Abstract
In this paper we introduce a new procedure for spatial sampling design. It is found in previous studies (Zhu and Stein in J Agric Biol Environ Stat 11:24–44, 2006) that the optimal sampling design for spatial prediction with estimated parameters is nearly regular with a few clustered points. The pattern is similar to a generalization of the Neyman–Scott (GNS) process (Yau and Loh in Statistica Sinica 22:1717–1736, 2012) which allows for regularity in the parent process. This motivates the use of a realization of the GNS process as sampling design points. This method translates the high-dimensional optimization problem of selecting sampling sites into a low-dimensional optimization problem of searching for the optimal parameter sets in the GNS process. Simulation studies indicate that the proposed sampling design algorithm is more computationally efficient than traditional methods while achieving similar minimization of the criterion functions. While the traditional methods become computationally infeasible for sample size larger than a hundred, the proposed algorithm is applicable to a size as large as \(n=1024\). A real data example of finding the optimal spatial design for predicting sea surface temperature in the Pacific Ocean is also considered.
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References
Auligne, T., McNally, A.P., and Dee, D.P..(2007). Adaptive bias correction for satellite data in a numerical weather prediction system. Quarterly Journal of the Royal Meteorological Society 1332, 631–642.
Benham T., Duan, Q., Kroese D. and Liquet B. (2017). CEoptim: Cross-Entropy R Package for Optimization. Journal of Statistical Software 76(8), 1–29.
Bogaert, P. and Russo, D.(1999). Optimal spatial sampling design for the estimation of the variogram based on a least-squares approach. Water Resources Research. 35, 1275–1289.
Chipeta, M.G., Terlouw, D.J., Phiri, K.S. and Diggle, P.J. (2017). Inhibitory geostatistical designs for spatial prediction taking account of uncertain covariance structure. Environmetrics 28, e2425.
Cressie, N.A.C. (1993). Statistics for Spatial Data. Wiley, New York.
Croft, H.T., Falconer, K.J. and Guy, R.K. (1991). Unsolved Problems in Geometry. Springer-Verlag, New York.
Davison, A.C. and Hinkley D.V. (1997). Bootstrap Methods and their Applications. Cambridge University Press, Cambridge.
Diggle, P. J. and Lophaven, S. (2005). Bayesian Geostatistical Design Scandinavian Journal of Statistics 33(1), 53–64.
Irvine, K. M., Gitelman A. I., and Hoeting J. A. (2007). Spatial designs and properties of spatial correlation: Effects on covariance estimation. Journal of Agricultural, Biological, and Environmental Statistics 12(4), 450–469.
Lark, R. M. (2002). Optimized Spatial Sampling of Soil for Estimation of the Variogram by Maximum Likelihood. Geoderma. 105, 49–80.
McBratney, A.B., Webster, R. and Burgess, T.M.(1981). The design of optimal sampling schemes for local estimation and mapping of regionalized variables. I. Theory and method. Computer & Geosciences 7, 331–334.
Møller, J. and Torrisi, G.L. (2005). Generalized shot noise Cox processes. Advances in Applied Probability 37, 48–74.
Møller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.
Müller, W.G. and Zimmerman, D.L. (1999). Optimal designs for variogram estimation. Environmetrics. 10, 23–37.
Kiefer, J.; Wolfowitz, J. (1952). Stochastic Estimation of the Maximum of a Regression Function. The Annals of Mathematical Statistics 23 462–466.
Neyman, J. and Scott, E. L.(1958). Statistical approach to problems of cosmology. Journal of the Poyal Statistical Society B 20, 1–29.
Ohser, J. (1983). On estimators for the reduced second moment measure of point processes. Mathematische Operationsforschung und Statistik 14, 63–71.
Price, W.L. (1976). A controlled random search procedure for global optimization. The Computer Journal 20(4), 367–370.
Ritter, K. (1996). Asymptotic Optimality of Regular Sequence Designs. The Annals of Statistics. 24, 2081–2096.
Rubinstein, R.Y. (1997). Optimization of Computer Simulation Models with Rare Events. European Journal of Operational Research. 99, 89–112.
Rubinstein, R.Y. (1999). The Cross-Entropy Method for Combinatorial and Continuous Optimization. Methodology and Computing in Applied Probability. 1, 127–190.
Su, Y. and Cambanis, S. (1993). Sampling designs for the estimation of a random process. Stochastic Processes and their Applications. 46, 47–89.
van Groenigen, J. W., and Stein, A. (1998). Constrained Optimization of Spatial Sampling using Continuous Simulated Annealing. Journal of Environomental Quality. 43, 684–691.
van Lieshout, M.N.M. and Baddeley, A.J. (1996). A nonparametric measure of spatial interaction in point patterns. Statistica Neerlandica 50, 344–361
Warrick, A.W. and Myers, D.E.(1987). Optimization of sampling locations for variogram calculations. Water Resources Research. 23, 496–500.
Wiens, D. P. (2004). Robustness in Spatial Studies I: Minimax Design. Environmetrics. 16, 191–203.
Xiang, Y., Gubian, S., Suomela, B., and Hoeng, J. (2013). Generalized Simulated Annealing for Global Optimization: The GenSA Package. The R Journal 5, 13–28.
Yau, C.Y. and Loh J.M. (2012). A generalization of Neyman-Scott processes. Statistica Sinica 22, 1717–1736.
Yfantis, E. A., Flatman, G. T., and Behar, J. V.(1987). Efficiency of Kriging Estimation for Square, Triangular, and Hexagonal Grids. Mathematical Geology 19, 183–205.
Zhu, Z. (2002). Optimal sampling design and parameter estimation of Gaussian random fields. PhD Thesis. Department of Statistics, University of Chicago.
Zhu, Z. and Stein, M. (2005). Spatial sampling design for parameter estimation of the covariance function. Journal of statistical planning and inference. 134, 583–603.
Zhu, Z. and Stein, M. (2006). Spatial sampling design for prediction with estimated parameters. Journal of Agricultural, Biological and Environmental Statistics 11, 24–44.
Zimmerman, D. L. (2006). Optimal network design for spatial prediction, covariance parameter estimation, and empirical prediction. Environmetrics 17, 635–652.
Acknowledgements
We would like to thank the associate editor and the two referees for their insightful and constructive suggestions. Research was supported in part by HKSAR-RGC-GRF Nos 14601015 and 14305517 (Yau), and NSF Grant TRIPODS CCF-1934884 (Zhu).
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Leung, S.H., Loh, J.M., Yau, C.Y. et al. Spatial Sampling Design Using Generalized Neyman–Scott Process. JABES 26, 105–127 (2021). https://doi.org/10.1007/s13253-020-00413-3
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DOI: https://doi.org/10.1007/s13253-020-00413-3