Dissimilarity indices

Momentum-space similarity and dissimilarity indices tend to be especially useful in situations for which the bonding topology is less important than the long-range valence electron density. In addition to model systems, the approach has now been applied successfully to a number of real problems, such as... 

All these peculiar characteristics have been commented here to introduce as soon as possible the reader to the evidence that similarity and dissimilarity indices are connected to each other in an inverse manner and are somehow interchangeable. Such a connection has been studied several years ago. ... 

As has been pointed out previously, both similarity-dissimilarity indices can be computed just knowing the corresponding QS measures. The CSI can be written using a generalized cosine expression of the angle subtended by two DFs ... 

It is trivial to see that a scheme about behavior and interconnection between both similarity-dissimilarity indices can be easily obtained. For example, (1) when = 1 = min, this corresponds to a collinear DF situation and results into maxi-... 

This new quantum similarity-dissimilarity index formulation does not necessarily have to coincide with the original similarity-dissimilarity indices (Equations 17.4 and 17.5). Therefore, both matrices (the original metric of the QOS DF Z and the metric of the DQOS tag set elements Z ) might provide complementary geometrical and topological information about the associated QO point cloud. A discussion on the nature of the QS metric matrices has been recently published. More information on this QS feature can be obtained. 

The technology of proximity indices has been available and in use for some time. There are two general types of proximity indices (Jain and Dubes, 1988) that can be distinguished based on how changes in similarity are reflected. The more closely two patterns resemble each other, the larger their similarity index (e.g., correlation coefficient) and the smaller their dissimilarity index (e.g., Euclidean distance). A proximity index between the ith and th patterns is denoted by D(i, j) and obeys the following three relations ... 

The Euclidean distance is the dissimilarity index most frequently used. It is characterized by invariance to translation and rotation,... 

The application of Eq. (9) to differential profiles is illustrated in the left-hand plot of Figure 5. With i = 1, it represents the wedge area between the two curves. If the curves do not cross each other, the nominator directly represents the difference of the two AUCs. If they intersect as shown in the example, the choice of absolute differences computes a general dissimilarity index the area difference would be obtained by using signed differences instead of absolute differences. 

If the two profiles do not cross, the nominator of Eq. (9) directly represents the difference of the two mean times. If they intersect at some time, signed differences compute the difference of the means and absolute differences provide the more general dissimilarity index. 

Snb-stmctnre diversity is most easily defined using metrics such as the Tanimoto Dissimilarity Index. These metrics are based on linear bitmaps (fingerprints) generated from the molecnlar fragments or compound sub-structures (Figure 3). This approach has been developed extensively by Daylight Chemical Information Systems. ... 

There are many different types of similarity indexes, including the association coefficients (e.g., Tanimoto coefficient [27], Jaccard coefficient [38], Hodgkin-Richards coefficient [39,40]), the correlation coefficients or cosinelike indexes, and the distance coefficients or dissimilarity indexes (e.g., Hamming distance) [26],... 

Values of Dab — 1) for this model series, computed using the same ab initio SCF wavefunctions, are also listed in Table 2. The same conclusions are obtained from this distance-like dissimilarity index as with the three similarity indices, but there is a very much larger variation (45.4). We have found that Dab is most useful in situations in which all the molecules are very closely related. There is no upper limit to r>iB( )-... 

Table 6. Values of the dissimilarity index Dab -1) for the C-H bonds in the series...

Although QS has started within such similarity-dissimilarity index premises, essentially the fact is that the elementary QS computational element building block reduces to the well-known scalar product of two DFs, a so-called similarity measure. Indeed, given two quantum systans, say [A,B), the familiar quantum mechanical theoretical basis permits to obtain their attached wavefunctions via solving the respective Schrbdinger equations. From the system wavefunctions, a pair of associated DF p,4(r),pg(r) can be simply set up, with the vector r representing some number of particle coordinates. In molecular QS studies, the usual DF chosen is the first-order one thus, vector r = (x,y,z) corresponds to one-electron position coordinate only. Then, the similarity measure between the system pair of DF is simply defined as the overlap similarity integral ... 

After collecting spectral data at different locations in the blender and at various hme points, Sanchez et al. [7] evaluated several qualita-hve blend endpoint methods. A mean standard deviation for the spectra collected at each hme point was compared over time to determine homogeneity. This method allowed authors to determine the character-ishcs of the mixture spectrum, corresponding to the desired blend state. Using that mixture spectrum, they defined a dissimilarity index based on the orthogonal projection of spectra collected at various hmes points onto the mixture spectrum. The norm of the resulting vector was then employed to evaluate the blend state. 

Ob-6 AvatyCO 1) In the QSAR-context, it is necessary and sufficient that p verifies conditions C2), i.e., to be a dissimilarity index. In this sense MSD was used as a steric parameter (instead of the Taft Eg constant, for example) in Hansch type structure-biological activity correlations. 2) These results obtained for MSD are valid... 

The refractive index of the sample can be written as a complex number 2 = n2 — ik2. At wavelengths where the sample is not absorbing, 2, the absorption constant, equals zero. However, kj is non-zero at wavelengths where the sample is absorbing. In transmission spectroscopy, the intensity of an absorption band depends almost entirely on k2 while in ATR the intensity of the same band is a complex function of 2 and 2- Nevertheless, the statement made previously still holds. There will be absorption bands in ATR at wavelengths where 2 0. Thus, bands are expected at the same wavelengths in transmission and in ATR but their intensities may be dissimilar. 

Both definitions are identical, but Eq. (12a) expresses all relative quantities (R, T, f2) as fractions, whereas the original definition according to Eq. (12b) expresses them as percentages. This index has found much attention in the subsequent literature (20,21), but some objections have been raised against the use of percentages and the similarity scale in the definition of f2, which is in opposition to the dissimilarity scale used generally in statistics (22). 

When a combination of CE and HPLC systems would be considered, the most dissimilar from the global set could be selected according to the above approach. For the CE methods, a response should be selected and applied with values in the same order of magnitude as the retention factors of the CSs, e.g., the migration times. Another possibility would be to use the so-called normalized migration indexes (see further Section III.C) for both the CE and the HPLC measurements. ... 

Fig. 3. Coverage of chemistry space by four overlapping sublibraries. (A) Different diversity libraries cover similar chemistry space but show little overlap. This shows three libraries chosen using different dissimilarity measures to act as different representations of the available chemistry space. The compounds from these libraries are presented in this representation by first calculating the intermolecular similarity of each of the compounds to all of the other compounds using fingerprint descriptors and the Tanimoto similarity index. Principal component analysis was then conducted on the similarity matrix to reduce it to a series of principal components that allow the chemistry space to be presented in three dimensions.

Besides having a convenient measure of the interrelationship between two variables, it is also useful to develop procedures to describe the relationships betweeen samples so that subsequently the samples can be grouped according to how similar or dissimilar they are to one another. One set of possible functions to describe the relationship between samples is the correlation and covariance function defined in the equations 6 to 9 with the meaning of the indexes changed so that j and k refer to different samples and the summations are taken over the n variables in the system. The use of such functions will be explored later in this chapter. 

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