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Symmetric measures via moments. (English) Zbl 1155.62001

Summary: Algebraic tools in statistics have recently been receiving special attention and a number of interactions between algebraic geometry and computational statistics have been rapidly developing. This paper presents another such connection, namely, one between probabilistic models invariant under a finite group of (non-singular) linear transformations and polynomials invariant under the same group. Two specific aspects of the connection are discussed: generalization of the (uniqueness part of the multivariate) problem of moments and log-linear, or toric, modeling by expansion of invariant terms. A distribution of minuscule subimages extracted from a large database of natural images is analyzed to illustrate the above concepts.

MSC:

62A01 Foundations and philosophical topics in statistics
20G99 Linear algebraic groups and related topics
62H35 Image analysis in multivariate analysis

Software:

Magma; Macaulay2; INVAR; GAP
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References:

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