Skip to main content
SpringerLink
Log in

Mark Correlations: Relating Physical Properties to Spatial Distributions

  • Chapter
  • First Online:
Morphology of Condensed Matter

Part of the book series: Lecture Notes in Physics ((LNP,volume 600))

Abstract

Mark correlations provide a systematic approach to look at objects both distributed in space and bearing intrinsic information, for instance on physical properties. The interplay of the objects’ properties (marks) with the spatial clustering is of vivid interest for many applications; are, e.g., galaxies with high luminosities more strongly clustered than dim ones? Do neighbored pores in a sandstone have similar sizes? How does the shape of impact craters on a planet depend on the geological surface properties? In this article, we give an introduction into the appropriate mathematical framework to deal with such questions, i.e. the theory of marked point processes. After having clarified the notion of segregation effects, we define universal test quantities applicable to realizations of a marked point processes. We show their power using concrete data sets in analyzing the luminosity-dependence of the galaxy clustering, the alignment of dark matter halos in gravitational N-body simulations, the morphology- and diameter-dependence of the Martian crater distribution and the size correlations of pores in sandstone. In order to understand our data in more detail, we discuss the Boolean depletion model, the random field model and the Cox random field model. The first model describes depletion effects in the distribution of Martian craters and pores in sandstone, whereas the last one accounts at least qualitatively for the observed luminosity-dependence of the galaxy clustering.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adler, R. J. (1981): The Geometry of Random Fields (John Wiley & Sons, Chichester)

    MATH  Google Scholar 

  2. Arns, C., M. Knackstedt, W. Pinczewski, K. Mecke (2001): ‘Characterisation of irregular spatial structures by prallel sets’, In press

    Google Scholar 

  3. Arns, C., M. Knackstedt, W. Pinczewski, K. Mecke (2001): ‘Euler-poincaré characteristics of classes of disordered media’, Phys. Rev. E 63, p. 31112

    Article  ADS  Google Scholar 

  4. Baddeley, A. J. (1999): ‘Sampling and censoring’. In: Stochastic Geometry, Likelihood and Computation, ed. by O. Barndorff-Nielsen, W. Kendall, M. van Lieshout, volume 80 of Monographs on Statistics and Applied Probability, chapter 2 (Chapman and Hall, London)

    Google Scholar 

  5. Balian, R., R. Schaeffer (1989): ‘Scale-invariant matter distribution in the Universe I. counts in cells’, Astronomy & Astrophysics 220, pp. 1–29

    ADS  MathSciNet  Google Scholar 

  6. Barlow, N. G., T. L. Bradley (1990): ‘Martian impact craters: Correlations of ejecta and interior morphologies with diameter, latitude, and terrain’, Icarus 87, pp. 156–179

    Article  ADS  Google Scholar 

  7. Beisbart, C., M. Kerscher (2000): ‘Luminosity-and morphology-dependent clustering of galaxies’, Astrophysical Journal 545, pp. 6–25

    Article  ADS  Google Scholar 

  8. Benoist, C., A. Cappi, L. Da Costa, S. Maurogordato, F. Bouchet, R. Schaeffer (April 1999): ‘Biasing and high-order statistics from the southern-sky redshift survey’, Astrophysical Journal 514, pp. 563–578

    Article  ADS  Google Scholar 

  9. Benoist, C., S. Maurogordato, L. Da Costa, A. Cappi, R. Schaeffer (December 1996): ‘Biasing in the galaxy distribution’, Astrophysical Journal 472, p. 452

    Article  ADS  Google Scholar 

  10. Bertschinger, E. (1998): ‘Simulations of structure formation in the universe’, Ann. Rev. Astron. Astrophys. 36, pp. 599–654

    Article  ADS  Google Scholar 

  11. Binggeli, B. (1982): ‘The shape and orientation of clusters of galaxies’, Astronomy & Astrophysics 107, pp. 338–349

    ADS  Google Scholar 

  12. Böhringer, H., P. Schuecker, L. Guzzo, C. Collins, W. Voges, S. Schindler, D. Neumann, G. Chincharini, R. Cruddace, A. Edge, H. MacGillivray, P. Shaver (2001): ‘The ROSTAESO flux limited X-ray (REFLEX) galaxy cluster survey I: The construction of the cluster sample’, Astronomy & Astrophysics, p. 826

