Comparative modeling errors

Although comparative modeling is the most accurate modeling approach, it is limited by its absolute need for a related template structure. For more than half of the proteins and two-thirds of domains, a suitable template structure cannot be detected or is not yet known [9,11]. In those cases where no useful template is available, the ab initio methods are the only alternative. These methods are currently limited to small proteins and at best result only in coarse models with an RMSD error for the atoms that is greater than 4 A. However, one of the most impressive recent improvements in the field of protein structure modeling has occurred in ab initio prediction [155-157]. 

The errors in comparative models can be divided into five categories [58] (Fig. 1) ... 

To put the errors in comparative models into perspective, we list the differences among strucmres of the same protein that have been detennined experimentally (Fig. 9). The 1 A accuracy of main chain atom positions corresponds to X-ray structures defined at a low resolution of about 2.5 A and with an / -factor of about 25% [192], as well as to medium resolution NMR structures determined from 10 interproton distance restraints per residue [193]. Similarly, differences between the highly refined X-ray and NMR structures of the same protein also tend to be about 1 A [193]. Changes in the environment... 

System Representation Errors. System representation errors refer to differences in the processes and the time and space scales represented in the model, versus those that determine the response of the natural system. In essence, these errors are the major ones of concern when one asks "How good is the model ". Whenever comparing model output with observed data in an attempt to evaluate model capabilities, the analyst must have an understanding of the major natural processes, and human impacts, that influence the observed data. Differences between model output and observed data can then be analyzed in light of the limitations of the model algorithm used to represent a particularly critical process, and to insure that all such critical processes are modeled to some appropriate level of detail. For example, a... 

Table 20.1 summarises the model errors from the validation trials and shows that the model is successful in predicting the steady-state condition of the plant. Errors in waste brine strength and temperature must be compared with the total change across the cell which is about 13% for brine strength and 40°C for temperature. This is because the plant is a waste brine process changes in brine temperature and strength are much smaller for a resaturation process. 

There are a number of ways to model calibration data by regression. Host researchers have attempted to describe data with a linear function. Others ( 4,5 ) have chosen a higher order or a polynomial method. One report ( 6 ) compared the error in the interpolation using linear segments over a curved region verses using a curvilinear regression. Still others ( 7,8 ) chose empirical or spline functions. Mixed model descriptions have also been used ( 4,7 ). 

The f-test is similar to the t-test, but is used to determine whether two different standard deviations are statistically different. In the context of chemometrics, the f-test is often used to compare distributions in regression model errors in order to assess whether one model is significantly different than another. The f-statistic is simply the ratio of the squares of two standard deviations obtained from two different distributions ... 

Calibration Measurement Residuals Plot (Model Diagnostic) The calibration spectral residuals shown in Figure 5-53 are still structured, but are a factor of 4 smaller than the residuals when temperature was not part of the model Comparing with Figure 5-51, the residuals structure resembles the estimated pure spectrum of temperature. Recall that the calibration spectral residuals are a function of model error as well as errors in the concentration matrix (see Equation 5.18). Either of these errors can cause nonrandom features in the spectral residuals. The temperature measurement is less precise relative to the chemical concentrations and, therefore, the hypothesis is that the structure in the residuals is due to temperature errors rather than an inadequacy in the model. 

ANOVA of the data confirms that there is a statistically significant relationship between the variables at the 99% confidence level. The i -squared statistic indicates that the model as fitted explains 96.2% of the variability. The adjusted P-squared statistic, which is more suitable for comparing models with different numbers of independent variables, is 94.2%. The prediction error of the model is less than 10%. Results of this protocol are displayed in Table 18.1. Validation results of the regression model are displayed in Table 18.2. 

Only in situations where numerical uncertainty is small compared to modeling uncertainty we can successfully validate a calculation. After minimizing numerical errors there will still be other uncertainties in calculations due to for example variations in inlet conditions or due to inherent uncertainty in tabulated material properties, etc. These can be best handled by repeating the calculations with appropriate variations in the uncertain input quantities, thus resulting in say nc calculations with seemingly n, random outcomes the mean and variance of which are donated by Xc and. S 2, Similarly there would be ne repeated experiments of the same phenomenon with a, random outcomes with the corresponding mean and variance, Xe and S2e, respectively. The estimated modeling error is by definition the difference between the experimental mean and calculation mean, i.e. 

Here, we want to emphasize that one is able to calculate the fraction of the experimental error only if replicate measurements (at least at one point x ) have been taken. It is then possible to compare model and experimental errors and to test the sources of residual errors. Then, in addition to the GOF test one can perform the test of lack of fit, LOF, and the test of adequacy, ADE, (commonly used in experimental design). In the lack of fit test the model error is tested against the experimental error and in the adequacy test the residual error is compared with the experimental error. 

The variable sample time control algorithm was tested experimentally and the results compared with computer simulations. Tests were made with and without modeling error (parameter shift) for set point and load changes. 

Fig. 3.2 Correlation coefficient, mean biases, and Root Mean Square Errors(RMSE) for WRF/ Chem, comparing model forecasts of 8-h averaged peak ozone mixing ratios with those observed by surface monitoring stations. The statistics span a time period of 30 days. The model was run once a day at OOOOUTC...

The first requirement is generally easily met as the error in the equation solution typically becomes small compared to overall modeling error for moderate values of the error control tolerances. The tradeoff between the second and third requirements is more difficult and depends on the particular numerical characteristics of the system equations and the particular values of the optimization parameters. It is desirable to have some means of estimating and adjusting the precision error to optimize this tradeoff. This requires that the precision error be estimated, its effect on the optimization assessed, and the integrator tolerances adjusted appropriately. 

To evaluate the efficiency of a classification model the error rate can be compared with the no-model error rate (NOMER), that represents the error rate without a clas-... 

To be of any practical value, a response surface model should give a satisfactory description of the variation of y in the experimental domain. This means that the model error R(x) should be negligible, compared to the experimental error. By multiple linear regression, least squares estimates of the model parameters would minimize the model error. Model fitting by least squares multiple linear regression is described in the next section. 

A couple of additional checks are performed for the adequacy of the model. The coefficient of determination, or value, is 96.8%, which is very good. A lack-of-fit test is performed comparing the error in the estimated values at each data point with the estimated noise obtained from the repeated trials. The lack-of-fit test passes, indicating that the model s fit to the data is within the accuracy expected based on the data s noise. 

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