Asymmetric similarity

Asymmetry in a similarity measure is the result of asymmetrical weighing of a dissimilarity component - multiplication is commutative by definition, difference is not. By weighing a and h, one obtains asymmetric similarity measures, including the Tversky similarity measure c j aa 4- fih + c), where a and fi are user-defined constants. The Tversky measure can be regarded as a generalization of the Tanimoto and Dice similarity measures like them, it does not consider the absence matches d. A particular case is c(a + c), which measures the number of common features relative to all the features present in A, and gives zero weight to h. 

A similarity index called subsimilaritywhich is short for substructure similarity, has been developed by developed by Hagadone (37). In form it is identical to one of the family of asymmetric similarity indices developed by Tversky (6) that is discussed in Subheading 2.2.2.,... 

In analogy to Eq. 2.21 the asymmetric similarity indices are given, respectively, by... 

Asymmetric similarity can provide some benefits in similarity searches not afforded by its symmetric competitors. For example, consider as in Subheading 2.2.1., the query and target molecules, Q and T, respectively, and the asymmetric similarity coefficients given in Eqs. 2.26 and 2.27. If Q is relatively small (N.B. small and large are used here refer to the size of the set and not to the size of the corresponding molecule), that is, if Q T, then target molecules for which Q is an approximate subset will be selected using Eq. 2.26, that is,... 

This result is approximately independent of the size of T given that Q is an approximate subset of T. A comparable selection of molecules would not be obtained using the symmetric similarity coefficient in Eq. 2.19 or the asymmetric similarity coefficient given by Eq. 2.27 because as the target molecule increased in size the denominator would reduce the overall similarity values making selection less likely. If, on the other hand, Q is a relatively large, that is, if IQ I I T, then using the lower expression for asymmetric similarity in Eq. 2.27 will produce similar results... 

Fig. 3. Asymmetric similarity searching might provide some benefits not afforded by symmetric similarity searching. (A) Database searching using ISIS keys and symmetric similarity searching, SXan, will not yield enalapril as a database hit because the similarity value is too low, 0.58. (B) Whereas database searching using asymmetric similarity searching, S-jvc, could yield enalapril as a database hit because the asymmetric similarity value is 0.78.

The extreme forms, but not the intermediate forms, of asymmetric similarity defined by Tversky (6) given in Eqs. 2.26 and 2.27 can be transformed into two symmetric measures by taking the maximum and minimum of the set cardinalities in the denominators of the two equations. The forms of these equations are obtained in analogy to those developed by Petke (33) for vectors and field-based functions (see Subheadings 2.3. and 2.4. for further details) ... 

As is the case for asymmetric similarity indices, both SPetmax (A,B) and SPet. (A,B) are bounded by zero and unity, but are ordered with respect to each other and with respect to Tanimoto similarity, that is,... 

The asymmetric similarity coefficients become, in an analogous fashion (see Eq. 2.28)... 

Basis Product, symmetric similarity score, asymmetric similarity score, lead hopping. 

Asymmetric similarity measure has been first described by Tver-sky (22) to provide a general mathematical framework for the perception of similarity and later adapted to molecular similarity by Bradshaw (23). The mathematical formula for both similarity measurements against BPs are shown below ... 

Asymmetric similarity (AS) favors retrieval of basis products with the most features embedded within the query. 

Fig. 13.3. Comparison of symmetric and asymmetric similarity scores. A virtual product from VRXN-2-00051 is used as a query molecule. The two corresponding Basis Products are VRXN-2-00051 A 1 and VRXN-2-00051 B 1. In reference to the query molecule, their corresponding similarity scores are listed under SS and AS (see equations [1] and...

Search a database of Basis Products using Asymmetric Similarity measure. Here this search is done using the query molecule against a database of 106 explicit enumerated Basis Products. The asymmetry similarity search in the BP database is implemented using MDL Keys finger print (24) with ISIS host technology (25). 

The output is a set of Basis Products with high asymmetric similarity (AS) values (the default cutoff value is set to 90%) when they are mapped into the query molecule. The reaction schemes and reactants encoded by those Basis Products are then extracted, ranked, and used to form sub-regions of PGVL for subsequent just-in-time enumeration and symmetric similarity search against the query molecule. 

