Geodesic
From similarity to distance: axiom set,monotonic transformations and metric determinacy
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 11 (2018) no. 3, pp. 331-341

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How to normalise similarity metric to a metric space for a clusterization? A new system of axioms describes the known generalizations of distance metrics and similarity metrics, the Pearson correlation coefficient and the cosine metrics. Equivalent definitions of order-preserving transformations of metrics (both monotonic and pivot-monotonic) are given in various terms. The metric definiteness of convex metric subspaces $\mathbb{R}^n$ and $\mathbb{Z}$ among the pivot-monotonic transformations is proved. Faster formulas for the monotonic normalization of metrics are discussed.
Keywords: metric space, similarity axioms, similarity normalization, metric determinacy, longest common subsequence. 
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     author = {Sergej V. Znamenskij},
     title = {From similarity to distance: axiom set,monotonic transformations and metric determinacy},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {331--341},
     publisher = {mathdoc},
     volume = {11},
     number = {3},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2018_11_3_a9/}
}
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Sergej V. Znamenskij. From similarity to distance: axiom set,monotonic transformations and metric determinacy. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 11 (2018) no. 3, pp. 331-341. http://geodesic.mathdoc.fr/item/JSFU_2018_11_3_a9/