[Fwd: Applied Continuous Markov Random Fields?]

From: Roger L. Bagula (rlbtftn_at_netscape.net)
Date: 09/04/04 

Date: Sat, 04 Sep 2004 14:20:13 GMT

-------- Original Message --------
Subject: Applied Continuous Markov Random Fields?
Date: Sat, 4 Sep 2004 15:12:14 +0300
From: Simo Särkkä <simo.sarkka@hut.fi>
Organization: Helsinki University of Technology
Newsgroups: sci.math.num-analysis,sci.stat.math,sci.math

In filtering applications (continuous and continuous-discrete
filtering), it is common to model the process dynamics using linear
stochastic differential equations such as

   dx(t)/dt = F(t) x(t) + L(t) w(t) (1a)

or in more rigorous terms

   dx(t) = F(t) x(t) dt + L(t) dB(t) (1b)

where x(t) \in R^n is the state and w(t) = dB(t)/dt is a white noise
process with spectral density Q(t). In practical point of view a very
useful result is that the solution x(t) is a Gaussian process with mean
m(t) and covariance P(t) which are given by differential equations

   dm/dt = F(t) m (2)
   dP/dt = F(t) P + P F^T(t) + L(t) Q(t) L^T(t) (3)

Markov random fields, at least in discrete case, can be considered
as filtering problems with multidimensional time coordinates (e.g.,
the spatial coordinates). In discrete case the spatial dependencies
can be eliminated by using pseudo-measurements, but how about the
continuous time coordinate case?

- What are the Markov random field counterparts of the equations
   (1a) and (1b)? Some kind of stochastic partial differential
   equations?

- Is it possible to derive some kind of partial differential
   equations for mean and covariance of the field, similarly
   as equations (2) and (3) in filtering applications?

- Is it possible to generalize the Kalman-Bucy filter and smoother
   to estimation of MRF's?

I have seen probability theoretical constructions of Markov random
fields with continuous time coordinates, but I'm still looking for
this practical point of view. References to articles and books would
be very useful. 

-- 
- Simo
-- 
Respectfully, Roger L. Bagula
tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 
619-5610814 :
URL :  http://home.earthlink.net/~tftn
URL :  http://victorian.fortunecity.com/carmelita/435/

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