Empirical objective functions

Whether simulation should be used to obtain the values of the empirical objective function, in which generations should it be used, and which particular simulation program should be employed to this end. 

How the actual values of the empirical objective function, which has been obtained either from experimental testing or from simulation, can be got from the database. 

The aim of this chapter is to show that the time and costs needed for the evaluation of empirical objective functions can be substantially reduced by means of an approach that is in optinusation referred to as surrogate modelling (Ratle, 2001 Ulmer et al., 2003 Ong et al., 2005 Zhou et al.. 

The traditional approach to optimize a process is schematically shown in Figure 2 its principle elements are the development of a model, model validation, definition of an objective function and an optimizing algorithn. The "model" can be (a) theoretical, (b) empirical or (c) a combination of the two. 

The minimization of the expected risk given by Eq. (1) cannot be explicitly performed, because P(, y) is unknown and data are not available in the entire input space. In practice, an estimate of 7(g) based on the empirical observations is used instead with the hope that the function that minimizes the empirical risk 7g p(g) (or objective function, as it is most commonly referred) will be close to the one that minimizes the real risk 7(g). 

There is one additional reason why the IT empirical risk is a better objective function to use. With the empirical risk given by Eq. (6), which is by definition a pointwise measure, it is clear how to define in practice the... 

Constraints in optimization arise because a process must describe the physical bounds on the variables, empirical relations, and physical laws that apply to a specific problem, as mentioned in Section 1.4. How to develop models that take into account these constraints is the main focus of this chapter. Mathematical models are employed in all areas of science, engineering, and business to solve problems, design equipment, interpret data, and communicate information. Eykhoff (1974) defined a mathematical model as a representation of the essential aspects of an existing system (or a system to be constructed) which presents knowledge of that system in a usable form. For the purpose of optimization, we shall be concerned with developing quantitative expressions that will enable us to use mathematics and computer calculations to extract useful information. To optimize a process models may need to be developed for the objective function/, equality constraints g, and inequality constraints h. 

The NONMEM program implements two alternative estimation methods, the first-order conditional estimation and the Laplacian methods. The first-order conditional estimation (FOCE) method uses a first-order expansion about conditional estimates (empirical Bayes estimates) of interindividual random effects, rather than about zero. In this respect, it is like the conditional first-order method of Lindstrom and Bates.f Unlike the latter, which is iterative, a single objective function is minimized, achieving a similar effect as with iteration. The Laplacian method uses second-order expansions about the conditional estimates of the random effects. ... 

To obtain the empirical estimates of a, Kowalski and Hutmacher (33) simulated 300 chnical trials for each combination of sample size and p, where the proportional reduction in CUP (0) was fixed to zero. Covariate and base models were fitted to each of the trials and the likelihood ratio tests were performed at the 5% level of significance. The percentage of trials where a statistically significant difference in CUP was observed provided an empirical estimate of a (i.e, PIoi = 0 is rejected when i/o is true). The data were analyzed with the NONMEM population phar-macokinetics/pharmacodynamics analysis software. The results suggested that an approximate nine-point change in the objective function should be used to assess statistical significance at the 5% level rather than the commonly used critical value of 3.84 for one degree of freedom. 

The Type I error (rejection of the reduced model in favor of the full model) that would result from the use of the theoretical critical value was assessed for each of the designs considered, and for three alternative NONMEM linearization methods first-order (FO), first-order conditional estimation (FOCE), and first-order conditional estimation with interaction (FOCEI). Type I error rates were assessed by empirical determination of the probability of rejection of the reduced model, given that the reduced model was the correct model. Data sets were simulated with the reduced model (FO, 1000 data sets FOCE/FOCEI, 200 data sets) and fitted using the full and reduced models. The empirical Type I error was determined as the percentage of simulated data sets for which a LRT statistic of 3.84 or greater was obtained. The 3.84 critical value for the LRT statistic corresponds to a significance level of 5%, for a distribution with 1 degree of freedom (for the one extra parameter in the full model). The LRT statistic was calculated as the difference between the NONMEM objective function values of the reduced and full models. The results of these simulations were also used to determine an empirical critical value that would result in the Type I error rate equal to the nominal 5% value. 

The data obtained from many processes are multivariate in nature, and have an empirical or theoretical model that relates the variables. Such measurements can be denoised by minimizing a selected objective function subject to the process model as the constraint. This approach has been very popular in the chemical and minerals processing industries under the name data rectification, and in electrical, mechnical and aeronautical fields under the names estimation or filtering. In this chapter all the model-based denoising methods are referred to as data rectification. 

In a previous work [19] we showed through empirical tests that there are optimal configurations regarding these two variables. In order to find these configurations we can apply an optimization technique, such as the Nelder-Mead method [20] to approximate the optimal values of the two parameters jumpahead distance and workahead size given the objective function is the performance of the application. The performance we measure in terms of average execution time between two samplings. 

A semi-empirical structure function that describes the spatial correlation of colloidal spherical objects embedded in a homogeneous matrix, derived using the Born-Cjreen approximation, is given by (Guinier, 1955) ... 

Thus we have a three dimensional global minimization problem for a complex density function (19). In literature two optimization functions has been proposed and used for this problem (Hanson, Westman, Zhu 2002, Hanson Westman 2002a). Both of these objective functions are based on the division of the log-return domain into bins and the calculation of theoretical and empirical bin frequencies. The weighted least squares function fit) is given by (Hanson, Westman, Zhu 2002) as follows ... 

In most other applications (including, for example, reaction kinetics), values of the objective function may be obtained analytically — that is, either as the result of setting the function input into a mathematical expression, or as the solution of an equation described with a mathematical expression (for example, of a differential equation). In contrast, values of functions describing the dependence of catalyst performance on its composition are obtained empirically, through experimental measurements. 

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