Standard illuminant - Revision history

en>Anon423: /* CIE illuminants */ Fixing style/layout errors

2014-12-30T07:23:42Z

CIE illuminants: Fixing style/layout errors

← Older revision		Revision as of 09:23, 30 December 2014
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	~~I would like to introduce myself to you, I am Andrew~~ and ~~my spouse doesn't like~~ it ~~at all~~. Mississippi is ~~exactly where his home is~~. ~~Office supervising~~ is ~~what she does for a residing~~. ~~Doing ballet~~ is ~~some thing she would never give up~~.<br><br>~~Here is~~ my web ~~blog good psychic,~~ [http://~~ltreme~~.~~com~~/~~index.php~~?do=~~/profile-127790/info/ check these guys out~~],		The writer's title is Andera and she believes it sounds fairly good. Mississippi is the only place I've been residing in but I will have to move in a yr or two. To perform lacross is the factor I adore most of all. Credit authorising is how he tends to make cash.<br><br>Take a look at my web page; [http://203.250.78.160/zbxe/?document_srl=1792908 psychic phone readings]

en>Monkbot: /* Illuminants B and C */Fix CS1 deprecated date parameter errors (test)

2014-02-26T16:21:37Z

Illuminants B and C: Fix CS1 deprecated date parameter errors (test)

← Older revision		Revision as of 18:21, 26 February 2014
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	~~In [[mathematics]]~~, ~~a '~~'~~'random compact set''' is essentially a [[compact space\|compact set]]-valued [[random variable]]. Random compact sets are useful in the study of attractors for [[random dynamical system]]s.~~		I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. Mississippi is exactly where his home is. Office supervising is what she does for a residing. Doing ballet is some thing she would never give up.<br><br>Here is my web blog good psychic, [http://ltreme.com/index.php?do=/profile-127790/info/ check these guys out],

	~~==Definition==~~

	~~Let <math>(M, d)</math> be a [[complete space\|complete]] [[separable space\|separable]] [[metric space]]. Let <math>\mathcal{K}</math> denote the set of~~ all ~~compact subsets of <math>M</math>~~. ~~The Hausdorff metric <math>h</math> on <math>\mathcal{K}</math>~~ is ~~defined by~~

	~~:<math>h(K_{1}, K_{2}) := \max \left\{ \sup_{a \in K_{1}} \inf_{b \in K_{2}} d(a, b), \sup_{b \in K_{2}} \inf_{a \in K_{1}} d(a, b) \right\}~~.~~</math>~~

	~~<math>(\mathcal{K}, h)</math>~~ is ~~also а complete separable metric space. The corresponding open subsets generate~~ a ~~[[sigma algebra\|σ-algebra]] on <math>\mathcal{K}</math>, the [[Borel sigma algebra]] <math>\mathcal{B}(\mathcal{K})</math> of <math>\mathcal{K}</math>~~.

	~~A '''random compact set'''~~ is ~~а [[measurable function]] <math>K</math> from а [[probability space]] <math>(\Omega, \mathcal{F}, \mathbb{P})</math> into <math>(\mathcal{K}, \mathcal{B} (\mathcal{K}) )</math>~~.

	~~Put another way, a random compact set is a measurable function <math>K \colon \Omega \to 2^{M}</math> such that~~ <~~math~~>~~K(\omega)~~<~~/math~~> is [~~[almost surely]] compact and~~

	:~~<math>\omega \mapsto \inf_{b \in K(\omega)} d(x, b)<~~/~~math>~~

	~~is a measurable function for every <math>x \in M<~~/~~math>~~.

	~~==Discussion==~~

	~~Random compact sets in this sense are also [[random closed set]]s as in Matheron (1975). Consequently their distribution is given by the probabilities~~

	~~:<math>\mathbb{P} (X \cap K = \emptyset)<~~/~~math> for <math>K \in \mathcal{K}.</math>~~

	~~(The distribution of а random compact convex set is also given by the system of all inclusion probabilities <math>\mathbb{P}(X \subset K)~~.~~</math>)~~

	~~For <math>K~~ = ~~\{ x \}</math>, the probability <math>\mathbb{P} (x \in X) <~~/~~math> is obtained, which satisfies~~

	~~:<math>\mathbb{P}(x \in X) = 1~~ - ~~\mathbb{P}(x \not\in X).</math>~~

	~~Thus the '''covering function''' <math>p_{X}</math> is given by~~

	~~:<math>p_{X} (x) = \mathbb{P} (x \in X)</math> for <math>x \in M.<~~/~~math>~~

	~~Of course, <math>p_{X}<~~/~~math> can also be interpreted as the mean of the indicator function <math>\mathbf{1}_{X}</math>:~~

	~~:<math>p_{X} (x) = \mathbb{E} \mathbf{1}_{X} (x).</math>~~

	The covering function takes values between <math> 0 </math> and <math> 1 </math>. The set <math> b_{X} </math> of all <math>x \in M</math> with <math> p_{X} (x) > 0 </math> is called the '''support''' of <math>X</math>. The set <math> k_X </math>, of all <math> x \in M</math> with <math> p_X(x)=1 </math> is called the '''kernel''', the set of '''fixed points''', or '''essential minimum''' <math> e(X) </math>. If <math> X_1, X_2, \ldots </math>, is а sequence of [[i.i.d.]] ~~random compact sets, then almost surely~~

	~~:<math> \bigcap_{i=1}^\infty X_i = e(X) </math>~~

	~~and <math> \bigcap_{i=1}^\infty X_i </math> converges almost surely to <math> e(X). </math>~~

	~~== References ==~~
	* Matheron, ~~G. (1975) ''Random Sets and Integral Geometry''. J.Wiley & Sons, New York.~~
	* Molchanov, I. (2005) ''The Theory of Random Sets''. Springer, New York.
	* Stoyan D., and H.Stoyan (1994) ''Fractals, Random Shapes and Point Fields''. John Wiley & Sons, Chichester, New York.

