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.
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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)
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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
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DOI: https://doi.org/10.1007/s10955-014-1088-6