) , then A(a,(S,v) can be safely approximated to 2 ... 

It is often necessary to include at least 50 samples in the calibration and prediction sets. Sometimes, measurement of the primary analytical data of so many samples is excessively time consuming. The number of samples can be approximately halved, at the cost of computation time, by using only one calibration set and calculating the root-mean-square error of cross validation (RMSECV), as described in Section 9.9. In general, however, it is preferable to use an independent prediction set to investigate the validity of the calibration but the leave-one-out method significantly reduces the number of samples for which primary analytical data are required. 

For the approximately 1,500 points in this range of uniform heat flux CHF experiments, the root-mean-square error was -10%. 

The BC and K0 models, on the other hand, show much smaller root mean square errors, typically in the 1 % range, over an amazingly substantial range x of intensities fitted, Fig. 5.8, lower set of data points. Maximal deviations from the exact profiles amount to no more than twice the root mean square errors shown, that is well within the experimental uncertainties of the best measurements. The BC model is especially well suited to approximate quadrupole-induced profiles. The K0 model, on the other... 

The root mean square error per molecule is ca. 0.6 kJ/mol larger when the Hartree-Fock and (T) components were not improved. This effect is even more pronounced for the CCSD-F12b approximation, where this difference is ca. 2.8 kJ/mol. In the case of the CCSD(F12/fixed) approximation, one observes significant improvement of the statistics, which is likely due to a fortunate cancellation of the errors. 

CAMD = computer-aided molecular design ES = evolutionary strategies GA = genetic algorithm GFA = genetic function approximation LOF = lack of fit LSE = least squares error MARS = multivariate adaptive regression spline PLS = partial least squares QSAR = quantitative structure-activity relationships RMSE = root mean squared error. 

However, while the SFD method is the best of the approximate procedures considered, the level energies it yields are typically in error by more than a full level width, and the root mean square relative error associated with its widths is ca. 352. Thus, while the SFD method does qualitatively explain the bulk of the differences among the various values in Table 11, its inadequacies are unfortunately too large to allow its predictions to be the basis of attempts to refine the potential energy surface. A more detailed discussion of the nature and weaknesses of the various approximate methods may be found in Ref.(19). 

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