    Google Scholar 

  13. Capobianco, R., E. Renshaw (1998): ‘The autocovariance function of marked point processes: A comparison between two different approaches’, Biom. J. 40, pp. 431–446

    Article  MATH  Google Scholar 

  14. Coles, P., B. Jones (January 1991): ‘A lognormal model for the cosmological mass distribution’, MNRAS 248, pp. 1–13

    ADS  Google Scholar 

  15. Coles, P., F. Lucchin (1994): Cosmology: The Origin and Evolution of Cosmic Structure (John Wiley & Sons, Chichester)

    Google Scholar 

  16. Cox, D., V. Isham (1980): Point Processes (Chapman and Hall, London)

    MATH  Google Scholar 

  17. Cressie, N. (1991): Statistics for Spatial Data (John Wiley & Sons, Chichester)

    MATH  Google Scholar 

  18. da Costa, L. N., C. N. A. Willmer, P. Pellegrini, O. L. Chaves, C. Rite, M. A. G. Maia, M. J. Geller, D.W. Latham, M. J. Kurtz, J. P. Huchra, M. Ramella, A. P. Fairall, C. Smith, S. Lipari (1998): ‘The Southern Sky Redshift Survey’, AJ 116, pp. 1–7

    Article  ADS  Google Scholar 

  19. Daley, D. J., D. Vere-Jones (1988): An Introduction to the Theory of Point Processes (Springer, Berlin)

    MATH  Google Scholar 

  20. Diggle, P. J. (1983): Statistical Analysis of Spatial Point Patterns (Academic Press, New York and London)

    MATH  Google Scholar 

  21. Djorgovski, S. (1987): ‘Coherent orientation effects of galaxies and clusters’. In: Nearly Normal Galaxies. From the Planck Time to the Present, ed. by S. M. Faber (Springer, New York), pp. 227–233

    Google Scholar 

  22. Faltenbacher, A., S. Gottlöber, M. Kerscher, V. Müller (2002): ‘Correlations in the orientation of galaxy clusters’, submitted to Astronomy & Astrophysics

    Google Scholar 

  23. Flannery, B. P., H.W. Deckman, W. G. Roberge, K. L. D’amico (1987): ‘Three-dimensional X-ray microtomography’, Science 237, pp. 1439–1444

    Article  ADS  Google Scholar 

  24. Fuller, T. M., M.J. West, T. J. Bridges (1999): ‘Alignments of the dominant galaxies in poor clusters’, Astrophysical Journal 519, pp. 22–26

    Article  ADS  Google Scholar 

  25. Gottlöber, S., M. Kerscher, A.V. Kravtsov, A. Faltenbacher, A. Klypin, V. Müller (2002): ‘Spatial distribution of galactic halos and their merger histories’, Astronomy & Astrophysics 387, pp. 778

    Article  MATH  ADS  Google Scholar 

  26. Gull, S., A. Lasenby, C. Doran (1993): ‘Imaginary numbers are nor real. — the geometric algebra of spacetime’, Found. Phys. 23(9), p. 1175

    Article  ADS  MathSciNet  Google Scholar 

  27. Guzzo, L., J. Bartlett, A. Cappi, S. Maurogordato, E. Zucca, G. Zamorani, C. Balkowski, A. Blanchard, V. Cayatte, G. Chincarini, C. Collins, D. Maccagni, H. MacGillivray, R. Merighi, M. Mignoli, D. Proust, M. Ramella, R. Scaramella, G. Stirpe, G. Vettolani (2000): ‘The ESO Slice Project (ESP) galaxy redshift survey. VII. the redshift and real-space correlation functions’, Astronomy & Astrophysics 355, pp. 1–16

