The Kriged Kalman filter

In recent years there has been growing interest in spatial-temporal modelling, partly due to the potential of large scale data in pollution and global climate monitoring to answer important environmental questions. We consider the Kriged Kalman filter (KKF), a powerful modelling strategy which combines the two wellestablished approaches of (a) Kriging, in the field of spatial statistics, and (b) the Kalman filter, in general state space formulations of multivariate time series analysis. We give a brief introduction to the model and describe its various properties, and highlight that the model allows prediction in time as well as in space, simultaneously. Some special cases of the time series model are considered. We give some procedures to implement the model, also illustrated through a practical example. The paper concludes with a discussion.

[1]  Ayala Cohen,et al.  Regression on a Random Field , 1969 .

[2]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[3]  R. Mehra,et al.  Computational aspects of maximum likelihood estimation and reduction in sensitivity function calculations , 1974 .

[4]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[5]  Richard H. Jones,et al.  Maximum Likelihood Fitting of ARMA Models to Time Series With Missing Observations , 1980 .

[6]  S. Cohn,et al.  Applications of Estimation Theory to Numerical Weather Prediction , 1981 .

[7]  R. Shumway,et al.  AN APPROACH TO TIME SERIES SMOOTHING AND FORECASTING USING THE EM ALGORITHM , 1982 .

[8]  C. Obled,et al.  Objective analyses and mapping techniques for rainfall fields: An objective comparison , 1982 .

[9]  R. Engle,et al.  Alternative Algorithms for the Estimation of Dynamic Factor , 1983 .

[10]  Nozer D. Singpurwalla,et al.  Understanding the Kalman Filter , 1983 .

[11]  K. Mardia,et al.  Maximum likelihood estimation of models for residual covariance in spatial regression , 1984 .

[12]  Robert Kohn,et al.  On the estimation of ARIMA Models with Missing Values , 1984 .

[13]  Guanrong Chen,et al.  Kalman Filtering with Real-time Applications , 1987 .

[14]  David R. Cox,et al.  A simple spatial-temporal model of rainfall , 1988, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[15]  J. Lewins Contribution to the Discussion , 1989 .

[16]  Fred L. Bookstein,et al.  Principal Warps: Thin-Plate Splines and the Decomposition of Deformations , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[17]  Andrew Harvey,et al.  Forecasting, Structural Time Series Models and the Kalman Filter , 1990 .

[18]  D. Myers,et al.  Problems in space-time kriging of geohydrological data , 1990 .

[19]  P. Guttorp,et al.  Nonparametric Estimation of Nonstationary Spatial Covariance Structure , 1992 .

[20]  Hong Chang,et al.  Model Determination Using Predictive Distributions with Implementation via Sampling-Based Methods , 1992 .

[21]  C. A. Glasbey A Reduced Rank Regression Model for Local Variation in Solar Radiation , 1992 .

[22]  M. Stein,et al.  A Bayesian analysis of kriging , 1993 .

[23]  Robert Haining,et al.  Statistics for spatial data: by Noel Cressie, 1991, John Wiley & Sons, New York, 900 p., ISBN 0-471-84336-9, US $89.95 , 1993 .

[24]  Nicholas I. Fisher,et al.  On the Nonparametric Estimation of Covariance Functions , 1994 .

[25]  Peter Guttorp,et al.  20 Methods for estimating heterogeneous spatial covariance functions with environmental applications , 1994, Environmental Statistics.

[26]  J. R. Wallis,et al.  An Approach to Statistical Spatial-Temporal Modeling of Meteorological Fields , 1994 .

[27]  Peter Hall,et al.  Properties of nonparametric estimators of autocovariance for stationary random fields , 1994 .

[28]  Noel A Cressie,et al.  Statistics for Spatial Data, Revised Edition. , 1994 .

[29]  P. Guttorp,et al.  A space-time analysis of ground-level ozone data , 1994 .

[30]  H. Storch,et al.  Principal oscillation patterns: a review , 1995 .

[31]  P. Green Reversible jump Markov chain Monte Carlo computation and Bayesian model determination , 1995 .

[32]  C. A. Glasbey Imputation of missing values in spatio‐temporal solar radiation data , 1995 .

[33]  Wenceslao González-Manteiga,et al.  Predicting using Box-Jenkins, nonparametric, and bootstrap techniques , 1995 .

[34]  Alexey Kaplan,et al.  Mapping tropical Pacific sea level : Data assimilation via a reduced state space Kalman filter , 1996 .

[35]  K. Mardia,et al.  Kriging and splines with derivative information , 1996 .

[36]  N. Cressie,et al.  Spatio-temporal prediction of snow water equivalent using the Kalman filter , 1996 .

[37]  J. Durbin,et al.  Monte Carlo maximum likelihood estimation for non-Gaussian state space models , 1997 .

[38]  Robert N. Miller,et al.  Applications of Data Assimilation to Analysis of the Ocean on Large Scales , 1997 .

[39]  H. J. Thiébaux The Power of the Duality in Spatial–Temporal Estimation , 1997 .

[40]  T. C. Haas,et al.  Model-based geostatistics. Discussion. Authors' reply , 1998 .

[41]  Olaf Berke,et al.  On spatiotemporal prediction for on-line monitoring data , 1998 .

[42]  David R. Brillinger,et al.  Modelling Longitudinal and Spatially Correlated Data: Methods, Applications, and Future Directions , 1998 .

[43]  T. C. Haas,et al.  Model-based geostatistics - Discussion , 1998 .

[44]  E. Demidenko Modelling Longitudinal and Spatially Correlated Data : Methods , Applications , and Future Directions , 1998 .

[45]  C. Wikle,et al.  A Dimension-Reduction Approach to Space-Time Kalman Filtering , 1999 .

[46]  N. Cressie,et al.  A dimension-reduced approach to space-time Kalman filtering , 1999 .