Abstract
The problem of estimation and prediction of a spatial-temporal stochastic process, observed at regular times and irregularly in space, is considered. A mixed formulation involving a non- parametric component, accounting for a deterministic trend and the effect of exogenous variables, and a parametric component representing the purely spatio-temporal random variation is proposed. Correspondingly, a two-step procedure, first addressing the estimation of the non- parametric component, and then the estimation of the parametric component is developed from the residual series obtained, with spatial-temporal prediction being performed in terms of suitable spatial interpolation of the temporal variation structure. The proposed model formula-tion, together with the estimation and prediction procedure, are applied using a Gaussian ARMA structure for temporal modelling to space-time forecasting from real data of air pollution concentration levels in the region surrounding a power station in northwest Spain.
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References
Bennett, R.J. (1979) Spatial Time Series, Pion Limited, London.
Besag, J. (1974) Spatial interaction and the statistical analysis of lattice systems (with discussion). Journal of the Royal Statistical Society, Series B, 36, 192–236.
Bilonick, R.A. (1983) Risk qualified maps of hydrogen ion concentration for the New York state area for 1966–1978. Atmospheric Environment, 17, 2513–24.
Box, G.E.P. and Jenkins, G.M. (1974) Time Series Analysis: Forecasting and Control, Holden-Day, San Francisco.
Brockwell, P.J. and Davis, R.A. (1991) Time Series: Theory and Methods, Springer-Verlag, New York.
Carroll, R.J., Chen, R., George, E.I., Li, T.H., Newton, H.J., Shimiediche, H. and Wang, W. (1997) Ozone exposure and population density in Harris County, Texas. Journal of the American Statistical Association, 92(438), 392–415.
Cliff, A.D. and Ord, J.K. (1981) Spatial Processes: Models and Applications, Pion, London.
Cressie, N. (1993) Statistics for Spatial Data, Wiley & Sons, New York.
Deutsch, S.J. and Pfeifer, P.E. (1981) Space-time ARMA modelling with contemporaneously correlated innovations, Technometrics, 23(4), 401–9.
García-Jurado, I., González-Manteiga, W., Prada-Sánchez, J.M., Febrero-Bande, M. and Cao, R. (1995) Predicting using Box-Jenkins, nonparametric, and bootstrap techniques, Technometrics, 37(3), 303–10.
Gasser, T. and Müller, H.G. (1979) Kernel estimation of regression functions, In Smoothing Techniques for Curve Estimation, T. Gasser and M. Rosenblatt (eds), Springer-Verlag, Heidelberg.
Goodall, C.R. and Mardia, K.V. (1994) Challenges in multivariate spatio-temporal modelling, Technical Report, STAT 94/07, University of Leeds, U.K.
Guttorp, P., Meiring, W. and Sampson, P.D. (1994) A space-time analysis of ground-level ozone data, Environmetrics, 5, 241–54.
Guttorp, P. and Sampson, P.D. (1994) Methods for estimating heterogeneous spatial covariance functions with environmental applications. In Handbook of Statistics, G.P. Patil and C.R. Rao, (eds) Elsevier Science, 12, 661–89.
Guyon, X. (1995) Random Fields on a Network: Modelling, Statistics, and Applications, Springer-Verlag, New York.
Haas, T.C. (1995) Local prediction of a spatio-temporal process with an application to wet sulfate deposition. Journal of the American Statistical Association, 90(432), 1189–99.
Härdle, W. (1990) Applied Nonparametric Regression, Cambridge University Press, London.
Haslett, J. (1989) Space time modelling in meteorology: a review. Bulletin of the I.S.I., 51, 229–46.
Haslett, J. and Raftery, A.E. (1989) Space-time modelling with long-memory dependence: assessing Ireland's wind power resource. Applied Statistics, 38, 1–21.
Hastie, T. and Tibshirani, R. (1990) Generalized Additive Models, Chapman & Hall, New York.
