Variables, error thresholds

Figure 11. The error threshold of replication and mutation in genotype space. Asexually reproducing populations with sufficiently accurate replication and mutation, approach stationary mutant distributions which cover some region in sequence space. The condition of stationarity leads to a (genotypic) error threshold. In order to sustain a stable population the error rate has to be below an upper limit above which the population starts to drift randomly through sequence space. In case of selective neutrality, i.e. the case of equal replication rate constants, the superiority becomes unity, Om = 1, and then stationarity is bound to zero error rate, pmax = 0. Polynucleotide replication in nature is confined also by a lower physical limit which is the maximum accuracy which can be achieved with the given molecular machinery. As shown in the illustration, the fraction of mutants increases with increasing error rate. More mutants and hence more diversity in the population imply more variability in optimization. The choice of an optimal mutation rate depends on the environment. In constant environments populations with lower mutation rates do better, and hence they will approach the lower limit. In highly variable environments those populations which approach the error threshold as closely as possible have an advantage. This is observed for example with viruses, which have to cope with an immune system or other defence mechanisms of the host.

Appendix 6. Brillouin-Wigner Perturbation Theory of the Quasi-species. Appendix 7. Renormalization of the Perturbation Theory Appendix 8. Statistical Convergence of Perturbation Theory Appendix 9. Variables, Mean Rate Constants, and Mean Selective Values for the Relaxed Error Threshold... 

In Eq. (12), SE is the standard error, c is the number of selected variables, p is the total number of variables (which can differ from c), and d is a smoothing parameter to be set by the user. As was mentioned above, there is a certain threshold beyond which an increase in the number of variables results in some decrease in the quality of modeling. In fact, the smoothing parameter reflects the user s guess of how much detail is to be modeled in the training set. 

Here xik is an estimated value of a variable at a given point in time. Given that the estimate is calculated based on a model of variability, i.e., PCA, then Qi can reflect error relative to principal components for known data. A given pattern of data, x, can be classified based on a threshold value of Qi determined from analyzing the variability of the known data patterns. In this way, the -statistic will detect changes that violate the model used to estimate x. The 0-statistic threshold for methods based on linear projection such as PCA and PLS for Gaussian distributed data can be determined from the eigenvalues of the components not included in the model (Jack-son, 1992). 

Care needs to be taken if some components are present in trace quantities. If an estimated concentration is 0.5 ppm and the calculated value is 1 ppm, the scaled error is 100%. This is much too large an error for most variables and yet the absolute error might be acceptable for a trace component. In other situations, it might be necessary to define trace components with a high precision. A trace component threshold can be set, below which the convergence criterion is ignored. 

Additionally, many sources of variability, such as model misspecification, or dosing and sampling history, may lead to residual errors that are time dependent. For example, the residual variance may be larger in the absorption phase than in the elimination phase of a drug. Hence, it may be necessary to include time in the residual variance model. One can use a more general residual variance model where time is explicitly taken into account or one can use a threshold model where one residual variance model accounts for the residual variability up to time t, but another model applies thereafter. Such models have been shown to result in significant model improvements (Karlsson, Beal, and Sheiner, 1995). 

The threshold X from the Ej values is computed using a binary dummy variable matrix and the prediction errors from the previous variable selection dk vectors in step 1. 

Signal with deterministic changes. In this example, the noise-free signal is deterministic with some sudden changes in the mean. The variables are contaminated by iid Gaussian error of standard deviation 0.5, and the results are summarized in Fig. 11. Wavelet thresholding of the result of maximum... 

An essential step in FDI is the evaluation of the time history of residuals serving as fault indicators. To that end, one approach is to establish ARRs from a model of a system (cf. Fig. 1.3). These relations are algebraic or dynamic constraints between known continuous variables, i.e. system inputs u and measured output variables y that include known model parameters O. ARRs derived from a bond graph of a hybrid system model also depend on discrete switch state variables. For a healthy system, the time evolution of ARRs should ideally be identical to zero in all system modes. In practice, residuals will be within certain small error bounds due to measurement noise, parameter uncertainties, and numerical inaccuracies. If, however, faults in some system components occur, then the values of some residuals will be outside given thresholds and can serve as fault indicators. The sfructure of ARRs expressed in the FSM will then indicate whether faults can be isolated. Let m t) denote the vector... 

Figure 8.14 Illustration of a typical Youden plot (fictitious data) in which the independent variable is the quantity of (unspiked) analytical sample taken through the entire analytical method, keeping the volumes v and Y constant. The threshold value for is Cy = —Ay/By, and reflects the magnitude of bias errors arising from a fixed analyte loss on active sites during either or both of the extraction/clean-up stage (L ) and in the final analytical train (L). Modified from Boyd, Rapid Commun. Mass Spectrom. 7, 257 (1993), with permission of John Wiley Sons, Ltd.

Analytical redundancy relations are balance equations of effort or flow variables, in which unknown variables have been replaced by input variables and measured output variables and in which parameters are known. Evaluation of an ARR provides a residual that theoretically should be zero. In practice, however, the residual of an ARR is within certain error bounds as long as no faults occur during system operation. The value is not exactly zero over some time interval due to noise in measurement, parameter uncertainties, and numerical inaccuracies. If, however, the numerical value of a residual exceeds certain thresholds, then this is an indicator to a fault in one of the system s components. Noise in measured output variables may result in residual values indicating a fault that does not exist. Hence, measured data should pass appropriate filters before being used in ARRs. 

The effects of the statistical uncertainties of the loading and system parameters on the mean exceedance rate of a particular threshold are investigated for a linear SDOF-system with viscous damping. For this purpose the structural loading is described by the well-known stationary Kanai-Tajimi-earthquake-model. The analysis is simplified by utilizing an approximate solution for the threshold-crossing rate, for which the error with respect to the exact solution is shown to be small. Each of the parameters involved in the expression for the mean exceedance rate of the stationary response of the structure is considered a random variable. The respective effects of the statistical uncertainties of the parameters on the threshold-crossing rate, as expressed by the first- and second moments, are shown explicitely in the numerical examples. 

Variable step size codes adjust the step size in such a way, that the global error increment is kept below a certain tolerance threshold TOL. This requires a good estimation of this quantity. Like in the multistep case, the error can be estimated by comparing two different methods. Here, a Runge-Kutta method of order p and another method of order p -f 1 is taken to perform a step from tn to tn+i, say. The global error increment of the p order method is... 

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