Abstract
Contemporary real-time problems like CAPTCHA generation and optical character recognition can be solved effectively using correlated random fields. These random fields should be produced on a graph in order that problems of any dimension and shape can be handled. However, traditional solutions are often too slow, inaccurate or both. Herein, the Quick Simulation Random Field algorithm to produce correlated random fields on general undirected graphs is introduced. It differs from prior algorithms by completing the graph and setting the unspecified covariances to zero, which facilitates analytic study. The Quick Simulation Random Field graph distribution is derived within and the following questions are studied: (1) For which marginal pmfs and covariances will this algorithm work? (2) When does the marginal property hold, where the sub-graph distribution of an algorithm-simulated field matches the distribution of the algorithm-simulated field on the subgraph? (3) When does the permutation property hold, where the vertex simulation order does not affect the joint distribution?
Highlights
Correlated random fields are used in science and technology to model spatially distributed random objects
The complete joint distribution of the field may be unknown or even irrelevant as enough meaningful information is captured by marginal distributions and pairwise covariances between random variables
This paper focuses on the constraints and properties of the random field generated by this quick simulation algorithm
Summary
Correlated random fields are used in science and technology to model spatially distributed random objects. We investigate the permutation property that makes sure the random field simulated from all topological sorts corresponding to the same complete undirected graph are the same in the sense of probability distribution. We show how to compute these probabilities so that the pmfs and covariances are preserved in general as well as establish the conditions for the marginality and permutation properties above to hold.
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