Encyclopedia of Distances

Measuring the class-imbalance extent of multi-class problems

“…Although a measure is always a finite positive value, it is not necessarily upper bounded. For example, Kullback-Leibler divergence may be unbounded, and Manhattan and Euclidean distances [9], in this context, are upper bounded by the values 2 and 1, respectively.…”

“…Since ID can be instantiated with any chosen distance/similarity function, we first introduce the measures used in the experiments: from the metrics in the vector space [9], Manhattan 2 , Euclidean and Chebyshev distances are chosen. Together, the fdivergences [7], the most utilised measures for probability distributions, are also included.…”

Monotonicity-based consensus states for the monometric rationalisation of ranking rules with application in decision making

“…Different transformations of distance functions have called the attention of the scientific community [44]. Here, we consider a natural transformation that takes the value of the original function whenever this value does not exceed a fixed lower bound, but is penalized and set to a fixed upper bound when it does.…”

“…Truncation [44] is a common transformation of a distance function, resulting in a so-called truncated distance function, which can be been as an (l, l)-penalized distance function.…”

Information, possible worlds and the cooptation of scepticism

“…So, a very simple option is to choose the dual notion of close, i.e., distant, and hence of similarity, i.e., dissimilarity. Dissimilarity relations (including, for 14 There are many equivalent sets of axioms for a proximity space, see for example the one provided here tends to be more common and is only slightly modified from Deza and Deza (2009), 70, in order to match more closely the previous list of metric axioms.…”