ADAPTIVE PARAMETRIC MODEL FOR NONSTATIONARY SPATIAL COVARIANCE
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Published:
Jun 30, 2020
Keywords:
Adaptive Locally Nonstationary Spatial Covariance Variability
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Abstract
In modelling environment processes, multi-disciplinary methods are used to explain, explore and predict how the earth responds to natural human-induced environmental changes over time. Consequently, when analyzing spatial processes in environmental and ecological studies, the spatial parameters of interest are always heterogeneous. This often negates the stationarity assumption. In this article, we propose the adaptive parametric nonstationary covariance structure for spatial processes. The adaptive turning parameter for this model was also proposed for nonstationary processes. The flexibility and efficiency of the propose model was examined through simulation. A real life data was also use to examine the efficiency of the propose model. The results show that the propose model perform competitively with existing models.
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Eghwerido, J. T., & Mbegbu, J. I. (2020). ADAPTIVE PARAMETRIC MODEL FOR NONSTATIONARY SPATIAL COVARIANCE. Malaysian Journal of Science, 39(2), 51–70. https://doi.org/10.22452/mjs.vol39no2.4
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Licensee MJS, Universiti Malaya, Malaysia. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
References
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Cherry, S., Banfield, J. and Quimby,W. F. (1996). An Evaluation of a Nonparametric Method of Estimating Semi-variograms of Isotropic Spatial Process. Journal of Applied Statistics, 23(4), 435-449.
Choi, I., Li, B., and Wang, X. (2013). Nonparametric estimation of spatial
and spacetime covariance function. Journal of Agricultural, Biological, and Environmental Statistics, 18(4), 1-20.
Cressie, N. (1993). Statistics for Spatial Data. New York. Wiley.
Damian, D., Sampson, P., and Guttorp, P. (2003). Variance Modeling for Nonstationary Spatial Processes with Temporal Replication.Journal of Geophysical Research Atmospheres, 108, (D24) A rt. No. 8778.
David, J. G. and Genton, M. G. (2000). Variogram Model Selection via Nonparametric Derivative Estimation. Mathematical Geology, 32 (3), 249-270.
David, N. J. and William, D. T. M. (2002). Estimation of Nonstationary Spatial Covariance Structure. Biometrika Trust, 89(4), 819-829.
Fuentes, M. (2001). A High Frequency Kriging Approach for Nonstationary Environmental Processes. Environmetrics, 12, 469-483.
Fuentes, M. (2002). Spectral Methods for Nonstationary Spatial Processes. Biometrika, 89, 197-210.
Gilmoura, A., Cullis, B., Welham, S., Gogel, B., and Thompson, R. (2004). An Efficient Computing Strategy for Prediction in Mixed Linear Models. Journal of Computational Statistics and Data Analysis, 44, 571-586.
Guttorp, P. and Sampson, P. D. (1994). Methods for Estimating Heterogeneous Spatial Covariance Functions with Environmental Applications. In Handbook of Statistics X II: Environmental Statistics, E d. G. P. Patil and C. R. Rao, pp. 663-90. New York: Elsevier/North Holland.
Guttorp, P. Fuentes, and M. Sampson, P. (2007). Using Transforms to Analyze Space-time Processes. In Statistics Methods of Spatio-Temporal Systems, V. Isham, B. Finkelstadt, L. Held (eds). Chapman and Hall/CRC: Boca Raton, 77-150.
Haas, T. (1990a). Kriging and Automated Variogram Modeling within a Moving Window, Atmospheric Environment. Part A. General Topics, 24(7), 1759-1769.
Haas, T. (1990b). Lognormal and Moving Window Methods of Estimating Acid Deposition. Journal of the American Statistical Association, 85(412), 950--963.
Haskard, K. A., Rawlins, B. G., and Lark, R. M. (2010). A Linear Mixed Model with Nonstationary mean and Covariance for Soil potassium based on Gamma Radiometry. Journal of Biogeosciences, 7, 2081-2089.
Higdon, D. (1998). A Process Convolution Approach to Modelling Temperatures in the North-Atlantic. Journal of Environmental Engineering and Science, 5,173-190.
Higdon, D. Swall, J. and Kern, J. (1999). Non-stationary spatial modeling. In Bayesian Statistics 6, Bernardo J, Berger J, Dawid A, Smith A (eds). Oxford University Press: Oxford, UK; 761–768.
Hsu, N., Chang, Y. and Huang, H. (2012). A group Lasso Approach for Nonstationary Spatial Temporal Covariance Estimation. Environmetrics, 23, 12-23.
