Probabilistic metric space

2020 Mathematics Subject Classification: Primary: 54E70 [MSN][ZBL]

Generalizations of metric spaces (cf. Metric space), in which the distances between points are specified by probability distributions (cf. Probability distribution) rather than numbers. The general notion was introduced by K. Menger in 1942 and has since been developed by a number of authors. A treatment, comprehensive up to 1983, may be found in [SSk].

Let $ \Delta ^ {+} $ be the set of all functions $ F $ from the real line $ \mathbf R $ into the unit interval $ I $ = $ [ 0,1 ] $ that are non-decreasing and left-continuous on $ [ 0, \infty ) $, and such that $ F ( 0 ) = 0 $ and $ F ( \infty ) = 1 $, i.e., the set of all probability distribution functions whose support lies in the extended half-line $ \mathbf R ^ {+} = [ 0, \infty ] $. For any $ a \in [ 0, \infty ) $, let $ \epsilon _ {a} \in \Delta ^ {+} $ be defined by $ \epsilon _ {a} ( x ) = 0 $ for $ x \leq a $ and $ \epsilon _ {a} ( x ) = 1 $ for $ x > a $; and let $ \epsilon _ \infty \in \Delta ^ {+} $ be defined by $ \epsilon _ \infty ( x ) = 0 $ for all $ x < \infty $ and $ \epsilon _ \infty ( \infty ) = 1 $. Then, under the usual pointwise ordering of functions, given by $ F \leq G $ if and only if $ F ( x ) \leq G ( x ) $ for all $ x \in \mathbf R $, the set $ \Delta ^ {+} $ is a complete lattice with maximal element $ \epsilon _ {0} $ and minimal element $ \epsilon _ \infty $. There is a natural topological structure (topology) on $ \Delta ^ {+} $, namely, the topology of weak convergence (cf. also Weak topology), where $ F _ {n} \rightarrow F $ if and only if $ F _ {n} ( x ) \rightarrow F ( x ) $ at every point of continuity of $ F $. Under this topology $ \Delta ^ {+} $ is compact and connected (cf. Compact space; Connected space); moreover, this topology can be metrized (cf. Metrizable space), e.g., by a variant of the Lévy metric.

A triangle function is a binary operation $ \tau $ on $ \Delta ^ {+} $ satisfying the following conditions:

a) $ \tau ( F, \epsilon _ {0} ) = F $ for all $ F \in \Delta ^ {+} $;

b) $ \tau ( E,F ) \leq \tau ( G,H ) $ whenever $ E \leq G $, $ F \leq H $;

c) $ \tau ( E,F ) = \tau ( F,E ) $;

d) $ \tau ( E, \tau ( F,G ) ) = \tau ( \tau ( E,F ) ,G ) $.

It is also often required that $ \tau $ be continuous with respect to the topology of weak convergence, or that $ \tau $ satisfies the condition:

e) $ \tau ( \epsilon _ {a} , \epsilon _ {b} ) \geq \epsilon _ {a + b } $ for all $ a,b \in \mathbf R ^ {+} $.

Examples of triangle functions are convolution and the functions $ \tau _ {T} $ given by

$$ \tau _ {T} ( F,G ) ( x ) = \sup _ {u + v = x } T ( F ( u ) ,G ( v ) ) . $$

Here $ T $ is a $ t $- norm, i.e., a binary operation on $ I $ that, like $ \tau $, has an identity element (the number $ 1 $ in this case) and is non-decreasing, commutative, and associative. Particular $ t $- norms are the functions $ W $, $ \Pi $, and $ M $ given, respectively, by $ W ( a,b ) = \max ( a + b - 1,0 ) $, $ \Pi ( a,b ) = ab $, and $ M ( a,b ) = \min ( a,b ) $. The corresponding triangle functions $ \tau _ {W} $, $ \tau _ \Pi $, and $ \tau _ {M} $ are continuous and satisfy e).

A probabilistic metric space is a triple $ ( S, {\mathcal F}, \tau ) $, where $ S $ is a set, $ {\mathcal F} $ is a function from $ S \times S $ into $ \Delta ^ {+} $, $ \tau $ is a triangle function, such that for any $ p,q,r \in S $,

I) $ {\mathcal F} ( p,p ) = \epsilon _ {0} $;

II) $ {\mathcal F} ( p,q ) \neq \epsilon _ {0} $ if $ p \neq q $;

III) $ {\mathcal F} ( p,q ) = {\mathcal F} ( q,p ) $;

IV) $ {\mathcal F} ( p,r ) \geq \tau ( {\mathcal F} ( p,q ) , {\mathcal F} ( q,r ) ) $.

