Distance-based measures

A mathematically very simple classification procedure is the nearest neighbour method. In this method one computes the distance between an unknown object u and each of the objects of the training set. Usually one employs the Euclidean distance D (see Section 30.2.2.1) but for strongly correlated variables, one should prefer correlation based measures (Section 30.2.2.2). If the training set consists of n objects, then n distances are calculated and the lowest of these is selected. If this is where u represents the unknown and I an object from learning class L, then one classifies u in group L. A three-dimensional example is given in Fig. 33.11. Object u is closest to an object of the class L and is therefore considered to be a member of that class. 

Good, A.G. and Kuntz, I.D. Investigating the extension of pairwise distance pharmacophore measures to triplet-based descriptors./. Comput.-Aided Mol. Des. 1995, 9, 373-379. 

An efhcient implementation of distance-based diversity measures based on k-d trees./. Chem. Inf. Comput. Sci. 1999,... 

In order to apply the SA protocol, one of the keys is to design a mathematical function that adequately measures the diversity of a subset of selected molecules. Because each molecule is represented by molecular descriptors, geometrically it is mapped to a point in a multidimensional space. The distance between two points, such as Euclidean distance, Tanimoto distance, and Mahalanobis distance, then measures the dissimilarity between any two molecules. Thus, the diversity function to be designed should be based on all pairwise distances between molecules in the subset. One of the functions is as follows ... 

To classify a new sample, fc-NN computes its distances (usually, the multivariate Euclidean distances, see Eq. 7) from each of the samples of a training set, whose class membership is known. The k nearest samples are then taken into consideration to perform the classification generally, a majority vote is employed, meaning that the new object is classified into the class mostly represented within the k selected objects. Being a distance-based method, it is sensitive to the measurement units and to the scaling procedures applied. 

There are many ways to obtain distance-like measures among flavor compounds. First, stimuli are generally equated in subjective intensity so that judgments are based on quality rather than intensity. Odorants are diluted in an odorless grade of diethyl phthalate and tastants, in deionized water. Then subjects rate all the n(n-l)/2 possible pairs for a set of n stimuli along an undifferentiated 5" line ... 

Peak height. The distance between the peak (band) maximum and the peak base, measured in a direction parallel to the detector response axis and perpendicular to the time axis. 

The unique ligating behavior of the bridging 2,6-dimethoxyphcnyl ligand with respect to promoting a substantial decrease in the metal atom separation for molybdenum(II) dimers is even more prominent in the case of chromium. The chromium-chromium distance of 1.847(1) A in Cr2(DMP)4 (90) is more than 0.1 A less than the corresponding value in any other chromous dimer yet reported. To compare homonuclear multiple bonds among elements with inherently different atomic radii, Cotton, Koch, and Millar proposed a normalized value for intemuclear distances based on Pauling s atomic radius of the element in question (209). A simple definition of formal shortness as t/(M—M)/2r(M) then follows as a measure of the relative compactness of the attractive interaction (90). The formal shortness ratio of 0.778 for the quadruple bond in... 

The considerable efficiencies obtained apparently result from the suppression of the self-quenching of Chi a by separating the Chl-Chl intermolecular distance. Based on photoelectrochemical measurements, an extention of this study may elucidate the energy interactions between Chi and other photosynthetic pigments in the mixed monolayer systems. 

Fig. 3 Coulombic coupling between the lowest tttt transition moments of Watson Crick AT base pair (left) and two stacked and parallel Ts (right) as a function of the distance between the two bases. The distance is measured between N1 and N3 atoms of A and T, respectively, for the AT pair (left) and between the centers of the mases of the two Ts (right). (From Ref. [23])...

Several types of similarity measures can be used to compare the similarity between such objects [4]. Two types of the most often used measures are distance-based similarity coefficients and correlation coefficients. Representatives of them are as follows ... 

More recently, additional 2H "I J REDOR measurements on new PTX derivatives bound to MT provided three new interatomic distances that helped to define more precisely the bound conformation of PTX [87], The two PTX analogues labeled with fluorine and deuterium were designed to determine the three key distances based on the measurement of 2II— "I distances, as indicated in Fig. 9 (compounds 4 and 5). These interatomic distances rule out the polar and nonpolar conformers and further support the T-taxol conformation (Tablel). The polar conformation is rejected on the basis of distance IV, as the 4.5 A is much shorter than the > 8 A determined by REDOR. The nonpolar conformation can be ruled out on the basis of distances I and II, that are too short in the nonpolar conformer compared to the experimental ones (1.8 and 3.6 A short). Besides, bridged taxanes resembling the nonpolar conformation displayed no activity in the tubulin assembly assay [88], PTX-NY closely matches the REDOR distances. Nevertheless, it is discarded as the bioactive form of PTX due to its poor fit to the EC density [87],... 

Using characteristic values is a widespread and helpful means. An example for material characteristics is the specific energy input based measured in kWh/kg (see Section 6.5.1), while the available torque/axis distance measured in Nm/m3 is a machine characteristic. Dimensionless parameters (with the unit 1) play an important role for scale down/scale up considerations. The Reynolds number is a known dimensionless parameter, determining whether flow is laminar or turbulent. However, it is of little importance for extruder scale downs/scale ups. 

Diversity is typically measured using a distance-based or cell-based method cost is typically given as reactant cost/gm and physiochemical properties such as AMW are typically measured as the difference in the distribution of the property in the library compared to the distribution of the same property in a collection of known drugs. The weights, wl5 w2j w3> are user-defined and are typically set so that diversity is maximized, while the cost and physicochemical properties are minimized. This weighted-sum approach leads to a single solution that represents one particular compromise in the objectives. Several other groups have also adopted this approach [67, 71 73]. 

Distance-based methods require a definition of molecular similarity (or distance) in order to be able to select subsets of molecules that are maximally diverse with respect to each other or to select a subset that is representative of a larger chemical database. Ideally, to select a diverse subset of size k, all possible subsets of size k would be examined and a diversity measure of a subset (for example, average near neighbor similarity) could be used to select the most diverse subset. Unfortunately, this approach suffers from a combinatoric explosion in the number of subsets that must be examined and more computationally feasible approximations must be considered, a few of which are presented below. 

A diversity metric is a function to aid the quantification of the diversity of a set of compounds in some predefined chemical space. Diversity metrics fall into three main classes (1) Distance-based methods, which express diversity as a function of the pairwise molecular dissimilarities defined through measurement. (2) Cell-based methods, which define diversity in terms of occupancy of a finite number of cells that represent disjoint regions of chemical space. (3) Variance-based methods, which quantify diversity based on the degree of correlation between a compound s important features. 

After a distance function is defined, the diversity of a compound collection can be measured in a number of ways. Minimum intermolecular dissimilarity (9) (where is the distance between the tth and yth compounds in the collection C), and average nearest neighbor distance, (10), are two common examples of distance-based diversity measures. Figure 1 illustrates examples of compound subsets using a nearest-neighbor design metric. 

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