Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T20:21:18.419Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometrical measures of the smoothness of random functions

Published online by Cambridge University Press:  14 July 2016

Paul Switzer
Paul Switzer*
Affiliation:
Stanford University
Get access Rights & Permissions [Opens in a new window]

Abstract

For stationary isotropic random functions on a Euclidean space, we characterize and compare the mean values of certain geometric measures of the smoothness of realizations. In particular we examine mean properties of the contours and gradients of the random function, and the effect of local averaging on smoothness in special cases.

Keywords

SAMPLE PATHSSMOOTHINGCONTOURSGRADIENTSSURFACE AREA
Type
Research Papers
Information
Journal of Applied Probability , Volume 13 , Issue 1 , March 1976 , pp. 86 - 95
Copyright
Copyright © Applied Probability Trust 1976 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Federer, H. (1959) Curvature measures. Trans. Amer. Math. Soc. 93, 418–491.Google Scholar
Hammersley, J. M. (1950) The distribution of distance in a hypersphere. Ann. Math. Statist. 21, 447–452.Google Scholar
Ivanov, V. A. (1960) On the average number of crossings of a level by sample functions of a stochastic process. Theor. Prob. Appl. 5, 319–323.Google Scholar
Leadbetter, M. R. (1966) On crossings of levels and curves by a wide class of stochastic processes. Ann. Math. Statist. 37, 260–267.Google Scholar
Longuet-Higgins, M. S. (1958) Statistical properties of an isotropic random surface. Phil. Trans. Roy. Soc. London A 250, 157–174.Google Scholar
Matérn, B (1960) Spatial Variation. Statens Skogsforskningsinstitut 49, 1–144.Google Scholar
Miller, I. and Freund, J. E. (1956) Expected arc length of a Gaussian process on a finite interval. Ann. Math. Statist. 27, 257–258.Google Scholar