Abstract
In this paper we explore a covariance-spectral modelling strategy for spatial-temporal processes which involves a spectral approach for time but a covariance approach for space. It facilitates the analysis of coherence between the temporal frequency components at different spatial sites. Stein (J R Stat Soc Ser B (Statistical Methodology) 67:667–687, 2005) developed a semi-parametric model within this framework. The purpose of this paper is to give a deeper insight into the properties of his model and to develop simpler and more intuitive methods of estimation and testing. A very neat estimation for drift direction is proposed while Stein assumes it is known. An example is given using the Irish wind speed data. Stein constructed various plot to assess the goodness of fit of the model, we use similar plots to estimates the parameters.
Similar content being viewed by others
References
Beran J (1994) Statistics for long memory processes. Chapman and Hall, New York
Bloomfield P (1973) An exponential model for the spectrum of a scalar time series. Biometrika 60:217–226
Bloomfield P (1976) Fourier analysis of time series: an introduction. Wiley, New York
Cox DR, Isham V (1988) A simple spatial-temporal model of rainfall. Proc R Soc Lond Ser A Math Phys Sci 415(1849):317–328
Cressie NAC (1993) Statistics for spatial data (wiley series in probability and statistics). Wiley-Interscience, London
Cressie NAC, Huang H (1999) Classes of non-separable, spatio-temporal stationary covariance functions. J Am Stat Assoc 94:1330–1340
De Luna X, Genton M (2005) Predictive spatio-temporal models for spatially sparse environmental data. Stat Sinica 15:547–568
Fuentes M (2006) Testing for separability of spatial-temporal covariance functions. J Stat Plan Inference 136(2):447–466
Gneiting T (2002) Non-separable, stationary covariance functions for space-time data. J Am Stat Assoc 97(458):590–600
Haslett J, Raftery AE (1989) Space-time modelling with long-memory dependence: assessing Ireland’s wind power resource. J Appl Stat 38(1):1–50
Lu N, Zimmerman DL (2002) Testing for directional symmetry in spatial dependence using the periodogram. J Stat Plan Inference 129(1–2):369–385
Ma C (2003) Families of spatio-temporal stationary covariance models. J Stat Plan Inference 116(2):489–501
Mardia KV, Kent JT, Bibby JM (1979) Multivariate analysis. Academic Press, London
Mitchell MW, Genton MG, Gumpertz ML (2006) A likelihood ratio test for separability of covariances. J Multivar Anal 97(5):1025–1043
Nadaraya EA (1964) On estimating regression. Theory Probab Appl 10:186–190
Priestley MB (1981) Spectral analysis and time series, volume I and II. Academic Press, London
Ruppert D, Sheather SJ, Wand MP (1995) An effective bandwidth selector for local least squares regression. J Am Stat Assoc 90:1257–1270
Scaccia L, Martin RJ (2005) Testing axial symmetry and separability of lattice processes. J Stat Plan Inference 131(1):19–39
Stein ML (2005) Statistical methods for regular monitoring data. J R Stat Soc Ser B (Statistical Methodology) 67:667–687
Stein ML (2009) Spatial interpolation of high frequency monitoring data. Ann Appl Stat 3:272–291
Subba Rao T, Das S, Boshnakov GN (2014) A frequency domain approach for the estimation of parameters of spatio-temporal stationary random processes. J Time Ser Anal 35(4):357–377
Watson GS (1964) Smooth regression analysis. Sankhya A26:359–372
Acknowledgments
The authors would like to thank two referees and the associate editors whose comments have been very helpful in improving the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Handling Editor: Bryan F. J. Manly.
Rights and permissions
About this article
Cite this article
Mosammam, A.M., Kent, J.T. Estimation and testing for covariance-spectral spatial-temporal models. Environ Ecol Stat 23, 43–64 (2016). https://doi.org/10.1007/s10651-015-0322-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10651-015-0322-y