# Considering groups in the statistical modeling of spatio-temporal data

## DOI:

https://doi.org/10.6092/issn.1973-2201/3600## Abstract

Spatio-temporal statistical methods are developing into an important research topic that goes beyond the study of processes that generate independent, identically distributed observations. Hierarchical models are a suitable proposal for both continuous and discrete spatio-temporal domains. They are flexible and permit separation of the various sources of uncertainty by means of a sequence of conditional models. In this work, we expanded on spatio-temporal data modeling by considering data categorization with respect to certain differentiating features. We studied the impact of the presence of subgroups on model building, with emphasis on Bayesian modeling. We discussed how differences in spatial locations can be reflected in a hierarchical model and assessed the performances of different models via a simulation study.## References

G. ADELFIO, M. CHIODI, (2009), Second-order diagnostics for space-time point processes with application to seismic events, “Environmetrics” 20, pp. 895-911.

S. BANERJEE, B.P. CARLIN, A.E. GELFAND, (2004), Hierarchical Modeling and Analysis for Spatial Data, Chapman and Hall, CRC Press.

L. BERNARDINELLI, D. CLAYTON, C. PASCUTTO, C. MONTOMOLI, M. GHISLANDI, M. SONGINI, (1995), Bayesian analysis of space-time variation in disease risk, “Statistics in Medicine” 14, pp. 2433-2443.

D. BOHNING, (2003), Empirical Bayes estimators and non-parametric mixture models for space and time-space disease mapping and surveillance, “Environmetrics” 14, pp. 431-451.

P.E. BROWN, K.F. KARESEN, G.O. ROBERTS, S. TONELLATO, (2000), Blur-generated non-separable spacetime models, “Journal of the Royal Statistical Society” Ser. B, 62, pp. 847-860.

F. BRUNO, D. COCCHI, (2006), Seasonal spatio-temporal non-separability: the case of Ozone in Emilia Romagna, in B. Cafarelli, G. Jona Lasinio, A. Pollice (a cura di) SPATIAL-Spatial Data Methods for Environmental and Ecological Processes, Book of Abstracts, ISBN: 88-8459-078-7, WIP Edizioni.

M. CAMELETTI, R. IGNACCOLO, S. BANDE, (2010), Comparing air-quality statistical models, GRASPA Working paper 40.

G. CHRISTAKOS, (2000), Modern spatiotemporal geostatistics, Oxford University Press, Oxford.

D. COCCHI, F. GRECO, C. TRIVISANO, (2006), Displaced calibration of PM10 measurements using spatio- temporal models, “STATISTICA” 66, pp. 127-138.

D. COCCHI, F. GRECO, C. TRIVISANO, (2007), Hierarchical space-time modelling of PM10 pollution, “Atmospheric Environment” 41, pp. 532-542.

N. CRESSIE, (1993), Statistics for Spatial Data. John Wiley & Sons, London.

N. CRESSIE, (1998), Aggregation and interaction issues in statistical modelling of spatiotemporal processes, “Geoderma” 85, 133-140.

N. CRESSIE, H.C. HUANG, (1999), Classes of nonseparable, spatiotemporal stationary covariance functions, “Journal of the American Statistical Association” 94, pp. 1330-1340.

R. M. CRUJEIRAS, R. FERNANDEZ-CASAL, W. GONZALEZ-MANTEIGA, (2010), Nonparametric test for separability of spatio-temporal processes, “Environmetrics” 21, pp. 382-399.

L. DE CESARE, D. MYERS, D. POSA, (2001), Product-sum covariance for spacetime modeling: an environmental application, “Environmetrics” 12, pp. 11-23.

P.J. DIGGLE, (1983), Statistical Analysis of Spatial Point Patterns, Academic Press, London.

R. FERNANDEZ-CASAL, W. GONZALEZ-MANTEIGA, M. FEBRERO-BANDE, (2003), Flexible spatiotemporal stationary variogram models, “Statistics and Computing” 13, pp. 127-136.

B.F. FINKENSTÄDT, L. HELD, V. ISHAM, (2007), Statistical Methods for Spatio-temporal systems, CRC Press, Chapman and Hall.

A.S FOTHERINGHAM, M. CHARLTON, C. BRUNSDON, (1998), Geographically weighted regression: a natural evolution of the expansion method for spatial data analysis, “Environment and Planning” 30, pp. 1905-1927.

M. FUENTES, (2006), Testing for separability of spatial-temporal covariance functions, “Journal of Statistical Planning and Inference” 136, pp. 447-466.

C. GAETAN, X. GUYON, (2010), Spatial Statistics and Modeling, Springer, New York.

A. GELMAN, J.B. CARLIN, H.S. STERN, D.B. RUBIN, (2004), Bayesian Data Analysis, Second Edition, CRC Press, Chapman and Hall.

T. GNEITING, (2002), Stationary covariance functions for spacetime data, “Journal of the American Statistical Association” 97, pp. 590-600.

T. GNEITING, M.G. GENTON, P. GUTTORP, (2007), Geostatistical space-time models, stationarity, separability and full symmetry, in B.F. FINKENSTÄDT, L. HELD, V. ISHAM, (2006), Statistical Methods for Spatio-temporal systems, CRC Press, Chapman and Hall, pp. 151-175.