If $ {\mathcal F} $ satisfies only I), III) and IV), then $ ( S, {\mathcal F}, \tau ) $ is a probabilistic pseudo-metric space.

For any $ x \in \mathbf R ^ {+} $ and any $ p,q \in S $, the value of $ {\mathcal F} ( p,q ) $ at $ x $, usually denoted by $ F _ {pq } ( x ) $, is often interpreted as "the probability that the distance between p and q is less than x" .

Thus, the generalization from ordinary to probabilistic metric spaces consists of:

1) replacing the range $ \mathbf R ^ {+} $ of the ordinary metric by the space of probability distributions $ \Delta ^ {+} $;

2) replacing the operation of addition on $ \mathbf R ^ {+} $, which plays the pivotal role in the ordinary triangle inequality, by a triangle function. Note that for a function $ d $ from $ S \times S $ into $ \mathbf R ^ {+} $, if $ {\mathcal F} $ is defined via $ F _ {pq } = \epsilon _ {d ( p,q ) } $ and if $ \tau $ is a triangle function satisfying e), then $ ( S,d ) $ is an ordinary metric space; and conversely. If $ \tau = \tau _ {T} $ for some $ t $- norm $ T $, then the probabilistic metric space is a Menger space.

There is a natural topology on a probabilistic metric space, determined by the system of neighbourhoods $ N _ {p} ( \epsilon, \delta ) = \{ {q \in S } : {F _ {pq } ( \epsilon ) > 1 - \delta } \} $. However, a more interesting class of topological structures is obtained by designating a particular $ \phi \in \Delta ^ {+} $ as a profile function, interpreting $ \phi ( x ) $ as the maximum confidence associated with distances less than $ x $, and considering the system of neighbourhoods

$$ N _ {p} ( \phi, \epsilon ) = \left \{ q : {F _ {pq } ( x + \epsilon ) \geq \phi ( x ) - \epsilon , x \in ( 0, {1 / \epsilon } ) } \right \} . $$

These determine a generalized topology (specifically, a closure space in the sense of E. Čech). There is also an associated indistinguishability relation, defined by $ p ( { \mathop{\rm ind} } \phi ) q $ if and only if $ F _ {pq } \geq \phi $. This relation is a tolerance relation, i.e., is reflexive and symmetric, but not necessarily transitive.

Let $ ( \Omega, {\mathcal A},P ) $ be a probability space, $ ( M,d ) $ a metric space, and $ S $ the set of all functions from $ \Omega $ into $ M $. For any $ p,q \in S $, define $ {\mathcal F} ( p,q ) $ via

$$ F _ {pq } ( t ) = {\mathsf P} \left \{ {\omega \in \Omega } : {d ( p ( \omega ) ,q ( \omega ) ) < t } \right \} . $$

Then IV) holds with $ \tau = \tau _ {W} $. The resultant probabilistic pseudo-metric space is called an $ E $- space. For any $ \omega \in \Omega $, the function $ d _ \omega $ from $ S \times S $ into $ \mathbf R ^ {+} $ given by $ d _ \omega ( p,q ) = d ( p ( \omega ) ,q ( \omega ) ) $ is a pseudo-metric on $ S $, and the $ E $- space is pseudo-metrically generated in the sense that

$$ F _ {pq } ( t ) = {\mathsf P} \left \{ {\omega \in \Omega } : {d _ \omega ( p,q ) < t } \right \} . $$

Conversely, any such pseudo-metrically generated space is an $ E $- space. An important class of $ E $- spaces is obtained when $ ( M,d ) $ is the Euclidean $ n $- dimensional space and $ S $ is the set of all non-degenerate $ n $- dimensional spherically symmetric Gaussian vectors.

The idea behind the construction of an $ E $- space has been generalized. For example, if $ \Sigma $ is a set with some structure, e.g., a normed, inner product or topological space, then the set of all functions from $ ( \Omega, {\mathcal A}, {\mathsf P} ) $ into $ \Sigma $ yields a space in which that structure is probabilistic. This idea has recently been applied in cluster analysis, where the numerical dissimilarity coefficient has been replaced by an element of $ \Delta ^ {+} $. The result is a theory of percentile clustering [JS]. The principal advantage of percentile clustering methods is that, when working with distributed data, they permit one to classify first and then summarize, instead of summarizing first and then classifying.