One further fact may be deduced from the results of experiments with glycophorin its disposition in the membrane is asymmetric. Similar studies of other membrane proteins show that each has a specific orientation... 

Asymmetric centres are only possible on sp3 carbons. An sp2 centre is planar and cannot be asymmetric. Similarly, an sp centre cannot be asymmetric. 

These asymmetric similarity measures are also called semisimilarity measures or one-sided similarity measures. Note, however, that if the shapes of A and B are identical, then both scaling factors and are equal to 1 ... 

Section 15.4 provides a discussion of similarity measures, which depend on three factors (1) the representation used to encode the desired molecular and chemical information, (2) whether and how much information is weighted, and (3) the similarity function (sometimes called the similarity coefficient) that maps the set of ordered pairs of representations onto the unit interval of the real line. Each of these factors is discussed in separate subsections. Section 15.5 presents a discussion of a number of questions that address significant issues associated with MSA Does asymmetric similarity have a role to play Do two-dimensional (2D) similarity methods perform better than three-dimensional (3D) methods Do data fusion and consensus similarity methods exhibit improved results Are different similarity measures statistically independent How do we compare similarity methods Can similarity measures be validated S ection 15.6 provides a discussion of activity landscapes... 

Tversky [45] addressed this problem using a feature-based approach, which admits the concept of asymmetric similarities (cf. the discussion of asymmetric similarities related to Tversky s work [46]). As will be seen in the sequel, asymmetric similarities also have a role to play when comparing molecules, albeit a relatively small one currently. In a manner that is quite like that used in molecular applications of similarity, (yide infra) lists of features are used to characterize the mental representations being compared. Some type of relationship is then used to assess the similarity based on the number of features that are common and different between two representations and their relative importance. The form of the functions investigated by Tversky [45] are closely related to some of those used in molecular similarity studies (see also Sections 15.4.2-15.4.4 and Table 15.3 and Table 15.4). 

Because they map onto [0,1], similarity relations are fuzzy relations [50], which differ from classical relations that map pairs of elements onto the set of binary values 0,1. Similarity relations satisfy two mathematical properties, namely, they are reflexive, S(i,j)=0 if m =m., and generally are symmetric, S(iJ)=S(j,i) for l ij n, but they are generally intransitive. Asymmetric similarities, which will be discussed in Section 15.5.1, have been employed in MSA, but the number of applications is relatively small to date. 

A number of workers have dealt with various aspects of the size dependency issue (vide supra). Bajorath and coworkers have addressed it from the stand point of the Tversky asymmenic similarity function [84, 96, 97]. Mestres and Maggiora [46] have discussed size dependency in the context of asymmetric similarity functions associated with vector- and function-based representations. 

Clearly, Equation 15.5.1 only strictly applies to set-based representations, although closely related asymmetric similarity functions can also be defined for graph-, vector-, and function-based representations (see Table 15.3 and Table 15.4 and the associated discussions in Sections 15.4.2 and 15.4.4). Because most applications employ set-based similarity functions and binary molecular fingerprints, the discussion in this section focuses on this category of MSA. Equivalent analyses can, however, be carried out with respect to other similarity measures (see e.g., [46]). 

D Asymmetric Similarity Searching In what follows, a brief description of an alternative approach will be outlined that does not combine the similarity values of the two asymmetric similarity functions [109], leading to what might be called 2D asymmetric similarity searches. Since there are now two asymmetric similarity functions rather than a single symmetric function, will the added information lead to more effective similarity searches ... 

FIGURE 15.4 2D plot of asymmetric similarity functions. The red dot corresponds to a pair of molecules where the ith molecule, which is taken to be active, is smaller than the jtb molecule with which it is paired. The blue dots located within the gray shaded region of the plot are also associated with the ith molecule but the molecules with which it is paired decrease in size (relative to the ith molecule) as one moves vertically up the gray shaded region. Dots located near the diagonal correspond to cases where the asymmetric similarities are approximately equal in value. For color details, please see color plate section. 

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