Huang, C., Hsing, T., and Cressie, N. (2011). Nonparametric Estimation of the Variogram and Its Spectrum. Biometrika, 98, 775-789.
Huser, R. and Genton, M. G. (2016). Nonstationary Dependence Structures for Spatial Extremes. \textit{ Journal of Agricultural, Biological, and Environmental Statistics, 12(3), doi: 10.1007/s13253-016-0247-4.
Hyoung-Moon, K., Mallick, B. K., and Holmes, C. C. (2005). Analyzing Nonstationary Spatial Data Using Piecewise Gaussian Processes. Journal of the American Statistical Association, 100(470), 653-668.
Ingebrigtsen, R., Finn, L. and Ingelin, S. (2014). Spatial Models with Explanatory Variables in the Dependence Structure. Spatial Statistics, 8, 20-38.
Jaehong, J., Mikyoung, J. and Genton, M. G. (2017). Spherical Process Models for Global Spatial Statistics. Journal of Statistical Science, 32(4), 501-513.
Katzfuss, M. (2013). Bayesian Nonstationary Spatial Modeling for very Large Datasets, Environmetrics, 24, 189-200.
Katzfuss, M., and Cressie, N. (2012). Bayesian Hierarchical Spatio-temporal Smoothing for very Large Datasets, Environmetrics, 23(1), 94-107.
Le, N., Sun, L., and Zidek, J. (2001). Spatial Prediction and Temporal Backcasting for Environmental Fields having Monotone Data Patterns. The Canadian Journal of Statistics, 29, 529--554.
Meiring, W., Guttorp, P., and Sampson, P. (1998). Space-time Estimation of Grid-cell hourly Ozone Levels for Assessment of a Deterministic Model. Environmental and Ecological Statistics, 5, 197-222.
Matheron, G. (1963). Principles of Geostatistics, Econom. Geol. 58, 1246-1266.
Nychka, D. and Saltzman, N. (1998). Design of Air Quality Monitoring Networks. Case Studies in Environmental Statistics, 132, 51-76.
Nychka D., Wikle C., and Royle, J. (2002). Multiresolution Models for Nonstationary Spatial Sovariance Functions, Statistical Modelling, 2(4), 315.
Paciorek, C. and Schervish, M. (2006). Spatial modelling using a new class of nonstationary covariance functions. Environmetrics Journal, 17, 483-506.
Parker, R. J., Reich, B. J. and Eidsvik, J. (2016). A Fused Lasso Approach to Nonstationary Spatial Covariance Estimation. Journal of Agricultural, Biological, and Environmental Statistics, DOI: 10.1007/s13253-016-0251-8
Pickle, S. M., Robinson, T. J., Birich, J. B. and Anderson-Cook, C. M. (2008). Semi-parametric Approach to Robust Parameter Design, Journal of Statistical Planning and Inference, 138, 114-131.
Risser, D. M. and Calder, A. C. (2015). Regression Based Covariance Functions for Nonstationary Spatial Modeling. Environmetrics, 26, 284--297.
Sampson, P., Damian, D., Guttorp, P., and Holland, D. M. (2001). Deformation Based Nonstationary Spatial Covariance Modelling and Network Design. In Spatio-temporal Modelling of Environmental processes, Colecion Treballs D Informatica I Technologia, Num. 10, J Mateu,M Fuentes (eds). Universitat Jaume I:Castellon, Spain; 125-132.
Sampson, P. and Guttorp, P. (1992). Nonparametric Estimation of Nonstationary Spatial Covariance Structure. Journal of the American Statistical Association, 87,108-119.
Schmidt, M. A., Guttorp, P. and O Hagan, A. (2011). Considering Covariates in the Covariance Structure of Spatial Processes. Environmetrics Journal, 22, 487-500.
Shand, L. and Li, B. (2017). Modeling Nonstationarity in Space and Time. Biometrics, 1-10.
Shapiro, A. and Botha, J. D. (1991) Variogram fitting with a general class of conditionally nonnegative definite functions. Comput. Statist. Data Anal., 11, 87 -96.
Cherry, S., Banfield, J. and Quimby,W. F. (1996). An Evaluation of a Nonparametric Method of Estimating Semi-variograms of Isotropic Spatial Process. Journal of Applied Statistics, 23(4), 435-449.
Choi, I., Li, B., and Wang, X. (2013). Nonparametric estimation of spatial
and spacetime covariance function. Journal of Agricultural, Biological, and Environmental Statistics, 18(4), 1-20.
Cressie, N. (1993). Statistics for Spatial Data. New York. Wiley.
