Nearest neighbour distance and dimension of intensity measure of Poisson point process
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Abstract:
We prove that the upper and lower local dimensions of a finite measure $\mu$ are equal to the upper and lower limit of $-\log \alpha / \log \hat {r}_{\alpha \mu }(x)$, where $\hat {r}_{\alpha \mu }(x)$ is the mean distance to the closest point for the Poisson point processes with intensity measure $\alpha \mu$. Moreover the upper local dimension of $\mu$ is a.e. bounded from above by the limit superior of $-\log \alpha / \log \hat {r}_{\alpha \mu }$, where $\hat {r}_{\alpha \mu }$ denotes the expected nearest-neighbour distance.References
- P. Asvestas, S. Golemati, G. K. Matsopoulos, K. S. Nikita, and A. N. Nicolaides, Fractal dimension estimation of carotid atherosclerotic plaques from B-mode ultrasound: a pilot study, Ultrasound in Medicine and Biology 28 (2002), no. 9, 1129–1136.
- R. Badii and A. Politi, Hausdorff dimension and uniformity factor of strange attractors, Phys. Rev. Lett. 52 (1984), no. 19, 1661–1664. MR 741988, DOI 10.1103/PhysRevLett.52.1661
- Remo Badii and Antonio Politi, Statistical description of chaotic attractors: the dimension function, J. Statist. Phys. 40 (1985), no. 5-6, 725–750. MR 806722, DOI 10.1007/BF01009897
- Heinz Bauer, Measure and integration theory, De Gruyter Studies in Mathematics, vol. 26, Walter de Gruyter & Co., Berlin, 2001. Translated from the German by Robert B. Burckel. MR 1897176, DOI 10.1515/9783110866209
- P.J. Clark and F.C. Evans, Distance to nearest neighbor as a measure of spatial relationships in populations, Ecology 35 (1954), 445–453.
- C. D. Cutler and D. A. Dawson, Estimation of dimension for spatially distributed data and related limit theorems, J. Multivariate Anal. 28 (1989), no. 1, 115–148. MR 996987, DOI 10.1016/0047-259X(89)90100-0
- D. J. Daley and D. Vere-Jones, An introduction to the theory of point processes. Vol. I, 2nd ed., Probability and its Applications (New York), Springer-Verlag, New York, 2003. Elementary theory and methods. MR 1950431
- S. G. De Bartolo, M. Veltri, and L. Primavera, Estimated generalized dimensions of river networks, Journal of Hydrology 322 (2006), no. 1-4, 181–191.
- Kenneth Falconer, Fractal geometry, 2nd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2003. Mathematical foundations and applications. MR 2118797, DOI 10.1002/0470013850
- Moisey Guysinsky and Serge Yaskolko, Coincidence of various dimensions associated with metrics and measures on metric spaces, Discrete Contin. Dynam. Systems 3 (1997), no. 4, 591–603. MR 1465128, DOI 10.3934/dcds.1997.3.591
- P. A. Henderson, Practical methods in ecology, Wiley-Blackwell, 2003.
- H. G. E. Hentschel and Itamar Procaccia, The infinite number of generalized dimensions of fractals and strange attractors, Phys. D 8 (1983), no. 3, 435–444. MR 719636, DOI 10.1016/0167-2789(83)90235-X
- Janine Illian, Antti Penttinen, Helga Stoyan, and Dietrich Stoyan, Statistical analysis and modelling of spatial point patterns, Statistics in Practice, John Wiley & Sons, Ltd., Chichester, 2008. MR 2384630
- J. F. C. Kingman, Poisson processes, Oxford Studies in Probability, vol. 3, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1207584
- E. J. Kostelich and H. L. Swinney, Practical considerations in estimating dimension from time series data, Physica Scripta 40 (1989), no. 3, 436–441.
- R. Lopes and N. Betrouni, Fractal and multifractal analysis: A review, Medical Image Analysis 13 (2009), no. 4, 634–649.
- Ya. B. Pesin, On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions, J. Statist. Phys. 71 (1993), no. 3-4, 529–547. MR 1219021, DOI 10.1007/BF01058436
- Yakov B. Pesin, Dimension theory in dynamical systems, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. Contemporary views and applications. MR 1489237, DOI 10.7208/chicago/9780226662237.001.0001
- Ryszard Rudnicki, Pointwise dimensions and Rényi dimensions, Proc. Amer. Math. Soc. 130 (2002), no. 7, 1981–1982. MR 1896030, DOI 10.1090/S0002-9939-02-06519-X
- Dietrich Stoyan and Helga Stoyan, Fractals, random shapes and point fields, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1994. Methods of geometrical statistics; Translated from the 1992 German original by N. Bamber and R. B. Johnson. MR 1297125
- Yves Termonia and Zeev Alexandrowicz, Fractal dimension of strange attractors from radius versus size of arbitrary clusters, Phys. Rev. Lett. 51 (1983), no. 14, 1265–1268. MR 718464, DOI 10.1103/PhysRevLett.51.1265
Additional Information
- Radosław Wieczorek
- Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Bankowa 14, 40-007 Katowice, Poland
- Email: r.wieczorek@impan.gov.pl
- Received by editor(s): October 5, 2009
- Received by editor(s) in revised form: November 6, 2009, February 6, 2010, and February 18, 2010
- Published electronically: June 29, 2010
- Additional Notes: This research was partially supported by the State Committee for Scientific Research (Poland) Grant No. N N201 0211 33 and by EC FP6 Marie Curie ToK programme SPADE2, MTKD-CT-2004-014508 and Polish MNiSW SPB-M
- Communicated by: Edward C. Waymire
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 139-152
- MSC (2010): Primary 28A80; Secondary 60G55
- DOI: https://doi.org/10.1090/S0002-9939-2010-10467-7
- MathSciNet review: 2729078