Coregionalization analysis with a drift for multi-scale assessment of spatial relationships between ecological variables 1. Estimation of drift and random components

  • Bernard Pelletier
  • Pierre Dutilleul
  • Guillaume Larocque
  • James W. Fyles
Article

Abstract

In this and a second article, we propose ‘coregionalization analysis with a drift’ (CRAD), as a method to assess the multi-scale variability of and relationships between ecological variables from a multivariate spatial data set. CRAD is carried out in two phases: (I) a deterministic component representing the large-scale pattern (called ‘drift’) and a random component modeled as a second-order stationary process are estimated for each variable separately; (II) a linear model of coregionalization is fitted to the direct and cross experimental variograms of residuals (i.e., after removing the estimated drifts) to assess relationships at smaller scales, while the estimated drifts are used to study relationships at large scale. In this article, we focus on phase I of CRAD, by addressing the questions of the choice of the drift estimation procedure, which is linked to the estimation of random components, and of the presence of a bias in the direct experimental variogram of residuals. In this phase, both the estimation of the drift and the fitting of a model to the direct experimental variogram of residuals are performed iteratively by estimated generalized least squares (EGLS). We use theoretical calculations and a Monte Carlo study to demonstrate that complex large-scale patterns, such as patchy drifts, are better captured with local drift estimation procedures using low-order polynomials within a moving window, than with global procedures. Furthermore, despite the bias in direct experimental variograms of residuals, good estimates of spatial autocovariance parameters are obtained with the double iterative EGLS procedure in the conditions of application of CRAD. An example with forest soil property and tree species diversity data is presented to discuss the choice of drift estimation procedure in practice.

Keywords

Bias in variogram of residuals Bounded sampling domain Estimated generalized least squares Geostatistical models Local versus global drift estimation Small versus large scales of variability 

