proximity measures can be metrics or non-metrics.

the following criteria defines a metric dissimilarity measurement: enter image description here

here is for a metric similarity measurement

  1. I would like to know the consequences of knowing whether such a proximity measure is metric or non-metric? what do we do if such measurement is not metric?
  2. why is triangle inequality not a criterion to prove similarity but dissimilarity?
  • 2
    $\begingroup$ A metric space is an idea from mathematical analysis and topology. The gist is that metric spaces have a notion of distance (the “metric”) that satisfies certain familiar properties (the three you posted). For a non-metric distance, consider the drive time between the suburbs and the city during the morning rush hour. It takes longer to drive into the city in the traffic than to go the opposite direction, right? This violates condition #2. $\endgroup$
    – Dave
    Jun 19, 2021 at 4:20
  • $\begingroup$ thank you @Dave, what I understood that we are questioning about credibitlity of the measurement. since the distance result is not symmetric then we cannot rely on the measurement as the case in your example. then in this case we should find some other way to calculate the distance, am i right? $\endgroup$ Jun 19, 2021 at 10:08
  • $\begingroup$ As Dave mentioned, a measure that satisfies the metric properties (ie proper distance metric) has some nice properties which can be exploited and arrive at or use theoretical mathematical results. Else a uniform approach cannot be followed and already made mathematical results about metric spaces cannot be used $\endgroup$
    – Nikos M.
    Jun 19, 2021 at 12:38


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