Abstract
When a large amount of spatial data is available computational and modeling challenges arise and they are often labeled as “big n problem”. In this work we present a brief review of the literature. Then we focus on two approaches, respectively based on stochastic partial differential equations and integrated nested Laplace approximation, and on the tapering of the spatial covariance matrix. The fitting and predictive abilities of using the two methods in conjunction with Kriging interpolation are compared in a simulation study.
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Banerjee S, Fuentes M: Bayesian modeling for large spatial datasets. WIREs Comput Stat 4, 59–66 (2012)
Banerjee S, Carlin BP, Gelfand AE (2004) Hierarchical modeling and analysis for spatial data. Chapman and Hall, London
Banerjee S, Gelfand A, Finley A, Sang H: Gaussian predictive process models for large spatial data sets. J R Stat Soc Ser B (Stat Methodol) 70, 825–848 (2008). doi:10.1111/j.1467-9868.2008.00663.x
Bolin D, Lindgren F (2011) Spatial wavelet Markov models are more efficient than covariance tapering and process convolutions. arXiv:11061980v1. http://arxiv.org/abs/1106.1980,1106.1980
Brenner SC, Scott R: The mathematical theory of finite element methods. Springer, Berlin (2007)
Cressie N, Johannesson G: Fixed rank kriging for very large spatial data sets. J R Stat Soc Ser B (Stat Methodol) 70, 209–226 (2008). doi:10.1111/j.1467-9868.2007.00633.x
Finley A, Sang H, Banerjee S, Gelfand A: Improving the performance of predictive process modeling for large datasets. Computat Stat Data Anal 53, 2873–2884 (2009). doi:10.1016/j.csda.2008.09.008
Fuentes M: Approximate likelihood for large irregularly spaced spatial data. J Am Stat Assoc 102, 321–331 (2007). doi:10.1198/016214506000000852
Furrer R, Genton MG, Nychka D: Covariance tapering for interpolation of large spatial datasets. J Comput Graph Stat 15, 502–523 (2006). doi:10.1198/106186006X132178
Haas T: Local prediction of a spatio-temporal process with an application to wet sulfate deposition. J Am Stat Assoc 90, 1189–1199 (1995)
Higdon D, Swall J, Kern J: Non-stationary spatial modeling. Bayesian Stat 6, 761–768 (1998)
Ji WY, Simon W, Koray K, Ercan EK: Variant functional approximations for latent Gaussian models. Technical Report of Statistics Department. Trinity College, Dublin (2011)
Kaufman C, Schervish M, Nychka D: Covariance tapering for likelihood based estimation in large spatial data set. J Am Stat Assoc 103, 1545–1555 (2008)
Lindgren F, Rue H: Explicit construction of GMRF approximations to generalised Matérn fields on irregular grids. Scand J Stat 35, 691–700 (2007)
Lindgren F, Rue H, Lindström J: An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach. J R Stat Soc Ser B (Stat Methodol) 73, 423–498 (2011). doi:10.1111/j.1467-9868.2011.00777.x
Mardia K, Goodall C, Refern E, Alonso F: The kriged Kalman filter. Test 7, 217–285 (1998)
Matsuda Y, Yajima Y: Fourier analysis of irregularly spaced data on R d. J R Stat Soc Ser B (Stat Methodol) 71, 191–217 (2009). doi:10.1111/j.1467-9868.2008.00685.x
Rubinstein BY: Simulation and the Monte Carlo method. Wiley, New York (1981)
Rue H, Held L: Gaussian Markov random fields theory and applications, 1st edn. Chapman and Hall, London (2005)
Rue H, Tjelmeland H (2002) Fitting Gaussian Markov random fields to Gaussian fields. Scand J Stat, pp 31–49. doi:10.1111/1467-9469.00058
Rue H, Martino S, Chopin N: Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. J R Stat Soc Ser B (Stat Methodol) 71, 319–392 (2009). doi:10.1111/j.1467-9868.2008.00700.x
Sampson P, Guttorp P: Nonparametric estimation of nonstationary spatial covariance structure. J Am Stat Assoc 87, 108–119 (1992). doi:10.2307/2290458
Shaby B, Ruppert D (2012) Tapered covariance: Bayesian estimation and asymptotics. J Comput Graph Stat 21:433–452
Stein ML, Chi Z, Welty LJ: Approximating likelihoods for large spatial data sets. J R Stat Soc Ser B (Stat Methodol) 66, 275–296 (2004)
Sun Y, Li B, Genton MG: Geostatistics for large datasets. In: Montero, J, Porcu, E, Schlather, M (eds) Advances and challenges in space-time modelling of natural events volume 207 of lecture notes in statistics chap 3, pp. 55–77. Springer, Berlin (2012)
Wendland H: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv Comput Math 4, 389–396 (1995). doi:10.1007/BF02123482
Whittle P (1954) On stationary processes in the plane. Biometrika 41. doi:10.2307/2332724
Whittle P: Stochastic processes in several dimensions. Bull Int Stat Inst 40, 974–994 (1963)
Zhang H: Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics. J Am Stat Assoc 99, 250–261 (2004)
Zhang H, Du J: Covariance tapering in spatial statistics. In: Mateu, E, Porcu, J (eds) Positive definite functions from Schoenberg to space-time challenges, Gráficas Castañ, s.l. (2008)
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Jona Lasinio, G., Mastrantonio, G. & Pollice, A. Discussing the “big n problem”. Stat Methods Appl 22, 97–112 (2013). https://doi.org/10.1007/s10260-012-0207-2
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DOI: https://doi.org/10.1007/s10260-012-0207-2