|InterJournal Complex Systems, 923
|Manuscript Number: |
Submission Date: 2004
|Metrics for sets of more than two points|
As conventionally defined, a metric is a function of two points that, intuitively, tells us how "spread out" from one another those two points are. However there are many scenarios where we need a function that tells us how spread out from one another a set of more than two points is. Potential applications of such an extension range throughout machine learning and statistical analysis. In particular, both supervised and unsupervised learning, as well clustering, are conventionally done using metrics that only take two arguments at a time. Accordingly, such techniques need to somehow combine values of a metric for all pairs of elements of a data set. The generalization of a metric to more than two points would allow such techniques to instead be applied to entire data sets at once. This paper shows how the conventional definition of a metric, and in particular the triangle inequality, can be extended to apply to collections of more than two points. This extension allows some of the points to be duplicates of one another. According, a natural generalization of it allows points that occur fractionally. This allows the extension to tell us how "spread out" a probability distribution is. It is also shown how the extended definition of a metric can be used to "bootstrap" from a measure of how spread out a set of points is to a measure of how spread out a set of such sets is. This can then be used to give a metric for how different from one another two probability distributions are.
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