, Volume 19, Issue 3, pp 417–451 | Cite as

A general science-based framework for dynamical spatio-temporal models

  • Christopher K. WikleEmail author
  • Mevin B. Hooten
Invited Paper


Spatio-temporal statistical models are increasingly being used across a wide variety of scientific disciplines to describe and predict spatially-explicit processes that evolve over time. Correspondingly, in recent years there has been a significant amount of research on new statistical methodology for such models. Although descriptive models that approach the problem from the second-order (covariance) perspective are important, and innovative work is being done in this regard, many real-world processes are dynamic, and it can be more efficient in some cases to characterize the associated spatio-temporal dependence by the use of dynamical models. The chief challenge with the specification of such dynamical models has been related to the curse of dimensionality. Even in fairly simple linear, first-order Markovian, Gaussian error settings, statistical models are often over parameterized. Hierarchical models have proven invaluable in their ability to deal to some extent with this issue by allowing dependency among groups of parameters. In addition, this framework has allowed for the specification of science based parameterizations (and associated prior distributions) in which classes of deterministic dynamical models (e.g., partial differential equations (PDEs), integro-difference equations (IDEs), matrix models, and agent-based models) are used to guide specific parameterizations. Most of the focus for the application of such models in statistics has been in the linear case. The problems mentioned above with linear dynamic models are compounded in the case of nonlinear models. In this sense, the need for coherent and sensible model parameterizations is not only helpful, it is essential. Here, we present an overview of a framework for incorporating scientific information to motivate dynamical spatio-temporal models. First, we illustrate the methodology with the linear case. We then develop a general nonlinear spatio-temporal framework that we call general quadratic nonlinearity and demonstrate that it accommodates many different classes of scientific-based parameterizations as special cases. The model is presented in a hierarchical Bayesian framework and is illustrated with examples from ecology and oceanography.


Bayesian Hierarchical Nonlinear Quadratic State-space SST 

Mathematics Subject Classification (2000)

35Q62 37M10 62F15 62H11 62M30 91B72 


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  1. Andrieu C, Doucet A, Holenstein R (2010) Particle Markov chain Monte Carlo methods. J R Stat Soc, Ser B 72:1–33 CrossRefGoogle Scholar
  2. Barnston AG, Glantz MH, He Y (1999) Predictive skill of statistical and dynamical climate models in forecasts of SST during the 1998–1997 El Niño episode and the 1998 La Niña onset. Bull Am Meteorol Soc 80:217–244 CrossRefGoogle Scholar
  3. Barry RP, Ver Hoef J (1996) Blackbox kriging: spatial prediction without specifying variogram models. J Agric Biol Ecol Stat 1:297–322 CrossRefGoogle Scholar
  4. Berliner LM (1996) Hierarchical Bayesian time series models. In: Hanson KM, Silver RN (eds) Maximum entropy and Bayesian methods. Kluwer Academic, Dordrecht, pp 15–22 Google Scholar
  5. Berliner LM (2003) Physical-statistical modeling in geophysics. J Geophys Res Atmos 108 (D24). doi: 10.1029/2002JD002865
  6. Berliner LM, Wikle CK, Cressie N (2000) Long-lead prediction of Pacific SSTs via Bayesian dynamic modeling. J Climate 13:3953–3968 CrossRefGoogle Scholar
  7. Berliner LM, Milliff RF, Wikle CK (2003) Bayesian hierarchical modeling of air-sea interaction. J Geophys Res, Oceans 108 (C4) Google Scholar
