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Spectral theory for random closed sets and estimating the covariance via frequency space

Part of: Harmonic analysis in several variables Inference from stochastic processes Geometric probability and stochastic geometry Nonparametric inference

Published online by Cambridge University Press:  01 July 2016

Karsten Koch ,
Joachim Ohser  and
Katja Schladitz
Karsten Koch*
Affiliation:
Philipps University of Marburg
Joachim Ohser*
Affiliation:
Fraunhofer ITWM
Katja Schladitz*
Affiliation:
Fraunhofer ITWM
*
∗ Postal address: Philipps University of Marburg, Faculty of Mathematics and Informatics, Hans-Meerwein Str., Lahnberge, D-35032 Marburg, Germany.
∗∗ Postal address: Fraunhofer ITWM, Gottlieb-Daimler-Str. 49, D-67663 Kaiserslautern, Germany.
∗∗ Postal address: Fraunhofer ITWM, Gottlieb-Daimler-Str. 49, D-67663 Kaiserslautern, Germany.
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Abstract

A spectral theory for stationary random closed sets is developed and provided with a sound mathematical basis. The definition and a proof of the existence of the Bartlett spectrum of a stationary random closed set as well as the proof of a Wiener-Khinchin theorem for the power spectrum are used to two ends. First, well-known second-order characteristics like the covariance can be estimated faster than usual via frequency space. Second, the Bartlett spectrum and the power spectrum can be used as second-order characteristics in frequency space. Examples show that in some cases information about the random closed set is easier to obtain from these characteristics in frequency space than from their real-world counterparts.

Keywords

Random setBartlett spectrumfast Fourier transformpower spectrum

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 62G05: Estimation 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 62M40: Random fields; image analysis
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 3 , September 2003 , pp. 603 - 613
Copyright
Copyright © Applied Probability Trust 2003 

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