en>EmilJ: Undid revision 563845812 by Bubba73 (talk) Not at all. You are confusing this notion of completeness with what is called “semantical completeness” in the completeness article.

2013-07-11T18:10:05Z

Undid revision 563845812 by Bubba73 (talk) Not at all. You are confusing this notion of completeness with what is called “semantical completeness” in the completeness article.

← Older revision		Revision as of 20:10, 11 July 2013
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	~~Hi there. Let me start by introducing~~ the ~~author, her name~~ is ~~Sophia~~. ~~North Carolina is~~ the ~~location he loves most but now he is contemplating other options~~. ~~Invoicing~~ is ~~what I do~~ for a ~~living but I've always wanted my own business~~. ~~It's not~~ a ~~common factor but what live psychic reading (~~[~~http~~:/~~/bigpolis.com/blogs/post/6503 http~~://~~bigpolis~~.~~com~~/~~blogs~~/~~post~~/~~6503]) I like performing~~ is to ~~climb but I don't have~~ the ~~time lately~~.~~<br><br>Here~~ is ~~my web blog~~ [http://www.~~herandkingscounty~~.~~com~~/~~content~~/~~information~~-~~and~~-~~facts~~-~~you~~-~~must~~-~~know~~-~~about~~-~~hobbies accurate psychic predictions] ([~~http://~~myfusionprofits~~.~~com~~/~~groups~~/~~find~~-~~out~~-~~about~~-~~self~~-~~improvement~~-~~tips~~-~~here/ click the up coming website page~~])		In [[Filtering problem (stochastic processes)\|filtering theory]] the '''Zakai equation''' is a linear stochastic partial differential equation for the un-normalized density of a hidden state. In contrast, the [[Kushner equation]] gives a non-linear stochastic partial differential equation for the normalized density of the hidden state. In principle either approach allows one to estimate a quantity (the state of a dynamical system) from noisy measurements, even when the system is non-linear (thus generalizing the earlier results of Wiener and Kalman for linear systems and solving a central problem in [[estimation theory]]). The application of this approach to a specific [[control engineering\|engineering]] situation may be problematic however, as these equations are quite complex. The Zakai equation is a bilinear [[stochastic partial differential equation]].

			== Overview ==

			Assume the state of the system evolves according to

			:<math>dx = f(x,t) dt + dw</math>

			and a noisy measurement of the system state is available:

			:<math>dz = h(x,t) dt + dv</math>

			where <math>dw, dv</math> are independent [[Wiener process]]es. Then the unnormalized conditional probability density <math>p(x,t)</math> of the state at time t is given by the Zakai equation:

			:<math>dp = L(p) dt + p h^T dz</math>

			where the operator <math>L = -\sum \frac{\partial (f_i p)}{\partial x_i} + \frac12 \sum \frac{\partial^2 p}{\partial x_i \partial x_j}</math>

			As previously mentioned p is an unnormalized density, i.e. it does not necessarily integrate to 1. After solving for p we can integrate it and normalize it if desired (an extra step not required in the Kushner approach).

			Note that if the last two terms on the right hand side are omitted (by choosing h identically zero), we are left with a nonstochastic PDE: the familiar [[Kolmogorov Forward Equation]], which describes the evolution of the state when no measurement information is available.

			== See also ==

			*[[Kushner Equation]]
			*[[Kalman filter]]
			*[[Wiener filter]]

			== References ==

			* {{cite journal \|last1=Zakai \|first1=M. \|year=1969 \|title=On the optimal filtering of diffusion processes \|journal=[[Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete]] \|volume=11 \|issue=3 \|pages=230 \|mr=242552 \|zbl=0164.19201 \|doi=10.1007/BF00536382}}
			* {{cite book \|last=Sritharan \|first=S. S. \|year=1994 \|chapter=Nonlinear filtering of stochastic Navier–Stokes equations \|url=http://www.nps.edu/Academics/Schools/GSEAS/SRI/BookCH9.pdf \|editor1-last=Funaki \|editor1-first=T. \|editor2-last=Woyczynski \|editor2-first=W. A. \|title=Nonlinear Stochastic PDEs: Burgers Turbulence and Hydrodynamic Limit \|pages=247–260 \|publisher=[[Springer-Verlag]] \|isbn=0-387-94624-1}}
			* {{cite book \|last=Hobbs \|first=S. L. \|last2=Sritharan \|first2=S. S. \|year=1996 \|chapter=Nonlinear filtering theory for stochastic reaction–diffusion equations \|url=http://www.nps.edu/Academics/Schools/GSEAS/SRI/BookCH10.pdf \|editor1-last=Gretsky \|editor1-first=N. \|editor2-last=Goldstein \|editor2-first=J. \|editor3-last=Uhl \|editor3-first=J. J. \|title=Probability and Modern Analysis \|pages=219–234 \|publisher=[[Marcel Dekker]] \|isbn= }}

			[[Category:Signal processing]]
			[[Category:Estimation theory]]
			[[Category:Stochastic differential equations]]
			[[Category:Stochastic processes]]


			{{mathapplied-stub}}

en>ZéroBot: r2.7.1) (Robot: Adding fr:Théorie complète

2012-06-21T05:49:38Z

r2.7.1) (Robot: Adding fr:Théorie complète

New page

Hi there. Let me start by introducing the author, her name is Sophia. North Carolina is the location he loves most but now he is contemplating other options. Invoicing is what I do for a living but I've always wanted my own business. It's not a common factor but what live psychic reading ([http://bigpolis.com/blogs/post/6503 http://bigpolis.com/blogs/post/6503]) I like performing is to climb but I don't have the time lately.<br><br>Here is my web blog [http://www.herandkingscounty.com/content/information-and-facts-you-must-know-about-hobbies accurate psychic predictions] ([http://myfusionprofits.com/groups/find-out-about-self-improvement-tips-here/ click the up coming website page])

Complete theory - Revision history

en>EmilJ: Undid revision 563845812 by Bubba73 (talk) Not at all. You are confusing this notion of completeness with what is called “semantical completeness” in the completeness article.

en>ZéroBot: r2.7.1) (Robot: Adding fr:Théorie complète