  8. Beverton RJH, Holt SJ (1957) On the dynamics of exploited fish populations. In: Fisheries investigations, series 2, vol 19. HM Stationery Office, London Google Scholar
  9. Bisgaard S, Kulahci M (2005) Quality quandaries: interpretation of time series models. Qual Eng 17:653–658 CrossRefGoogle Scholar
  10. Box GEP, Jenkins GM (1970) Time series analysis, forecasting, and control. Holden–Day, Oakland zbMATHGoogle Scholar
  11. Brix A, Diggle PJ (2001) Spatiotemporal prediction for log-Gaussian Cox processes. J R Stat Soc, Ser B 63:823–841 zbMATHMathSciNetCrossRefGoogle Scholar
  12. Brown PE, Karesen KF, Roberts GO, Tonellato S (2000) Blur-generated non-separable space-time models. J R Stat Soc, Ser B 62:847–860 zbMATHMathSciNetCrossRefGoogle Scholar
  13. Burgers G, Stephenson DB (1999) The “normality” of El Niño. Geophys Res Lett 26:1027–1030 CrossRefGoogle Scholar
  14. Calder CA (2007) A dynamic process convolution approach to modeling ambient particulate matter concentrations. Environmetrics 19:39–48 MathSciNetCrossRefGoogle Scholar
  15. Calder CA, Holloman C, Higdon D (2002) Exploring space-time structure in ozone concentration using a dynamic process convolution model. In: Case studies in Bayesian statistics, vol 6. Springer, Berlin, pp 165–176 Google Scholar
  16. Cangelosi AR, Hooten MB (2009) Models for bounded systems with continuous dynamics. Biometrics 65:850–856 zbMATHCrossRefGoogle Scholar
  17. Carlin BP, Polson NG, Stoffer DS (1992) A Monte Carlo approach to nonnormal and nonlinear state space modeling. J Am Stat Assoc 87:493–500 CrossRefGoogle Scholar
  18. Caswell H (2001) Matrix population models, 2nd edn. Sinauer Associates, Sunderland Google Scholar
  19. Chatfield C (2004) The analysis of time series: an introduction, 6th edn. Chapman & Hall/CRC Press, Boca Raton zbMATHGoogle Scholar
  20. Cressie NAC (1993) Statistics for spatial data, revised edn. Wiley, New York Google Scholar
  21. Cressie N, Huang N-C (1999) Classes of nonseparable, spatiotemporal stationary covariance functions. J Am Stat Assoc 94:1330–1340 zbMATHMathSciNetCrossRefGoogle Scholar
  22. Cressie N, Shi T, Kang EL (2009) Fixed rank filtering for spatio-temporal data. Technical report number 819, Department of Statistics, The Ohio State University Google Scholar
  23. Christen JA, Fox C (2010) A general purpose sampling algorithm for continuous distributions (the t-walk). Bayesian Anal 5:1–20 CrossRefGoogle Scholar
  24. de Luna X, Genton MG (2005) Predictive spatio-temporal models for spatially sparse environmental data. Stat Sin 15:547–568 zbMATHGoogle Scholar
  25. Delicado P, Giraldo R, Comas R, Mateu J (2009) Statistics for spatial functional data; some recent contributions. Envirometrics. doi: 10.1102/env.1003 Google Scholar
  26. Dewar M, Scerri K, Kadirkamanathan V (2009) Data-driven spatio-temporal modeling using the integro-difference equation. IEEE Trans Signal Process 57(1):83–91 MathSciNetCrossRefGoogle Scholar
  27. Diggle PJ, Tawn JA, Moyeed RA (1998) Model-based geostatistics (with discussion). Appl Stat 47:299–350 zbMATHMathSciNetGoogle Scholar
  28. Doucet A, de Freitas N, Gordon N (eds) (2001) Sequential Monte Carlo methods in practice. Springer, New York zbMATHGoogle Scholar
  29. Evensen G (1994) Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte-Carlo methods to forecast error statistics. J Geophys Res 99:10143–10162 CrossRefGoogle Scholar
  30. Fan J, Yao Q (2005) Nonlinear time series. Springer, New York Google Scholar
  31. Fahrmeir L (1992) Posterior mode estimation by extended Kalman filtering for a multivariate dynamic generalized linear models. J Am Stat Assoc 87:501–509 zbMATHCrossRefGoogle Scholar
  32. Fahrmeir L, Kaufmann H (1991) On Kalman filtering, posterior mode estimation and Fisher scoring in dynamic exponential family regression. Metrika 38:37–60 zbMATHMathSciNetCrossRefGoogle Scholar
