Hostname: page-component-cb9f654ff-5jtmz Total loading time: 0 Render date: 2025-08-21T11:37:20.279Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-parametric estimation of the directional distribution of stationary line and fibre processes

Part of: Geometric probability and stochastic geometry Nonparametric inference Inference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

Markus Kiderlen
Markus Kiderlen*
Affiliation:
Universität Karlsruhe
*
∗ Postal address: Universität Karlsruhe, Mathematisches Institut II, D-76128 Karlsruhe, Germany. Email address: kiderlen@math.uni-karlsruhe.de
Get access Rights & Permissions [Opens in a new window]

Abstract

Two non-parametric methods for the estimation of the directional measure of stationary line and fibre processes in d-dimensional space are presented. The input data for both methods are intersection counts with finitely many test windows situated in hyperplanes. The first estimator is a measure valued maximum likelihood estimator, if applied to Poisson line processes. The second estimator uses an approximation of the associated zonoid (the Steiner compact) by zonotopes. Consistency of both estimators is proved (without use of the Poisson assumption). The estimation methods are compared empirically by simulation.

Keywords

rose of directionsplanar sectionfibre processSteiner compactzonoidestimationmaximum likelihood

MSC classification

Primary: 62M30: Spatial processes
Secondary: 62G05: Estimation 60D05: Geometric probability and stochastic geometry

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 33 , Issue 1 , March 2001 , pp. 6 - 24
Copyright
Copyright © Applied Probability Trust 2001 

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.)

Article purchase

Temporarily unavailable

References

[1] Benes, V. and Gokhale, A. M. (2000). Planar anisotropy revisited. Kybernetika 36, 149–164.Google Scholar
[2] Campi, S., Haas, D. and Weil, W. (1994). Approximation of zonoids by zonotopes in fixed directions. Discrete Comput. Geom. 11, 419–431.CrossRefGoogle Scholar
[3] Conway, J. H., Hardin, R. H. and Sloane, N. J. A. (1996). Packing lines, planes, etc., packings in Grassmannian spaces. Exp. Math. 5, 139–159.CrossRefGoogle Scholar
[4] Cruz-Orive, L. M., Hoppeler, H., Mathieu, O. and Weibel, E. R. (1985). Stereological analysis of anisotropic structures using directional statistics. Appl. Statist. 34, 14–32.CrossRefGoogle Scholar
[5] Hoffmann-Jörgensen, J., (1994). Probability With a View Toward Statistics, Vol. 1. Chapman and Hall, New York.CrossRefGoogle Scholar
[6] Kanatani, K. I. (1984). Stereological determination of structural anisotropy. Int. J. Eng. Sci. 22, 531–546.CrossRefGoogle Scholar
[7] Kiderlen, M. (1999). Schnittmittelungen und äquivariante Endomorphismen konvexer Körper. , University of Karlsruhe.Google Scholar
[8] Kullback, S. (1968). Information Theory and Statistics. Dover Publications, New York.Google Scholar
[9] Mair, B. A., Rao, M. and Anderson, J. M. M. (1996). Positron emission tomography, Borel measures and weak convergence. Inverse Problems 12, 965–976.CrossRefGoogle Scholar
[10] Marriott, F. H. (1971). Buffon's problem for non-random directions. Biometrics 27, 233.CrossRefGoogle Scholar
[11] McLachlan, G. J. and Krishnan, T. (1997). The EM Algorithm and Extensions. John Wiley, New York.Google Scholar
[12] Mecke, J. and Nagel, W. (1980). Stationäre räumliche Faserprozesse und ihre Schnittzahlrosen. Elektron. Informationsverarb. Kybernet 16, 475–483.Google Scholar
[13] Mülthei, H. N. (1989). On properties of the iterative maximum likelihood reconstruction method. Math. Meth. Appl. Sci. 11, 331–342.CrossRefGoogle Scholar
[14] Rataj, J. and Saxl, I. (1989). Analysis of planar anisotropy by means of the Steiner compact. J. Appl. Prob. 26, 490–502.CrossRefGoogle Scholar
[15] Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory (Encyclopedia Math. Appl. 44). Cambridge University Press.Google Scholar
[16] Serra, J. (1975). Anisotropy fast characterization (Cahiers du Centre de morphologie mathématique 15(8)). École de Mines de Paris, Fontainebleau.Google Scholar
[17] Shepp, L. A. and Vardi, Y. (1982). Maximum likelihood reconstruction for emission tomography. IEEE Trans. Medical Imaging MI-1, 113–122.CrossRefGoogle Scholar
[18] Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and its Applications, 2nd edn. John Wiley, New York.Google Scholar
[19] Weil, W. (1987). Point processes of cylinders, particles and flats. Acta Appl. Math. 9, 103–136.CrossRefGoogle Scholar