Over-parametrizations

Ishikawa etal. proposed an approach for the determination of the ligand-field (LF) parameters of a set of isostructural lanthanide complexes. This method consists of a simultaneous fit of the temperature dependence of magnetic susceptibilities and NMR spectra for the whole isostructural series [18]. In order to avoid over-parametrization a key restriction is imposed each parameter is expressed as a linear function of the number of f electrons, n ... 

However, for this potential to be fulfilled, the developed models must themselves fulfill a number of requirements. For instance, the models must be trustworthy, and contain the necessary degree of mechanistic details. These two requirements are difficult to fulfill simultaneously, since a mechanistically detailed model typically becomes unidentifiable (i.e., over-parametrized with respect to existing data), which means that the model predictions are the result of arbitrary choices on a parameter manifold, and thus much less trustworthy. 

Nonparametric statistics are often applied to interval data when sample sizes are very small. When using very small sample sizes, the variable data distribution often cannot be assured to be normal, a requisite for using parametric statistics. A normal, bell curve distribution is not a requirement of nonparametric models. Hence, they are preferred in this area over parametric models. Common nonparametric models follow. 

D2) Over-parametrization The more ligator atoms are involved in mixed-ligand complexes the larger is the number of AOM parameters to be determined with the inherent uncertainty in their determination from experiment. This may lead to an over-parametrized scheme, in particular when experimental data are scarce. 

Data Envelopment Analysis (DEA) is a nonparametric, deterministic performance analysis tool. DEA is a "data oriented" approach for evaluating the performance of a set of peer units called Decision Making Units (DMUs) which convert multiple inputs into multiple outputs (Cooper et al., 2000). DEA is among the highly preferred methods of performance or efficiency analysis basically due to a number of advantages over parametric methods. Unlike most other approaches like regression analysis that need a priori assumptions, DEA requires very few assumptions. It does not attempt to explain the nature of the relations between the multiple inputs and multiple outputs that belong to the analysis units. 

Application of the TST to complex reactions is extremely important, allowing to predict theoretically the values of pre-exponential factors for multistep reaction mechanisms (Fig. 3.14), where due to over-parametrization, direct numerical fitting of rate constants results in values with a large error interval. 

The OPA should not be confiised with an optical parametric oscillator (OPO), a resonant-cavity parametric device that is syncln-onously pumped by a femtosecond, mode-locked oscillator. 14 fs pulses, tunable over much of the visible regime, have been obtained by Hache and co-workers [49, with a BBO OPO pumped by a self-mode-locked Ti-sapphire oscillator. 

This implies that, at constant k, the line integral of the differential form s de, parametrized by time t, taken over the closed curve h) zero. This is the integrability condition for the existence of a scalar function tj/ e) such that s = d j//de (see, e.g., Courant and John [13], Vol. 2, 1.10). This holds for an elastic closed cycle at any constant values of the internal state variables k. Therefore, in general, there exists a function ij/... 

Despite their popularity, these methods normally have an inherent limitation—the fluid dynamics information they generate is usually described in global parametric form. Such information conceals local turbulence and mixing behavior that can significantly affect vessel performance. And because the parameters of these models are necessarily obtained and fine-tuned from a given set of experimental data, the validity of the models tends to extend over only the range studied in that experimental program. 

The angle bending in H9O occurs without breaking any bonds, and the electron correlation energy is therefore relatively constant over the whole curve. The HF, MP2 and MP4 bending potentials are shown in Figure 11.14, where the reference curve is taken from a parametric fit to a large number of spectroscopic data. ... 

Such Bayesian models could be couched in terms of parametric distributions, but the mathematics for real problems becomes intractable, so discrete distributions, estimated with the aid of computers, are used instead. The calculation of probability of outcomes from assumptions (inference) can be performed through exhaustive multiplication of conditional probabilities, or with large problems estimates can be obtained through stochastic methods (Monte Carlo techniques) that sample over possible futures. 

In their work [58], GY demonstrated that a standard Lennard-Jones model grossly over-predicted the well-depth of rare gas-halide ion dimer potential energy curves when they were parametrized to reproduce the neutral rare gas-halide dimer curves. They further showed that the OPNQ model performed just as badly when the charge dependence of the expressions were ignored, but the potential energy curves for both the neutral and ionic dimers could be simultaneously be reproduced if the charge dependence is considered. 

The experiments of Dou et al. (1991) also indicate that the heat transfer coefficient varied with radial position across the bed, even for a given cross-sectional-averaged suspension density. Their data, as shown in Fig. 20, clearly indicate that the heat transfer coefficient at the bed wall is significantly higher than that for vertical surfaces at the centerline of the bed, over the entire range of suspension densities tested. Almost certainly, this parametric effect can be attributed to radial variations in local solid concentration which tends to be high at the bed wall and low at the bed centerline. 

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