Model metrics

Inclndes candidate measures and examples of model metrics processes and... 

The enterprise shall estabhsh, and place under control, the system baseline. This system-level configuration baseline shonld inclnde the system interface specification, the prodnet interface specifications, the system specification, the product specifications, and the integrated database that captures the design, data, models, metrics, changes, design rationale, and other pertinent information on decisions or clarification made to system requirements. 

When using multivariate analyses, there are a variety of model metrics which evaluate the goodness of fit of a model to an individual spectrum (not to be confused with spectral band metrics previously discussed, which are simply some observed property of a band in the spectrum) [8]. These metrics should give a good indication of when the shift of a spectrum has passed outside of an acceptable range for an individual model. 

In simple relaxation (the fixed approximate Hessian method), the step does not depend on the iteration history. More sophisticated optimization teclmiques use infonnation gathered during previous steps to improve the estimate of the minunizer, usually by invoking a quadratic model of the energy surface. These methods can be divided into two classes variable metric methods and interpolation methods. 

By changing from the simplest to larger aliphatic and cyclic ketones, structural factors may be introduced which favor alternative unimolecular primary photoprocesses or provide pathways to products not available to the simple model compound. In addition, both the increase in molecular size and irradiation in solution facilitate rapid vibrational relaxation of the electronically excited reactant as well as the primary products to thermally equilibrated species. In this way the course of primary and secondary reactions will also become increasingly structure-selective. In a,a -unsym-metrically substituted ketones, the more substituted bond undergoes a-cleavage preferentially. 

One could view the occurrence of the metric terms in the equations of motion as an annoying complication, but we hold a more positive view. First they assure that whatever the choice of parameters to be used as dynamical variables, that choice will not introduce unphysical artifacts. Second, the metric terms are another component of the theory with potential for providing guiding principles for development of XC models. Those terms also allow the mathematical origin of physical affects to be assigned. 

Approximations thus must be introduced that involve modeling both the XC potential and the metric tensor, and a truncation of the space within which to choose the unknown functions v, to finite dimension r < >. The modeling is based on the restt-icted ansatz chosen for the form of states used to determine paths that approximate D (p), D](p) and ). It can be carried... 

Fig. 2. Metrical data for [Fe3S4] clusters taken from the high-resolution crystsd structures of D. gigas Fdll (1.7 A resolution) (19, 61, 166), A. vinelandii Fdl (1.35 A resolution) (165), Emd the model complex (Et4N)3[Fe3S4(LSs)] (50).

Metrical parameters for the [FesSJ cluster in A. vinelandii Fdl and the [Fe3S4]° in the model complex have been determined to an accu-... 

The identities of 12 and 13 were confirmed by single crystal X-ray diffraction studies. Structural models of 12 and 13 along with selected metrical parameters are presented in Figs, 10 and 11 and Table III, respectively. 

A solution space, a, consisting of hyperrectangles defined in the decision space, X, is a basic characteristic common to all the learning methodologies that will be described in subsequent sections. The same does not happen with the specific performance criteria tfi, mapping models /, and search procedures 5, which obviously depend on the particular nature of the systems under analysis, and the type of the corresponding performance metric, y. 

In Section IV we considered a categorical performance metric y. Although that represents a common practice, especially when y defines the quality of a product or process operation, there are many instances where system performance is measured by a continuous variable. Even when y is quality-related, it is becoming increasingly clear that explicit continuous quality cost models should be adopted and replace evaluations of performance based on categorical variables. 

This Section addresses cases with a continuous performance metric, y. We identify the corresponding problem statements and results, which are compared with conventional formulations and solutions. Then Taguchi loss functions are introduced as quality cost models that allow one to express a quality-related y on a continuous basis. Next we present the learning methodology used to solve the alternative problem statements and uncover a set of final solutions. The section ends with an application case study. 

To illustrate how different m(X ) and x may happen to be, let s consider as a specific example (others can be found in Saraiva and Stephanopoulos, 1992c) a Kraft pulp digester. The performance metric y, that one wishes to minimize, is determined by the kappa index of the pulp produced and the cooking yield. Two decision variables are considered H-factor (xj), and alkali charge (X2). Furthermore, we will assume as perfect an available deterministic empirical model (Saraiva and Stephanopoulos, 1992c), /, which expresses y as function of x, i.e., that y =/(xi, X2) is perfectly known. 

Both situations with categorical and continuous, real-valued performance metrics will be considered and analyzed. Since Taguchi loss functions provide quality cost models that allow the different objectives to be expressed on a commensurate basis, for continuous performance variables only minor modifications in the problem definition of the approach presented in Section V are needed. On the other hand, if categorical variables are chosen to characterize the system s multiple performance metrics, important modifications and additional components have to be incorporated into the basic learning methodology described in Section IV. 

Given a space G, let g (x) be the closest model in G to the real function, fix). As it is shown in Appendbc 1, if /e G and the L°° error measure [Eq. (4)] is used, the real function is also the best function in G, g = f, independently of the statistics of the noise and as long as the noise is symmetrically bounded. In contrast, for the measure [Eq. (3)], the real function is not the best model in G if the noise is not zero-mean. This is a very important observation considering the fact that in many applications (e.g., process control), the data are corrupted by non-zero-mean (load) disturbances, in which cases, the error measure will fail to retrieve the real function even with infinite data. On the other hand, as it is also explained in Appendix 1, if f G (which is the most probable case), closeness of the real and best functions, fix) and g (x), respectively, is guaranteed only in the metric that is used in the definition of lig). That is, if lig) is given by Eq. (3), g ix) can be close to fix) only in the L -sense and similarly for the L definition of lig). As is clear,... 

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