RMS fitting error

Figure 3.10 The number of configurations and RMS fitting error as a function of the energy cut-off, as explained in the text.

The efficiency of the methods outlined above has been tested by calculating the intermolecular Coulomb energies and forces for a series of water boxes (64,128,256, 512 and 1024) under periodic boundary conditions [15, 62], The electron density of each monomer is expanded on five sites (atomic positions and bond mid-points) using two standard ABSs, A2 and PI.These sets were used to fit QM density of a single water molecule obtained at the B3LYP/6-31G level. We have previously shown that the A1 fitted density has an 8% RMS force error with respect to the corresponding ab initio results. In the case of PI, this error is reduced to around 2% [15, 16], Table 6-1 shows the results for the 5 water boxes using both ABSs (Table 6-7). 

As mentioned above, this is the covariance matrix not only of the coordinate increments of the last iteration but also of the final coordinates as determined by the fit, rm = Ari i = g. By means of rm, the errors and correlations of the independent and dependent coordinates can be assessed. Because of the presence of E in Eq. 59, the matrix 0Arm = 0rw will contain an empty (zero-filled) row and column where a coordinate has been kept fixed. 

Concentration and molecular weight dependences of the probe radius of gyration Rg for molecular weight P probes in solutions of molecular weight M matrix polymers as functions of matrix concentration c. The fits are to stretched exponentials Rgo exp(—ac"), with the percent root-mean-square fractional fit error %RMS, the materials, and the reference. Materials include EB-ethyl benzoate, PMMA-polymethylmethacrylate, pS-polystyrene. 

Listing 9.11 shows flie code changes to Listing 9.6 for this example. The best fit values of the parameters as shown in the selected ou ut of the listing and have a very small standard error. This arises because the model fits very accurately the data and there is very little random error associated with the data points. In fact almost all the fitting error occurs in the two data points with the smallest time value. The RMS error in the fitting is only 0.0019 miits along the dependent variable axis. With a model fliat provides an excellent functional fit to the data and... 

Equation (21) fits the published data belonging to the low-velocity regime (Fig. 27), with an rms error of 5.5 %, and is applicable to any system pressure. 

Equation 8 yielded a fine correlation with rms error of 0.88 kcal/mol a, (3 and y were -0.150, 0.114, and 0.0286, though the ASASA term could be replaced by a constant with little degradation of the fit. Nevertheless, we remained concerned about the size of the data set (it is statistically desirable to have at least 5 data points per descriptor) and slow convergence for the Coulombic energy components. 

Root Mean Square Error of Prediction (RMSEP) Plot (Model Diagnostic) The validation set is employed to determine the optimum number of variables to use in the model based on prediction (RMSEP) rather than fit (RMSEO- RM-SEP as a function of the number of variables is plotted in Figure 5.7S for the prediction of the caustic concentration in the validation set, Tlie cuive levels off after three variables and the RMSEP for this model is 0.053 Tliis value is within the requirements of the application (lcr= 0.1) and is not less than the error in the reported concentrations. 

Figure 3-28 H2O diffusion profile for a diffusion-couple experiment. Points are data, and the solid curve is fit of data by (a) error function (i.e., constant D) with 167 /irn ls, which does not fit the data well and (b) assuming D = Do(C/Cmax) with Do = 409 /im ls, which fits the data well, meaning that D ranges from 1 /rm /s at minimum H2O content (0.015 wt%) to 409 firn ls at maximum H2O content (6.2 wt%). Interface position has been adjusted to optimize the fit. Data are adapted from Behrens et al. (2004), sample DacDC3.

Applying the model to the high-pressure data also distinctly improved the quality of the fits for both samples and at all pressures. A striking example is the D2 multiplet of Pr3+, where the rms error rises from 14.8 cm-1 at ambient pressure to a maximum value of 20.1 cm-1 at 8 GPa, when using solely the one-electron crystal-field parameters. On the contrary, if the 5-function model is included, a maximum rms error of only 0.9 cm-1 is found at ambient pressure and even smaller values are found at higher pressures. 

As mentioned above, the expanded covariance matrix rni of the results contains a zero-filled row and column when a Cartesian coordinate has been kept fixed. A judiciously chosen variance G2 11) can be entered on the respective diagonal of rm, prior to the transformation (Eq. 61) if the fixed coordinate is afflicted with an (estimated) error and the propagation of this error to any of the derived internal coordinates is to be studied. For this reason it may even be practical to carry along in the expanded vector of variables jj an atom that has never been substituted (and will hence drop out of the fit by the application of E), but whose position can be estimated and is required for the calculation of certain bond lengths and angles involving atoms that were substituted. 

In the analysis of the ENDOR data for Radical I all three hyperfine coupling tensors fit the expected directions of the crystal structure very closely. These tensors were produced by fitting >90 accurately measured data points to theoretical equations with a total rms error of ca. 0.25 MHz. For Radical II the tensors are just as accurate, but the expected directions are off by ca. 10°. It can be seen that the Amid direction for both the >N6-H and C8-H couplings are both 8.3° from the computed ring perpendicular. This suggests that there is some slight deviation from planarity for this radical. 

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