Cosine similarities

For attractive interactions (actually, for negative gj) a similar cosine term appears in the spin equation. The spins develop a gap of order gi /27r. The CDW and SS susceptibilities keep their power law behavior with... 

For an isotropic material, all orientations are equally probable and all such products that have an odd number of Tike direction cosines will vanish upon averaging-. This restricts the nonvanishing tensor elements to those such as xVaaa abba - Similarly for the elements Such orientational averaging is crucial in... 

For examples of different types of similarity measures, see Table 6-2. The Tanimoto similarity measure is monotonic with that of Dice (alias Sorensen, Czekanowski), which uses an arithmetic-mean normaJizer, and gives double weight to the present matches. Russell/Rao (Table 6-2) add the matching absences to the nor-malizer in Tanimoto the cosine similarity measure [19] (alias Ochiai) uses a geometric mean normalizer. 

The resulting similarity measures are overlap-like Sa b = J Pxi ) / B(r) Coulomblike, etc. The Carbo similarity coefficient is obtained after geometric-mean normalization Sa,b/ /Sa,a Sb,b (cosine), while the Hodgkin-Richards similarity coefficient uses arithmetic-mean normalization Sa,b/0-5 (Saa+ b b) (Dice). The Cioslowski [18] similarity measure NOEL - Number of Overlapping Electrons (Eq. (10)) - uses reduced first-order density matrices (one-matrices) rather than density functions to characterize A and B. No normalization is necessary, since NOEL has a direct interpretation, at the Hartree-Fodt level of theory. 

Empirically, the Dice coefficient has worked better than cosine similarity in retrieving actives and is the standard choice for use with the ap and tt descriptors. 

As the scalar product of two vectors is related to the cosine of the angle included by these vectors by Eq. (4), a frequently used similarity measure is the cosine coefficient (Eq. (5)). 

Cosine similarity coefficient Also known as the Carbo index c. EllWiB s... 

The hyperbolic sine, hyperbolic cosine, etc. of any number x are functions related to the exponential function e . Their definitions and properties are very similar to the trigonometric functions and are given in Table 1-5. 

The E-state indices may define chemical spaces that are relevant in similarity/ diversity search in chemical databases. This similarity search is based on atom-type E-state indices computed for the query molecule [55]. Each E-state index is converted to a z score, Z =(% -p )/0 , where is the ith E-state atomic index, p is its mean and O is its standard deviation in the entire database. The similarity was computed with the EucHdean distance and with the cosine index and the database used was the Pomona MedChem database, which contains 21000 chemicals. Tests performed for the antiinflamatory drug prednisone and the antimalarial dmg mefloquine as query molecules demonstrated that the chemicals space defined by E-state indices is efficient in identifying similar compounds from drug and drug-tike databases. 

The application of Eq. (6) to predict lipophilicity for compounds with several functional groups runs into problems. The difficulties are associated with intramolecular interactions, which could not be addressed by addihve schemes as used in the SLIPPER model. Therefore, the authors correct the logP prediction of a given molecule according to the lipophilicity values of the nearest neighbors by using cosine similarity measures and molecular fragments [13, 14]. 

Because x, as well as w are normalized, represents the cosine or correlation coefficient between the two vectors. In a variant of ART, Fuzzy ART, a fuzzy similarity measure is used instead of the cosine similarity measure [14]. 

The pragmatic beauty of the chemical fingerprint is that the more common features of two molecules that there are, the more common bits are set. The mathematic approach used to translate the fingerprint comparison data into a measure of similarity tunes the molecular comparison [5]. The Tanimoto similarity index works well when a relatively sparse fingerprint is used and when the molecules to be compared are broadly comparable in size and complexity [5]. If the nature of the molecules or the comparison desired is not adequately met by the Tanimoto index, multiple other indices are available to the researcher. For example, the Daylight software offers the user over ten similarity metrics, and the Pipeline Pilot as distributed offers at least three. Some of these metrics (e.g., Tversky, Cosine) offer better behavior if the query molecule is significantly smaller than the molecule compared to it. 

Similarly, many different types of functions can be used. Arden discusses, for example, the use of Chebyshev polynomials, which are based on trigonometric functions (sines and cosines). But these polynomials have a major limitation they require the data to be collected at uniform -intervals throughout the range of X, and real data will seldom meet that criterion. Therefore, since they are also by far the simplest to deal with, the most widely used approximating functions are simple polynomials they are also convenient in that they are the direct result of applying Taylor s theorem, since Taylor s theorem produces a description of a polynomial that estimates the function being reproduced. Also, as we shall see, they lead to a procedure that can be applied to data having any distribution of the X-values. 

Without repeating the work, a similar result is obtained for S(x) = cos m. Combination of sine and cosine series leads to the even more general exponential form, such that... 

Finally, Fig. 5.13 shows the adiabatic bending potential for nonlinear deformations. The angular dependence appears similar to a cosine-like (or dipole-dipole) behavior near equilibrium, but departs conspicuously from this mathematical form at larger deformation angles. The potential shows the strong propensity for linear F H F H-bonding arrangements consistent with maximization of n-o donor-acceptor overlap. 

Similar orthogonality relationships can be shown for cosines. In addition, sines and cosines are mutually orthogonal, i.e., for any m and n... 

The propagator nature of the Chebyshev operator is not merely a formality it has several important numerical implications.136 Because of the similarities between the exponential and cosine propagators, any formulation based on time propagation can be readily transplanted to one that is based on the Chebyshev propagation. In addition, the Chebyshev propagation can be implemented easily and exactly with no interpolation errors using Eq. [56], whereas in contrast the time propagator has to be approximated. 

FIGURE 2.10 Euclidean distance and city block distance (Manhattan distance) between objects represented by vectors or points xA and xB. The cosine of the angle between the object vectors is a similarity measure and corresponds to the correlation coefficient of the vector... 

On the other hand, factor analysis involves other manipulations of the eigen vectors and aims to gain insight into the structure of a multidimensional data set. The use of this technique was first proposed in biological structure-activity relationship (i. e., SAR) and illustrated with an analysis of the activities of 21 di-phenylaminopropanol derivatives in 11 biological tests [116-119, 289]. This method has been more commonly used to determine the intrinsic dimensionality of certain experimentally determined chemical properties which are the number of fundamental factors required to account for the variance. One of the best FA techniques is the Q-mode, which is based on grouping a multivariate data set based on the data structure defined by the similarity between samples [1, 313-316]. It is devoted exclusively to the interpretation of the inter-object relationships in a data set, rather than to the inter-variable (or covariance) relationships explored with R-mode factor analysis. The measure of similarity used is the cosine theta matrix, i. e., the matrix whose elements are the cosine of the angles between all sample pairs [1,313-316]. 

Screens are simple structural fragments, centroids, with the topological distance equal to 1 bond length between the central atom and the atoms maximally remote from it Cosine coefficients are calculated, and the sums of nondiagonal similarity matrix elements are used in ChemoSoffi program as a diversity measure the diversity coefficient can possess the value from 0 to 1, which correspond to minimal and maximal possible diversity of a selection. 

Identification involves the confirmation of a certain chemical entity from its spectrum by matching against the components of a spectral library using an appropriate measure of similarity such as the correlation coefficient, also known as the spectral match value (SMV). SMV is the cosine of the angle formed by the vectors of the spectram for the sample and the average spectrum for each product included in the library. 

The most prevalent among the similarity coefficients is the so-called cosine similarity index or correlation coefficient. For the field functions discussed in Subheading 2.4. it is usually called the Carbo similarity index, and this nomenclature will be used here as well,... 

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