Abstract
Measures of the non-Gaussianity of a random field depend on how accurately one is able to measure the field. If a signal measured at a certain point is to be averaged with its surroundings, or coarse-grained, the magnitude of its non-Gaussian component can vary. In this article, we investigate the variation of the “apparent” non-Gaussianity, as a function of the coarse-graining length, when we measure non-Gaussianity using the statistics of extrema in the field. We derive how the relative difference between maxima and minima—which is a geometrical measure of the field’s non-Gaussianity—behaves as the field is coarse-grained over increasingly larger length scales. Measuring this function can give extra information about the non-Gaussian statistics and facilitate its detection.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berry, M.V., Hannay, J.H.: Umbilic points on Gaussian random surfaces. J. Phys. A 10, 1809 (1977)
Beuman, T.H., Turner, A.M., Vitelli, V.: Stochastic geometry and topology of non-gaussian fields. Proc. Natl. Acad. Sci. USA 109(49), 19943–19948 (2012)
Beuman, T.H., Turner, A.M., Vitelli, V.: Critical and umbilical points of a non-gaussian random field. Phys. Rev. E 88, 012115 (2013)
Beuman, T.H., Turner, A.M., Vitelli, V.: Extrema statistics in the dynamics of a non-gaussian random field. Phys. Rev. E 87, 022142 (2013)
Dennis, M.R.: Correlations and screening of topological charges in Gaussian random fields. J. Phys. A 36, 6611–6628 (2003)
Dennis, M.R.: Polarization singularity anisotropy: determining monstardom. Opt. Lett. 33, 2572–2574 (2008)
Dodelson, S.: Modern Cosmology. Academic Press, Amsterdam (2003)
Flossmann, F., O’Holleran, K., Dennis, M.R., Padgett, M.J.: Polarization singularities in 2d and 3d speckle fields. Phys. Rev. Lett. 100, 203902 (2008)
Hoekstra, H., Jain, B.: Weak gravitational lensing and its cosmological applications. Ann. Rev. Nucl. Part. Sci. 58, 99–123 (2008)
Hu, W.: Power spectrum tomography with weak lensing. ApJL 522, L21 (1999)
Longuet-Higgins, M.: The statistical analysis of a random, moving surface. Phil. Trans. R. Soc. Lond. A 249, 321–387 (1957)
Longuet-Higgins, M.S.: Statistical properties of an isotropic random surface. Phil. Trans. R. Soc. Lond. A 250, 157–174 (1957)
Vitelli, V., Jain, B., Kamien, R.D.: Topological defects in gravitational lensing shear fields. JCAP 09, 034 (2009)
Worsley, K.J., Marrett, S., Neelin, P., Vandal, A.C., Friston, K.J., Evans, A.C.: A unified statistical approach for determining significant signals in location and scale space images of cerebral activation. Human Brain Mapp. 4, 58–73 (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Beuman, T.H., Turner, A.M. & Vitelli, V. Geometrical Detection of Weak Non-Gaussianity upon Coarse-Graining. J Stat Phys 157, 571–581 (2014). https://doi.org/10.1007/s10955-014-1088-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-014-1088-6