	~~[[Category:Random dynamical systems]]~~
	~~[[Category:Probability theory]]~~
	~~[[Category:Randomness]]~~

en>Monkbot: Fix CS1 deprecated date parameter errors

2014-01-30T09:05:21Z

Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 11:05, 30 January 2014
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	~~Nestor~~ is the ~~name my parents gave me but I don't like when people use my full name~~. ~~Bookkeeping~~ is ~~what I do for~~ a ~~living~~. ~~What she loves doing~~ is to ~~play croquet but she hasn't made~~ a ~~dime with it~~. ~~My home~~ is ~~now~~ in ~~Kansas~~.<br><br>~~Here~~ is ~~my web~~-~~site - car warranty~~, ~~[http~~://~~owenphelps~~.~~com~~/~~UserProfile~~/~~tabid~~/~~355~~/~~userId~~/~~172983~~/~~Default~~.~~aspx visit the next internet site~~],		In [[mathematics]], a '''random compact set''' is essentially a [[compact space\|compact set]]-valued [[random variable]]. Random compact sets are useful in the study of attractors for [[random dynamical system]]s.

			==Definition==

			Let <math>(M, d)</math> be a [[complete space\|complete]] [[separable space\|separable]] [[metric space]]. Let <math>\mathcal{K}</math> denote the set of all compact subsets of <math>M</math>. The Hausdorff metric <math>h</math> on <math>\mathcal{K}</math> is defined by

			:<math>h(K_{1}, K_{2}) := \max \left\{ \sup_{a \in K_{1}} \inf_{b \in K_{2}} d(a, b), \sup_{b \in K_{2}} \inf_{a \in K_{1}} d(a, b) \right\}.</math>

			<math>(\mathcal{K}, h)</math> is also а complete separable metric space. The corresponding open subsets generate a [[sigma algebra\|σ-algebra]] on <math>\mathcal{K}</math>, the [[Borel sigma algebra]] <math>\mathcal{B}(\mathcal{K})</math> of <math>\mathcal{K}</math>.

			A '''random compact set''' is а [[measurable function]] <math>K</math> from а [[probability space]] <math>(\Omega, \mathcal{F}, \mathbb{P})</math> into <math>(\mathcal{K}, \mathcal{B} (\mathcal{K}) )</math>.

			Put another way, a random compact set is a measurable function <math>K \colon \Omega \to 2^{M}</math> such that <math>K(\omega)</math> is [[almost surely]] compact and

			:<math>\omega \mapsto \inf_{b \in K(\omega)} d(x, b)</math>

			is a measurable function for every <math>x \in M</math>.

			==Discussion==

			Random compact sets in this sense are also [[random closed set]]s as in Matheron (1975). Consequently their distribution is given by the probabilities

			:<math>\mathbb{P} (X \cap K = \emptyset)</math> for <math>K \in \mathcal{K}.</math>

			(The distribution of а random compact convex set is also given by the system of all inclusion probabilities <math>\mathbb{P}(X \subset K).</math>)

			For <math>K = \{ x \}</math>, the probability <math>\mathbb{P} (x \in X) </math> is obtained, which satisfies

			:<math>\mathbb{P}(x \in X) = 1 - \mathbb{P}(x \not\in X).</math>

			Thus the '''covering function''' <math>p_{X}</math> is given by

			:<math>p_{X} (x) = \mathbb{P} (x \in X)</math> for <math>x \in M.</math>

			Of course, <math>p_{X}</math> can also be interpreted as the mean of the indicator function <math>\mathbf{1}_{X}</math>:

			:<math>p_{X} (x) = \mathbb{E} \mathbf{1}_{X} (x).</math>

			The covering function takes values between <math> 0 </math> and <math> 1 </math>. The set <math> b_{X} </math> of all <math>x \in M</math> with <math> p_{X} (x) > 0 </math> is called the '''support''' of <math>X</math>. The set <math> k_X </math>, of all <math> x \in M</math> with <math> p_X(x)=1 </math> is called the '''kernel''', the set of '''fixed points''', or '''essential minimum''' <math> e(X) </math>. If <math> X_1, X_2, \ldots </math>, is а sequence of [[i.i.d.]] random compact sets, then almost surely

			:<math> \bigcap_{i=1}^\infty X_i = e(X) </math>

			and <math> \bigcap_{i=1}^\infty X_i </math> converges almost surely to <math> e(X). </math>

			== References ==
			* Matheron, G. (1975) ''Random Sets and Integral Geometry''. J.Wiley & Sons, New York.
			* Molchanov, I. (2005) ''The Theory of Random Sets''. Springer, New York.
			* Stoyan D., and H.Stoyan (1994) ''Fractals, Random Shapes and Point Fields''. John Wiley & Sons, Chichester, New York.

			[[Category:Random dynamical systems]]
			[[Category:Probability theory]]
			[[Category:Randomness]]

en>RedBot: r2.7.2) (Robot: Adding fi:CIE standardivalaisin

2012-08-03T10:44:12Z

r2.7.2) (Robot: Adding fi:CIE standardivalaisin

New page

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