  33. Fuentes M, Chen L, Davis JM (2008) A class of nonseparable and nonstationary spatio temporal covariance functions. Environmetrics 19:487–507 MathSciNetCrossRefGoogle Scholar
  34. Gamerman D (1998) Markov chain Monte Carlo for dynamic generalized linear models. Biometrika 85:215–227 zbMATHMathSciNetCrossRefGoogle Scholar
  35. Gelfand AE, Kim H-J, Sirmans CF, Banerjee S (2003) Spatial modeling with spatially varying coefficient processes. J Am Stat Assoc 98:387–396 zbMATHMathSciNetCrossRefGoogle Scholar
  36. Gelfand AE, Banerjee S, Gamerman D (2005) Spatial process modelling for univariate and multivariate dynamic spatial data. Environmetrics 16:465–479 MathSciNetCrossRefGoogle Scholar
  37. Giraldo R, Delicado P, Mateu J (2009) Continuous time-varying kriging for spatial prediction of functional data: An environmental application. J Agric Biol Environ Stat. doi: 10.1198/jabes.2009.08051 Google Scholar
  38. Gneiting T (2002) Nonseparable, stationary covariance functions for space-time data. J Am Stat Assoc 97:590–600 zbMATHMathSciNetCrossRefGoogle Scholar
  39. Gordon NJ, Salmond DJ, Smith AFM (1993) Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proc F Radar Signal Process 140:107113 Google Scholar
  40. Goulard M, Voltz M (1993) Geostatistical interpolation of curves: a case study in soil science. In: Soares A (ed) Geostatistics Troia 92, vol 2. Kluwer Academic, Dordrecht, pp 805–816 Google Scholar
  41. Graybill FA (1969) Matrices with applications in statistics. Wadsworth, Belmont Google Scholar
  42. Gregori P, Porcu E, Mateu J, Sasvàri Z (2008) Potentially negative space time covariances obtained as a sum of products of marginal ones. Ann Inst Stat Math 60:865–882 zbMATHCrossRefGoogle Scholar
  43. Haario H, Laine M, Mira A, Saksman E (2006) DRAM: efficient adaptive MCMC. Stat Comput 16:339–354 MathSciNetCrossRefGoogle Scholar
  44. Hastings A (1997) Population biology: concepts and models. Springer, New York Google Scholar
  45. Heine V (1955) Models for two-dimensional stationary stochastic processes. Biometrika 42:170–178 zbMATHMathSciNetGoogle Scholar
  46. Higdon D (1998) A process-convolution approach to modelling temperatures in the North Atlantic ocean. Environ Ecol Stat 5:173–190 CrossRefGoogle Scholar
  47. Higdon D, Reese CS, Moulton JD, Vrugt JA, Fox C (2009) Posterior exploration for computationally intensive forward models. Technical report LAUR 08-05905, Statistical Sciences Group, Los Alamos National Lab Google Scholar
  48. Holton JR (2004) An introduction to dynamic meteorology, 4th edn. Elsevier, Boston Google Scholar
  49. Hooten MB, Wikle CK (2007) Shifts in the spatio-temporal growth dynamics of shortleaf pine. Environ Ecol Stat 14:207–227 MathSciNetCrossRefGoogle Scholar
  50. Hooten MB, Wikle CK (2008) A Hierarchical Bayesian non-linear spatio-temporal model for the spread of invasive species with application to the Eurasian Collared-Dove. Environ Ecol Stat 15:59–70 MathSciNetCrossRefGoogle Scholar
  51. Hooten MB, Wikle CK, Dorazio RM, Royle JA (2007) Hierarchical spatio-temporal matrix models for characterizing invasions. Biometrics 63:558–567 zbMATHMathSciNetCrossRefGoogle Scholar
  52. Hoerling MP, Kumar A, Zhong M (1997) El Niño, La Niña, and the nonlinearity of their teleconnections. J Climate 10:1769–1786 CrossRefGoogle Scholar
  53. Hotelling H (1927) Differential equations subject to error, and population estimates. J Am Stat Assoc 22:283–314 CrossRefGoogle Scholar
  54. Huang H-C, Cressie N (1996) Spatio-temporal prediction of snow water equivalent using the Kalman filter. Comput Stat Data Anal 22:159–175 MathSciNetCrossRefGoogle Scholar
  55. Huang H-C, Hsu N-J (2004) Modeling transport effects on ground-level ozone using a non-stationary space-time model. Environmetrics 15:5–25 CrossRefGoogle Scholar
