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Non-linear optimal multivariate spatial design using spatial vine copulas

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Abstract

A multivariate spatial sampling design that uses spatial vine copulas is presented that aims to simultaneously reduce the prediction uncertainty of multiple variables by selecting additional sampling locations based on the multivariate relationship between variables, the spatial configuration of existing locations and the values of the observations at those locations. Novel aspects of the methodology include the development of optimal designs that use spatial vine copulas to estimate prediction uncertainty and, additionally, use transformation methods for dimension reduction to model multivariate spatial dependence. Spatial vine copulas capture non-linear spatial dependence within variables, whilst a chained transformation that uses non-linear principal component analysis captures the non-linear multivariate dependence between variables. The proposed design methodology is applied to two environmental case studies. Performance of the proposed methodology is evaluated through partial redesigns of the original spatial designs. The first application is a soil contamination example that demonstrates the ability of the proposed methodology to address spatial non-linearity in the data. The second application is a forest biomass study that highlights the strength of the methodology in incorporating non-linear multivariate dependence into the design.

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References

  • Aas K, Czado C, Frigessi A, Bakken H (2009) Pair-copula constructions of multiple dependence. Insurance 44(2):182–198. doi:10.1016/j.insmatheco.2007.02.001

    Google Scholar 

  • Atteia O, Dubois JP, Webster R (1994) Geostatistical analysis of soil contamination in the Swiss Jura. Environ Pollut 86(3):315–327. doi:10.1016/0269-7491(94)90172-4

    Article  CAS  Google Scholar 

  • Bandarian EM, Bloom LM, Mueller UA (2008) Direct minimum/maximum autocorrelation factors within the framework of a two structure linear model of coregionalisation. Comput Geosci 34(3):190–200. doi:10.1016/j.cageo.2007.03.015

    Article  Google Scholar 

  • Bárdossy A (2006) Copula-based geostatistical models for groundwater quality parameters. Water Resour Res 42(11):W11,416. doi:10.1029/2005WR004754

  • Bárdossy A, Li J (2008) Geostatistical interpolation using copulas. Water Resour Res 44(7):W07,412. doi:10.1029/2007WR006115

  • Barnett RM, Deutsch CV (2012) Practical implementation of non-linear transforms for modeling geometallurgical variables. In: Abrahamsen P, Hauge R, Kolbjørnsen O (eds) Geostatistics Oslo 2012, Springer, Dordrecht, pp 409–422. doi:10.1007/978-94-007-4153-9_33

  • Barnett RM, Manchuk JG, Deutsch CV (2014) Projection pursuit multivariate transform. Math Geosci 46(3):337–359. doi:10.1007/s11004-013-9497-7

    Article  Google Scholar 

  • Bayraktar H, Turalioglu FS (2005) A Kriging-based approach for locating a sampling site—in the assessment of air quality. Stoch Environ Res Risk Assess 19(4):301–305. doi:10.1007/s00477-005-0234-8

    Article  Google Scholar 

  • Bohorquez M, Giraldo R, Mateu J (2016) Multivariate functional random fields: prediction and optimal sampling. Stoch Environ Res Risk Assess. doi:10.1007/s00477-016-1266-y

  • Bonneau M, Gaba S, Peyrard N, Sabbadin R (2014) Reinforcement learning-based design of sampling policies under cost constraints in Markov random fields: application to weed map reconstruction. Computat Stat Data Anal 72:30–44. doi:10.1016/j.csda.2013.10.002

    Article  Google Scholar 

  • Brown PJ, Le ND, Zidek JV (1994) Multivariate spatial interpolation and exposure to air pollutants. Can J Stat 22(4):489–509. doi:10.2307/3315406

    Article  Google Scholar 

  • Bueso MC, Angulo JM, Cruz-Sanjulián J, García-Aróstegui JL (1999) Optimal spatial sampling design in a multivariate framework. Math Geol 31(5):507–525. doi:10.1023/A:1007511923053

    Article  Google Scholar 

  • Desbarats AJ, Dimitrakopoulos R (2000) Geostatistical simulation of regionalized pore-size distributions using min/max autocorrelation factors. Math Geol 32(8):919–942. doi:10.1023/A:1007570402430

    Article  Google Scholar 

  • Diggle P, Lophaven S (2006) Bayesian geostatistical design. Scand J Stat 33(1):53–64

    Article  Google Scholar 

  • Diggle PJ, Ribeiro PJ (2007) Geostatistical design. Model-based geostatistics. Springer, New York, pp 199–212

