Difference between revisions of "Choquet-Kendall-Matheron theorem"
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A theorem characterizing the distribution of a random closed set in terms of the Choquet capacity functional [[#References|[a1]]]. This theorem was established independently by D.G. Kendall [[#References|[a2]]] and G. Matheron [[#References|[a3]]] in their work on random closed sets. | A theorem characterizing the distribution of a random closed set in terms of the Choquet capacity functional [[#References|[a1]]]. This theorem was established independently by D.G. Kendall [[#References|[a2]]] and G. Matheron [[#References|[a3]]] in their work on random closed sets. | ||
− | Let | + | Let $ {\mathcal F} $ |
+ | and $ {\mathcal K} $ | ||
+ | be the family of closed and compact subsets of $ \mathbf R ^ {d} $, | ||
+ | respectively. The family $ {\mathcal F} $ | ||
+ | is endowed with the [[Hit-or-miss topology|hit-or-miss topology]], which is generated by | ||
− | + | $$ | |
+ | {\mathcal F} _ {G _ {1} \dots G _ {n} } ^ {K} = | ||
+ | $$ | ||
− | + | $$ | |
+ | = | ||
+ | \left \{ {F \in {\mathcal F} } : {F \cap G _ {i} \neq \emptyset ( i = 1 \dots n ) , F \cap K = \emptyset } \right \} , | ||
+ | $$ | ||
− | for a compact set | + | for a compact set $ K $ |
+ | and open sets $ G _ {i} $. | ||
+ | Let $ \Sigma _ {\mathcal F} $ | ||
+ | be the Borel $ \sigma $- | ||
+ | algebra generated by the hit-or-miss topology. Then $ \Sigma _ {\mathcal F} $ | ||
+ | is the smallest $ \sigma $- | ||
+ | algebra of subsets of $ {\mathcal F} $ | ||
+ | containing the sets | ||
− | + | $$ | |
+ | {\mathcal F} _ {K} = \left \{ {F \in {\mathcal F} } : {F \cap K \neq \emptyset } \right \} , K \in {\mathcal K}. | ||
+ | $$ | ||
− | Now, a random closed set | + | Now, a random closed set $ \Xi $ |
+ | is an $ {\mathcal F} $- | ||
+ | valued random element. Its distribution is described by the corresponding [[Probability measure|probability measure]] $ {\mathsf P} $ | ||
+ | on $ \Sigma _ {\mathcal F} $: | ||
− | + | $$ | |
+ | {\mathsf P} ( {\mathcal F} _ {G _ {1} \dots G _ {n} } ^ {K} ) = | ||
+ | $$ | ||
− | + | $$ | |
+ | = | ||
+ | {\mathsf P} ( \Xi \cap K = \emptyset, \Xi \cap G _ {i} \neq \emptyset, i = 1 \dots n ) . | ||
+ | $$ | ||
This distribution can also be characterized by the functional | This distribution can also be characterized by the functional | ||
− | + | $$ | |
+ | T _ \Xi ( K ) = {\mathsf P} ( \Xi \cap K \neq \emptyset ) , K \in {\mathcal K}. | ||
+ | $$ | ||
− | The functional | + | The functional $ T _ \Xi $ |
+ | is an alternating Choquet capacity of infinite order. This means that: | ||
− | i) | + | i) $ T _ \Xi $ |
+ | is upper semi-continuous ( $ K _ {n} \downarrow K $ | ||
+ | implies $ T _ \Xi ( K _ {n} ) \downarrow T _ \Xi ( K ) $; | ||
+ | cf. also [[Semi-continuous function|Semi-continuous function]]); | ||
− | ii) | + | ii) $ S _ {n} ( K;K _ {1} \dots K _ {n} ) \geq 0 $, |
+ | $ n \geq 0 $, | ||
+ | where | ||
− | + | $$ | |
+ | S _ {0} ( K ) = 1 - T _ \Xi ( K ) , | ||
+ | $$ | ||
− | + | $$ | |
+ | S _ {n} ( K;K _ {1} \dots K _ {n} ) = S _ {n - 1 } ( K;K _ {1} \dots K _ {n - 1 } ) - | ||
+ | $$ | ||
− | + | $$ | |
+ | - | ||
+ | S _ {n - 1 } ( K \cup K _ {n} ;K _ {1} \dots K _ {n - 1 } ) . | ||
+ | $$ | ||
− | The values | + | The values $ S _ {n} ( K;K _ {1} \dots K _ {n} ) $ |
+ | can be interpreted as the probability that $ \Xi $ | ||
+ | does not intersect $ K $ | ||
+ | but does intersect $ K _ {1} \dots K _ {n} $. | ||
− | The Choquet–Kendall–Matheron theorem asserts that given a functional | + | The Choquet–Kendall–Matheron theorem asserts that given a functional $ T $ |
+ | on $ {\mathcal K} $, | ||
+ | there exists a distribution $ {\mathsf P} $ | ||
+ | on $ {\mathcal F} $ | ||
+ | with | ||
− | + | $$ | |
+ | {\mathsf P} ( {\mathcal F} _ {K} ) = T ( K ) , K \in {\mathcal K}, | ||
+ | $$ | ||
− | if and only if | + | if and only if $ T $ |
+ | is an alternating Choquet capacity of infinite order with $ 0 \leq T ( K ) \leq 1 $ | ||
+ | and $ T ( \emptyset ) = 0 $. | ||
+ | This distribution is necessarily unique. | ||
See [[#References|[a4]]] for applications. | See [[#References|[a4]]] for applications. |
Latest revision as of 16:44, 4 June 2020
A theorem characterizing the distribution of a random closed set in terms of the Choquet capacity functional [a1]. This theorem was established independently by D.G. Kendall [a2] and G. Matheron [a3] in their work on random closed sets.
