95.89.211.247: Each channel is normalized to 3/N where N is total pixel count such

2014-08-01T22:53:48Z

Each channel is normalized to 3/N where N is total pixel count such

en>Monkbot: Fix CS1 deprecated date parameter errors

2014-02-04T02:24:49Z

Fix CS1 deprecated date parameter errors

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	~~Florist King Jernigan from Richmond~~, ~~has hobbies~~ which ~~include animals~~, ~~como ganhar dinheiro na internet~~ and ~~collecting artwork~~. ~~Was exceptionally encouraged after visiting Group~~ of ~~Monuments~~ at ~~Pattadakal~~.<br><br>~~Feel free to surf to my homepage~~; [http://www.~~comoganhardinheiro101~~.~~com~~/~~como~~-~~ganhar~~-~~dinheiro~~-~~pela~~-~~internet~~/ ~~ganhando dinheiro na internet~~]		In [[probability theory]], '''reflected Brownian motion''' (or '''regulated Brownian motion''',<ref>{{cite doi\|10.1002/9780470400531.eorms0711}}</ref><ref name="harrison-book" /> both with the acronym '''RBM''') is a [[Wiener process]] in a space with reflecting boundaries.<ref>{{cite doi\|10.1023/B:CSEM.0000049491.13935.af}}</ref>

			RBMs have been shown to describe [[queueing model]]s experiencing [[heavy traffic approximation\|heavy traffic]]<ref name="harrison-book" /> as first proposed by [[John Kingman\|Kingman]]<ref>{{cite jstor\|2984229}}</ref> and proven by Iglehart and [[Ward Whitt\|Whitt]].<ref>{{cite jstor\|3518347}}</ref><ref>{{cite jstor\|1426324}}</ref>

			==Definition==

			A ''d''–dimensional reflected Brownian motion ''Z'' is a [[stochastic process]] on <math>\scriptstyle \mathbb R^d_+</math> uniquely defined by
			* a ''d''–dimensional drift vector ''μ''
			* a ''d''×''d'' non-singular covariance matrix ''Σ'' and
			* a ''d''×''d'' reflection matrix ''R''.<ref name="open">{{cite doi\|10.1080/17442508708833469}}</ref>
			where ''X''(''t'') is an unconstrained [[Brownian motion]] and<ref name="Bramson" />
			::<math>Z(t) = X(t) + R Y(t)</math>
			with ''Y''(''t'') a ''d''–dimensional vector where
			* ''Y'' is continuous and non–decreasing with ''Y''(0) = 0
			* ''Y''<sub>''j''</sub> only increases at times for which ''Z''<sub>''j''</sub> = 0 for ''j'' = 1,2,...,''d''
			* ''Z''(''t'') ∈ S, t ≥ 0.

			The reflection matrix describes boundary behaviour. In the interior of <math>\scriptstyle \mathbb R^d_+</math> the process behaves like a [[Wiener process]], on the boundary "roughly speaking, ''Z'' is pushed in direction ''R''<sup>''j''</sup> whenever the boundary surface <math>\scriptstyle \{ z \in \mathbb R^d_+ : z_j=0\}</math> is hit, where ''R''<sup>''j''</sup> is the ''j''th column of the matrix ''R''."<ref name="Bramson" />

			==Stability conditions==

			Stability conditions are known for RBMs in 1, 2, and 3 dimensions. "The problem of recurrence classification for SRBMs in four and higher dimensions remains open."<ref name="Bramson">{{cite doi\|10.1214/09-AAP631}}</ref> In the special case where ''R'' is an [[M-matrix]] then necessary and sufficient conditions for stability are<ref name="Bramson" />
			# ''R'' is a [[non-singular matrix]] and
			# ''R''<sup>−1</sup>''μ'' < 0.

			==Stationary distribution==

			===One dimension===
			The transient distribution of a one-dimensional Brownian motion starting at 0 restricted to positive values (a single reflecting barrier at 0) with drift ''μ '' and variance ''σ''<sup>2</sup> is
			::<math>\mathbb P(Z \leq z) = \Phi \left(\frac{z-\mu t}{\sigma t^{1/2}} \right) - e^{2 \mu z /\sigma^2} \Phi \left( \frac{-z-\mu t}{\sigma t^{1/2}} \right)</math>
			for all ''t'' ≥ 0, (with Φ the [[cumulative distribution function of the normal distribution]]) which yields (for ''μ'' < 0) when taking t → ∞ an [[exponential distribution]]<ref name="harrison-book">{{cite book \| title = Brownian Motion and Stochastic Flow Systems \| first = J. Michael \| last = Harrison \| authorlink = J. Michael Harrison \| year = 1985 \| publisher = John Wiley & Sons \| isbn = 0471819395 \| url = http://faculty-gsb.stanford.edu/harrison/Documents/BrownianMotion-Stochasticms.pdf}}</ref>
			::<math>\mathbb P(Z<z) = 1-e^{2\mu z/\sigma^2}.</math>

			===Multiple dimensions===

			The stationary distribution of a reflected Brownian motion in multiple dimensions is tractable analytically when there is a [[product form stationary distribution]],<ref>{{cite doi\|10.1214/aoap/1177005704}}</ref> which occurs when the process is stable and<ref>{{cite doi\|10.1137/0141030}}</ref>
			::<math>2 \Sigma = RD + DR'</math>
			where ''D'' = [[diagonal matrix\|diag]](''Σ''). In this case the [[probability density function]] is<ref name="open" />
			::<math>p(z_1,z_2,\ldots,z_d) = \prod_{k=1}^d \eta_k e^{-\eta_k z_k}</math>
			where ''η''<sub>''k''</sub> = 2''μ''<sub>''k''</sub>''γ''<sub>''k''</sub>/''Σ''<sub>''kk''</sub> and ''γ'' = ''R''<sup>−1</sup>''μ''. [[Closed-form expression]]s for situations where the product form condition does not hold can be computed numerically as described below in the simulation section.

