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ON MARKOVIAN LATTICES,
HAWAII UNIV HONOLULU INFORMATION SCIENCES PROGRAM
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In many information processing problems one is confronted with modeling a random quantity that depends upon two, or more, parameters. A natural model for such a quantity is a random function of multi-dimensional argument i.e., a random field. In this paper we consider random fields that are defined on only a discrete set of points the points from a rectangular lattice in an n-dimensional space. The Markovian property of processes is extended to include such random fields. The extension is non-trivial since the concept of a preferential direction is lost in the transition to a multi-dimensional argument. In addition, Levys definition of a Markov random field on all of n-space is not easily extended to a field defined on a lattice. Particular emphasis is given to homogeneous random lattices a homogenous lattice is one with probabilities that are invariant to all coordinate translations. Discrete, homogeneous, Markov lattices are practically determined by n commutative transition matrices -- one for each principle coordinate of the space -- and an initial condition. Another useful property of random fields is isotropy, or invariance to all coordinate rotations, as well as to all coordinate translations. It is shown that there are no isotropic, Markov lattices other than the trivial white one. Author
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