Linear systems approach errors

Let us now consider replication with errors. For sufficiently accurate replication the system approaches a stationary mutant distribution. We do not observe selection of a single species. The target of the selection process is a unique combination of species determined by the dominant eigenvector of the value matrix W. We perform a linear transformation of variables and choose the eigenvectors of the matrix W as the new basis of the coordinate system ... 

Analytic expressions are available for assessing the propagation of errors through linear systems. Such approaches can be used as well when the variances are sufficiently small that the system can be linearized about its expectation value. Numerical techniques are generally needed to assess the propagation of errors through nonlinear systems. 

The concept of statistical linearization replaces a non-linear dynamical model with a linear model, whose, coefficients are explicitly formulated. However, since these coefficients must be evaluated in terms of statistics that are generated by the non-linear system and, therefore, unknown the usual approach is to evaluate these coefficients based upon assumed statistics. In the gaussian excitation case, the assumed statistics are gaussian. Thus, this lead to a further error in the approximation. 

FIGURE 7.10 Pseudo-binary phase diagram of the water, HNO3/DMDBTDMA, dodecane system to identify the third-phase limit. Experimental points (circles) and theory (lines) obtained from Baxter sticky hard-sphere approach. The theoretical line is obtained with the experimental determination of the linear variation of the stickiness parameter tt1 versus [HN03]/[DMDBTDMA]. The different lines illustrate the impact of the error in this T-1 experimental linear law. (From C. Erlinger, L. Belloni, T. Zemb, and C. Madic, Langmuir, 15 2290-2300, 1999. With permission). 

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