  56. Hughes JP, Guttorp P (1994) A class of stochastic models for relating synoptic atmospheric patterns to regional hydrologic phenomena. Water Resour Res 30:1535–1546 CrossRefGoogle Scholar
  57. Jazwinski AH (1970) Stochastic processes and filtering theory. Academic Press, New York zbMATHGoogle Scholar
  58. Johannesson G, Cressie NAC, Huang H-C (2007) Dynamic multi-resolution spatial models. Environ Ecol Stat 14:5–25 MathSciNetCrossRefGoogle Scholar
  59. Jones RH, Zhang Y (1997) Models for continuous stationary space-time processes. In: Gregoire TG, Brillinger DR, Diggle PJ, Russek-Cohen E, Warren WG, Wolfinger RD (eds) Modelling longitudinal and spatially correlated data. Lecture Notes in Statistics, vol 122. Springer, Berlin, pp 289–298 Google Scholar
  60. Jungbacker B, Koopman SJ (2007) Monte Carlo estimation for nonlinear non-Gaussian state space models. Biometrika 94(4):827–839 zbMATHMathSciNetCrossRefGoogle Scholar
  61. Kitagawa G (1987) Non-Gaussian state-space modeling of non-stationary time series. J Am Stat Assoc 82:1032–1041 zbMATHMathSciNetCrossRefGoogle Scholar
  62. Kondrashov D, Kravtsov S, Robertson AW, Ghil M (2005) A hierarchy of data-based ENSO models. J Climate 18:4425–4444 CrossRefGoogle Scholar
  63. Kot M (1992) Discrete-time travelling waves: Ecological examples. J Math Biol 30(4):413–436 zbMATHMathSciNetCrossRefGoogle Scholar
  64. Kot M, Lewis MA, van den Briessche P (1996) Dispersal data and the spread of invading organisms. Ecology 77(7):2027–2042 CrossRefGoogle Scholar
  65. Lemos RT, Sansó B (2009) A spatio-temporal model for mean, anomaly and trend fields of North Atlantic sea surface temperature. J Am Stat Assoc 104:5–25 CrossRefGoogle Scholar
  66. Liu JS (2004) Monte Carlo strategies in scientific computing. Springer, New York Google Scholar
  67. Lopes HF, Salazar E, Gamerman D (2008) Spatial dynamic factor analysis. Bayesian Anal 3(4):759–792 MathSciNetGoogle Scholar
  68. Ma C (2003) Families of spatio-temporal stationary covariance models. J Stat Plan Inference 116:489–501 zbMATHCrossRefGoogle Scholar
  69. Majda AJ, Wang X (2006) Nonlinear dynamics and statistical theories for basic geophysical flows. Cambridge University Press, Cambridge zbMATHCrossRefGoogle Scholar
  70. Malmberg A, Arellano A, Edwards DP, Flyer N, Nychka D, Wikle CK (2008) Interpolating fields of carbon monoxide data using a hybrid statistical-physical model. Ann Appl Stat 2:1231–1248 zbMATHCrossRefMathSciNetGoogle Scholar
  71. Mardia K, Goodall C, Redfern E, Alonso F (1998) The kriged Kalman filter. Test 7:217–285 (with discussion) zbMATHMathSciNetCrossRefGoogle Scholar
  72. Martino S, Rue H (2009) INLA: Functions which allow to perform a full Bayesian analysis of structured additive models using integrated nested laplace approximaxion, R Package Google Scholar
  73. McCulloch CE, Searle SR (2001) Generalized, linear and mixed models. Wiley, New York zbMATHGoogle Scholar
  74. Niemi J, West M (2010) Adaptive mixture modeling Metropolis methods for Bayesian analysis of nonlinear state-space models. J Comput Graph Stat 19:260–280 CrossRefGoogle Scholar
  75. Paciorek CJ (2007) Computational techniques for spatial logistic regression with large datasets. Comput Stat Data Anal 51:3631–3653 zbMATHMathSciNetCrossRefGoogle Scholar
  76. Pedlosky J (1987) Geophysical fluid dynamics, 2nd edn. Springer, New York zbMATHGoogle Scholar
  77. Penland C, Magorian T (1993) Prediction of Niño 3 sea surface temperatures using linear inverse modeling. J Climate 6:1067–1076 CrossRefGoogle Scholar
  78. Philander SG (1990) El Niño, La Niña, and the southern oscillation. Academic Press, San Diego Google Scholar
  79. Priestly MB (1988) Non-linear and non-stationary time series analysis. Academic Press, London Google Scholar
  80. Raiko T, Tornio M (2009) Variational Bayesian learning of nonlinear hidden state-space models for model predictive control. Neurocomputing 72:3704–3712 CrossRefGoogle Scholar