    ADS  Google Scholar 

  28. Hamilton, A. J. S. (August 1988): ‘Evidence for biasing in the cfa survey’, Astrophysical Journal 331, pp. L59–L62

    Article  ADS  Google Scholar 

  29. Heavens, A. F., A. Refregier, C. Heymans (2000): ‘Intrinsic correlation of galaxy shapes: implications for weak lensing measurements’, MNRAS 319, pp. 649–656

    Article  ADS  Google Scholar 

  30. Hermit, S., B. X. Santiago, O. Lahav, M. A. Strauss, M. Davis, A. Dressler, J. P. Huchra (1996): ‘The two-point correlation function and the morphological segregation in the optical redshift survey’, MNRAS 283, p. 709

    ADS  Google Scholar 

  31. Hestens, D. (1986): New Foundations for Classical Mechanics (D. Reidel Publishing Company, Dordrecht, Holland)

    Google Scholar 

  32. Huchra, J. P., M. J. Geller, V. De Lapparent, H. G. Corwin Jr. (1990): ‘The CfA redshift survey — data for the NGP + 30 zone’, Astrophysical Journal Supplement 72, pp. 433–470

    Article  ADS  Google Scholar 

  33. Isham, V. (1985): ‘Marked point processes and their correlations’. In: Spatial Processes and Spatial Time Series Analysis, ed. by F. Droesbeke (Publications des Facultés universitaires Sain-Louis, Bruxelles)

    Google Scholar 

  34. Kerscher, M. (2000): ‘Statistical analysis of large-scale structure in the Universe’. In: Statistical Physics and Spatial Statistics: The Art of Analyzing and Modeling Spatial Structures and Pattern Formation, ed. by K. R. Mecke, D. Stoyan, Number 554 in Lecture Notes in Physics (Springer, Berlin), astro-ph/9912329

    Google Scholar 

  35. Kerscher, M. (2001): ‘Constructing, characterizing and simulating Gaussian and high-order point processes’, Phys. Rev. E 64(5), p. 056109, astro-ph/0102153

    Article  ADS  Google Scholar 

  36. Klypin, A. A. (2000): ‘Numerical simulations in cosmology i: Methods’, in ‘Lecture at the Summer School “Relativistic Cosmology: Theory and Observations”’, Astro-ph/0005502

    Google Scholar 

  37. Lambas, D. G., E. J. Groth, P. Peebles (1988): ‘Statistics of galaxy orientations: Morphology and large-scale structure’, Astronomical Journal 95, pp. 975–984

    Article  ADS  Google Scholar 

  38. Lasenby, J., A. N. Lasenby, C. J. Doran (2000): ‘A unified mathematical language for physics and engineering in the 21st century’, Phil. Trans. R. Soc. London A 358, pp. 21–39

    Article  MATH  ADS  MathSciNet  Google Scholar 

  39. Löwen, H. (2000): ‘Fun with hard spheres’. In: Statistical Physics and Spatial Statistics: The Art of Analyzing and Modeling Spatial Structures and Pattern Formation, ed. by K. R. Mecke, D. Stoyan, Number 554 in Lecture Notes in Physics (Springer, Berlin)

    Google Scholar 

  40. Mecke, K. (2000): ‘Additivity, convexity, and beyond: Application of minkowski functionals in statistical physics’. In: Statistical Physics and Spatial Statistics: The Art of Analyzing and Modeling Spatial Structures and Pattern Formation, ed. by K. R. Mecke, D. Stoyan, Number 554 in Lecture Notes in Physics (Springer, Berlin)

    Google Scholar 

  41. Melott, A. L., S. F. Shandarin (1990): ‘Generation of large-scale cosmological structures by gravitational clustering’, Nature 346, pp. 633–635.

