Plot, of residuals

Fig. 18. Plot of residual depth parameter as a function of the ratio H/E according to equation 10. A value of v = 0.41 is taken for PE. Data for die-drawn PE (O) and POM ( ) and for soda-glass (A) and hard-steel (A) are shown. H/E values for lamellar isotropic PE with a 200 A thick surface (ft) and isotropic chain-extended material (1 2 x 103 A) (f) are also indicated...

Parity diagrams the quantity calculated y, fc vs. the quantity observed yexp or plots of residual deviations (>> ,/, - y, xp) vs. predicted values should show uniform bands the scatter of points should be uniform any systematic deviations disqualify the model, which should then be rejected. The data points on plots of linearized equations should scatter uniformly. 

For these reasons, it is desirable to perform a series of simple calculations to determine if the field capacity for a given depth of soil is ever exceeded, rather than simply overlaying water inputs over plots of residue data. The following series of calculations addresses the primary issue of whether sufficient water was applied to the test system at appropriate intervals to create leaching opportunities ... 

As shown in Fig. 10, a semilog plot of residuals versus time is a straight line with a slope of —ka. 

Fig. 6.8. Typical plots of residual deviations random scattering (a), systematic deviations indicating nonlinearity (b), and trumpet-like form of heteroscedasticity (c)...

Fig. 16. Plot of residual rates versus predicted rate for Eq. (8).

The plot of residuals versus some measure of the time at which experiments were run can also be informative. If the number of hours on stream or the cumulative volume of feed passed through the reactor is used, nonrandom residuals could indicate improper treatment of catalyst-activity decay. In the same fashion that residuals can indicate variables not taken into account in predicting reaction rates, variables not taken into account as affecting activity decay can thus be ascertained. 

Further analysis of linearity data typically involves inspection of residuals for fit in the linear regression form and to verify that the distribution of data points around the line is random. Random distribution of residuals is ideal however, non-random patterns may exist. Depending on the distribution of the pattern seen in a plot of residuals, the results may uncover non-ideal conditions within the separation that may then help define the range of the method or indicate areas in which further development is required. An example of residual plot is shown in Figure 36. There was no apparent trend across injection linearity range. 

Plots of Residuals. Residuals can be plotted in many ways overall against a linear scale versus time that the observations were made versus fitted values versus any independent variable (3 ). In every case, an adequate fit provides a uniform, random scatter of points. The appearance of any stematic trend warns of error in the fitting method. Figures 4 and 5 shows a plot of area versus concentration and the associated plot of residuals. Also, the lower part of Figure 2 shows a plot of residuals (as a continuous line because of the large number of points) for the fit of the Gaussian shape to the front half of the experimental peak. In addition to these examples, plots of residuals have been used in SBC to examine shape changes in consecutive uv spectra from a diode array uv/vis spectrophotometer attached to an SBC euid the adequacy of linear calibration curve fits (1). 

Figure 5 Plot of residuals for the linear regression of Figure 4. (Reproduced with permission from Ref. 1. Copyright 1984, Elsevier.)...

The plot of residuals below shows that the procedure used (Barker s method with regression for H2) is not in this case very satisfactory, no doubt because the data do not extend close enough to xl = 0. 

Another result of interest is the plot of residual char vs. water influx ratio in Figure 4. At the conditions investigated here, the model predicts that as much as half of the char content of the raw coal can be left in the aftermath of the combustion... 

FIGURE 4.17 Plot of residual spectra from the training set (solid line) and the residual spectrum for the unknown spectrum (dashed line) shown in Figure 4.16. 

Example residual plots are depicted in Figure 5.9. If all assumptions about the model are correct, a plot of residuals (computed by Equation 5.14) against the estimated y, values should show a horizontal band, as illustrated in Figure 5.9a. A plot similar to Figure 5.9b indicates a dependence on the predicted value, suggesting that numerical calculations are incorrect or an intercept term has been omitted from the... 

Figure 4. Plot of residuals versus fitted values from the 2s experiment on hue. The location model includes the main effects of factors A and F only.