F. GRECO, C. TRIVISANO, (2009), A multivariate CAR model for improving the estimation of relative risks, “Statistics in Medicine” 28, pp. 1707-1724.

G. JONA LASINIO, A. ORASI, F. DIVINO, P.L. CONTI, (2007), Statistical contributions to the analysis of environmental risks along the coastline. In Società Italiana di Statistica - rischio e previsione. Venezia, 6-8 giugno,cleup, pp. 255-262.

R.H. JONES, Y. ZHANG, (1997), Models for continuous stationary spacetime processes, in T.G. Gregoire, D.R. Brillinger, P.J. Diggle, E. Russek-Cohen, W.G. Warren, R.D. Wolfinger (eds) “Modelling longitudinal and spatially correlated data. Lecture Notes in Statistics” 122, Springer, Berlin Heidelberg New York, pp. 289-298.

M. JUN, M. L STEIN, (2004), Statistical comparison of observed and CMAQ modeled daily sulfate levels, “Atmospheric Environment” 38, pp. 4427-4436.

A. KOLOVOS, G. CHRISTAKOS, D.T. HRISTOPULOS, M.L. SERRE, (2004), Methods for generating nonseparable spatiotemporal covariance models with potential environmental applications, “Advances in Water Resources” 27, pp. 815-830.

P.C. KYRIAKIDIS, A.G. JOURNEL, (1999), Geostatistical Space-Time Models: A Review, “Mathematical Geology” 31, pp. 651-684.

D. LEE, G. SHADDICK, (2007), Time-varying coefficient models for the analysis of air pollution and health outcome data, “Biometrics” 63, pp. 1253-1261.

B. LI, M.G. GENTON, M. SHERMAN, (2007), A nonparametric assessment of properties of space-time covariance functions, “Journal of the American Statistical Association” 102, pp. 736-744.

N. LU, D.L. ZIMMERMAN, (2005), The likelihood ratio test for a separable covariance matrix, “Statistics and Probability Letters” 73, pp. 449-457.

C. MA, (2002), Spatio-temporal covariance functions generated by mixtures, “Mathematical Geology” 34, pp. 965-974.

M. MITCHELL, M.G. GENTON, M. GUMPERTZ, (2006), A likelihood ratio test for separability of covariances, “Journal of Multivariate Analysis” 97, pp. 1025-1043.

L. PACI, (2010), Hierarchical Bayesian space-time model: the case of Ozone in Emilia Romagna, Thesis of master in statistics (in Italian).

A. POLLICE, G. JONA LASINIO, (2009), Two approaches to imputation and adjustment of air quality data from a composite monitoring network, “Journal of Data Science” 7, pp. 43-59.

A. POLLICE, G. JONA LASINIO, (2010), Spatiotemporal analysis of the PM10 concentration over the Taranto area, “Environmental Monitoring and Assessment” 162 (1-4), pp. 177-190.

E. PORCU, P. GREGORI, J. MATEU, (2006), Nonseparable stationary anisotropic spacetime covariance functions, “Stochastic Environmental Research and Risk Assessment” 21, pp. 113-122.

H. SANG, A.E. GELFAND, (2009), Hierarchical modeling for extreme values observed over space and time, “Environmental and Ecological Statistics” 16, pp. 407-426.

S.K. SAHU, A.E. GELFAND, D.M. HOLLAND, (2007), High Resolution Space-Time Ozone Modeling for Assessing Trends, “Journal of the American Statistical Association” 102, pp. 1221-1234.

S.K SAHU, O. NICOLIS, (2009), An evaluation of European air pollution regulations for particulate matter monitored from a heterogeneous network, “Environmetrics” 20, pp. 943-961.

D.J. SPIEGELHALTER, N.G, BEST, B.P, CARLIN, A. VAN DER LINDE, (2002), Bayesian Measures of Model Complexity and Fit (with Discussion), “Journal of the Royal Statistical Society, Series B” 64, pp. 583-616.

M.L. STEIN, (2005), Spacetime covariance functions, “Journal of the American Statistical Association” 100, pp. 310-321.

J. WANG, G. CHRISTAKOS, M-G. HU, (2009), Modeling spatial means of surfaces with stratified nonhomgeneity, “IEEE Transactions on Geoscience and Remote Sensing” 47, 4167-4174.

C.K. WIKLE, (2003), Hierarchical models in environmental science, “International Statistical Review” 71, pp. 181-199.

C.K WIKLE, L.M. BERLINER, N. CRESSIE, (1998), Hierarchical Bayesian space-time models, “Environmental and Ecological Statistics” 5, pp. 117-154.

A.M. ZASLAVSKY, (2003), Hierarchical Bayesian modeling, in S.J. PRESS, Subjective and objective Bayesian statistics: principles, models and application, New York, John Wiley and Sons, pp. 336-358.

H. ZHANG, (2004), Inconsistent Estimation and Asymptotically Equal Interpolations in Model-Based Geostatistics, “Journal of the American Statistical Association” 99, pp. 250-261.

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*70*(4), 511–527. https://doi.org/10.6092/issn.1973-2201/3600

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