  81. Raiko T, Tornio M, Honkela A, Karhunen J (2006) State inference in variational Bayesian nonlinear state-space models. In: Proceedings of the 6th international conference on independent component analysis and blind source separation (ICA, 2006), Charelston, South Carolina, March 2006, pp 222–229 CrossRefGoogle Scholar
  82. Ricker WE (1954) Stock and recruitment. J Fish Res Board Can 11:559–623 Google Scholar
  83. Royle JA, Berliner LM (1999) A hierarchical approach to multivariate spatial modeling and prediction. J Agric Biol Environ Stat 4:29–56 MathSciNetCrossRefGoogle Scholar
  84. Royle JA Wikle CK, (2005) Efficient statistical mapping of avian count data. Environ Ecol Stat 12:225–243 MathSciNetCrossRefGoogle Scholar
  85. Rubin DB (1987) A noniterative sampling/importance resampling alternative to the data augmentation algorithm for creating a few imputations when fractions of missing information are modest: the SIR algorithm. J Am Stat Assoc 52:543–546 CrossRefGoogle Scholar
  86. Rue H, Held L (2005) Gaussian Markov random fields—theory and applications. Monographs on statistics and applied probability, vol 104. Chapman & Hall, London zbMATHGoogle Scholar
  87. Rue H, Martino S, Chopin N (2009) Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. J R Stat Soc, Ser B 71:319–392 CrossRefGoogle Scholar
  88. Sansó B, Guenni L (1999) Venezuelan rainfall data analyzed by using a Bayesian space-time model. Appl Stat 48:345–362 zbMATHGoogle Scholar
  89. Sansó B, Schmidt A, Nobre A (2008) Spatio-temporal models based on discrete convolutions. Can J Stat 36(2):239–258 zbMATHCrossRefGoogle Scholar
  90. Scerri K, Dewar M, Kadirkamanathan K (2009) Estimation and model selection for an IDE-based spatio-temporal model. IEEE Trans Signal Process 57(2):482–492 MathSciNetCrossRefGoogle Scholar
  91. Shumway RH, Stoffer DS (2006) Time series analysis and its applications with R examples, 2nd edn. Springer, New York zbMATHGoogle Scholar
  92. So MKP (2003) Posterior mode estimation for nonlinear and non-Gaussian state space models. Stat Sin 13:255–274 zbMATHMathSciNetGoogle Scholar
  93. Stein M (2005) Space-time covariance functions. J Am Stat Assoc 100:310–321 zbMATHCrossRefGoogle Scholar
  94. Storvik G, Frigessi A, Hirst D (2002) Stationary space-time Gaussian random fields and their time autoregressive representation. Stat Model 2:139–161 zbMATHMathSciNetCrossRefGoogle Scholar
  95. Stroud J, Mueller P, Sansó B (2001) Dynamic models for spatio-temporal data. J R Stat Soc, Ser B 63:673–689 zbMATHCrossRefGoogle Scholar
  96. Stroud JR, Stein ML, Lesht BM, Schwab DJ, Beletsky D (2010) An ensemble Kalman filter and smoother for satellite data assimilation. J Am Stat Assoc. doi: 10.1198/jasa.2010.ap07636 Google Scholar
  97. Tang BY, Hsieh WW, Monahan AH, Tangang FT (2000) Skill comparisons between neural networks and canonical correlation analysis in predicting the equatorial Pacific sea surface temperatures. J Climate 13:287–293 CrossRefGoogle Scholar
  98. Tangang FT, Tang B, Monahan AH, Hsieh WW (1998) Forecasting ENSO events: A neural network-extended EOF approach. J Climate 11:29–41 CrossRefGoogle Scholar
  99. Timmermann A, Voss HU, Pasmanter R (2001) Empirical dynamic system modeling of ENSO using nonlinear inverse techniques. J Phys Oceanogr 31:1579–1598 CrossRefGoogle Scholar
  100. Tong H (1990) Non-linear time series: a dynamical systems approach. Oxford University Press, Oxford zbMATHGoogle Scholar
  101. Villagran A, Huerta G, Jackson CS, Sen MK (2008) Computational methods for parameter estimation in climate models. Bayesian Anal 3:823–850 CrossRefMathSciNetGoogle Scholar
  102. Vrugt JA, ter Braak CJF, Dilks CGH, Robinson BA, Hyman JM, Higdon D (2009) Accelerating Markov Chain Monte Carlo simulation by differential evolution with self-adaptive randomized subspace sampling. Int J Nonlinear Sci Numer Simul 10(3):273–290 Google Scholar