References

  1. Alpargu G and Dutilleul P (2006). Stepwise regression in mixed quantitative linear models with autocorrelated errors. Commun Stat Simul Comput 35(1): 79–104 CrossRefGoogle Scholar
  2. Altman NS (1990). Kernel smoothing of data with correlated error. J Am Stat Assoc 85(411): 749–759 CrossRefGoogle Scholar
  3. Altman N (2000). Krige, smooth, both or neither? (with discussion). Aust NZ J Stat 42(4): 441–461 CrossRefGoogle Scholar
  4. Armstrong M (1984). Problems with universal kriging. Math Geol 16(1): 101–108 CrossRefGoogle Scholar
  5. Atkinson PM and Tate NJ (2000). Spatial scale problems and geostatistical solutions: a review. Prof Geogr 52(4): 607–623 CrossRefGoogle Scholar
  6. Beckers F and Bogaert P (1998). Nonstationarity of the mean and unbiased variogram estimation: extension of the weighted least-squares method. Math Geol 30(2): 223–240 CrossRefGoogle Scholar
  7. Berke O (2001). Modified median-polish kriging and its application to the Wolfcamp-Aquifer data. Environmetrics 12: 731–748 CrossRefGoogle Scholar
  8. Bjørke JT and Nilsen S (2005). Trend extraction using average interpolation subdivision. Math Geol 37(6): 615–634 CrossRefGoogle Scholar
  9. Borcard D and Legendre P (2002). All-scale spatial analysis of ecological data by means of principal coordinates of neighbour matrices. Ecol Model 152: 51–68 CrossRefGoogle Scholar
  10. Borcard D, Legendre P and Drapeau P (1992). Partialling out the spatial component of ecological variation. Ecology 73: 1045–1055 CrossRefGoogle Scholar
  11. Box GEP and Cox DR (1964). An analysis of transformations. J Roy Stat Soc, Ser B 26: 211–243 Google Scholar
  12. Chilès JP and Delfiner P (1999). Geostatistics: modeling spatial uncertainty. Wiley, New York Google Scholar
  13. Cleveland WS and Devlin SJ (1988). Locally weighted regression: an approach to regression analysis by local fitting. J Am Stat Assoc 83(403): 596–610 CrossRefGoogle Scholar
  14. Cressie N (1986). Kriging nonstationary data. J Am Stat Assoc 81(395): 625–634 CrossRefGoogle Scholar
  15. Cressie NAC (1993). Statistics for spatial data, Revised edition. Wiley, New York Google Scholar
  16. Delfiner P, Matheron G (1980) Les fonctions aléatoires intrinsèques d’ordre k. Note C-84. Centre de Géostatistique, Ecole des Mines de Paris, Fontainebleau, Paris.Google Scholar
  17. Denny MW, Helmuth B, Leonard GH, Harley CDG, Hunt LJH and Nelson EK (2004). Quantifying scale in ecology: lessons from a wave-swept shore. Ecol Monogr 74(3): 513–532 CrossRefGoogle Scholar
  18. Diggle PJ, Liang KY and Zeger SL (1996). Analysis of longitudinal data. Oxford University Press, Oxford Google Scholar
  19. Draper NR and Smith H (1981). Applied regression analysis, 2nd edn. Wiley, New York Google Scholar
  20. Fan J and Gijbels I (1995a). Data-driven bandwidth selection in local polynomial fitting: variable bandwidth and spatial adaptation. J Roy Stat Soc, Ser B 57(2): 371–394 Google Scholar
  21. Fan J and Gijbels I (1995b). Adaptive order polynomial fitting: bandwidth robustification and bias reduction. J Comput Graph Stat 4: 213–227 CrossRefGoogle Scholar
  22. Goovaerts P (1992). Factorial kriging analysis: a useful tool for exploring the structure of multivariate spatial information. J Soil Sci 43: 597–619 CrossRefGoogle Scholar
  23. Goovaerts P (1994). Study of spatial relationships between two sets of variables using multivariate geostatistics. Geoderma 62: 93–107 CrossRefGoogle Scholar
  24. Goovaerts P (1997). Geostatistics for natural resources evaluation. Oxford University Press, Oxford Google Scholar
  25. Hart JD and Yi S (1998). One-sided cross-validation. J Am Stat Assoc 93: 620–631 CrossRefGoogle Scholar
  26. Hastie T and Loader C (1993). Local regression: automatic kernel carpentry. Stat Sci 8(2): 120–143 CrossRefGoogle Scholar
  27. Helterbrand JD and Cressie N (1994). Universal cokriging under intrinsic coregionalization. Math Geol 26(2): 205–226 CrossRefGoogle Scholar
  28. Hill MO (1973). Diversity and evenness: a unifying notation and its consequences. Ecology 54(2): 427–432 CrossRefGoogle Scholar
  29. Høst G (1999). Kriging by local polynomials. Comput Stat Data Anal 29: 295–312 CrossRefGoogle Scholar
  30. Hurvich CM, Simonoff JS and Tsai CL (1998). Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion. J Roy Stat Soc, Ser B 60(2): 271–293 CrossRefGoogle Scholar
  31. Jost L (2006). Entropy and diversity. Oikos 113(2): 363–375 CrossRefGoogle Scholar
  32. Journel AG and Huijbregts CJ (1978). Mining geostatistics. Academic Press, London Google Scholar
  33. Journel AG and Rossi ME (1989). When do we need a trend model in kriging?. Math Geol 21(7): 715–739 CrossRefGoogle Scholar
  34. Keitt TH and Urban DL (2005). Scale-specific inference using wavelets. Ecology 86(9): 2497–2504 CrossRefGoogle Scholar
  35. Keylock C (2005). Simpson diversity and Shannon-Wiener index as special cases of a generalized entropy. Oikos 109: 203–207 CrossRefGoogle Scholar
  36. Künsch HR, Papritz A and Bassi F (1997). Generalized cross-covariances and their estimation. Math Geol 29(6): 779–799 CrossRefGoogle Scholar
  37. Lark RM, Cullis BR and Welham SJ (2006). On spatial prediction of soil properties in the presence of a spatial trend: the empirical best linear unbiased predictor (E-BLUP) with REML. Eur J Soil Sci 57(6): 787–799 CrossRefGoogle Scholar