    Article  ADS  Google Scholar 

  42. Møller, J., A. R. Syversveen, R. P. Waagepetersen (1998): ‘Log Gaussian cox processes’, Scand. J. Statist. 25, pp. 451–482

    Article  MATH  MathSciNet  Google Scholar 

  43. Ogata, Y., K. Katsura (1988): ‘Likelihood analysis of spatial inhomogeneity for marked point pattersn’, Ann. Inst. Statist. Math. 40, pp. 29–39

    Article  MATH  MathSciNet  Google Scholar 

  44. Ogata, Y., M. Tanemura (1985): ‘Estimation of interaction potentials of marked spatial point patterns through the maximum likelihood method’, Biometrics 41, pp. 421–433

    Article  MATH  Google Scholar 

  45. Ohser, J., D. Stoyan (1981): ‘On the second-order and orientation analysis of planar stationary point processes’, Biom. J. 23, pp. 523–533

    Article  MATH  MathSciNet  Google Scholar 

  46. Onuora, L. I., P. A. Thomas (2000): ‘The alignment of clusters using large-scale simulations’, MNRAS 319, pp. 614–618

    Article  ADS  Google Scholar 

  47. Peebles, P. J. E. (1980): The Large Scale Structure of the Universe (Princeton University Press, Princeton, New Jersey)

    Google Scholar 

  48. Peebles, P. J. E. (1993): Principles of Physical Cosmology (Princeton University Press, Princeton, New Jersey)

    Google Scholar 

  49. Penttinen, A., D. Stoyan (1989): ‘Statistical analysis for a class of line segment processes’, Scand. J. Statist. 16, pp. 153–168

    MATH  MathSciNet  Google Scholar 

  50. Reichert, H., O. Klein, H. Dosch, M. Denk, V. Honkimäki, T. Lippmann, G. Reiter (2000): ‘Observation of five-fold local symmetry in liquid lead’, Nature 408, p. 839

    Article  ADS  Google Scholar 

  51. Schlather, M. (2001): ‘On the second-order characteristics of marked point processes’, Bernoulli 7(1), pp. 99–107

    Article  MATH  MathSciNet  Google Scholar 

  52. Schlather, M. (2002): ‘Characterization of point processes with Gaussian marks independent of locations’, Math. Nachr. Accepted

    Google Scholar 

  53. Sok, R. M., M. A. Knackstedt, A. P. Sheppard, W. V. Pinczewski, W. B. Lindquist, A. V. A, L. Paterson (2000): ‘Direct and stochastic generation of network models from tomographic images; effect of topology on two-phase flow properties’. In: Proc. Upscaling Downunder, ed. by L. Paterson, (Kluwer Academic, Dordrecht)

    Google Scholar 

  54. Spanne, P., J. Thovert, C. Jacquin, W. Lindquist, K. Jones, P. Adler (1994): ‘Synchrotron computed microtomography of porous media: Topology and transport’, Phys. Rev. Lett. 73, pp. 2001–2004

    Article  ADS  Google Scholar 

  55. Stoyan, D. (1984): ‘On correlations of marked point processes’, Math. Nachr. 116, pp. 197–207

    Article  MATH  MathSciNet  Google Scholar 

  56. Stoyan, D. (2000): ‘Basic ideas of spatial statistics’. In: Statistical Physics and Spatial Statistics: The Art of Analyzing and Modeling Spatial Structures and Pattern Formation, ed. by K. R. Mecke, D. Stoyan, Number 554 in Lecture Notes in Physics (Springer, Berlin)

    Google Scholar 

  57. Stoyan, D. (2000): ‘Recent applications of point process methods in forestry statistics’, Statistical Sciences 15, pp. 61–78

    Article  MathSciNet  Google Scholar 

  58. Stoyan, D., W. S. Kendall, J. Mecke (1995): Stochastic Geometry and its Applications (John Wiley & Sons, Chichester), 2nd edition

    MATH  Google Scholar 

  59. Stoyan, D., H. Stoyan (1985): ‘On one of matérn’s hard-core point process models’, Biom. J. 122, p. 205

    MATH  MathSciNet  Google Scholar 

  60. Stoyan, D., H. Stoyan (1994): Fractals, Random Shapes and Point Fields (John Wiley & Sons, Chichester)

    MATH  Google Scholar 

  61. Stoyan, D., O. Wälder (2000): ‘On variograms in point process statistics, ii: Models for markings and ecological interpretation’, Biom. J. 42, pp. 171–187

    Article  MATH  Google Scholar 

  62. Struble, M. F., P. Peebles (1985): ‘Erratum: A new application of Binggeli’s test for large-scale alignment of clusters of galaxies’, Astronomical Journal 90, pp. 582–589