  103. Wakefield JC, Smith AFM, Racine-Poon A, Gelfand AE (1994) Bayesian analysis of linear and non-linear population models by using the Gibbs sampler. Appl Stat 43(1):201–221 zbMATHCrossRefGoogle Scholar
  104. West M, Harrison J (1997) Bayesian forecasting and dynamic models, 2nd edn. Springer, New York zbMATHGoogle Scholar
  105. Whittle P (1954) On stationary processes in the plane. Biometrika 44:434–449 MathSciNetGoogle Scholar
  106. Whittle P (1962) Topographic correlation, power-law covariance functions, and diffusion. Biometrika 49:305–314 zbMATHMathSciNetGoogle Scholar
  107. Wikle CK (1996) Spatio-temporal statistical models with applications to atmospheric processes. PhD dissertation, Iowa State University, Ames, IA Google Scholar
  108. Wikle CK (2002a) A kernel-based spectral model for non-Gaussian spatial processes. Stat Model, Int J 2:299–314 zbMATHMathSciNetCrossRefGoogle Scholar
  109. Wikle CK (2002b) Spatial modeling of count data: a case study in modelling breeding bird survey data on large spatial domains. In Lawson A, Denison D (eds) Spatial cluster modelling. Chapman & Hall, London, pp 199–209 Google Scholar
  110. Wikle CK (2003) Hierarchical Bayesian models for predicting the spread of ecological processes. Ecology 84:1382–1394 CrossRefGoogle Scholar
  111. Wikle CK, Berliner LM (2005) Combining information across spatial scales. Technometrics 47:80–91 MathSciNetCrossRefGoogle Scholar
  112. Wikle CK, Cressie N (1999) A dimension-reduced approach to space-time Kalman Filtering. Biometrika 86:815–829 zbMATHMathSciNetCrossRefGoogle Scholar
  113. Wikle CK, Hooten MB (2006) Hierarchical Bayesian spatio-temporal models for population spread. In: lark JS, Gelfand A (eds) Applications of computational statistics in the environmental sciences: hierarchical Bayes and MCMC methods. Oxford University Press, Oxford, pp 145–169 Google Scholar
  114. Wikle CK, Berliner LM, Cressie N (1998) Hierarchical Bayesian space-time models. Environ Ecol Stat 5:117–154 CrossRefGoogle Scholar
  115. Wikle CK, Milliff R, Nychka D, Berliner LM (2001) Spatiotemporal hierarchical Bayesian modeling: tropical ocean surface winds. J Am Stat Assoc 96:382–397 zbMATHMathSciNetCrossRefGoogle Scholar
  116. Wikle CK, Berliner LM, Milliff RF (2003) Hierarchical Bayesian approach to boundary value problems with stochastic boundary conditions. Mon Weather Rev 131:1051–1062 CrossRefGoogle Scholar
  117. Xu K, Wikle CK (2007) Estimation of parameterized spatio-temporal dynamic models. J Stat Plan Inference 137:567–588 zbMATHMathSciNetCrossRefGoogle Scholar
  118. Xu K, Wikle CK, Fox NI (2005) A kernel-based spatio-temporal dynamical model for nowcasting radar precipitation. J Am Stat Assoc 100(472):1133–1144 zbMATHMathSciNetCrossRefGoogle Scholar
  119. Young P (2000) Stochastic, dynamic modelling and signal processing: time variable and state dependent parameter estimation. In: Fitzgerald WJ, Smith RL, Walden AT, Young P (eds) Nonlinear and nonstationary signal processing. Cambridge University Press, Cambridge, pp 74–114 Google Scholar
  120. Yule GU (1927) On a method of investigating periodicities in disturbed series, with reference to Wolfer’s sunspot numbers. Philos Trans R Soc Lond, Ser A 226:267–298 CrossRefGoogle Scholar
  121. Zastavnyi VP, Porcu E (2009) Space-time covariance functions with compact support. arXiv:0902.3656v1

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© Sociedad de Estadística e Investigación Operativa 2010

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of MissouriColumbiaUSA
  2. 2.USGS Colorado Cooperative Fish and Wildlife Research Unit, Department of Fish, Wildlife, and Conservation BiologyColorado State UniversityFort CollinsUSA

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