Multidimensional objects distance-based

Distance-based metrics quantify the diversity of a set of compounds as a function of their pairwise (dis)similarities in a descriptor space. It is important to mention that distance coefficients are analogous to distances in multidimensional geometric space, although they are usually not equivalent to such distances. For a distance coefficient to be described as a metric, it must possess the following four properties (1) Distance values must be nonzero and the distance from an object to itself must be zero. (2) Distance values must be symmetric. (3) Distance values must obey the triangular inequality. (4) Distances between nonidentical objects must be greater than zero. A coefficient containing only the first three properties is dubbed a pseudometric, and one without the third property is a nonmetric. 

Before delving into the specific similarity calculation, we start our discussion with the characteristics of attributes in multidimensional data objects. The attributes can be quantitative or qualitative, continuous or binary, nominal or ordinal, which determines the corresponding similarity calculation (Xu and Wunsch, 2005). Typically, distance-based similarity measures are used to measure continuous features, while matching-based similarity measures are more suitable for categorical variables. 

Distance-Based Similarity Measures Similarity measm-es determine the proximity (or distance) between two data objects. Multidimensional objects can be formalized as numerical vectors O, = oy = 1 data object and p is the number of dimensions for the data object Oy. Figure 5.1 provides an intuitive view of multidimensional data. The similarity between two objects can be measured by a distance function of corresponding vectors Oj and (. ... 

Since each molecule is represented by a vector of molecular descriptors, geometrically it is mapped to a point in a multidimensional space. The distance between two points, such as Euclidean distance, then measures the dissimilarity between the two molecules. Thus, the diversity function should be based on pair-wise distances between molecules in the subset. Another requirement for the diversity function is that, after the diversity value has been maximized by choosing dilferent subsets of molecules, the final subset that corresponds to the maximum function value is most diverse. An objective function was proposed in SAGE. In a related publication, an S-optimal function was used as the diversity function." Then, SA optimization techniques were used to optimize the diversity function by choosing different combinations of compounds or building blocks. 

It is generally accepted without proof that similarity and distance are complementary objects close together in multidimensional space are more alike than those further apart. Most of the similarity measures used in practice are based on some distance function, and whilst many such functions are referenced in the literature the most common is the simple Euclidean distance metric. 

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