Reduction of dimensionality

An important aspect of biological transport is that nature makes extensive use of the reduction of dimensionality to speed up search and discovery (SD) (see also section C2.14.6.2). SD is enonnously enlranced upon moving from tliree to two or one dimensions, because the spatial extent to be explored is drastically reduced. Affinity follows kinetics in being enlranced upon moving from tliree dimensions to two dimensions 1791. 

Adam G and Delbrtick M 1968 Reduction of dimensionality in biological diffusion processes Structural Chemistry and Molecular Biology ed A Rich and N Davidson (San Francisco Freeman)... 

The principle of reduction of dimensionality will be illustrated schematically. In case that the property (+/—) of an object depends mainly on one... 

In this way, the discrimination problem by means of two variables which are represented in two dimensions is reduced to one dimension by means of a new variable dv = f(xi,x2). All the reductions of dimensionality, from m to graphically presentable three- or two dimensions, happen according to this principle where uncorrelated variables are generated. 

FIGURE 22.4. Decision tree for selection of reduction of dimensionality procedures. 

Techniques for the reduction of dimensionality are those that simplify the understanding of data, either visually or numerically, while causing only minimal reductions in the amount of information present. These techniques operate primarily by pooling or combining groups of variables into single variables, but may also entail the identification and elimination of low-information-content (or irrelevant) variables. 

The final pair of methods for reduction of dimensionality which will be tackled in this chapter are Fourier analysis and the life table analysis. Fourier analysis seeks to identify cyclic patterns in data and then either analyze the patterns or the residuals after the patterns are taken out. Life table analysis techniques are directed to identifying and quantitating the time course of risks (such as death, or the occurrence of tumors). 

Biomedical spectra are often extremely complex. Hyphenated techniques such as MS-MS can generate databases that contain hundreds of thousands or millions of data points. Reduction of dimensionality is then a common step preceding data analysis because of the computational overheads associated with manipulating such large datasets.9 To classify the very large datasets provided by biomedical spectra, some form of feature selection10 is almost essential. In sparse data, many combinations of attributes may separate the samples, but not every combination is plausible. 

In an interesting analysis of the effects of reduction of dimensionality on rates of adsorption/desorption reactions (26), the bimolecular rate of 10 M- s- has been reported as the lower limit of diffusion control. Based on this value, the rates given in Table III indicate the desorption step is chemical-reaction-controlled, likely controlled by the chemical activation energy of breaking the surface complex bond. On the other hand, the coupled adsorption step is probably diffusion controlled. 

One of the Important functions of principal component analysis Is the reduction of dimensionality so that an overview or graphical... 

Both graphics and reduction of dimensionality, and also the use of eigenvectors in some classification methods (e.g., SIMCA) require knowledge of the number of significant components. 

D. Axelrod and M. D. Wang, Reduction of dimensionality kinetics at reaction limited cell surface receptors, Biophys. J. 66, 588-600 (1994). 

Murray J. D. (1977) Reduction of dimensionality in diffusion processes antenna receptors of moths. In Lectures on Nonlinear-Differential-Equation Models in Biology, pp. 83-127. Oxford University Press, Oxford. 

On the other hand, diethyl l,4-butanediylbis(aminomethylene)-bis(cyanoacetate) H2(L24) (61) reacts with copper(II) acetate to give coordination polymer J [Cu(L24)J (62). In contrast to the hexacoordinate examples discussed so far, in 62 copper is only pentacoordinate. This leaves one cyano group of monomer [Cu(L24)] (63) unoccupied and as in stair-like 59 leads to reduction of dimensionality resulting in a zig-zag lD-stmcture for 62 (Scheme 23, Fig. 23) [168]. 

Astumian RD, Schelly ZA. (1984) Geometric effects of reduction of dimensionality in interfacial reactions. / Am Chem Soc 106 304-308. 

The master problem (M-MIP) can be rather large in the OA method. One option is to keep only the last linearization point, but this may lead to nonconvergence even in convex problems (e.g., see Bremicker et al, 1990). A rigorous reduction of dimensionality without greatly sacrificing the strength of the lower bound can be achieved in the case of the largely linear MINLP problem... 

Selective experiments offer a significant simplification of experiments, and a reduction of dimensionality, instrumentation requirements and data size, and in some cases also of the experimental time. For instance, measurement of spin-spin couplings between rare spin nuclei may require instrumentation for triple resonance experiments, including a special probe. In many cases selective measurements offer simple solutions for these demanding experiments. 

This procedure was used mainly to aehieve a reduction of dimensionality, i.e., to fit a A -dimensional subspace to the original / -variate observations p k). The statistics used to summarize the most important results was the percent of the total variation explained by the first k (usually two or three) components. In the case at hand the interpretation of PCA results was based on the diagrams of coefficients of variables (total concentrations of metals in the samples) and the scatter plot of samples (the stations separated in coastal, intermediate and offshore stations). 

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