Environmental and Ecological Statistics

, Volume 15, Issue 4, pp 469–489 | Cite as

Evaluating the sampling pattern when detecting Zones of Abrupt Change

  • Edith Gabriel
  • Denis Allard


We present a method for detecting the zones where an irregularly sampled variable changes abruptly in the plane. Such zones are called Zones of Abrupt Change (ZACs). This method not only allows estimation of ZACs, but also testing of their statistical significance against the null hypothesis of a stationary correlated random field. The sampling pattern, in particular its local density, is crucial in the detection of potential ZACs. In this paper, we address the problem of evaluating the sampling pattern by assessing the power of the local test used for detecting ZACs. It is shown that mapping the power allows us to identify zones where ZACs may or may not be detected. The methodology is applied to a soil data set sampled at eight different dates in an agricultural field. Detecting ZACs for the soil water content allowed us to identify permanent structures in the agricultural field related to the boundaries between different soil types. Mapping the power for various sampling densities proved to be useful to determine the minimal sampling density necessary for detecting ZACs.


Boundary detection Power Test Wombling 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adler R (2000) On excursion sets, tube formulas and maxima of random fields. Ann Appl Probab 10: 1–74Google Scholar
  2. Allard D, Gabriel E, Bacro JN (2005) Estimating and testing zones of abrupt change for spatial data. Research Report 2, Unité Biostatistique et Processus Spatiaux, INRA-Avignon. Available at:
  3. Aronowich M, Adler R (1988) Sample path behaviour of χ 2 surfaces at extrema. Adv Appl Probab 18: 901–920CrossRefGoogle Scholar
  4. Banerjee S, Gelfand A, Sirmans C (2003) Directional rates of change under spatial process models. J Am Stat Assoc 98: 946–954CrossRefGoogle Scholar
  5. Barbujani G, Oden N, Sokal R (1989) Detecting areas of abrupt change in maps of biological variables. Syst Zool 38: 376–389CrossRefGoogle Scholar
  6. Bocquet-Appel J-P, Bacro J-N (1994) Generalized wombling. Syst Biol 43: 442–448CrossRefGoogle Scholar
  7. Cao J (1999) The size of the connected components of excursion sets of χ 2, t and F fields. Adv Appl Probab (SGSA) 31: 579–595CrossRefGoogle Scholar
  8. Chaudhuri P, Marron J (1999) SiZer for exploration of structures in curves. J Am Stat Assoc, Theory Method 94: 807–823CrossRefGoogle Scholar
  9. Chilès J-P, Delfiner P (1999) Geostatistics: modeling spatial uncertainty. Wiley, New-YorkGoogle Scholar
  10. Cressie N (1993) Statistics for spatial data, revised edition. Wiley, New-YorkGoogle Scholar
  11. Dudoit S, Shaffer J-P, Boldrick J (2003) Multiple hypothesis testing in microarray experiments. Stat Sci 18: 71–103CrossRefGoogle Scholar
  12. Fortin M-J (1994) Edge detection algorithms for two-dimensional ecological data. Ecology 75: 956–965CrossRefGoogle Scholar
  13. Fortin M-J, Drapeau P (1995) Delineation of ecological boundaries: comparisons of approaches and significance tests. Oikos 72: 323–332CrossRefGoogle Scholar
  14. Gabriel E (2004) Détection de zones de changement abrupt dans des données spatiales et application à l’agriculture de précision. Ph.D. thesis, University of Montpellier. Available at:
  15. Gabriel E, Allard D, Bacro J-N (2004) Detecting Zones of Abrupt Change: application to soil data. In: Sanchez-Vila X, Carrera J, Froidevaux R (eds) Proceedings of the IV European conference on geostatistics for environmental applications, pp 437–448Google Scholar
  16. Gabriel E, Allard D, Mary B, Guérif M (2007) Detecting zones of abrupt change in soil data, with an application to an agricultural field. Eur J Soil Sci 58: 1273–1284CrossRefGoogle Scholar
  17. Gleyze J-F, Bacro J-N, Allard D (2001) Detecting regions of abrupt change: wombling procedure and statistical significance. In: Monestiez P, Allard D, Froidevaux R(eds) geoENV III: geostatistics for environmental applications. Kluwer, The Netherlands, pp 311–322Google Scholar
  18. Godtliebsen F, Marron J, Pizer S (2002) Significance in scale-space for clustering. In: Lawson AB, Denison DGT (eds) Spatial clustering modeling. Chapman and Hall/CRC, pp 24–36Google Scholar
  19. Guérif M, Beaudoin C, Durr V, Houlès V, Machet J-M, Mary B, Moulin S, Richard G (2001) Designing a field experiment for assessing soil and crop spatial variability and defining site-specific management strategies. In: Grenier G, Blackmore S (eds) Proceedings of the third European conference on precision agriculture, pp 677–682Google Scholar
  20. Hall P, Rau C (2001) Local likelihood tracking of fault lines and boundaries. J R Stat Soc B 63: 569–582CrossRefGoogle Scholar
  21. Jacquez G, Maruca S (1998) Geographic boundary detection. In: Poiker TK, Chrisman N (eds) Proceedings of the 8th international symposium on spatial data handling. International Geographical Union, pp 496–509Google Scholar
  22. Jacquez G, Maruca S, Fortin M-J (2000) From fields to objects: a review of geographic boundary analysis. J Geogr Syst 2: 221–241CrossRefGoogle Scholar
  23. Lantuéjoul C (1991) Ergodicity and integral range. J Microsc 161: 387–403Google Scholar
  24. Mary B, Beaudoin N, Machet J-M, Bruchou C, Ariès F (2001) Characterization and analysis of soil variability within two agricultural fields: the case of water and mineral N profiles. In: Grenier G, Blackmore S (eds) Proceedings of the 3rd European conference on precision agriculture, pp 431–436Google Scholar
  25. Oden N, Sokal R, Fortin M-J, Goebl H (1993) Categorical wombling: detecting regions of significant change in spatially located categorical variables. Geogr Anal 25: 315–336CrossRefGoogle Scholar
  26. Pagel M, Mace R (2004) The cultural wealth of nations. Nature 428: 275–278PubMedCrossRefGoogle Scholar
  27. Womble W (1951) Differential systematics. Science 114: 315–322PubMedCrossRefGoogle Scholar
  28. Worsley K (1994) Local maxima and the expected Euler characteristic of excursion sets of χ 2, F and t fields. Adv Appl Probab 26: 13–42CrossRefGoogle Scholar
  29. Worsley K (2001) Testing for signals with unknown location and scale in a χ 2 random field, with application to fMRI. Adv Appl Probab (SGSA) 33: 773–793CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.IUT STIDUniversité d’ Avignon, Site AgroparcAvignonFrance
  2. 2.Unité de Biostatistique et Precessus SpatiauxINRAAvignonFrance

Personalised recommendations