Distance measures Hamming

Euclidean and Hamming distance measures of torsional similarity. 

A very useful distance measure between ordered sets of discrete valued elements is the Hamming distance, Bn- Given two sets, x = (xi,...,Xn) and y= yi, .., yn),... 

Because these vectors live in an -dimensional hypercubic space, the use of non-integer distance measures is inappropriate, although in this special case the square of the Euclidean distance is equal to the Hamming distance. 

Other similarity coefficients used in similarity studies include the cosine coefficient, and the Hamming and Euclidean distance measures [7], Similarity coefficients can also be applied to vectors of attributes where the attributes are real numbers, for example, topological indices or physiochemical properties. 

Hamming distance The Hamming distance between two sequences of equal length is the number of character positions in which they differ. The Hamming Distance is a distance measure. 

Distance measures give 0 for identical structures and have an upper bound defined by the property space. The Euclidean and Hamming distances are the most common ... 

The most popular measures are Hamming and Tanimoto coefficients, which are listed below, together with other important distance measures on binary variables ... 

The most popular distance binary measures are Hamming and Tanimoto distances that are listed below (Table SIO), together with other distance measures on binary vectors. 

Distance measures between two sets of variables are important to avoid, for example, the selection of models, which are only seemingly diverse due to the presence of different descriptors, but closely correlated among themselves. Distance between the sets of variables can be measured by the Hamming distance where the total distance is the sum of the variables that differ in the two sets. However, the Hamming distance usually overestimates the distance between the two sets of variables, neglecting the variable correlations. 

Fig. 12.28 The differences between the Soergel and Hamming distance measures for various molecules (see text)...

To construct dissimilarity measures, one uses mismatches Here a + b is the Hamming (Manhattan, taxi-cab, city-block) distance, and a + h) is the Euclidean distance. 

If the binary descriptors for the objects s and t are substructure keys the Hamming distance Eq. (6)) gives the number of different substructures in s and t (components that are 1 in either s or but not in both). On the other hand, the Tanimoto coefficient (Eq. (7)) is a measure of the number of substructures that s and t have in common (i.e., the frequency a) relative to the total number of substructures they could share (given by the number of components that are 1 in either s or t). 

Bit vectors live in an -dimensional, discrete hypercubic space, where each vertex of the hypercube corresponds to a set. Figure 2 provides an example of sets with three elements. Distances between two bit vectors, vA and vB, measured in this space correspond to Hamming distances, which are based on the city-block Zj metric... 

As well as the Euclidean distance, other metrics have been proposed and employed to measure similarities of pattern vectors between objects. One method used for comparing and classifying spectroscopic data is the Hamming distance. For two pattern vectors i and X2 defined by Equation (38), the... 

Here we will consider the chirality measure of the second kind for planar benzenoids. Our measure of chirality will be based on the degree of similarity/dissimi-larity among the pair of enantiomers. We will consider all chiral benzenoids with the periphery P = 22, i.e., a total of 28 pairs which are shown in Figure 34. Clearly by inspection of Figure 34 it is hard to guess which pair or pairs of benzenoids are the most chiral. It is even difficult to speculate which pair or pairs of benzenoids would be the least chiral. We will therefore use Hamming distances between the codes of the two enantiomers to find the most dissimilar (hence, most chiral) and the most similar (i.e., the least chiral). 

A typical dissimilarity measure is the so-called Hamming distance based on the exclusive OR (cf. Figure 7.7) calculated as follows ... 

MCD is a 3D-measure of steric misfit between the most active compound and the others within a given series of ligands under study. It translates the topological similarity/dissimilarity MSD parameter, which is an extended Hamming distance, from 2D space into a 3D space (Ciubotariu et al. 1990). 

Since the set of cells in a cell-based CS are analogous to binaiy structural FPs, other similarity measures such as those based on the Tanimoto or Dice similarity coefficients given in Eqs. (1.8) and (1.9) can be used. Alternatively, the corresponding dissimilarity coefficients given in Eqs. (1.21) and (1.22) also can be used. As noted in Sect. 1.2.1.3, the numerator of the Tanimoto dissimilarity coefficient is just the Hamming distance, which is a measure of the number of differences between the two DB FPs. 

The second, the WSA approach proposed by Hamly et al. [50,51] for use with CCD detectors, in contrast, appears to be much more practical. In this case narrower entrance slits may be used and, in addition to the evaluation of the CP only, i.e. the line core (or the CP 1 or 2 etc.), other pairs of pixels, such as +3 and —3 or +4 and -4 are used for evaluation. These latter pixels measure the absorbance at the wings only, at a given distance from the line center, where the absorbance is still increasing linearly with the concentration. In this case the measurement is not influenced by line broadening, as the absorbance around the line center is not considered. This means that a set of linear calibration curves with a slope of 1.0 on a log-log scale is obtained. 

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