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In the theory of [[stochastic processes]], the '''filtering problem''' is a mathematical model for a number of filtering problems in [[signal processing]] and the like. The general idea is to form some kind of "best estimate" for the true value of some system, given only some (potentially noisy) observations of that system.  The problem of optimal non-linear filtering (even for the non-stationary case) was solved by [[Ruslan L. Stratonovich]] (1959,<ref>Stratonovich, R. L. (1959). ''Optimum nonlinear systems which bring about a separation of a signal with constant parameters from noise''. Radiofizika, 2:6, pp. 892-901.</ref> 1960<ref>Stratonovich, R.L. (1960). ''Application of the Markov processes theory to optimal filtering''. Radio Engineering and Electronic Physics, 5:11, pp.1-19.</ref>), see also [[Harold J. Kushner]]'s work <ref>Kushner, Harold. (1967). Nonlinear filtering: The exact dynamical equations satisfied by the conditional mode. Automatic Control, IEEE Transactions on Volume 12, Issue 3, Jun 1967 Page(s): 262 - 267</ref> and [[Moshe Zakai]]'s, who introduced a simplified dynamics for the unnormalized conditional law of the filter<ref>Zakai, Moshe (1969), On the optimal filtering of diffusion processes. Zeit. Wahrsch. 11  230–243. {{MR|242552}}, {{Zbl|0164.19201}}, {{doi|10.1007/BF00536382}}</ref> known as [[Zakai equation]].  The solution, however, is infinite-dimensional in the general case.<ref>[[Mireille Chaleyat-Maurel]] and Dominique Michel. Des resultats de non existence de filtre de dimension finie. Stochastics, 13(1+2):83-102, 1984.</ref> Certain approximations and special cases are well-understood: for example, the linear filters are optimal for Gaussian random variables, and are known as the [[Wiener filter]] and the [[Kalman-Bucy filter]]. More generally, as the solution is infinite dimensional, it requires finite dimensional approximations to be implemented in a computer with finite memory. A finite dimensional approximated [[nonlinear filter]] may be more based on heuristics, such as the [[Extended Kalman Filter]] or the Assumed Density Filters,<ref>Maybeck, Peter S., Stochastic models, estimation, and control, Volume 141, Series Mathematics in Science and Engineering, 1979, Academic Press</ref> or more methodologically oriented such as for example the Projection Filters,<ref>Damiano Brigo, Bernard Hanzon and François LeGland, A Differential Geometric approach to nonlinear filtering: the Projection Filter, I.E.E.E. Transactions on Automatic Control Vol. 43, 2 (1998), pp 247--252.</ref> some sub-families of which are shown to coincide with the Assumed Density Filters.<ref>[[Damiano Brigo]], [[Bernard Hanzon]] and [[François Le Gland]], Approximate Nonlinear Filtering by Projection on Exponential Manifolds of Densities, Bernoulli, Vol. 5, N. 3 (1999), pp. 495--534</ref>


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In general, if the [[separation principle]] applies, then filtering also arises as part of the solution of an [[optimal control]] problem. For example, the [[Kalman filter]] is the estimation part of the optimal control solution to the [[Linear-quadratic-Gaussian control]] problem.
 
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Consider a [[probability space]] ,&nbsp;Σ,&nbsp;'''P''') and suppose that the (random) state ''Y''<sub>''t''</sub> in ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup> of a system of interest at time ''t'' is a [[random variable]] ''Y''<sub>''t''</sub>&nbsp;:&nbsp;Ω&nbsp;→&nbsp;'''R'''<sup>''n''</sup> given by the solution to an [[Kiyoshi Itō|Itō]] [[stochastic differential equation]] of the form
 
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:<math>\mathrm{d} Y_{t} = b(t, Y_{t}) \, \mathrm{d} t + \sigma (t, Y_{t}) \, \mathrm{d} B_{t},</math>


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:<math>H_{t} = c(t, Y_{t}) + \gamma (t, Y_{t}) \cdot \mbox{noise}.</math>


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Adopting the Itō interpretation of the stochastic differential and setting
 
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:<math> Z_{t} = \int_{0}^{t} H_{s} \, \mathrm{d} s,</math>


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this gives the following stochastic integral representation for the observations ''Z''<sub>''t''</sub>:
 
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:<math>\mathrm{d} Z_{t} = c(t, Y_{t}) \, \mathrm{d} t + \gamma (t, Y_{t}) \, \mathrm{d} W_{t},</math>


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where ''W'' denotes standard ''r''-dimensional [[Brownian motion]], independent of ''B'' and the initial condition ''X''<sub>0</sub>, and ''c''&nbsp;:&nbsp;[0,&nbsp;+∞)&nbsp;&times;&nbsp;'''R'''<sup>''n''</sup>&nbsp;→&nbsp;'''R'''<sup>''n''</sup> and ''γ''&nbsp;:&nbsp;[0,&nbsp;+∞)&nbsp;&times;&nbsp;'''R'''<sup>''n''</sup>&nbsp;→&nbsp;'''R'''<sup>''n''&times;''r''</supsatisfy
 
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The space ''K''(''Z'',&nbsp;''t'') of candidates is a [[Hilbert space]], and the general theory of Hilbert spaces implies that the solution ''Ŷ''<sub>''t''</sub> of the minimization problem (M) is given by