    Google Scholar 

  • Erhardt TM, Czado C, Schepsmeier U (2015a) R-vine models for spatial time series with an application to daily mean temperature. Biometrics 71(2):323–332. doi:10.1111/biom.12279

    Article  Google Scholar 

  • Erhardt TM, Czado C, Schepsmeier U (2015b) Spatial composite likelihood inference using local C-vines. J Multivar Anal 138(C):74–88. doi:10.1016/j.jmva.2015.01.021

  • Finley AO, Banerjee S, Carlin P (2007) spBayes: an R package for univariate and multivariate hierarchical point-referenced spatial models. J Stat Softw 19(4):1–24. doi:10.18637/jss.v019.i04

    Article  Google Scholar 

  • Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, New York

    Google Scholar 

  • Gräler B (2014) Modelling skewed spatial random fields through the spatial vine copula. Spat Stat 10:87–102. doi:10.1016/j.spasta.2014.01.001

    Article  Google Scholar 

  • Gräler B, Pebesma E (2011) The pair-copula construction for spatial data: a new approach to model spatial dependency. Procedia Environ Sci 7:206–211. doi:10.1016/j.proenv.2011.07.036

    Article  Google Scholar 

  • Harris P, Clarke A, Juggins S, Brunsdon C, Charlton M (2014) Geographically weighted methods and their use in network re-designs for environmental monitoring. Stoch Environ Res Risk Assess 28(7):1869–1887. doi:10.1007/s00477-014-0851-1

    Article  Google Scholar 

  • Haslauer CP, Li J, Bárdossy A (2010) Application of copulas in geostatistics. In: Atkinson PM, Lloyd CD (eds) geoENV VII—geostatistics for environmental applications, Springer, Dordrecht, pp 395–404. doi:10.1007/978-90-481-2322-3_34

  • Hassanipak AA, Sharafodin M (2004) GET: A function for preferential site selection of additional borehole drilling. Explor Mining Geol 13(1–4):139–146. doi:10.2113/gsemg.13.1-4.139

    Article  Google Scholar 

  • Joe H (1996) Families of m-variate distributions with given margins and m(m-1)/2 bivariate dependencee parameters. In: Distributions with fixed marginal and related topics, lecture notes-monograph series, vol 28. Institute of Mathematical Statistics, pp 120–141

  • Kramer MA (1991) Nonlinear principal component analysis using autoassociative neural networks. AIChE J 37(2):233–243. doi:10.1002/aic.690370209

    Article  CAS  Google Scholar 

  • Kurowicka D, Cooke R (2006) Uncertainty analysis with high dimensional dependence modelling. Wiley, New York, NY, doi:10.1002/0470863072

  • Lark RM (2002) Optimized spatial sampling of soil for estimation of the variogram by maximum likelihood. Geoderma 105(1–2):49–80. doi:10.1016/S0016-7061(01)00092-1

    Article  Google Scholar 

  • Leuangthong O, Deutsch CV (2003) Stepwise conditional transformation for simulation of multiple variables. Math Geol 35(2):155–173. doi:10.1023/A:1023235505120

    Article  Google Scholar 

  • Li J, Zimmerman DL (2015) Model-based sampling design for multivariate geostatistics. Technometrics 57(1):75–86. doi:10.1080/00401706.2013.873003

    Article  Google Scholar 

  • Li J, Bárdossy A, Guenni L, Liu M (2011) A copula based observation network design approach. Environ Modell Softw 26(11):1349–1357. doi:10.1016/j.envsoft.2011.05.001

    Article  Google Scholar 

  • Lischeid G (2009) Non-linear visualization and analysis of large water quality data sets: a model-free basis for efficient monitoring and risk assessment. Stoch Environ Res Risk Assess 23(7):977–990. doi:10.1007/s00477-008-0266-y

    Article  Google Scholar 

  • Melles SJ, Heuvelink GBM, Twenhöfel CJW, van Dijk A, Hiemstra PH, Baume O, Stöhlker U (2011) Optimizing the spatial pattern of networks for monitoring radioactive releases. Comput Geosci 37(3):280–288. doi:10.1016/j.cageo.2010.04.007

    Article  CAS  Google Scholar 

  • Moon CJ, Whateley MKG (2006) From prospect to prefeasibility. In: Moon CJ, Whateley MKG, Evans AM (eds) Introduction to mineral exploration, 2nd edn. Wiley-Blackwell, New York, pp 70–103

