Errors deterministic

CCF means different things to different people. Smith and Watson (1980) define CCF as the inability of multiple components to perform when needed to cause the loss of one or moi e systems. Virolainen (1984) criticizes some CCF analyses for including design errors and poor quality as CCF and points out that the phenomenological methods do not address physical and statistical dependencies. Here, CCF is classed as known deterministic coupling (KDC), known stochastic coupling (KSC), and unknown stochastic coupling (USC). 

A single experiment consists of the measurement of each of the m response variables for a given set of values of the n independent variables. For each experiment, the measured output vector which can be viewed as a random variable is comprised of the deterministic part calculated by the model (Equation 2.1) and the stochastic part represented by the error term, i.e.,... 

Finally, we should refer to situations where both independent and response variables are subject to experimental error regardless of the structure of the model. In this case, the experimental data are described by the set (yf,x,), i=l,2,...N as opposed to (y,Xj), i=l,2,...,N. The deterministic part of the model is the same as before however, we now have to consider besides Equation 2.3, the error in Xj, i.e., x, = Xj + ex1. These situations in nonlinear regression can be handled very efficiently using an implicit formulation of the problem as shown later in Section 2.2.2... 

We have seen that Lagrangian PDF methods allow us to express our closures in terms of SDEs for notional particles. Nevertheless, as discussed in detail in Chapter 7, these SDEs must be simulated numerically and are non-linear and coupled to the mean fields through the model coefficients. The numerical methods used to simulate the SDEs are statistical in nature (i.e., Monte-Carlo simulations). The results will thus be subject to statistical error, the magnitude of which depends on the sample size, and deterministic error or bias (Xu and Pope 1999). The purpose of this section is to present a brief introduction to the problem of particle-field estimation. A more detailed description of the statistical error and bias associated with particular simulation codes is presented in Chapter 7. 

Note that, by construction, all notional particles are identically distributed. Thus, in the absence of deterministic errors caused by using

true mean field, the Lagrangian PDF (/ w.xw) found from (6.194) and (6.195) would be equal... 

For multi-dimensional flows, the mean velocity field may be a function of x. For this case, deterministic errors can be introduced through (V )(A-I" t), resulting in a non-uniform distribution for Xin, (t). This is an important source of numerical error in transported PDF codes, and we will look at this problem in more detail in Chapter 7. 

The statistical and deterministic errors resulting from this approximation are discussed next. 

As with all statistical methods, the mean-field estimate will have statistical error due to the finite sample size (X ), and deterministic errors due to the finite grid size (S ) and feedback of error in the coefficients of the SDEs Ui,p). Since error control is an important consideration in transported PDF simulations, we will now consider a simple example to illustrate the tradeoffs that must be made to minimize statistical error and bias. The example that we will use corresponds to (6.198), where the exact solution141 to the SDEs has the form ... 

The exact solution occurs when the deterministic errors are null and corresponds to (t) written in terms of... 

Note that the spatial dependence of the mixing time enters the deterministic error through the expected value. Using (6.231), we can rewrite (6.232) as... 

From this expression, it is now evident that, without the spatial dependence of the mixing time, D (x, t + At) would be null, and the deterministic errors would be independent of time.147... 

Thus, the final expression for the change in the deterministic error at time t + At is... 

By again neglecting statistical errors due to and the deterministic errors at time t + 2At could be estimated by using... 

The form of the spatial dependence is not particularly important. However, by choosing a simple form we are able to find analytical expressions for the deterministic errors. 

However, it should be obvious to the reader that the deterministic errors will continue to grow as time stepping proceeds. Eventually, these errors may reach statistically stationary values149 that can most easily be determined by numerical experiments. In order to control these errors, it will be necessary to choose Nv and M sufficiently large. Similar conclusions can be drawn for more complicated examples (Pope 1995 Welton and Pope 1997 Xu and Pope 1999), and should be carefully considered when developing a transported PDF code. 

However, use of the grid-cell kernel induces a deterministic error similar to numerical diffusion due to die piece-wise constant approximation. 

Deterministic trend models are based on the assumption that the trend of a time series can be approximated closely by simple mathematical functions of time over the entire span of the series. The most common representation of a deterministic trend is by means of polynomials or of transcendental functions. The time series from which the trend is to be identified is assumed to be generated by a nonstationary process where the nonstationarity results from a deterministic trend. A classical model is the regression or error model (Anderson, 1971) where the observed series is treated as the sum of a systematic part or trend and a random part or irregular. This model can be written as... 