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==References==
 
* {{cite book
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| title = Stochastic Processes and Filtering Theory
 
| publisher=Academic Press
  <li>[http://energiadiario.com/publicacion/spip.php?article19316/ http://energiadiario.com/publicacion/spip.php?article19316/]</li>
| location = New York
 
| year = 1970
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* {{cite book
| last = Øksendal
| first = Bernt K.
| authorlink = Bernt Øksendal
| title = Stochastic Differential Equations: An Introduction with Applications
| edition = Sixth edition
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In the theory of stochastic processes, the filtering problem is a mathematical model for a number of filtering problems in signal processing and the like. The general idea is to form some kind of "best estimate" for the true value of some system, given only some (potentially noisy) observations of that system. The problem of optimal non-linear filtering (even for the non-stationary case) was solved by Ruslan L. Stratonovich (1959,[1] 1960[2]), see also Harold J. Kushner's work [3] and Moshe Zakai's, who introduced a simplified dynamics for the unnormalized conditional law of the filter[4] known as Zakai equation. The solution, however, is infinite-dimensional in the general case.[5] Certain approximations and special cases are well-understood: for example, the linear filters are optimal for Gaussian random variables, and are known as the Wiener filter and the Kalman-Bucy filter. More generally, as the solution is infinite dimensional, it requires finite dimensional approximations to be implemented in a computer with finite memory. A finite dimensional approximated nonlinear filter may be more based on heuristics, such as the Extended Kalman Filter or the Assumed Density Filters,[6] or more methodologically oriented such as for example the Projection Filters,[7] some sub-families of which are shown to coincide with the Assumed Density Filters.[8]

In general, if the separation principle applies, then filtering also arises as part of the solution of an optimal control problem. For example, the Kalman filter is the estimation part of the optimal control solution to the Linear-quadratic-Gaussian control problem.

The mathematical formalism

Consider a probability space (Ω, Σ, P) and suppose that the (random) state Yt in n-dimensional Euclidean space Rn of a system of interest at time t is a random variable Yt : Ω → Rn given by the solution to an Itō stochastic differential equation of the form

dYt=b(t,Yt)dt+σ(t,Yt)dBt,

where B denotes standard p-dimensional Brownian motion, b : [0, +∞) × Rn → Rn is the drift field, and σ : [0, +∞) × Rn → Rn×p is the diffusion field. It is assumed that observations Ht in Rm (note that m and n may, in general, be unequal) are taken for each time t according to

Ht=c(t,Yt)+γ(t,Yt)noise.

Adopting the Itō interpretation of the stochastic differential and setting

Zt=0tHsds,

this gives the following stochastic integral representation for the observations Zt:

dZt=c(t,Yt)dt+γ(t,Yt)dWt,

where W denotes standard r-dimensional Brownian motion, independent of B and the initial condition X0, and c : [0, +∞) × Rn → Rn and γ : [0, +∞) × Rn → Rn×r satisfy

|c(t,x)|+|γ(t,x)|C(1+|x|)

for all t and x and some constant C.

The filtering problem is the following: given observations Zs for 0 ≤ s ≤ t, what is the best estimate Ŷt of the true state Yt of the system based on those observations?

By "based on those observations" it is meant that Ŷt is measurable with respect to the σ-algebra Gt generated by the observations Zs, 0 ≤ s ≤ t. Denote by K = K(Zt) be collection of all Rn-valued random variables Y that are square-integrable and Gt-measurable:

K=K(Z,t)=L2(Ω,Gt,P;Rn).

By "best estimate", it is meant that Ŷt minimizes the mean-square distance between Yt and all candidates in K:

E[|YtY^t|2]=infYKE[|YtY^|2].(M)

Basic result: orthogonal projection

The space K(Zt) of candidates is a Hilbert space, and the general theory of Hilbert spaces implies that the solution Ŷt of the minimization problem (M) is given by

Y^t=PK(Z,t)(Xt),

where PK(Z,t) denotes the orthogonal projection of L2(Ω, Σ, PRn) onto the linear subspace K(Zt) = L2(Ω, GtPRn). Furthermore, it is a general fact about conditional expectations that if F is any sub-σ-algebra of Σ then the orthogonal projection

PF:L2(Ω,Σ,P;Rn)L2(Ω,F,P;Rn)

is exactly the conditional expectation operator E[·|F], i.e.,

PF(X)=E[X|F].

Hence,

Y^t=PK(Z,t)(Xt)=E[Xt|Gt].

This elementary result is the basis for the general Fujisaki-Kallianpur-Kunita equation of filtering theory.

References

  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 (See Section 6.1)
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  2. Stratonovich, R.L. (1960). Application of the Markov processes theory to optimal filtering. Radio Engineering and Electronic Physics, 5:11, pp.1-19.
  3. Kushner, Harold. (1967). Nonlinear filtering: The exact dynamical equations satisfied by the conditional mode. Automatic Control, IEEE Transactions on Volume 12, Issue 3, Jun 1967 Page(s): 262 - 267
  4. Zakai, Moshe (1969), On the optimal filtering of diffusion processes. Zeit. Wahrsch. 11 230–243. Template:MR, Template:Zbl, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park.
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  7. Damiano Brigo, Bernard Hanzon and François LeGland, A Differential Geometric approach to nonlinear filtering: the Projection Filter, I.E.E.E. Transactions on Automatic Control Vol. 43, 2 (1998), pp 247--252.
  8. Damiano Brigo, Bernard Hanzon and François Le Gland, Approximate Nonlinear Filtering by Projection on Exponential Manifolds of Densities, Bernoulli, Vol. 5, N. 3 (1999), pp. 495--534