    Google Scholar 

  • Musafer GN, Thompson MH (2016) Optimal adaptive sequential sampling of soil using pair-copulas. Geoderma 271:124–133

  • Musafer GN, Thompson MH, Wolff RC, Kozan E (2015) Non-linear multivariate spatial modelling using NLPCA and pair-copulas. Technical report, Queensland University of Technology

  • Peyrard N, Sabbadin R, Spring D, Brook B, Mac Nally R (2011) Model-based adaptive spatial sampling for occurrence map construction. Stat Comput 23(1):29–42. doi:10.1007/s11222-011-9287-3

    Article  Google Scholar 

  • Quessy J, Rivest L, Toupin M (2014) Semi-parametric pariwise inference methods in spatial models based on copulas. Spatial Stat 14(Part C):472–490. doi:10.1016/j.spasta.2015.08.002

  • R Core Team (2016) R: a language and environment for statistical computing. R foundation for statistical computing, Vienna. http://www.R-project.org/

  • Ritchie MW, Zhang J, Hamilton TA (2013) Aboveground tree biomass for Pinus ponderosa in Northeastern California. Forests 4(1):179–196. doi:10.3390/f4010179

    Article  Google Scholar 

  • Shapiro A (2003) Monte Carlo sampling methods. In: Ruszczyriski A, Shapiro A (eds) Stochastic programming, handbooks in operations research and management science, vol 10, Elsevier, Amsterdam, pp 353–425. doi:10.1016/S0927-0507(03)10006-0

  • Spöck G, Pilz J (2010) Spatial sampling design and covariance-robust minimax prediction based on convex design ideas. Stoch Environ Res Risk Assess 24(3):463–482. doi:10.1007/s00477-009-0334-y

    Article  Google Scholar 

  • Vašát R, Heuvelink GBM, Borůvka L (2010) Sampling design optimization for multivariate soil mapping. Geoderma 155(3–4):147–153. doi:10.1016/j.geoderma.2009.07.005

    Google Scholar 

  • Wackernagel H (2003) Multivariate geostatistics, 3rd edn. Springer, London

    Book  Google Scholar 

  • Wang J, Stein A, Gao B, Ge Y (2012) A review of spatial sampling. Spat Stat 2:1–14. doi:10.1016/j.spasta.2012.08.001

    Article  Google Scholar 

  • Webster R, Oliver MA (1992) Sample adequately to estimate variograms of soil properties. J Soil Sci 43(1):177–192. doi:10.1111/j.1365-2389.1992.tb00128.x

    Article  Google Scholar 

  • Zhou X, Hemstrom MA (2009) Estimating aboveground tree biomass on forest land in the Pacific Northwest: a comparison of approaches. Research paper PNW-RP-584, United States Department of Agriculture Forest Service, Pacific Southwest Research Station. http://www.fs.fed.us/pnw/pubs/pnw_rp584.pdf

  • Zhu Z, Stein ML (2006) Spatial sampling design for prediction with estimated parameters. J Agric Biol Environ Stat 11(1):24–44. doi:10.1198/108571106X99751

    Article  Google Scholar 

  • Zimmerman DL (2006) Optimal network design for spatial prediction, covariance parameter estimation, and empirical prediction. Environmetrics 17(6):635–652. doi:10.1002/env.769

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to acknowledge the support of the Australian Government’s Cooperative Research Centre for Optimising Resource Extraction (Grant P3C-030). The authors would also like to acknowledge the United States Department of Agriculture Forest Service for assistance with the Bartlett Experimental Forest data. Thanks also to the reviewers, whose comments helped improve the introduction and to further clarify the model dependent nature of the designs presented.

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Authors and Affiliations

  1. School of Mathematical Sciences, Queensland University of Technology (QUT), Brisbane, Australia

    G. Nishani Musafer & M. Helen Thompson

  2. Cooperative Research Centre for Optimising Resource Extraction (CRC ORE), Brisbane, Australia

    G. Nishani Musafer & M. Helen Thompson

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  1. G. Nishani Musafer
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  2. M. Helen Thompson
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Correspondence to M. Helen Thompson.

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Musafer, G.N., Thompson, M.H. Non-linear optimal multivariate spatial design using spatial vine copulas. Stoch Environ Res Risk Assess 31, 551–570 (2017). https://doi.org/10.1007/s00477-016-1307-6

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