Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-15T09:30:37.601Z Has data issue: false hasContentIssue false
  • English
  • Français

On the fractal dimensions of point patterns

Part of:Stochastic processesInference 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
Get accessRights & Permissions [Opens in a new window]

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 processes62M09: Non-Markovian processes

Information

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 

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

Cutler, C. D. (1991). Some results on the behaviour and estimation of the fractal dimensions of distributions on attractors. J. Statist. Phys. 62, 651–708.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Denker, M. and Keller, G. (1986). Rigorous mathematical procedures for data from dynamical systems. J. Statist. Phys. 44, 67–93.CrossRefGoogle Scholar
Falconer, K. (1990). Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester, UK.Google Scholar
Falk, M. (1995). L. A. N. of extreme order statistics. Ann. Inst. Statist. Math. 47, 693–718.Google Scholar
Grassberger, P. and Procaccia, I. (1983). Measuring the strangeness of strange attractors. Physica D, 9, 189–208.Google Scholar
Harte, D. (1998). Dimension estimates of earthquake epicentres and hypocentres. J. Non-linear Sci. 8, 581–618.Google Scholar
Isham, V. (1992). Statistical aspects of chaos: a review. In Statistical Aspects of Neural Networks and Chaos. Seminaire Européenne de Statistique, Aarhus University, Denmark.Google Scholar
Kagan, Y. (1991). Fractal dimension of brittle fracture. J. Non-linear Sci. 1, 1–16.CrossRefGoogle Scholar
Kagan, Y. (1994). Observational evidence for earthquakes as a non-linear dynamic system. Physica D, 77, 160–192.Google Scholar
Kagan, Y. and Vere-Jones, D. (1996). Problems in the modelling and statistical analysis of earthquakes. In Athens Conference on Applied Probability and Time Series, ed. Heyde, C. C., Proborov, Yu V., Pyke, R. and Rachev, S. T., Vol 1: Applied Probability, pp. 398–425. (Lecture Notes in Statistics 114.) Springer, Berlin.Google Scholar
Mikosch, T. and Wang, Q. (1993). Some results on estimating Rényi-type dimensions. Preprint, ISOR, Victoria University of Wellington.Google Scholar
Mikosch, T. and Wang, Q. (1995). A Monte-Carlo method for estimating the correlation exponent. J. Statist. Phys. 78, 799–813.Google Scholar
Neyman, J. and Scott, E. L. (1958). Statistical approach to problems of cosmology. J. Roy. Statist. Soc. B, 20, 1–43.Google Scholar
Nguyen, X. X. and Zessin, H. (1979). Ergodic theorems for spatial point processes. Z. Wahrscheinlichkeitsth. 48, 133–158.Google Scholar
Ogata, Y. and Katsura, K. (1991). Maximum likelihood estimates of the fractal dimension Biometrika 78, 463–474.Google Scholar
Pesin, Y. B. (1993). On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions. J. Statist. Phys. 71, 529–547.CrossRefGoogle Scholar
Rényi, A., (1959). On the dimension and entropy of probability distributions. Acta Math. Acad. Sci. Hungar. 10, 193–215.Google Scholar
Serinko, R. J. (1994). A consistent approach to least squares estimation of correlation dimension in weak Bernoulli dynamical systems. Ann. Appl. Prob. 4, 1234–1254.Google Scholar
Smith, R. L. (1992). Estimating dimension in noisy chaotic time series. J. Roy. Statist. Soc. B, 54, 329–351.Google Scholar
Stoyan, D. (1994). Caution with fractal point patterns. Statistics 25, 267–270.Google Scholar
Stoyan, D. and Stoyan, H. (1994). Fractals, Random Shapes, and Point Fields. Wiley, New York.Google Scholar
Takens, F. (1985). On the numerical determination of the dimension of an attractor. In Dynamical Systems and Bifurcations, Proceedings Groningen (1984) (Lecture Notes in Mathematics 1125). Springer, Berlin, pp. 99–106.Google Scholar