, Volume 27, Issue 1, pp 221–245 | Cite as

Spatio-temporal analysis with short- and long-memory dependence: a state-space approach

  • Guillermo Ferreira
  • Jorge Mateu
  • Emilio Porcu
Original Paper


This paper deals with the estimation and prediction problems of spatio-temporal processes by using state-space methodology. The spatio-temporal process is represented through an infinite moving average decomposition. This expansion is well known in time series analysis and can be extended straightforwardly in space–time. Such an approach allows easy implementation of the Kalman filter procedure for estimation and prediction of linear time processes exhibiting both short- and long-range dependence and a spatial dependence structure given on the locations. Furthermore, we consider a truncated state-space equation, which allows to calculate an approximate likelihood for large data sets. The performance of the proposed Kalman filter approach is evaluated by means of several Monte Carlo experiments implemented under different scenarios, and it is illustrated with two applications.


Kalman filter algorithm Second-order stationary Space–time geostatistics Time series models 

Mathematics Subject Classification

62M10 62M20 62M30 



The first author would like to express his thanks for the support from DIUC 215.014.024-1.0, established by the Universidad de Concepción and Postdoctoral scholarship from Conicyt, Chile, 2014 (Folio 74150023). Jorge Mateu’s research was supported by Grant MTM2013-43917-P from the Spanish Ministry of Science and Education, and Grant P1-1B2015-40 and Emilio Porcu’s research was supported by Fondecyt Regular Project from Ministery of Science and Education, Chile.


