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Blur‐generated non‐separable space–time models

Author

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  • Patrick E. Brown
  • Gareth O. Roberts
  • Kjetil F. Kåresen
  • Stefano Tonellato

Abstract

Statistical space–time modelling has traditionally been concerned with separable covariance functions, meaning that the covariance function is a product of a purely temporal function and a purely spatial function. We draw attention to a physical dispersion model which could model phenomena such as the spread of an air pollutant. We show that this model has a non‐separable covariance function. The model is well suited to a wide range of realistic problems which will be poorly fitted by separable models. The model operates successively in time: the spatial field at time t +1 is obtained by ‘blurring’ the field at time t and adding a spatial random field. The model is first introduced at discrete time steps, and the limit is taken as the length of the time steps goes to 0. This gives a consistent continuous model with parameters that are interpretable in continuous space and independent of sampling intervals. Under certain conditions the blurring must be a Gaussian smoothing kernel. We also show that the model is generated by a stochastic differential equation which has been studied by several researchers previously.

Suggested Citation

  • Patrick E. Brown & Gareth O. Roberts & Kjetil F. Kåresen & Stefano Tonellato, 2000. "Blur‐generated non‐separable space–time models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(4), pages 847-860.
    • Handle: RePEc:bla:jorssb:v:62:y:2000:i:4:p:847-860
    DOI: 10.1111/1467-9868.00269
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    Citations

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    Cited by:

    1. Christopher K. Wikle, 2003. "Hierarchical Models in Environmental Science," International Statistical Review, International Statistical Institute, vol. 71(2), pages 181-199, August.
    2. Christopher Wikle & Mevin Hooten, 2010. "A general science-based framework for dynamical spatio-temporal models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(3), pages 417-451, November.
    3. Min Deng & Wentao Yang & Qiliang Liu & Yunfei Zhang, 2017. "A divide-and-conquer method for space–time series prediction," Journal of Geographical Systems, Springer, vol. 19(1), pages 1-19, January.
    4. Giacomini, Raffaella & Granger, Clive W.J., 2001. "Aggregationn of Space-Time Processes," University of California at San Diego, Economics Working Paper Series qt77f76455, Department of Economics, UC San Diego.
    5. Ma, Chunsheng, 2003. "Nonstationary covariance functions that model space-time interactions," Statistics & Probability Letters, Elsevier, vol. 61(4), pages 411-419, February.
    6. Alexandre Rodrigues & Peter J. Diggle, 2010. "A Class of Convolution‐Based Models for Spatio‐Temporal Processes with Non‐Separable Covariance Structure," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(4), pages 553-567, December.
    7. Giacomini, Raffaella & Granger, Clive W. J., 2004. "Aggregation of space-time processes," Journal of Econometrics, Elsevier, vol. 118(1-2), pages 7-26.
    8. Richardson, Robert & Kottas, Athanasios & Sansó, Bruno, 2017. "Flexible integro-difference equation modeling for spatio-temporal data," Computational Statistics & Data Analysis, Elsevier, vol. 109(C), pages 182-198.
    9. Serena Arima & Lorenza Cretarola & Giovanna Jona Lasinio & Alessio Pollice, 2012. "Bayesian univariate space-time hierarchical model for mapping pollutant concentrations in the municipal area of Taranto," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 21(1), pages 75-91, March.
    10. Sigrist, Fabio & Künsch, Hans R. & Stahel, Werner A., 2015. "spate: An R Package for Spatio-Temporal Modeling with a Stochastic Advection-Diffusion Process," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 63(i14).
    11. Tata Subba Rao & Sourav Das & Georgi N. Boshnakov, 2014. "A Frequency Domain Approach For The Estimation Of Parameters Of Spatio-Temporal Stationary Random Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(4), pages 357-377, July.
    12. Robert Richardson & Athanasios Kottas & Bruno Sansó, 2020. "Spatiotemporal modelling using integro‐difference equations with bivariate stable kernels," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(5), pages 1371-1392, December.
    13. Athanasios Kottas, 2018. "Discussion of paper “nonparametric Bayesian inference in applications” by Peter Müller, Fernando A. Quintana and Garritt L. Page," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 27(2), pages 219-225, June.
    14. Giorgos Sermaidis & Omiros Papaspiliopoulos & Gareth O. Roberts & Alexandros Beskos & Paul Fearnhead, 2013. "Markov Chain Monte Carlo for Exact Inference for Diffusions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(2), pages 294-321, June.
    15. A. E. Madrid & J. M. Angulo & J. Mateu, 2016. "Point Pattern Analysis of Spatial Deformation and Blurring Effects on Exceedances," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 21(3), pages 512-530, September.
    16. Kim, Yongku & Berliner, L. Mark, 2016. "Change of spatiotemporal scale in dynamic models," Computational Statistics & Data Analysis, Elsevier, vol. 101(C), pages 80-92.
    17. Harrison Quick & Sudipto Banerjee & Bradley P. Carlin, 2015. "Bayesian modeling and analysis for gradients in spatiotemporal processes," Biometrics, The International Biometric Society, vol. 71(3), pages 575-584, September.
    18. Ruiz-Medina, M.D. & Salmeron, R. & Angulo, J.M., 2007. "Kalman filtering from POP-based diagonalization of ARH(1)," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4994-5008, June.

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