[EMAIL PROTECTED] writes:
>
>I often see (and use) the term "metric" as a particular kind of
>measure. However, I have had difficulty in finding a clear definition.
>This is made more difficult because of the more common use of
>"metric" as an adjective denoting the system of measurement units.
>Please tell me what I mean when I call something "a metric."
>
>Thanks.
>Stephen Dubin VMD
>http://www.hometown.aol.com/dubinse
>[EMAIL PROTECTED]
>
>
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To qualify for a measure of distance to be called metric, one popular way
is to
see if the measure satisfies 'metric axioms,' which consist of the
following:
(1) d(jj)=0 ; the distance from point j to itself is zero.
(2) d(jk) is greater than or equal to zero; the distance is non-negative
(3) d(jk)=d(kj); the distance is symmetric.
(4) d(jk) is less than or equal to d(jp)+d(kp)' the triangular inequality,
that is,
the sum of any two segments of a triangle cannot be smaller than the
remaining segment.
(5) that d(jk)=0 means j=k; if the distance between the two points is zero,
the two points are the same.
If all the above are satisfied, the measure is called metric.
But, not all measures satify all of them.
(a) it is pseudometric if it satisfies only (1), (2), (3) and (4)
(b) it is semi-metric if it satisfies only (1), (2), (3) and (5)
(c) it is semi-pseudometric if it satisfies only (1), (2) and (3)
(d) it is ultrametric if (4) is replaced by (6), where
(6) d(jk) is less than or equal to the larger of d(jp) and d(kp).
There are a family of metrics for continuous variables, called the
Minkowski power metric,
d(jk)(p)={the sum of |x(ij)-x(ik)|, raised to the power p, over i}
provided that p is equal to or larger than 1.
When p=1, it is called the city-block metric.
When p=1 and the variables are binary, it is called the Hamming distance.
When p=2, it is the Euclidean distance.
I hope the above is sufficient as an answer to your inquiry.
Nishi
Shizuhiko Nishisato, Professor Emeritus, Measurement and Evaluation
Program,
CTL, OISE/UT, 252 Bloor Street West, Toronto, Ontario, Canada M5S 1V6
[EMAIL PROTECTED]
(Tel): 416-923-6641, X2696
http://fcis.oise.utoronto.ca/~snishisato
http://fcis.oise.utoronto.ca/~icmma
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