Abstract
When working with contemporary spatial ecological datasets, statistical modellers are often confronted by two major challenges: (I) the need for covariance models with the flexibility to accommodate directional patterns of anisotropy; and (II) the computational effort demanded by high-dimensional inverse and determinant problems involving the covariance matrix \(\vec {V}\). In the case of rectangular lattice data, the spatially separable covariogram is a longstanding but underused model that can reduce arithmetic complexity by orders of magnitude. We examine a class of covariograms for stationary data that extends the separable model through affine coordinate transformations, providing a far greater flexibility for handling anisotropy than that offered by the standard approach of using geometric anisotropy to extend an isotropic model. This motivates our development of an extremely fast estimator of the orientation of the axes of range anisotropy on spatial lattice data, and a powerful visual diagnostic for nonstationarity. In a case study, we demonstrate how these tools can be used to analyze and predict forest damage patterns caused by outbreaks of the mountain pine beetle.
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References
Allard D, Senoussi R, Porcu E (2016) Anisotropy models for spatial data. Math Geosci 48(3):305–328. https://doi.org/10.1007/s11004-015-9594-x
Ambikasaran S, Foreman-Mackey D, Greengard L, Hogg DW, O’Neil M (2016) Fast direct methods for Gaussian processes. IEEE Trans Pattern Anal Mach Intell 38(2):252–265. https://doi.org/10.1109/TPAMI.2015.2448083. arXiv:1403.6015
Aukema BH, Carroll AL, Zhu J, Raffa KF, Sickley TA, Taylor SW (2006) Landscape level analysis of mountain pine beetle in British Columbia, Canada: spatiotemporal development and spatial synchrony within the present outbreak. Ecography 29(3):427–441. https://doi.org/10.1111/j.2006.0906-7590.04445.x
Aukema BH, Carroll AL, Zheng Y, Zhu J, Raffa KF, Dan Moore R, Stahl K, Taylor SW (2008) Movement of outbreak populations of mountain pine beetle: influences of spatiotemporal patterns and climate. Ecography 31(3):348–358. https://doi.org/10.1111/j.0906-7590.2007.05453.x
Banerjee S, Gelfand AE, Finley AO, Sang H (2008) Gaussian predictive process models for large spatial data sets. J R Stat Soc Ser B Stat Methodol 70(4):825–848. https://doi.org/10.1111/j.1467-9868.2008.00663.x
Banerjee S, Carlin BP, Gelfand AE (2014) Hierarchical modeling and analysis for spatial data. CRC Press, Boca Raton. https://doi.org/10.1201/b17115
Beale CM, Lennon JJ, Yearsley JM, Brewer MJ, Elston DA (2010) Regression analysis of spatial data. Ecol Lett 13(2):246–264. https://doi.org/10.1111/j.1461-0248.2009.01422.x
Chen J, Yang B, Yang L (2019) To BE or not to BE, that is the question. Nat Biotechnol 37(5):520–522. https://doi.org/10.1038/s41587-019-0119-x
Chilès JP, Delfiner P (2012) Geostatistics: modeling spatial uncertainty, 2nd edn. Wiley, Hoboken. https://doi.org/10.1002/9781118136188
Cressie N (1992) Statistics for spatial data, vol 4. Wiley, New York. https://doi.org/10.1111/j.1365-3121.1992.tb00605.x
Cressie N, Johannesson G (2008) Fixed rank kriging for very large spatial data sets. J R Stat Soc Ser B Stat Methodol 70(1):209–226. https://doi.org/10.1111/j.1467-9868.2007.00633.x
Dale MR, Fortin MJ (2014) Spatial analysis: a guide for ecologists, 2nd edn. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511978913
Das A, Lele SR, Glass GE, Shields T, Patz J (2002) Modelling a discrete spatial response using generalized linear mixed models: application to Lyme disease vectors. Int J Geogr Inf Sci 16(2):151–166. https://doi.org/10.1080/13658810110099134
Davies S, Hall P (1999) Fractal analysis of surface roughness by using spatial data. J R Stat Soc Ser B Stat Methodol 61(1):3–37. https://doi.org/10.1111/1467-9868.00160
Dhar A, Parrott L, Heckbert S (2016) Consequences of mountain pine beetle outbreak on forest ecosystem services in western Canada. Can J For Res 46(8):987–999. https://doi.org/10.1139/cjfr-2016-0137
Dietrich CR (1993) Computationally efficient cholesky factorization of a covariance matrix with bedk toeplitz structure. J Stat Comput Simul 45(3–4):203–218. https://doi.org/10.1080/00949659308811481