Let $ {\mathcal F} $ and $ {\mathcal K} $ be the family of closed and compact subsets of $ \mathbf R ^ {d} $, respectively. The family $ {\mathcal F} $ is endowed with the hit-or-miss topology, which is generated by
$$ {\mathcal F} _ {G _ {1} \dots G _ {n} } ^ {K} = $$
$$ = \left \{ {F \in {\mathcal F} } : {F \cap G _ {i} \neq \emptyset ( i = 1 \dots n ) , F \cap K = \emptyset } \right \} , $$
for a compact set $ K $ and open sets $ G _ {i} $. Let $ \Sigma _ {\mathcal F} $ be the Borel $ \sigma $- algebra generated by the hit-or-miss topology. Then $ \Sigma _ {\mathcal F} $ is the smallest $ \sigma $- algebra of subsets of $ {\mathcal F} $ containing the sets
$$ {\mathcal F} _ {K} = \left \{ {F \in {\mathcal F} } : {F \cap K \neq \emptyset } \right \} , K \in {\mathcal K}. $$
Now, a random closed set $ \Xi $ is an $ {\mathcal F} $- valued random element. Its distribution is described by the corresponding probability measure $ {\mathsf P} $ on $ \Sigma _ {\mathcal F} $:
$$ {\mathsf P} ( {\mathcal F} _ {G _ {1} \dots G _ {n} } ^ {K} ) = $$
$$ = {\mathsf P} ( \Xi \cap K = \emptyset, \Xi \cap G _ {i} \neq \emptyset, i = 1 \dots n ) . $$
This distribution can also be characterized by the functional
$$ T _ \Xi ( K ) = {\mathsf P} ( \Xi \cap K \neq \emptyset ) , K \in {\mathcal K}. $$
The functional $ T _ \Xi $ is an alternating Choquet capacity of infinite order. This means that:
i) $ T _ \Xi $ is upper semi-continuous ( $ K _ {n} \downarrow K $ implies $ T _ \Xi ( K _ {n} ) \downarrow T _ \Xi ( K ) $; cf. also Semi-continuous function);
ii) $ S _ {n} ( K;K _ {1} \dots K _ {n} ) \geq 0 $, $ n \geq 0 $, where
$$ S _ {0} ( K ) = 1 - T _ \Xi ( K ) , $$
$$ S _ {n} ( K;K _ {1} \dots K _ {n} ) = S _ {n - 1 } ( K;K _ {1} \dots K _ {n - 1 } ) - $$
$$ - S _ {n - 1 } ( K \cup K _ {n} ;K _ {1} \dots K _ {n - 1 } ) . $$
The values $ S _ {n} ( K;K _ {1} \dots K _ {n} ) $ can be interpreted as the probability that $ \Xi $ does not intersect $ K $ but does intersect $ K _ {1} \dots K _ {n} $.
The Choquet–Kendall–Matheron theorem asserts that given a functional $ T $ on $ {\mathcal K} $, there exists a distribution $ {\mathsf P} $ on $ {\mathcal F} $ with
$$ {\mathsf P} ( {\mathcal F} _ {K} ) = T ( K ) , K \in {\mathcal K}, $$
if and only if $ T $ is an alternating Choquet capacity of infinite order with $ 0 \leq T ( K ) \leq 1 $ and $ T ( \emptyset ) = 0 $. This distribution is necessarily unique.
See [a4] for applications.
References
[a1] | G. Choquet, "Theory of capacities" Ann. Inst. Fourier , V (1953-1954) pp. 131–295 |
[a2] | D.G. Kendall, "Foundations of a theory of random sets" E.F. Harding (ed.) D.G. Kendall (ed.) , Stochastic Geometry , Wiley (1974) pp. 322–376 |
[a3] | G. Matheron, "Random sets and integral geometry" , Wiley (1975) |
[a4] | D. Stoyan, W.S. Kendall, J. Mecke, "Stochastic geometry and its applications" , Wiley (1995) (Edition: Second) |
Choquet-Kendall-Matheron theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Choquet-Kendall-Matheron_theorem&oldid=19235