Fit residuals

Figure 2. Diffuse reflectance UV spectra of dehydrated [1.5]TS-1 experimental data and fit residue (dotted lines), gaussian fit components (solid lines) insert details for two different TS-1.

The inspection of the fit residuals, that is, the (normalized) differences between the experimental and fitted data point, is a reliable tool to check for deviations from the fitted model. Residuals should be statistically noncorrelated and randomly distributed around zero. For example, if a bi-exponential decay is fitted to a single exponential function, the residuals will show systematic errors. Therefore, correlations in the residuals may indicate that another fit model should be used. 

Figure 1. Top portion shows a plot of the observed Fenvalerate response vs. the mass (ng). Lower plot gives ordered, normalized residuals from the fit of model-3 to the data (Table IV) using the weights given in Table III. (Symbols indicate the five replicates, and the plotted residuals are normalized by the standard deviations for these individual replicates. The "goodness of fit residuals of the model to the means of the replicates are larger by 1/5T because they are normalized by the standard errors at each concentration.)...

Figure 7.8. Is spectra of vacuum-cleaved, clean flat MgO(lOO) and various water-dosed MgO(lOO) surfaces (3 min exposures at the pressures indicated). One (clean surface) and two component (water-dosed surfaces) fits are shown, along with the fit residual. The incident photon energy was 620 eV. (from [144])...

Other parameters, such as CNLS-fit residuals (Are and Ajm), also indicate the goodness of fit . They are defined as... 

FIGURE 6.7 Comparison between the smoothed exp. and the fitted tr( )/ of SO2. Upper part the smoothed exp. cr( )/ of SO2. On this upper part, the fitted curve is not superimposed on the exp. curve because these two curves are too close. The two components of this XS have been fitted simultaneously the strong component is described with Formula (29) (4 parameters) and the weak component with a Gaussian (3 parameters). The fitted parameters are given in Table 6.4. Lower part the fit residual which is typically 1% of the max amplitude. Note that the typical experimental uncertainty is significantly larger than 1%. Then the residual in not necessarily due to the model but may be mostly due to the experimental XS. 

FIGURE 6.S A comparison between the exp. and the fitted fitted simultaneously, each components being described with the Formula (29). The fit residual is at most about 0.5% of the max amplitude. The superimposed fast oscillations are due to the vibronic structures which have not been completely smoothed. 

Although Foster et al. (2000) conservatively fit only 1 or 2 different model As spectra to each sample spectrum, least-squares fitting does not impose limitations on the number of models used in the fit, nor is their any parameter besides the fit residual to guide selection of the models used in the fits. Well-constrained fits can be obtained using principal component analysis (PCA) to guide selection of the type and number of components used in linear least-squares fits (Ressler et al., 2000). However, use of PCA... 

Figure 13.7 XRD pattern of a ceria stabilized zirconia powder mixed with 20 wt.% of standard silicon (asterisks). Experimental data (O) profile fitting result (line) and difference between data and fit (residual, lower line) (a) WH plot for the stabilized zirconia phase, with indication of Miller indices, regression line and 95% confidence range (b). (Reprinted from ref. 38 with the permission of the International Union of Crystallography.)...

Experimental data (O) profile fitting result (line) and difference between data and fit (residual, lower line). Insets Log scale plots and Miller indices of Bragg peaks. 

Fig 6 (a) The temporal evolution of the fit residual at 2152 em of (—) air dropped Type A Small and ( —) Type B large ordnanee, (b) Probability density funetions generated using FLD using three features taken from residual, namely the initial, maximal and t = 0.5 values of AI, with weights 0.522, 0.548, and 0.654, respectively. 

As in ordinary two-way analysis, the scree plot based on fit values can sometimes be misleading, if the model is overfitting. Exchanging fit residuals with cross-validated... 

The main goal of the data analysis is usually to find X, but the residual E can give important clues to the quality of this model. Possibly, residuals obtained from a test-set or from cross-validation can be used instead of fitted residuals. Random noise or some symmetrical type of distribution for the elements of E is normally expected and this can be verified from plotting the residuals and by the use of diagnostics. A good description of the use of residuals in three-way analysis is given by Kroonenberg [1983],... 

By comparing the size of the residuals of the test data with the residuals from the calibration samples, possible outlying behavior can be detected. For the three left-out samples, the sum-of-squared residuals are shown together with the sum-squared residuals obtained in the calibration model. For the calibration samples, the sum-of-squared residuals are given both as the fitted residuals and those obtained from leave-one-sample-out cross-validation. The fitted residuals are usually small because the model has been found specifically to minimize these. This makes them difficult to compare with residuals from new samples. By using residuals from leave-one-sample-out cross-validation, this overfit problem is minimized because the residuals from the calibration samples are obtained under similar conditions to the new samples. 

The above established ARIMA (2, 0, 2) model is used to fit residual error sequence and obtain the residual sequence of residuals, and its autocorre-... 

Fig. 6. Radial distribution function of counterions and exchange broadening in CW ESR spectra, (a) The experimental line shape of 0.5 mM FS / 8 mM PDADMAC in water-giyc-erol (solid line) cannot be fitted without considering exchange broadening (dotted line), (b) Fit residuals for the central line in the spectrum of 0.5 mM FS /7.5 mM PDADMAC in water-ethanol assuming different radial distribution functions c(r). (c) Spectrum (top trace) and fit residual (bottom trace) for c(r) r for 0.5 mM FS / 7.5 mM PDADMAC in...

These residuals will be referred to as fit residuals. The values of the velocities predicted, at each substrate concentration, used in the calculation of these fit residuals are shown in Table 3.4. Finally, the random distribution of fit residuals shown in Fig. 3.10 suggests that the model fits the data adequately. A systematic trend in the fit residuals would suggest a systematic error in the fit and possibly a failure of the model to describe the behavior of the system. It is important to remember that these fit residuals should not be used in determination of the error stmcture of the data or to make judgments on possible weighting strategies. This would be the case only if y, = y,.. 

Figure 3.10. Fit residual analysis for the experimental data set. The patterns obtained suggest that the model fits the data well.

Aniline itself gives the worst fit (residual, -0.68). This, as will be discussed later, may be related to its log P, which is the lowest value of the compounds in the data set. 

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