Dormann CF (2009) Response to comment on “methods to account for spatial autocorrelation in the analysis of species distributional data: a review”. Ecography 32(3):379–381. https://doi.org/10.1111/j.1600-0587.2009.05907.x
Duan JJ, Taylor PB, Fuester RW (2011) Biology and life history of balcha indica, an ectoparasitoid attacking the Emerald Ash Borer, Agrilus planipennis, in North America. J Insect Sci 11:1–9. https://doi.org/10.1673/031.011.12701
Genton MG (2007) Separable approximations of space–time covariance matrices. Environmetrics 18(7):681–695. https://doi.org/10.1002/env.854
Goodchild MF, Mark DM (1987) The fractal nature of geographic phenomena. Ann Assoc Am Geogr 77(2):265–278. https://doi.org/10.1111/j.1467-8306.1987.tb00158.x
Guan Y, Sherman M, Calvin JA (2004) A nonparametric test for spatial isotropy using subsampling. J Am Stat Assoc 99(467):810–821. https://doi.org/10.1198/016214504000001150
Guillot G, Schilling RL, Porcu E, Bevilacqua M (2014) Validity of covariance models for the analysis of geographical variation. Methods Ecol Evol 5(4):329–335. https://doi.org/10.1111/2041-210X.12167. arXiv:1311.4136
Guttorp P, Gneiting T (2006) Studies in the history of probability and statistics XLIX on the Matérn correlation family. Biometrika 93(4):989–995. https://doi.org/10.1093/biomet/93.4.989
Hawkins BA (2012) Eight (and a half) deadly sins of spatial analysis. J Biogeogr 39(1):1–9. https://doi.org/10.1111/j.1365-2699.2011.02637.x
Heagerty PJ, Lele SR (1998) A composite likelihood approach to binary spatial data. J Am Stat Assoc 93(443):1099–1111. https://doi.org/10.1080/01621459.1998.10473771
Higdon D (1998) A process-convolution approach to modelling temperatures in the North Atlantic Ocean. Environ Ecol Stat 5(2):173–190. https://doi.org/10.1023/A:1009666805688
Hirano T (2014) Pseudo best estimator by a separable approximation of spatial covariance structures. J Jpn Stat Soc 44(1):43–71. https://doi.org/10.14490/jjss.44.43
Hooke R, Jeeves TA (1961) “Direct search” solution of numerical and statistical problems. J ACM 8(2):212–229. https://doi.org/10.1145/321062.321069
Inchausti P (1998) In: Hilborn R, Mangel M (eds) The ecological detective: confronting models with data, vol 73. Princeton University Press, Princeton. https://doi.org/10.1086/420265
Jun M, Stein ML (2008) Nonstationary covariance models for global data. Ann Appl Stat 2(4):1271–1289. https://doi.org/10.1214/08-AOAS183
Kaufman CG, Schervish MJ, Nychka DW (2008) Covariance tapering for likelihood-based estimation in large spatial data sets. J Am Stat Assoc 103(484):1545–1555. https://doi.org/10.1198/016214508000000959
Keitt TH, Bjørnstad ON, Dixon PM, Citron-Pousty S (2002) Accounting for spatial pattern when modeling organism-environment interactions. Ecography 25(5):616–625. https://doi.org/10.1034/j.1600-0587.2002.250509.x
Klutsch JG, Negrón JF, Costello SL, Rhoades CC, West DR, Popp J, Caissie R (2009) Stand characteristics and downed woody debris accumulations associated with a mountain pine beetle (Dendroctonus ponderosae Hopkins) outbreak in Colorado. For Ecol Manage 258(5):641–649. https://doi.org/10.1016/j.foreco.2009.04.034
Legendre P (1993) Spatial autocorrelation: trouble or new paradigm? Ecology 74(6):1659–1673. https://doi.org/10.2307/1939924
Lindsay BG (1988) Composite likelihood methods. Contemp Math 80(1):221–239. https://doi.org/10.1090/conm/080/999014
Martin RJ (1979) A subclass of lattice processes applied to a problem in planar sampling. Biometrika 66(2):209–217. https://doi.org/10.1093/biomet/66.2.209
Matheron G (1962) Traité de Géostatistique Appliquée, vol 1. Editions Technip, Paris
Nakagawa S, Freckleton RP (2008) Missing inaction: the dangers of ignoring missing data. Trends Ecol Evol 23(11):592–596. https://doi.org/10.1016/j.tree.2008.06.014
Oliveira FMM, Dantas RT, Furtado DA, Nascimento JWB, Medeiros AN (2005) Parâmetros de conforto térmico e fisiológico de ovinos Santa Inês, sob diferentes sistemas de acondicionamento. Rev Bras Eng Agrícola e Ambient 9(4):631–635. https://doi.org/10.1590/s1415-43662005000400029
Paciorek CJ, Schervish MJ (2006) Spatial modelling using a new class of nonstationary covariance functions. Environmetrics 17(5):483–506. https://doi.org/10.1002/env.785
Pick G (1899) Geometrisches zur Zahlenlehre. Sitzenber Lotos 19:311–319
Rathbun SL, Stein ML (2000) Interpolation of spatial data: some theory for Kriging, vol 95. Springer, New York. https://doi.org/10.2307/2669494