  1. Bevilacqua M, Gaetan C, Mateu J, Porcu E (2012) Estimating space and space–time covariance functions for large data sets: a weighted composite likelihood approach. J Am Stat Assoc 107(497):268–280MathSciNetCrossRefMATHGoogle Scholar
  2. Bilonick RA (1985) The space–time distribution of sulfate deposition in the northeastern united states. Atmos Environ (1967) 19(11):1829–1845CrossRefGoogle Scholar
  3. Bivand R, Keitt T, Rowlingson B (2015) rgdal: bindings for the geospatial data abstraction library. R Foundation for Statistical Computing, ViennaGoogle Scholar
  4. Bocquet M, Elbern H, Eskes H, Hirtl M, Zabkar R, Carmichael G, Flemming J, Inness A, Pagowski M, Pérez Camaño J et al (2015) Data assimilation in atmospheric chemistry models: current status and future prospects for coupled chemistry meteorology models. Atmos Chem Phys 15(10):5325–5358CrossRefGoogle Scholar
  5. Brockwell PJ, Davis RA (1991) Time series: theory and methods, 2nd edn. Springer, New YorkCrossRefMATHGoogle Scholar
  6. Brown PJ, Le ND, Zidek JV (1994) Multivariate spatial interpolation and exposure to air pollutants. Can J Stat 22(4):489–509MathSciNetCrossRefMATHGoogle Scholar
  7. Broyden CG (1969) A new double-rank minimization algorithm. Not Am Math Soc 16:670Google Scholar
  8. Cameletti M, Lindgren F, Simpson D, Rue H (2013) Spatio-temporal modeling of particulate matter concentration through the spde approach. AStA Adv Stat Anal 97(2):109–131MathSciNetCrossRefGoogle Scholar
  9. Carroll SS, Cressie N (1997) Spatial modeling of snow water equivalent using covariances estimated from spatial and geomorphic attributes. J Hydrol (Amst) 190(1):42–59CrossRefGoogle Scholar
  10. Chan NH, Palma W (1998) State space modeling of long-memory processes. Ann Stat 26(2):719–740MathSciNetCrossRefMATHGoogle Scholar
  11. Cressie N, Wikle CK (2011) Statistics for spatio-temporal data. Wiley, New YorkMATHGoogle Scholar
  12. Daley D, Porcu E (2014) Dimension walks and schoenberg spectral measures. P Am Math Soc 142(5):1813–1824MathSciNetCrossRefMATHGoogle Scholar
  13. Durbin J, Koopman SJ (2012) Time series analysis by state space methods. Number 38 in Oxford statistical science series. Oxford University Press, OxfordMATHGoogle Scholar
  14. Eynon B, Switzer P (1983) The variability of rainfall acidity. Can J Stat 11:11–24CrossRefGoogle Scholar
  15. Fasso A, Cameletti M, Nicolis O (2007) Air quality monitoring using heterogeneous networks. Environmetrics 18(3):245–264MathSciNetCrossRefGoogle Scholar
  16. Ferreira G, Rodríguez A, Lagos B (2013) Kalman filter estimation for a regression model with locally stationary errors. Comput Stat Data Anal 62:52–69MathSciNetCrossRefMATHGoogle Scholar
  17. Fletcher R (1970) A new approach to variable metric methods. Comput J 13:317–322CrossRefMATHGoogle Scholar
  18. Gneiting T (2002) Nonseparable, stationary covariance functions for space–time data. J Am Stat Assoc 97(458):590–600MathSciNetCrossRefMATHGoogle Scholar
  19. Goldfarb D (1970) A family of variable metric methods derived by variational means. Math Comput 24:23–26MathSciNetCrossRefMATHGoogle Scholar
  20. Guyon X (1995) Random fields on a network: modeling, statistics, and applications. Springer, New YorkMATHGoogle Scholar
  21. Handcock MS, Wallis JR (1994) An approach to statistical spatial-temporal modeling of meteorological fields. J Am Stat Assoc 89(426):368–378MathSciNetCrossRefMATHGoogle Scholar
  22. Hannan EJ, Deistler M (1988) The statistical theory of linear systems. Wiley, New YorkMATHGoogle Scholar
  23. Harvey AC (1989) Forecasting structural time series and the Kalman filter. Cambridge University Press, CambridgeGoogle Scholar
  24. Haslett J, Raftery AE (1989) Space–time modelling with long-memory dependence: assessing Ireland’s wind power resource. J R Stat Soc C Appl 38(1):1–50Google Scholar
  25. Huang H-C, Cressie N (1996) Spatio-temporal prediction of snow water equivalent using the Kalman filter. Comput Stat Data Anal 22(2):159–175MathSciNetCrossRefGoogle Scholar
  26. Hughes JP, Guttorp P, Charles SP (1999) A non-homogeneous hidden markov model for precipitation occurrence. J R Stat Soc C Appl 48(1):15–30CrossRefMATHGoogle Scholar
  27. Ippoliti L (2001) On-line spatio-temporal prediction by a state space representation of the generalized space time autoregressive model. Metron Int J Stat LIX:157–169MathSciNetMATHGoogle Scholar
  28. Jun M, Stein ML (2008) Nonstationary covariance models for global data. Ann Appl Stat 2(4):1271–1289MathSciNetCrossRefMATHGoogle Scholar
  29. Kokoszka PS, Taqqu MS (1995) Fractional arima with stable innovations. Stoch Proc Appl 60(1):19–47MathSciNetCrossRefMATHGoogle Scholar
  30. Li B, Genton MG, Sherman M (2008) On the asymptotic joint distribution of sample space–time covariance estimators. Bernoulli 14(1):228–248MathSciNetCrossRefMATHGoogle Scholar
  31. Mardia KV, Goodall C, Redfern EJ, Alonso FJ (1998) The kriged Kalman filter. Test 7(2):217–282MathSciNetCrossRefMATHGoogle Scholar
  32. Matérn B (1986) Spatial variation, volume 36 of lecture notes in statistics. Springer, BerlinGoogle Scholar
  33. Mikosch T, Gadrich T, Kluppelberg C, Adler RJ (1995) Parameter estimation for arma models with infinite variance innovations. Ann Stat 23(1):305–326MathSciNetCrossRefMATHGoogle Scholar
  34. Militino A, Ugarte M, Goicoa T, Genton M (2015) Interpolation of daily rainfall using spatiotemporal models and clustering. Int J Climatol 35(7):1453–1464CrossRefGoogle Scholar
  35. Oehlert GW (1993) Regional trends in sulfate wet deposition. J Am Stat Assoc 88(422):390–399CrossRefGoogle Scholar
  36. Palma W (2007) Long-memory time series: theory and methods. Wiley series in probability and statistics. Wiley, HobokenCrossRefMATHGoogle Scholar
  37. Palma W, Olea R, Ferreira G (2013) Estimation and forecasting of locally stationary processes. J Forecast 32(1):86–96MathSciNetCrossRefGoogle Scholar
  38. Peng RD, de Leeuw J (2002) An introduction to the .C interface to R. UCLA, Academic Technology Services, Statistical Consulting GroupGoogle Scholar
  39. Porcu E, Bevilacqua M, Genton MG (2015) Spatio-temporal covariance and cross-covariance functions of the great circle distance on a sphere. J Am Stat Assoc. doi: 10.1080/01621459.2015.1072541 Google Scholar
  40. R Core Team (2017) R: a language and environment for statistical computing. R Foundation for Statistical Computing, ViennaGoogle Scholar
  41. Rao SS (2008) Statistical analysis of a spatio-temporal model with location-dependent parameters and a test for spatial stationarity. J Time Ser Anal 29(4):673–694MathSciNetCrossRefMATHGoogle Scholar
  42. Stein ML (2005) Space–time covariance functions. J Am Stat Assoc 100(469):310–321MathSciNetCrossRefMATHGoogle Scholar
  43. Stroud JR, Stein ML, Lesht BM, Schwab DJ, Beletsky D (2010) An ensemble Kalman filter and smoother for satellite data assimilation. J Am Stat Assoc 105(491):978–990MathSciNetCrossRefMATHGoogle Scholar
  44. Waller LA, Carlin BP, Xia H, Gelfand AE (1997) Hierarchical spatio-temporal mapping of disease rates. J Am Stat Assoc 92(438):607–617CrossRefMATHGoogle Scholar
  45. Wikle CK (2003) Hierarchical models in environmental science. Int Stat Rev 71(2):181–199CrossRefMATHGoogle Scholar
  46. Wikle CK, Cressie N (1999) A dimension-reduced approach to space-time Kalman filtering. Biometrika 86(4):815–829MathSciNetCrossRefMATHGoogle Scholar
  47. Wikle CK, Berliner LM, Cressie N (1998) Hierarchical Bayesian space–time models. Environ Ecol Stat 5(2):117–154CrossRefGoogle Scholar
  48. Xu K, Wikle CK (2007) Estimation of parameterized spatio-temporal dynamic models. J Stat Plan Inference 137(2):567–588MathSciNetCrossRefMATHGoogle Scholar
  49. Zes D (2014) Facile spatio-temporal modeling, forecasting with adaptive least squares and the Kalman filter. J Environ Stat 6(1).

Copyright information

© Sociedad de Estadística e Investigación Operativa 2017

Authors and Affiliations

  1. 1.Department of StatisticsUniversidad de ConcepciónConcepciónChile
  2. 2.Department of MathematicsUniversity Jaume ICastellónSpain
  3. 3.Department of MathematicsUniversity Federico Santa MaríaValparaisoChile

Personalised recommendations