Stochastic Systems Group  

Dewey S. Tucker
SSG, MIT
The class of Gaussian multiscale models has been shown to be wellsuited in representing a wide variety of random processes. These types of models have proven useful in a number of applications including image processing, remote sensing, and geophysics, due to the fact that multiscale models admit an extremely efficient estimation algorithm. This efficiency is tied to the factorized structure of a multiscale distribution and the Markov properties implied by this type of factorization.
In this talk, we discuss the Markov properties exhibited by multiscale models. It is wellknown that multiscale models satisfy a "global" Markov property, but here, we show that there exist reduced sets of independence constraints which are equivalent to the global Markov property. Furthermore, for the purpose of model realization, we discuss a further reduction in this set of conditions.
We then discuss a method for realizing approximate multiscale models. Approximations are necessary when constraints are placed on the dimension of the states in the model. One popular method for finding the "optimal" approximate multiscale model is the EM algorithm. We show how EM may be used to realize Gaussian multiscale models given a set of exact covariance statistics, and we interpret how the iterations of this algorithm are related to the Markov properties discussed in the first part of the talk.
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