Arithmetical descriptors

Arithmetical descriptors are, for example, the numbers of atoms of specihc elements, as given in the molecular formula, or the molecular mass. 

A molecular descriptor i5 is called an arithmetical descriptor if the value D(M) on M = (e.C.y) does not depend on y but solely on e and/or f. If this holds, then D (A1 ) depends on and alone, so thatD is also an arithmetical descriptor. In formal terms, D is arithmetical if and only... 

Basic arithmetical descriptors arise from the following mappings ... 

These are purely arithmetical descriptors. Arithmetical descriptors that are not purely arithmetical since they depend on come from the mappings... 

Example (Arithmetical descriptors) We recall a few basic arithmetical descriptors - Let M = (e, y) M be a molecular graph. The purely arithmetic indices A and A are the counts of all or of the non-H atoms, obtained from the function A with its values... 

These descriptors are not purely arithmetical. They may have different values for compounds of the same molecular formula (and of the same state distribution). They are nevertheless considered arithmetic descriptors since they are based merely on numbers of bonds. 

In contrast to the example in the previous section we now have compounds with several molecular formulas in our real library. It therefore makes sense to use arithmetical descriptors. In MOLGEN-QSPR there are 48 arithmetical descriptors available (see Appendix A.l). Since there are no compounds containing iodine, no free radicals and no charged species in the library, Nj, rel. Nj, rad and charge are useless. Further, there is only one compound containing a triple bond. Therefore the descriptors n and rel. n are also excluded. Fluorine, bromine, and sulfur-containing compounds are scarce, so that an even distribution over LS and TS is questionable. Therefore, we also exclude Np, rel. Np, Ns, rel. Ng, Np,., rel.Np. Thus, 35 arithmetical indices are retained. 

Some authors (Peinado et al. 2004a, 2004b Zea et al. 2007) have grouped aroma compounds in aromatic series based on their aroma descriptors. The overall value for each series is obtained by combining the odour activity value (OAV, which is defined as the ratio of the concentration of a compound to its perception thresholds) of the individual aroma compounds in the series. The combination of individual OAVs in order to calculate the overall value for a series cannot be interpreted as an arithmetic addition of odorant sensations. Some aspects of this classifying scheme can be subject to criticism, but in any case, the scheme is effective for comparing wines obtained with different aging methods inasmuch as the odour series always comprise the same compounds. 

The standard reference for molecular descriptors is the encyclopedic book [304]. Descriptors can be split into three main groups arithmetical, topological and geometrical descriptors. 

These descriptors are purely arithmetical, which is not the case for the descriptors arising from the following mappings ... 

We saw that the PD of propyl acrylates is modeled well using arithmetical and topological descriptors with OLS as the predicting method. Thus, computer-intensive... 

We started with the 48 arithmetical and 170 topological indices from Appendix A. Of these 218 descriptors, 25 ene removed as they are constant within our library... 

Figure 8.27 shows the arithmetic means of the number of nodes and descriptors depending on mindev. In addition, the corresponding numbers for the example property methyl ester (me-est) are shown. It is easy to see that CT complexity decreases with increasing mindev. 

Figure 8.30 shows the arithmetic means of misclassification rates for learning set and test set for various Z, depending on the number of descriptors. Obviously, for the learning set, MCE decreases monotonically with increasing number of descriptors and with increasing Z. The latter is also true for the test set. With respect to the number of... 

Tables 8.14 and 8.15 summarize the misclassification rates for the test sets. Table 8.14 contains results for variable selection using CART. Obviously, classification by LDA has better predictive ability than classification by CT for these descriptor sets. Looking at the arithmetic meem or medieui over all 77 structural properties, SVMs with radial kernel exhibit the lowest misclassification rates.

This appendix contains lists of molecular descriptors available in MOLGEN-QSPR, arithmetical, topological, and geometrical descriptors. Some were introduced in this book, the specifications of others can be found in [34,262). 

MOLGEN MOLGEN Freely available web-service that calculates 708 arithmetical, topological and geometrical descriptors http // molgen.de/... 

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