    Article  ADS  Google Scholar 

  63. Struble, M. F., P. Peebles (1986): ‘A new application of binggeli’s test for large-scale alignment of clusters of galaxies’, Astronomical Journal 91, p. 1474

    Article  ADS  Google Scholar 

  64. Sylos Labini, F., M. Montuori, L. Pietronero (1998): ‘Scale invariance of galaxy clustering’, Physics Rep. 293, pp. 61–226

    Article  ADS  Google Scholar 

  65. Szapudi, I., G. B. Dalton, G. Efstathiou, A. S. Szalay (1995): ‘Higher order statistics from the APM galaxy survey’, Astrophysical Journal 444, pp. 520–531

    Article  ADS  Google Scholar 

  66. Szapudi, I., A. S. Szalay (1993): ‘Higher order statistics of the galaxy distribution using generating functions’, Astrophysical Journal 408, pp. 43–56

    Article  ADS  Google Scholar 

  67. Ulmer, M., S. L.W. McMillan, M. P. Kowalski (1989): ‘Do the major axis of rich clusters of galaxies point toward their neighbors?’, Astrophysical Journal 338, pp. 711–717

    Article  ADS  Google Scholar 

  68. van Lieshout, M. N. M., A. J. Baddeley (1996): ‘A nonparametric measure of spatial interaction in point patterns’, Statist. Neerlandica 50, pp. 344–361

    Article  MATH  MathSciNet  Google Scholar 

  69. van Lieshout, M. N. M., A. J. Baddeley (1999): ‘Indices of dependence between types in multivariate point patterns’, Scand. J. Statist. 26, pp. 511–532

    Article  MATH  MathSciNet  Google Scholar 

  70. Wälder, O., D. Stoyan (1996): ‘On variograms and point process statistics’, Biom. J. 38, pp. 895–905

    Article  MATH  Google Scholar 

  71. Wälder, O., D. Stoyan (1997): ‘Models of markings and thinnings of poisson processes’, Statistics 29, pp. 179–202

    Article  MATH  MathSciNet  Google Scholar 

  72. Widom, B., J. Rowlinson (1970): ‘New model for the study of liquid-vapor phase transitions’, J. Chem. Phys. 52, pp. 1670–1684

    Article  ADS  Google Scholar 

  73. Willmer, C., L. N. da Costa, P. Pellegrini (March 1998): ‘Southern sky redshift survey: Clustering of local galaxies’, Astronomical Journal 115, pp. 869–884

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Nuclear & Astrophysics Laboratory, University of Oxford, Keble Road, Oxford, OX1 3RH, Great Britain

    Claus Beisbart

  2. Sektion Physik, Ludwig-Maximilians-Universität, Theresienstraße 37, D-80333, München, Germany

    Martin Kerscher

  3. Max-Planck-Institut für Metallforschung, Heisenbergstr. 1, D-70569, Stuttgart, Germany

    Klaus Mecke

  4. Institut für Theoretische und Angewandte Physik, Fakultät für Physik, Universität Stuttgart, Pfaffenwaldring 57, D-70569, Stuttgart, Germany

    Klaus Mecke

Authors
  1. Claus Beisbart
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Kerscher
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Klaus Mecke
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Max-Planck-Institut für Metallforschung, Heisenbergstrasse 1, 70569, Stuttgart, Germany

    Klaus Mecke

  2. Fakultät für Mathematik und Informatik Institut für Stochastik, TU Bergakademie Freiberg, 09596, Freiberg, Germany

    Dietrich Stoyan

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Beisbart, C., Kerscher, M., Mecke, K. (2002). Mark Correlations: Relating Physical Properties to Spatial Distributions. In: Mecke, K., Stoyan, D. (eds) Morphology of Condensed Matter. Lecture Notes in Physics, vol 600. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45782-8_15

Download citation

  • DOI: https://doi.org/10.1007/3-540-45782-8_15

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-44203-5

  • Online ISBN: 978-3-540-45782-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation