Scale-error complexity

The scale-error-complexity (SEC) surfaces. Instead of observing the prediction error with respect to resolution, it is also possible to monitor the complexity of the calibration/classification model. In PLS this can be measured by the number of PLS factors needed. How the error (e.g. RMSECV, RMSEP, PRESS) changes with varying the added scale and model complexity can be observed in scale-error-complexity (SEC) surfaces. In this case the first axis is the scales, the second axis is the model complexity (for PLS this is the number factors) and the third axis is the error. The complexity dimension is not limited to the number of PLS factors. For example classification and regression trees (CART) a measure based on tree depth and branching could be used [45],... 

Errors and confusion in modelling arise because the complex set of coupled, nonlinear, partial differential equations are not usually an exact representation of the physical system. As examples, first consider the input parameters, such as chemical rate constants or diffusion coefficients. These input quantities, used as submodels in the detailed model, must be derived from more fundamental theories, models or experiments. They are usually not known to any appreciable accuracy and often their values are simply guesses. Or consider the geometry used in a calculation. It is often one or two dimensions less than needed to completely describe the real system. Multidimensional effects which may be important are either crudely approximated or ignored. This lack of exact correspondence between the model adopted and the actual physical system constitutes the basic problem of detailed modelling. This problem, which must be overcome in order to accurately model transient combustion systems, can be analyzed in terms of the multiple time scales, multiple space scales, geometric complexity, and physical complexity of the systems to be modelled. 

Formation of polynuclear lead species with parameters close to isolated lead bromophenoxides during DPC synthesis was found by EXAFS of frozen active reaction mixtures (Pb-0 = 2.34 A, Pb Pb = 3.83 A). Noteworthy, in samples of final reaction mixtures, where catalyst was inactive, short Pb Pb distances were absent. These polynuclear compounds have been tested as lead sources in large-scale runs (small scale reactions were inconclusive due to heterogeneity of reaction mixtures because these compounds are less soluble than PbO). It was found that the use of lead bromophenoxides instead of PbO increases both Pd TON (by 25-35%), and reaction selectivity (from 65 - 67 % to 75 - 84 %). Activity of different lead bromophenoxides was about the same (within experimental error) but the best selectivity was observed for complex Pb602(0Ph)6Br2. Therefore, the gain in selectivity vs. loss due to additional preparation step should be analyzed for practical application. 

In dealing with the SGS terms, Revstedt et al. (1998, 2000) and Revstedt and Fuchs (2002) did not use any model rather, they assumed these terms were just as small as the truncation errors in the numerical computations. This heuristic approach lacks physics and does not deserve copying. A most welcome aspect of LES is that the SGS stresses may be conceived as being isotropic, i.e., insensitive to effects of the larger scales, to the way the turbulence is induced and to the complex and varying boundary conditions of the flow domain. Exactly this... 

Aside from the strained complexity of this sequence of actions, Charles Honorton [34] noted that Hansel s diagram, which is the main support of his conjectures, is labeled not to scale. If the drawing were to scale, it would be obvious that the two doors were actually too far apart for such a scenario to be possible. This certainly raises the question of whether the critics of parapsychology make motivated errors themselves. 

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