Here (AOH ) is the total amount of OH formed from photolyzed A per unit volume during the reaction time /. A condition for this integration is that E /c,(S,] remains constant this assumption is generally valid. Equation vii states that a semi-log plot of residual [M] should decline linearly with the amount of OH formed (AOH ), that is, with t)(AA), the slope being inversely proportional to the amount of scavenger present in the water. 

Fig. 5 shows the plot of residual colour and a value vs. co t values. It actually shows the distribution of sedimentation rates of particles formed and the changes of the rate for t = 0, 50 minutes, and 24 hours. Coagulant dose 17 mg/L is lower than the optimum dose for t = 0 (Fig. 4). 

The greatest sensitivity is observed for plots of residual errors. Residual errors normalized by the value of the impedance are presented in Figures 20.5(a) and (b), respectively, for the real and imaginary parts of the impedance. The experimentally measured standard deviation of the stochastic part of the measurement is presented as dashed lines in Figure 20.5. The interval between the dashed lines represents the 95.4 percent confidence interval for the data ( 2cr). Significant trending is observed as a function of frequency for residual errors of both real and imaginary parts of the impedance. 

Figure 7 shows the plot of residual potassium cyanide as parts per million of cyanide vs. reaction rate r. The slope of these lines is constant and is expressed as ... 

Figure i4-34 Top, Scatter plot showing an example of nonlinearity in the form of downward deviating x2 values at the upper part of the range. Bottom, Plot of residuals showing the effect of nonlinearity. At the upper end of the analytical measurement range, a sequence (run) of negative residuals is present from x= ISO to 200,... 

Figure 14-35 An example of weighted Deming regression analysis for the comparison of drug assays. A, The solid line is the estimated weighted Deming regression line, the dashed curves indicate the 95% confidence region, and the dotted line is the line of identity. B, A plot of residuals standardized to unit standard deviation.The homogeneous scatter supports the assumed proportional error model and the assumption of linearity.

The regression model vs. actual results scatter plot is shown in Figure 10 and the plot of residuals (y/ - ft) in Figure 11. Despite the apparent high correlation between tryptophan concentration and A12, the univariate model is a poor predictor, particularly at low concentrations. 

To verify the adequacy of the developed models of solvent-resistance in THF, chloroform, and MEK, normal probability plots were evaluated. Typical normal probability plots of residuals should be close to a straight line as shown in Fig. 19.7 because the underlying error distribution is expected to be normal.47 This means that the normality assumption is valid for the proposed model. Residuals that were intensified in the middle of straight line indicated that data were normally distributed. Also, there were no outliers in the model as indicated by absence of significant deviations from the straight line. A combination of the normal distribution of the model residuals (Fig. 19.7) and the very high values of adjusted R2 demonstrated a good quality of the model. 

Figure 4.5.4 Plot of residuals versus factor levels from an ANOVA of concentrations (pg L ) of boron...

Boron. There was no evidence of significant deviation from normality (p = 98%) of the B data. This was not the case for Zn and Sc where a significant (p = 3.78 10-10 and 1.25 10-6 respectively) deviation from normality was detected. The multifactor ANOVA was performed on the boron data, and a plot of residuals versus factor levels plot indicates a lack of homogeneity of the variances across different factor levels as can be seen in Figure 4.5.4. 

Zinc and scandium. The nonnormality of the distributions of Zn and Sc can be seen in the plots of residuals versus factors (Figure 4.5.5). Under these circumstances it is necessary to either apply a normalizing transformation, or to use nonparametric statistical procedures. Since the data distributions for Zn and Sc both demonstrate deviation from normality the statistical analyses of these two elements have been dealt with in the same paragraph. The box and whisker plots for these elements are presented in Figure 4.5.6 to provide further insight into their distributions. 

Prediction of the log reduction of an inoculated organism as a function of acid concentration, time, and temperature can also be done by a mathematical model developed for this purpose, using the second-order polynomial equation to fit the data. The following tests justified the reliability of the model the analysis of variance for the response variable indicated that the model was significant (P < 0.05 and R2 = 0.9493) and had no significant lack of fit (P > 0.05). Assumptions underlying the ANOVA test were also investigated and it was demonstrated that with the normal probability plot of residuals, plot of residuals versus estimated values for the responses, and plot of residuals versus random order of runs, that the residuals satisfied the assumptions of normality, independence, and randomness (Jimenez et al., 2005). 

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