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Volume 9, Issue 4


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Neighbourhoods of Independence for Random Processes via Information Geometry
Khadiga Arwini
C. T. J. Dodson

In this article we consider the Freund family of distributions as a 4-manifold equipped with Fisher information as Riemannian metric and derive the induced Alpha-geometry, that is, the Alpha-Ricci curvature, the Alpha-scalar curvature, and so forth. We show that the Freund manifold has a positive constant 0-scalar curvature, so geometrically it constitutes part of a sphere. We examine special cases as submanifolds and discuss their geometrical structures; via one submanifold we provide examples of neighbourhoods of the independent case for bivariate distributions having identical exponential marginals. Thus, since exponential distributions complement Poisson point processes, we obtain a means to discuss the neighbourhood of independence for random processes. Our approach using Mathematica handles the Alpha-geometry calculations and graphics effectively and could be transferred to other distribution families.




*1. Differential Geometry of the Freund 4-Manifold F

*2. Submanifolds of the Freund 4-Manifold F

*3. Concluding Remarks



Khadiga Arwini
C. T. J. Dodson

School of Mathematics
Manchester University
Sackville Street
Manchester M601QD, UK

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