Probabilistic metric space
This article needs additional citations for verification. (July 2023) |
In mathematics, probabilistic metric spaces are a generalization of metric spaces where the distance no longer takes values in the non-negative real numbers R ≥ 0, but in distribution functions.[1]
Let D+ be the set of all probability distribution functions F such that F(0) = 0 (F is a nondecreasing, left continuous mapping from R into [0, 1] such that max(F) = 1).
Then given a non-empty set S and a function F: S × S → D+ where we denote F(p, q) by Fp,q for every (p, q) ∈ S × S, the ordered pair (S, F) is said to be a probabilistic metric space if:
- For all u and v in S, u = v if and only if Fu,v(x) = 1 for all x > 0.
- For all u and v in S, Fu,v = Fv,u.
- For all u, v and w in S, Fu,v(x) = 1 and Fv,w(y) = 1 ⇒ Fu,w(x + y) = 1 for x, y > 0.[2]
History[edit]
Probabilistic metric spaces are initially introduced by Menger, which were termed statistical metrics.[3] Shortly after, Wald criticized the generalized triangle inequality and proposed an alternative one.[4] However, both authors had come to the conclusion that in some respects the Wald inequality was too stringent a requirement to impose on all probability metric spaces, which is partly included in the work of Schweizer and Sklar.[5] Later, the probabilistic metric spaces found to be very suitable to be used with fuzzy sets[6] and further called fuzzy metric spaces[7]
Probability metric of random variables[edit]
A probability metric D between two random variables X and Y may be defined, for example, as
One may easily show that such probability metrics do not satisfy the first metric axiom or satisfies it if, and only if, both of arguments X and Y are certain events described by Dirac delta density probability distribution functions. In this case:
For all other random variables X, Y the probability metric does not satisfy the identity of indiscernibles condition required to be satisfied by the metric of the metric space, that is:
Example[edit]
For example if both probability distribution functions of random variables X and Y are normal distributions (N) having the same standard deviation , integrating yields:
In this case:
Probability metric of random vectors[edit]
The probability metric of random variables may be extended into metric D(X, Y) of random vectors X, Y by substituting with any metric operator d(x, y):
References[edit]
- ^ Sherwood, H. (1971). "Complete probabilistic metric spaces". Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. 20 (2): 117–128. doi:10.1007/bf00536289. ISSN 0044-3719.
- ^ Schweizer, Berthold; Sklar, Abe (1983). Probabilistic metric spaces. North-Holland series in probability and applied mathematics. New York: North-Holland. ISBN 978-0-444-00666-0.
- ^ Menger, K. (2003), "Statistical Metrics", Selecta Mathematica, Springer Vienna, pp. 433–435, doi:10.1007/978-3-7091-6045-9_35, ISBN 978-3-7091-7294-0
- ^ Wald, A. (1943), "On a Statistical Generalization of Metric Spaces", Proceedings of the National Academy of Sciences, 29 (6): 196–197, Bibcode:1943PNAS...29..196W, doi:10.1073/pnas.29.6.196, PMC 1078584, PMID 16578072
- ^ Schweizer, B. and Sklar, A (2003), "Statistical Metrics", Selecta Mathematica, Springer Vienna, pp. 433–435, doi:10.1007/978-3-7091-6045-9_35, ISBN 978-3-7091-7294-0
- ^ Bede, B. (2013). Mathematics of Fuzzy Sets and Fuzzy Logic. Studies in Fuzziness and Soft Computing. Vol. 295. Springer Berlin Heidelberg. doi:10.1007/978-3-642-35221-8. ISBN 978-3-642-35220-1.
- ^ Kramosil, Ivan; Michálek, Jiří (1975). "Fuzzy metrics and statistical metric spaces" (PDF). Kybernetika. 11 (5): 336–344.