			==Hitting times==

			===One dimension===

			Write ''T''(''y'') for the first time a one dimensional RBM starting at 0 reaches the level ''y''. Then<ref name="harrison-book" />
			::<math>\mathbb P(T(y)>t) = \Phi\left(\frac{y-\mu t}{\sigma t^{1/2}}\right) -e^{2\mu y/\sigma^2}\Phi\left(\frac{-y-\mu t}{\sigma t^{1/2}}\right).</math>

			==Simulation==

			===One dimension===
			In one dimension the simulated process is the [[absolute value]] of a [[Wiener process]]. The following [[MATLAB]] program creates a sample path.<ref>{{cite book\| page = 202 \| title = Handbook of Monte Carlo Methods \| first1=Dirk P. \| last1= Kroese \| first2= Thomas \|last2=Taimre\|first3= Zdravko I.\|last3= Botev \| publisher = John Wiley & Sons \|year = 2011 \| isbn = 1118014952}}</ref>
			<source lang="matlab">
			%rbm.m
			n=10^4; h=10^(-3); t=h.*(0:n); mu=-1;
			X=zeros(1,n+1); M=X; B=X;
			B(1)=3; X(1)=3;
			for k=2:n+1
			Y=sqrt(h)*randn; U=rand(1);
			B(k)=B(k-1)+mu*h-Y;
			M=(Y + sqrt(Y^2-2hlog(U)))/2;
			X(k)=max(M-Y,X(k-1)+h*mu-Y);
			end
			subplot(2,1,1)
			plot(t,X,'k-');
			subplot(2,1,2)
			plot(t,X-B,'k-');
			</source>
			The error involved in discrete simulations has been quantified.<ref>{{cite doi\|10.1214/aoap/1177004597}}</ref> An exact simulation method for time-dependent RBM is also proposed.<ref>{{cite journal\|last1=Mousavi\|first1=Mohammad\|first2=Peter W.\|last2= Glynn\|title=Exact Simulation of Non-stationary Reflected Brownian Motion\|journal=arXiv preprint\|arxiv=1312.6456\|year=2013\|month=Dec}}</ref>

			===Multiple dimensions===
			[http://www2.isye.gatech.edu/~dai/QNET/ QNET] allows simulation of steady state RBMs.<ref>{{cite jstor\|2959623}}</ref><ref>{{cite journal\| title = Steady-state analysis of reflected Brownian motions: characterization, numerical methods and queueing applications (Ph. D. thesis) \| publisher = Stanford University. Dept. of Mathematics \| year = 1990\| first = Jiangang "Jim" \| last = Dai \| url = http://www2.isye.gatech.edu/~dai/publications/dai90Dissertation.pdf \| accessdate = 5 December 2012 \| chapter = Section A.5 (code for BNET)}}</ref><ref>{{cite jstor\| 2959654}}</ref>

			==Other boundary conditions==

			Feller described possible boundary condition for the process<ref name="skorokhod">{{cite doi\|10.1137/1107002}}</ref><ref>{{cite doi\|10.1090/S0002-9947-1954-0063607-6}}</ref><ref>{{cite journal \| url = http://www.maths.manchester.ac.uk/~goran/skorokhod.pdf \| title = Stochastic Differential Equations
			for Sticky Brownian Motion \| first1 = H. J. \| last1 = Engelbert \| first2 = G. \| last2 = Peskir \| journal = Probab. Statist. Group Manchester Research Report \| issue = 5 \| year = 2012}}</ref>
			* absorption<ref name="skorokhod" /> or killed Brownian motion,<ref>{{cite doi\|10.1007/978-3-642-57856-4_2}}</ref> a [[Dirichlet boundary condition]]
			* instantaneous reflection,<ref name="skorokhod" /> as described above a [[Neumann boundary condition]]
			* elastic reflection, a [[Robin boundary condition]]
			* delayed reflection<ref name="skorokhod" /> (the time spent on the boundary is positive with probability one)
			* partial reflection<ref name="skorokhod" /> where the process is either immediately reflected or is absorbed
			* sticky Brownian motion.<ref>{{cite doi\|10.1007/978-3-642-62025-6_6}}</ref>

			==See also==

			* [[Skorokhod problem]]

			==References==

			{{Reflist}}

			{{Queueing theory}}

			[[Category:Stochastic processes]]
			[[Category:Variants of random walks]]

en>AK456: WP:DPL

2012-07-30T09:52:18Z

WP:DPL

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Florist King Jernigan from Richmond, has hobbies which include animals, como ganhar dinheiro na internet and collecting artwork. Was exceptionally encouraged after visiting Group of Monuments at Pattadakal.<br><br>Feel free to surf to my homepage; [http://www.comoganhardinheiro101.com/como-ganhar-dinheiro-pela-internet/ ganhando dinheiro na internet]

Color normalization - Revision history

95.89.211.247: Each channel is normalized to 3/N where N is total pixel count such

en>Monkbot: Fix CS1 deprecated date parameter errors

en>AK456: WP:DPL