Nonparametric spatial covariance functions: Estimation and testing

Abstract

Spatial autocorrelation techniques are commonly used to describe genetic and ecological patterns. To improve statistical inference about spatial covariance, we propose a continuous nonparametric estimator of the covariance function in place of the spatial correlogram. The spline correlogram is an adaptation of a recent development in spatial statistics and is a generalization of the commonly used correlogram. We propose a bootstrap algorithm to erect a confidence envelope around the entire covariance function. The meaning of this envelope is discussed. Not all functions that can be drawn inside the envelope are candidate covariance functions, as they may not be positive semidefinite. However, covariance functions that do not fit, are not supported by the data. A direct estimate of the L0 spatial correlation length with associated confidence interval is offered and its interpretation is discussed. The spline correlogram is found to have high precision when applied to synthetic data. For illustration, the method is applied to electrophoretic data of an alpine grass (Poa alpina).

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References

  1. Albert, P.S. and McShane, L.M. (1995) A generalized estimating equations approach for spatially correlated binary data: Applications to the analysis of neuroimaging data. Biometrics, 51, 627–38.

    Google Scholar 

  2. Bayly, P.V., Johnson, E.E., Wolf, P.D., Greenside, H.D., Smith, W.M., and Ideker, E.E. (1993) A quantitative measurement of spatial order in ventricular fibrillation. Journal of Cardiovascular Electrophysiology, 4, 533–46

    Google Scholar 

  3. Bertorelle, G. and Barbujani, G. (1995) Analysis of DNA diversity by spatial autocorrelation. Genetics, 140, 811–19.

    Google Scholar 

  4. BjÙrnstad, O.N., Iversen, A., and Hansen, M. (1995) The spatial structure of the gene pool of a viviparous population of Poa alpinaÐenvironmental controls and spatial constraints. Nordic Journal of Botany, 15, 347–54.

    Google Scholar 

  5. BjÙrnstad, O.N. and Falck, W. (1997) Chapter 10: An extension of the spatial correlogram and the xintercept for genetic data. In Statistical Models for Fluctuating Populations: Patterns and Processes in Time and Space, O.N. BjÙrnstad, Dr Philos. Dissertation, University of Oslo, Oslo.

    Google Scholar 

  6. BjÙrnstad, O.N., Ims, R.A., and Lambin, X. (1999a) Spatial population dynamics: Analysing patterns and processes of population synchrony. Trends in Ecology and Evolution, 11,427–31.

    Google Scholar 

  7. BjÙrnstad, O.N., Stenseth, N.C., and Saitoh, T. (1999b) Synchrony and scaling in dynamics of voles and mice in northern Japan. Ecology, 80, 622–37.

    Google Scholar 

  8. BjÙrnstad, O. and Bolker, B. (2000) Canonical functions for dispersal-induced spatial covariance and synchrony. Proceedings of Royal Society London, B., 267, 1787–94.

    Google Scholar 

  9. Box, G.E.P. and Jenkins, G.M. (1970) Time Series Analysis: Forecasting and Control, Holden-Day, San Francisco.

    Google Scholar 

  10. Cressie, N. (1993) Statistics for Spatial Data, Wiley, New York.

    Google Scholar 

  11. Deutsch, C.V. and Journel, A.G. (1992) GSLIB: Geostatistical Software Library and User's Guide, Oxford University Press, New York.

    Google Scholar 

  12. Efron, B. and Tibshirani, R.J. (1993) An Introduction to the Bootstrap, Chapman and Hall, London.

    Google Scholar 

  13. Epperson, B.K. (1993a) Recent advances in correlation studies of spatial patterns of genetic variation. Evolutionary Biology, 27, 95–155.

    Google Scholar 

  14. Epperson, B.K. (1993b) Spatial and space-time correlations in systems of subpopulations with genetic drift and migration. Genetics, 133, 71–27.

    Google Scholar 

  15. Epperson, B.K. (1995) Fine-scale spatial structure: correlations for individual genotypes differ from those for local gene frequencies. Evolution, 49, 1022–26.

    Google Scholar 

  16. Epperson, B.K. and Li, T. (1997) Gene dispersal and spatial genetic structure. Evolution, 51, 672–81.

    Google Scholar 

  17. Green, P.J. and Silverman, B.W. (1994) Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach, Chapman and Hall, London.

    Google Scholar 

  18. Hall, P. (1993) On edgeworth expansion and bootstrap confidence bands in nonparametric curve estimation. Journal of Royal Statistical Society B, 55, 291–304.

    Google Scholar 

  19. Hall, P., Fisher, N.I., and Hoffmann, B. (1994) On the nonparametric estimation of covariance functions. Annals of Statistics, 22, 2115–34.

    Google Scholar 

  20. Härdle, W. (1990) Applied Nonparametric Regression, Cambridge University Press, Cambridge.

    Google Scholar 

  21. Hilgers, J.W., Reynolds, W.R., Strang, D.E., and McManamey, J.R. (1996) Correlation length used as a predictor of fundamental scale lengths for image characterization. Optical Engineering, 35, 786–93.