Let $ f $ be a function from a metric space $ ( S,d ) $ into itself, and, for any non-negative integer $ m $, let $ f ^ {m} $ denote the $ m $ th iterate of $ f $. For any $ p,q \in S $, define the sequence $ \delta _ {pq } $ by

$$ \delta _ {pq } ( m ) = d ( f ^ {m} ( p ) ,f ^ {m} ( q ) ) , $$

and for any positive integer $ n $ define $ F _ {pq } ^ {( n ) } \in \Delta ^ {+} $ via

$$ F _ {pq } ^ {( n ) } ( t ) = $$

$$ = { \frac{1}{n} } \# \left \{ m : {0 \leq m \leq n - 1, \delta _ {m} ( p,q ) < t } \right \} , $$

where $ \# $ denotes the cardinality of the set in question. The number $ F _ {pq } ^ {( n ) } ( t ) $ may be interpreted as the probability that the distance between the initial segments, of length $ n $, of the trajectories of $ p $ and $ q $ is less than $ t $. Let

$$ F _ {pq } ( t ) = {\lim\limits \inf } _ {n \rightarrow \infty } F _ {pq } ^ {( n ) } ( t ) , $$

$$ F _ {pq } ^ {*} ( t ) = {\lim\limits \sup } _ {n \rightarrow \infty } F _ {pq } ^ {( n ) } ( t ) , $$

and let $ F _ {pq } $ and $ F _ {pq } ^ {*} $ be normalized to be left-continuous, hence in $ \Delta ^ {+} $. If $ {\mathcal F} $ is defined via $ {\mathcal F} ( p,q ) = F _ {pq } $, then, again, IV) holds with $ \tau = \tau _ {W} $. The resultant probabilistic pseudo-metric space is a transformation generated space. Note that $ F _ {pq } = F _ {f ( p ) f ( q ) } $, so that $ f $ is (probabilistic) distance-preserving.

If $ f $ is measure-preserving (cf. Measure-preserving transformation) with respect to a probability measure $ {\mathsf P} $ on $ S $, then $ F _ {pq } = F _ {pq } ^ {*} $ for almost all pairs $ ( p,q ) $ in $ S \times S $; and if, in addition, $ f $ is mixing, then there is a $ G \in \Delta ^ {+} $ such that $ F _ {pq } = F _ {pq } ^ {*} = G $ for almost all pairs $ ( p,q ) $.

The above ideas play an important role in chaos theory. For example, if $ S $ is a closed interval $ [ a,b ] $, if $ f $ is continuous, and if there is a single pair of points $ p,q \in [ a,b ] $ for which $ F _ {pq } \neq F _ {pq } ^ {*} $, then $ f $ is chaotic in a very strong sense. This fact leads to a theory of distributional chaos. Specifically, if $ ( S,d ) $ is compact, then $ f $ is distributionally chaotic if and only if there is a pair of points $ p,q \in S $ for which $ F _ {pq } \neq F _ {pq } ^ {*} $. Furthermore, the number

$$ \mu ( f ) = \sup _ {p,q \in S } { \frac{1}{d _ {S} } } \int\limits _ { 0 } ^ \infty {( F _ {pq } ^ {*} ( t ) - F _ {pq } ( t ) ) } {dt } , $$

where $ d _ {S} $ is the diameter of $ S $, provides a useful measure of the degree of distributional chaos. For details see [SSm], [SSS].

References

[SSk]	B. Schweizer, A. Sklar, "Probabilistic metric spaces" , Elsevier & North-Holland (1983) MR0790314 MR0719088 Zbl 0546.60010 Zbl 0524.54005
[JS]	M.F. Janowitz, B. Schweizer, "Ordinal and percentile clustering" Math. Social Sci. , 18 (1989) pp. 135–186 MR1016504 Zbl 0695.62144
[SSm]	B. Schweizer, J. Smítal, "Measures of chaos and a spectral decomposition of dynamical systems on the interval" Trans. Amer. Math. Soc. , 344 (1994) pp. 737–754 MR1227094 Zbl 0812.58062
[SSS]	B. Schweizer, A. Sklar, J. Smítal, "Distributional (and other) chaos and its measurement" (to appear)