Environmental and Ecological Statistics

, Volume 16, Issue 4, pp 515–529 | Cite as

A simple non-separable, non-stationary spatiotemporal model for ozone

  • Francesca Bruno
  • Peter Guttorp
  • Paul D. Sampson
  • Daniela Cocchi


The past two decades have witnessed an increasing interest in the use of space-time models for a wide range of environmental problems. The fundamental tool used to embody both the temporal and spatial components of the phenomenon in question is the covariance model. The empirical estimation of space-time covariance models can prove highly complex if simplifying assumptions are not employed. For this reason, many studies assume both spatiotemporal stationarity, and the separability of spatial and temporal components. This second assumption is often unrealistic from the empirical point of view. This paper proposes the use of a model in which non-separability arises from temporal non-stationarity. The model is used to analyze tropospheric ozone data from the Emilia-Romagna Region of Italy.


Spatiotemporal process Temporal non-stationarity Non-separability Deformation Tropospheric ozone 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bakhsh A, Jaynes DB, Colvin TS, Kanwar RS (2000) Spatiotemporal analysis of yield variability for a corn soybean field in Iowa. Trans ASABE 43: 31–38Google Scholar
  2. Bochner S (1955) Harmonic analysis and the theory of probability. University of California, Berkeley and Los AngelesGoogle Scholar
  3. Bruno F (2004) Non-separability in space-time covariance functions, Ph.D thesis, Department of Statistics, University of BolognaGoogle Scholar
  4. Carroll RJ, Chen R, George EI, Li TH, Newton HJ, Schmiediche H, Wang N (1997) Ozone exposure and population density in Harris County, Texas. J Am Stat Assoc 92: 392–404CrossRefGoogle Scholar
  5. Cocchi D, Trivisano C (2002) Ozone. In: El-Sharaawi A, Piegorsch W(eds) Encyclopedia of environmetrics. Wiley, New York, pp 1518–1523Google Scholar
  6. Cocchi D, Greco F, Trivisano C (2006) Hierarchical space-time modelling of PM10 pollution. Atmos Environ 41: 532–541CrossRefGoogle Scholar
  7. Cox WM, Chu S (1993) Meteorologically adjusted ozone trends in urban areas: a probabilistic approach. Atmos Environ 27: 425–434CrossRefGoogle Scholar
  8. Cressie N (1993) Statistics for spatial data. Wiley, New YorkGoogle Scholar
  9. Cressie N, Huang HC (1999) Classes of nonseparable, spatio-temporal stationary covariance functions. J Am Stat Assoc 94: 1330–1340CrossRefGoogle Scholar
  10. Damian D, Sampson PD, Guttorp P (2000) Bayesian estimation of semi-parametric non-stationary spatial covariance structure. Environmetrics 12: 161–176CrossRefGoogle Scholar
  11. De Cesare L, Myers D, Posa D (1997) Spatial-temporal modeling of SO2 in Milan district. In: Baafi EY, Schofield NA(eds) Geostatistics wollongong ’96, vol 2. Kluwer Academic Publishers, Dordrecht, pp 1031–1042Google Scholar
  12. Diggle PJ, Tawn JA, Moyeed RA (1998) Model based geostatistics. Appl Stat 47: 299–326Google Scholar
  13. Dimitrakopoulos R, Luo X (1997) Joint space-time modeling in the presence of trends. In: Baafi EY, Schofield NA(eds) Geostatistics wollongong ’96, vol 1. Kluwer Academic Publishers, Dordrecht, pp 138–149Google Scholar
  14. Dryden IL, Márkus L, Taylor CC, Kovács J (2005) Non-stationary spatiotemporal analysis of karst water levels. Appl Stat 54: 673–690Google Scholar
  15. EPA Environmental Protection Agency (1998) EPA’s updated air quality standards for smog (ozone) and particulate matter. http://www.epa.gov/ozonedesignations/
  16. European Community (1996) Council directive on ambient air quality assessment and management 96/62/EC. OJ No L 296/55–63Google Scholar
  17. Fassó A, Negri I (2002a) Nonlinear statistical modelling of high frequency ground ozone data. Environmetrics 13: 225–241CrossRefGoogle Scholar
  18. Fassó A, Negri I (2002b) Multi step forecasting for nonlinear models of high frequency ground ozone data: a Monte Carlo approach. Environmetrics 13: 365–378CrossRefGoogle Scholar
  19. Fassó A, Cameletti M, Nicolis O (2007) Air quality monitoring using heterogeneous networks. Environmetrics 18: 245–264CrossRefGoogle Scholar
  20. Fuentes M (2005) Testing for separability of spatial-temporal covariance functions. J Stat Plan Inference 136: 447–466CrossRefGoogle Scholar
  21. Gneiting T (2002) Nonseparable, stationary covariance functions for space-time data. J Am Stat Assoc 97: 590–600CrossRefGoogle Scholar
  22. Guttorp P, Meiring W, Sampson PD (1994) A space-time analysis of ground-level ozone data. Environmetrics 5: 241–254CrossRefGoogle Scholar
  23. Jaynes DB, Colvin TS (1997) Spatiotemporal variability of corn and soybean yield. Agron J 89: 30–37CrossRefGoogle Scholar
  24. Kyriakidis PC, Journel AG (1999) Geostatistical space-time models: a review. Math Geol 31: 651–684CrossRefGoogle Scholar
  25. Mitchell MW, Genton MG, Gumpertz ML (2006) A likelihood ratio test for separability of covariances. J Multivar Anal 97: 1025–1043CrossRefGoogle Scholar
  26. Roberts GO, Kareson K, Brown P, Tonellato S (2000) Blur-generated non-separable space-time models. J R Stat Soc, Ser B 62: 847–860CrossRefGoogle Scholar
  27. Rodriguez-Iturbe I, Mejia JM (1974) The design of rainfall networks in time and space. Water Resour Res 10: 713–728CrossRefGoogle Scholar
  28. Rouhani S, Myers DE (1990) Problems in space-time kriging of geohydrological data. Math Geol 22: 611–623CrossRefGoogle Scholar
  29. Sampson PD, Guttorp P (1992) Nonparametric estimation of nonstationary spatial covariance structure. J Am Stat Assoc 87: 108–119CrossRefGoogle Scholar
  30. Scaccia L, Martin RJ (2005) Testing axial symmetry and separability of lattice processes. J Stat Plan Inference 131: 19–39CrossRefGoogle Scholar
  31. Smith RL, Huang LS (1993) Modeling high threshold exceedances of urban ozone. Technical Report, 6, National Institute of Statistical Sciences, Research Triangle parkGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Francesca Bruno
    • 1
  • Peter Guttorp
    • 2
  • Paul D. Sampson
    • 2
  • Daniela Cocchi
    • 1
  1. 1.Department of Statistics P. FortunatiUniversity of BolognaBolognaItaly
  2. 2.Department of StatisticsUniversity of WashingtonWashingtonUSA

Personalised recommendations