Roberts S, Osborne M, Ebden M, Reece S, Gibson N, Aigrain S (2013) Gaussian processes for time-series modelling. Philos Trans R Soc A 371(1984):20110550
Robertson C, Farmer CJ, Nelson TA, MacKenzie IK, Wulder MA, White JC (2009) Determination of the compositional change (1999–2006) in the pine forests of British Columbia due to mountain pine beetle infestation. Environ Monit Assess 158(1–4):593–608. https://doi.org/10.1007/s10661-008-0607-9
Sampson PD, Guttorp P (1992) Nonparametric estimation of nonstationary spatial covariance structure. J Am Stat Assoc 87(417):108–119. https://doi.org/10.1080/01621459.1992.10475181
Sherman M (2010) Spatial statistics and spatio-temporal data: covariance functions and directional properties. Wiley, West Sussex. https://doi.org/10.1002/9780470974391
Simpson D, Lindgren F, Rue H (2012) In order to make spatial statistics computationally feasible, we need to forget about the covariance function. Environmetrics 23(1):65–74. https://doi.org/10.1002/env.1137
Stein ML (2005) Space-time covariance functions. J Am Stat Assoc 100(469):310–321. https://doi.org/10.1198/016214504000000854
Stein ML, Chi Z, Welty LJ (2004) Approximating likelihoods for large spatial data sets. J R Stat Soc Ser B Stat Methodol 66(2):275–296. https://doi.org/10.1046/j.1369-7412.2003.05512.x
Van Loan CF (2000) The ubiquitous Kronecker product. J Comput Appl Math 123(1):85–100
Varadhan R, Borchers HW, Varadhan MR (2016) Package ‘dfoptim’
Varin C, Reid N, Firth D (2011) An overview of composite likelihood methods. Stat Sin 21(1):5–42
Vecchia AV (1988) Estimation and model identification for continuous spatial processes. J R Stat Soc Ser B 50(2):297–312. https://doi.org/10.1111/j.2517-6161.1988.tb01729.x
Ver Hoef JM, Peterson EE, Hooten MB, Hanks EM, Fortin MJ (2018) Spatial autoregressive models for statistical inference from ecological data. Ecol Monogr 88(1):36–59. https://doi.org/10.1002/ecm.1283
Wall MM (2004) A close look at the spatial structure implied by the CAR and SAR models. J Stat Plan Inference 121(2):311–324. https://doi.org/10.1016/S0378-3758(03)00111-3
Warton DI, Hui FK (2011) The arcsine is asinine: the analysis of proportions in ecology. Ecology 92(1):3–10. https://doi.org/10.1890/10-0340.1
Weller ZD, Hoeting JA (2016) A review of nonparametric hypothesis tests of isotropy properties in spatial data. Stat Sci 31(3):305–324. https://doi.org/10.1214/16-STS547. arXiv:1508.05973
Westfall J (2005) 2005 Summary of forest health conditions in British Columbia. Tech rep, British Columbia Ministry of Forests and Range, Forest Practices Branch, Victoria
Wilson AG, Gilboa E, Nehorai A, Cunningham JP (2014) Fast kernel learning for multidimensional pattern extrapolation. Adv Neural Inf Process Syst 4:3626–3634
Wulder MA, Dymond CC, White JC, Leckie DG, Carroll AL (2006) Surveying mountain pine beetle damage of forests: a review of remote sensing opportunities. For Ecol Manage 221(1–3):27–41. https://doi.org/10.1016/j.foreco.2005.09.021
Zimmerman DL (1989) Computationally exploitable structure of covariance matrices and generalized covariance matrices in spatial models. J Stat Comput Simul 32(1–2):1–15. https://doi.org/10.1080/00949658908811149
Zimmerman DL (1993) Another look at anisotropy in geostatistics. Math Geol 25(4):453–470. https://doi.org/10.1007/BF00894779
Acknowledgements
The authors thank the Lewis Research Group for providing expert advice and feedback. MAL is also grateful for support through NSERC and the Canada Research Chair Program. This research was supported by a grant to MAL from the Natural Science and Engineering Research Council of Canada (Grant No. NET GP 434810-12) to the TRIA Network, with contributions from Alberta Agriculture and Forestry, Foothills Research Institute, Manitoba Conservation and Water Stewardship, Natural Resources Canada-Canadian Forest Service, Northwest Territories Environment and Natural Resources, Ontario Ministry of Natural Resources and Forestry, Saskatchewan Ministry of Environment, West Fraser and Weyerhaeuser.
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Koch, D., Lele, S. & Lewis, M.A. Computationally simple anisotropic lattice covariograms. Environ Ecol Stat 27, 665–688 (2020). https://doi.org/10.1007/s10651-020-00456-2
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DOI: https://doi.org/10.1007/s10651-020-00456-2