  38. Larocque G, Dutilleul P, Pelletier B and Fyles JW (2006). Conditional Gaussian co-simulation of regionalized componenents of soil variation. Geoderma 134(1/2): 1–16 CrossRefGoogle Scholar
  39. Larocque G, Dutilleul P, Pelletier B and Fyles JW (2007). Characterization and quantification of uncertainty in coregionalization analysis. Math Geol 39(3): 263–288 CrossRefGoogle Scholar
  40. Legendre P and Legendre L (1998). Numerical ecology, 2nd English edn. Elsevier, Amsterdam Google Scholar
  41. Levin SA (1992). The problem of pattern and scale in ecology. Ecology 73(6): 1943–1967 CrossRefGoogle Scholar
  42. Loader C (1999). Local regression and likelihood. Springer, New York Google Scholar
  43. Matheron G (1971) La théorie des variables régionalisées, et ses applications. Cahiers du Centre de Morphologie Mathématique, No 5, Ecole des Mines de Paris, Fontainebleau, ParisGoogle Scholar
  44. Matheron G (1989). Estimating and choosing: an essay on probability in practice. Springer, Berlin Google Scholar
  45. McBratney AB and Webster R (1986). Choosing functions for semi-variograms of soil properties and fitting them to sampling estimates. J Soil Sci 37: 617–639 CrossRefGoogle Scholar
  46. Meisel JE and Turner MG (1998). Scale detection in real and artificial landscapes using semivariance analysis. Landscape Ecol 13: 347–362 CrossRefGoogle Scholar
  47. Myers DE (1989). To be or not to be... stationary? That is the question. Math Geol 21(3): 347–362 CrossRefGoogle Scholar
  48. Neuman SP and Jacobson EA (1984). Analysis of nonintrinsic spatial variability by residual kriging with application to regional groundwater levels. Math Geol 16(5): 499–521 CrossRefGoogle Scholar
  49. Opsomer J, Wang Y and Yang Y (2001). Nonparametric regression with correlated errors. Stat Sci 16(2): 134–153 CrossRefGoogle Scholar
  50. Pardo-Igúzquiza E and Dowd PA (1998). The second-order stationary universal kriging model revisited. Math Geol 30(4): 347–378 CrossRefGoogle Scholar
  51. Pardo-Igúzquiza E and Dowd PA (2001). Variance-covariance matrix of the experimental variogram: assessing variogram uncertainty. Math Geol 33(4): 397–419 CrossRefGoogle Scholar
  52. Patil GP and Taillie C (1982). Diversity as a concept and its measurement. J Am Stat Assoc 77(379): 548–561 CrossRefGoogle Scholar
  53. Pelletier B, Fyles JW and Dutilleul P (1999). Tree species control and spatial structure of forest floor properties in a mixed-species stand. Ecoscience 6(1): 79–91 Google Scholar
  54. Pelletier B, Dutilleul P, Larocque G and Fyles JW (2004). Fitting the linear model of coregionalization by generalized least squares. Math Geol 36(3): 323–343 CrossRefGoogle Scholar
  55. Platt T and Denman KL (1975). Spectral analysis in ecology. Annu Rev Ecol Syst 6: 189–210 CrossRefGoogle Scholar
  56. Ripley BD (1981). Spatial statistics. Wiley, New York CrossRefGoogle Scholar
  57. Russo D and Jury WA (1987). A theoretical study of the estimation of the correlation scale in spatially variable fields. 2. Nonstationary fields. Water Resour Res 23(7): 1269–1279 CrossRefGoogle Scholar
  58. Sclocco T and Di Marzio M (2004). A weighted polynomial regression method for local fitting of spatial data. Stat Methods Appl 13: 315–325 Google Scholar
  59. Searle SR (1971). Linear models. Wiley, New York Google Scholar
  60. Shannon CE (1948). A mathematical theory of communication. AT&T Tech J 27: 379–423 Google Scholar
  61. Spellerberg IF and Fedor PJ (2003). A tribute to Claude Shannon (1916–2003) and a plea for more rigorous use of species richness, species diversity and the ‘Shannon-Wiener’ index. Global Ecol Biogeogr 12: 177–179 CrossRefGoogle Scholar
  62. Stein ML (1999). Interpolation of spatial data: some theory for kriging. Springer-Verlag, New York Google Scholar
  63. The Math Works (2006) MATLAB Version R2006, The MathWorks, Inc., Natick, USAGoogle Scholar
  64. Tukey JW (1977). Exploratory data analysis. Addison-Wesley, Reading, USA Google Scholar
  65. Turner MG, O’Neill RV, Gardner RH and Milne BT (1989). Effects of changing spatial scale on the analysis of landscape pattern. Landscape Ecol 3(3/4): 153–162 CrossRefGoogle Scholar
  66. Wackernagel H (2003). Multivariate geostatistics: an introduction with applications, 3rd edn. Springer-Verlag, Berlin Google Scholar
  67. Wackernagel H, Petitgas P and Touffait Y (1989). Overview of methods for coregionalization analysis. In: Armstrong, M (eds) Geostatistics, vol 1, pp 409–420. Kluwer Academic, Dordrecht Google Scholar
  68. Wiens JA (1989). Spatial scaling in ecology. Funct Ecol 3: 385–397 CrossRefGoogle Scholar
  69. Wu J (2004). Effects of changing scale on landscape pattern analysis: scaling relations. Landscape Ecol 19: 125–138 CrossRefGoogle Scholar
  70. Zhang XF, Van Eijkeren JCH and Heemink AW (1995). On the weighted least-squares method for fitting a semivariogram model. Comput Geosci 21(4): 605–608 CrossRefGoogle Scholar
  71. Zimmerman DL and Zimmerman MB (1991). A comparison of spatial semivariogram estimators and corresponding ordinary kriging predictors. Technometrics 33(1): 77–91 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Bernard Pelletier
    • 1
    • 2
  • Pierre Dutilleul
    • 2
  • Guillaume Larocque
    • 1
  • James W. Fyles
    • 1
  1. 1.Department of Natural Resource SciencesMcGill UniversityMontrealCanada
  2. 2.Department of Plant ScienceMcGill UniversityMontrealCanada

Personalised recommendations