Høst, G., Omre, H. and Switzer, P. (1995) Spatial interpolation errors for monitoring data. Journal of the American Statistical Association, 90(431), 853–61.
Huang, H. and Cressie, N. (1996) Spatio-temporal prediction of snow water equivalent using the Kalman filter. Computational Statistics & Data Analysis, 22, 159–75.
Laslett, G.M. (1994) Kriging and splines: an empirical comparison of their predictive performance in some applications. Journal of the American Statistical Association, 89(426), 391–409.
Mardia, K.V. and Goodall, C.R. (1993a) Factorized models in spatial modeling. In Multivariate Environmental Statistics, N.K. Bose, G.P. Patil and C.R. Rao (eds), North Holland, Amsterdam.
Mardia, K.V. and Goodall, C.R. (1993b) Spatial-temporal analysis of multivariate environmental monitoring data. In Multivariate Environmental Statistics, G.P. Patil and C.R. Rao, (eds) North Holland, Amsterdam. pp. 347–86.
Mardia, K.V., Kent, J.T., Goodall, C.R. and Little, J.A. (1996) Kriging and splines with derivative information. Biometrika, 83(1), 207–21.
Müller, H.G., Stadtmüller, U. and Tabnak, F. (1997) Spatial smoothing of geographically aggregated data, with application to the construction of incidence maps. Journal of the American Statistical Association, 92(437), 61–71.
Niu, X.F. (1996) Nonlinear additive models for environmental time series, with applications to ground-level ozone data analysis. Journal of the American Statistical Association, 91(435), 1310–21.
Patil, G.P. and Rao, C.R. (eds) (1993) Multivariate Environmental Statistics, North Holland, Amsterdam.
Pfeifer, P.E. and Deutsch, S.J. (1980a) A three-stage iterative procedure for space-time modelling. Technometrics, 22(1), 35–47.
Pfeifer, P.E. and Deutsch, S.J. (1980b) Identification and interpretation of first order space-time ARMA models, Technometrics, 22(4), 397–408.
Posa, D. (1993) A simple description of spatial-temporal processes. Computational Statistics & Data Analysis, 15, 425–37.
Rouhani, S. and Wackernagel, H. (1990) Multivariate geostatistical approach to space-time data analysis. Water Resources Research, 26, 585–91.
Sampson, P.D. and Guttorp, P. (1992) Nonparametric estimation of nonstationary spatial covariance structure. Journal of the American Statistical Association, 87(417), 108–19.
Shumway, R.H. and Stoffer, D.S. (1982) An approach to time series smoothing and forecasting using the EM algorithm. Journal of Time Series Analysis, 3, 253–64.
Stein, M. (1986) A simple model for spatial-temporal processes. Water Resources Weather Review, 22, 2107–10.
Truong, Y.K. (1994) Nonparametric time series regression. Annals of the Institute of Statistical Mathematics, 46(2), 279–93.
Truong, Y.K. and Stone, C.J. (1994) Semiparametric time series regression. Journal of Time Series Analysis, 15(4), 405–28.
Wahba, G. (1990) Spline Models for Observational Data, SIAM, Philadelphia.
Wikle, C.K. and Cressie, N.A.C. (1997) A dimension-reduction approach to space-time Kalman filtering. Preprint 97–24.
Yakowitz, S.J. and Szidaroyszky, F. (1985) A comparison of kriging with nonparametric regression methods. Journal of Multivariate Analysis, 16, 21–53.
Zhang, B. and Stein, M. (1993) Kernel approximations for universal kriging predictors. Journal of Multivariate Analysis, 44, 286–313.
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Angulo, J., González-Manteiga, W., Febrero-Bande, M. et al. Semi-parametric statistical approaches for space-time process prediction. Environmental and Ecological Statistics 5, 297–316 (1998). https://doi.org/10.1023/A:1009670920927
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DOI: https://doi.org/10.1023/A:1009670920927