    Google Scholar 

  22. Hyndman, R.J. and Wand, M.P. (1997) Nonparametric autocovariance function estimation. Australian Journal of Statistics, 39, 313–24.

    Google Scholar 

  23. Jones, M.C., Davies, S.J., and Park, B.U. (1994) Versions of kernel-type regression estimators. Journal of the American Statistical Association, 89, 825–32.

    Google Scholar 

  24. Journel, A.G. and Huijbregts, C.J. (1978) Mining Geostatistics, Academic Press, London.

    Google Scholar 

  25. Koenig, W.D. (1999) Spatial autocorrelation of ecological phenomena. Trends in Ecology and Evolution, 14, 22–6.

    Google Scholar 

  26. Lande, R. (1991) Isolation by distance in a quantitative trait. Genetics, 128, 443–53.

    Google Scholar 

  27. Legendre, P. and Fortin, M.-J. (1989) Spatial pattern and ecological analysis. Vegetatio, 80, 107–38.

    Google Scholar 

  28. Linhart, Y.B. and Grant, M.C. (1996) Evolutionary significance of local genetic differentiation in plants. Annual Review of Ecology and Systematics, 27, 237–77.

    Google Scholar 

  29. MacDonald, I.L. and Zucchini, W. (1997) Hidden Markov and Other Models for Discrete-Valued Time Series, Chapman & Hall, London.

    Google Scholar 

  30. Manly, B.F.J. (1997) Randomization, Bootstrap and Monte Carlo Methods in Biology, (2nd ed.). Chapman and Hall, London.

    Google Scholar 

  31. McGraw, J.B. (1995) Patterns and causes of genetic diversity in arctic plants. In Arctic and Alpine Biodiversity: Patterns, Causes and Ecosystem Consequences, F.S. Chapin and C. Koèrner (eds), Springer-Verlag, Berlin. pp. 33–43.

    Google Scholar 

  32. Myers, R.A., Mertz, G., and Barrowman, N.J. (1995) Spatial scales of variability in cod recruitment in the North Atlantic. Canadian Journal of Fisheries and Aquatic Science, 52, 1849–62.

    Google Scholar 

  33. Nordal, I. and Iversen, A.P. (1993) Mictic and monomorphic versus parthenogenetic and polymorphicÐa comparison of two Scandinavia mountain grasses. Opera Botanica, 21, 19–27.

    Google Scholar 

  34. Nychka, D. (1995) Splines as local smoothers. Annals of Statistics, 23, 1175–97.

    Google Scholar 

  35. Oden, N.L. (1984) Assessing the significance of a spatial correlogram. Geographical Analysis, 16, 1–16.

    Google Scholar 

  36. Oden, N.L. and Sokal, R.R. (1986) Directional autocorrelation: an extension of spatial correlograms to two dimensions. Systematic Zoology, 35, 608–17.

    Google Scholar 

  37. Ripley, B.D. (1987). Stochastic Simulation, Wiley.

  38. Scott, D.W. (1992) Multivariate Density Estimation: Theory, Practice, and Visualization, John Wiley & Sons, New York.

    Google Scholar 

  39. Sokal, R.R. and Jacquez, G.M. (1991) Testing inferences about microevolutionary processes by means of spatial autocorrelation techniques. Evolution, 45, 152–68.

    Google Scholar 

  40. Sokal, R.R. and Oden, N.L. (1978a) Spatial autocorrelation in biology. I. Methodology. Biological Journal of the Linnean Society, 10, 199–228.

    Google Scholar 

  41. Sokal, R.R. and Oden, N.L. (1978b) Spatial autocorrelation in biology. II. Some biological implications and four applications of evolutionary and ecological interest. Biological Journal of the Linnean Society, 10, 229–49.

    Google Scholar 

  42. Sokal, R.R. and Wartenberg, D.E. (1983) A test for spatial autocorrelation analysis using an isolation-by-distance model. Genetics, 105, 219–37.

    Google Scholar 

  43. Tsay, R.S. (1992) Model checking via parametric bootstraps in time series analysis. Applied Statistics, 41, 1–15.

    Google Scholar 

  44. Young, G.A. (1994) Bootstrap: more than a stab in the dark? Statistical Science, 9, 382–415.

    Google Scholar 

  45. Zimmerman, D.L. (1989) Computationally exploitable structure of covariance matrices and generalized covariance matrices in spatial models. Journal of Statistics and Computer Simulation, 32, 1–15.

    Google Scholar 

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BjØrnstad, O.N., Falck, W. Nonparametric spatial covariance functions: Estimation and testing. Environmental and Ecological Statistics 8, 53–70 (2001). https://doi.org/10.1023/A:1009601932481

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  • bootstrapping dependent data
  • correlogram
  • geostatistis
  • nonparametric regression
  • population genetics
  • smoothing spline
  • spatial autocorrelation