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On the fractal dimensions of point patterns

Part of: Stochastic processes Inference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

David Vere-Jones
David Vere-Jones*
Affiliation:
Victoria University of Wellington
*
∗ Postal address: Institute of Statistics and Operations Research, Victoria University of Wellington, P.O. Box 600, Wellington, New Zealand. Email address: dvj@isor.vuw.ac.nz
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Abstract

When fractal dimensions are estimated from an observed point pattern, there is some ambiguity as to the interpretation of the quantity being estimated. (The point pattern itself has dimension zero.) Two possible interpretations are described. In the first of these, the observation region is regarded as being held fixed, while observations accumulate with time. In this case, provided the process is stationary and ergodic in time, and the cumulants satisfy certain regularity constraints, the dimension estimates consistently estimate the Rényi moment dimensions of the marginal distribution in space. If the regularity constraints are not satisfied, then different limits can be obtained according to the manner in which the limits are taken.

In the second case, the process is regarded as being stationary and ergodic in its spatial component, time being held fixed. In this case the estimates provide consistent estimates of the initial power-law rates of growth of the moment measures of the Palm distribution, the estimates for successively higher Rényi dimensions estimating the growth rates for successively higher-order moment measures of the Palm distribution.

Several examples are given, to illustrate the different types of behaviour which may occur, including the case where the points are generated by a dynamical system.

Keywords

Fractal dimensionmultifractalspoint patternsergodic theoryconsistent estimation

MSC classification

Primary: 60G55: Point processes
Secondary: 62M30: Spatial processes 62M09: Non-Markovian processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 3 , September 1999 , pp. 643 - 663
Copyright
Copyright © Applied Probability Trust 1999 

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