At the beginning of the twentiest century, the French mathematician, Henri Poincare, found that the solution of certain coupled nonlinear differential equations exhibits chaotic behavior although the underlying laws were fully deterministic. He pointed out that two systems starting with slightly different initial conditions would, after some time, move into very different directions. Since empirical observations are never exact in the mathematical sense but bear a finite error of measurement, the behavior of such systems could not be predicted beyond a certain point these systems seem to be of random nature. A random process - in contrast to a deterministic process - is characterized by From A follows B with probability pB, C with probability pc etc. 

In the introduction, we have already classified the optimization problems as deterministic and stochastic. It is evident that deterministic problems are based on functional models or models that disregard experiment error. Problems where one cannot neglect experiment error are stochastic ones and, as established, they are the subject of this book. Besides, optimization problems are by the number of factors divided into one-dimensional and more-dimensional. The Optimization problem grows with dimension. The problem becomes even more complicated if optimization is not done by one but by more responses simultaneously-multiple response processes. 

So, we find that the mean behavior of the stochastic model is described by the deterministic model we have already developed. The fundamental difference between the stochastic and the deterministic model arises from the chance mechanism in the stochastic model that generates so-called process uncertainty, or stochastic error. 

The stochastic error is expressed in (9.23) by the variance Var [Aj (t)] and co-variance Cov [Nj (t) Nk (t)] that did not exist in the deterministic model. This error could also be named spatial stochastic error, since it describes the process uncertainty among compartments for the same t and it depends on the number of drug particles initially administered in the system. For the sake of simplicity, assume riQi = uq for each compartment i. From the previous relations, the coefficient of variation CVj (t) associated with a time curve Nj (t) in compartment 3 is... 

Besides the hypothesis of spatially homogeneous processes in this stochastic formulation, the particle model introduces a structural heterogeneity in the media through the scarcity of particles when their number is low. In fact, the number of differential equations in the stochastic formulation for the state probability keeps track of all of the particles in the system, and therefore it accounts for the particle scarcity. The presence of several differential equations in the stochastic formulation is at the origin of the uncertainty, or stochastic error, in the process. The deterministic version of the model is unable to deal with the stochastic error, but as stated in Section 9.3.4, that is reduced to zero when the number of particles is very large. Only in this last case can the set of Kolmogorov differential equations be adequately approximated by the deterministic formulation, involving a set of differential equations of fixed size for the states of the process. 

Aside from the continuity assumption and the discrete reality discussed above, deterministic models have been used to describe only those processes whose operation is fully understood. This implies a perfect understanding of all direct variables in the process and also, since every process is part of a larger universe, a complete comprehension of how all the other variables of the universe interact with the operation of the particular subprocess under study. Even if one were to find a real-world deterministic process, the number of interrelated variables and the number of unknown parameters are likely to be so large that the complete mathematical analysis would probably be so intractable that one might prefer to use a simpler stochastic representation. A small, simple stochastic model can often be substituted for a large, complex deterministic model since the need for the detailed causal mechanism of the latter is supplanted by the probabilistic variation of the former. In other words, one may deliberately introduce simplifications or errors in the equations to yield an analytically tractable stochastic model from which valid statistical inferences can be made, in principle, on the operation of the complex deterministic process. 

Parametric population methods also obtain estimates of the standard error of the coefficients, providing consistent significance tests for all proposed models. A hierarchy of successive joint runs, improving an objective criterion, leads to a final covariate model for the pharmacokinetic parameters. The latter step reduces the unexplained interindividual randomness in the parameters, achieving an extension of the deterministic component of the pharmacokinetic model at the expense of the random effects. Recently used individual empirical Bayes estimations exhibit more success in targeting a specific individual concentration after the same dose. 

Since this monograph is devoted only to the conception of mathematical models, the inverse problem of estimation is not fully detailed. Nevertheless, estimating parameters of the models is crucial for verification and applications. Any parameter in a deterministic model can be sensibly estimated from time-series data only by embedding the model in a statistical framework. It is usually performed by assuming that instead of exact measurements on concentration, we have these values blurred by observation errors that are independent and normally distributed. The parameters in the deterministic formulation are estimated by nonlinear least-squares or maximum likelihood methods. 

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