Damian, D., Sampson, P., and Guttorp, P. (2003). Variance Modeling for Nonstationary Spatial Processes with Temporal Replication.Journal of Geophysical Research Atmospheres, 108, (D24) A rt. No. 8778.
David, J. G. and Genton, M. G. (2000). Variogram Model Selection via Nonparametric Derivative Estimation. Mathematical Geology, 32 (3), 249-270.
David, N. J. and William, D. T. M. (2002). Estimation of Nonstationary Spatial Covariance Structure. Biometrika Trust, 89(4), 819-829.
Fuentes, M. (2001). A High Frequency Kriging Approach for Nonstationary Environmental Processes. Environmetrics, 12, 469-483.
Fuentes, M. (2002). Spectral Methods for Nonstationary Spatial Processes. Biometrika, 89, 197-210.
Gilmoura, A., Cullis, B., Welham, S., Gogel, B., and Thompson, R. (2004). An Efficient Computing Strategy for Prediction in Mixed Linear Models. Journal of Computational Statistics and Data Analysis, 44, 571-586.
Guttorp, P. and Sampson, P. D. (1994). Methods for Estimating Heterogeneous Spatial Covariance Functions with Environmental Applications. In Handbook of Statistics X II: Environmental Statistics, E d. G. P. Patil and C. R. Rao, pp. 663-90. New York: Elsevier/North Holland.
Guttorp, P. Fuentes, and M. Sampson, P. (2007). Using Transforms to Analyze Space-time Processes. In Statistics Methods of Spatio-Temporal Systems, V. Isham, B. Finkelstadt, L. Held (eds). Chapman and Hall/CRC: Boca Raton, 77-150.
Haas, T. (1990a). Kriging and Automated Variogram Modeling within a Moving Window, Atmospheric Environment. Part A. General Topics, 24(7), 1759-1769.
Haas, T. (1990b). Lognormal and Moving Window Methods of Estimating Acid Deposition. Journal of the American Statistical Association, 85(412), 950--963.
Haskard, K. A., Rawlins, B. G., and Lark, R. M. (2010). A Linear Mixed Model with Nonstationary mean and Covariance for Soil potassium based on Gamma Radiometry. Journal of Biogeosciences, 7, 2081-2089.
Higdon, D. (1998). A Process Convolution Approach to Modelling Temperatures in the North-Atlantic. Journal of Environmental Engineering and Science, 5,173-190.
Higdon, D. Swall, J. and Kern, J. (1999). Non-stationary spatial modeling. In Bayesian Statistics 6, Bernardo J, Berger J, Dawid A, Smith A (eds). Oxford University Press: Oxford, UK; 761–768.
Hsu, N., Chang, Y. and Huang, H. (2012). A group Lasso Approach for Nonstationary Spatial Temporal Covariance Estimation. Environmetrics, 23, 12-23.
Huang, C., Hsing, T., and Cressie, N. (2011). Nonparametric Estimation of the Variogram and Its Spectrum. Biometrika, 98, 775-789.
Huser, R. and Genton, M. G. (2016). Nonstationary Dependence Structures for Spatial Extremes. \textit{ Journal of Agricultural, Biological, and Environmental Statistics, 12(3), doi: 10.1007/s13253-016-0247-4.
Hyoung-Moon, K., Mallick, B. K., and Holmes, C. C. (2005). Analyzing Nonstationary Spatial Data Using Piecewise Gaussian Processes. Journal of the American Statistical Association, 100(470), 653-668.
Ingebrigtsen, R., Finn, L. and Ingelin, S. (2014). Spatial Models with Explanatory Variables in the Dependence Structure. Spatial Statistics, 8, 20-38.
Jaehong, J., Mikyoung, J. and Genton, M. G. (2017). Spherical Process Models for Global Spatial Statistics. Journal of Statistical Science, 32(4), 501-513.
Katzfuss, M. (2013). Bayesian Nonstationary Spatial Modeling for very Large Datasets, Environmetrics, 24, 189-200.
Katzfuss, M., and Cressie, N. (2012). Bayesian Hierarchical Spatio-temporal Smoothing for very Large Datasets, Environmetrics, 23(1), 94-107.
Le, N., Sun, L., and Zidek, J. (2001). Spatial Prediction and Temporal Backcasting for Environmental Fields having Monotone Data Patterns. The Canadian Journal of Statistics, 29, 529--554.
Meiring, W., Guttorp, P., and Sampson, P. (1998). Space-time Estimation of Grid-cell hourly Ozone Levels for Assessment of a Deterministic Model. Environmental and Ecological Statistics, 5, 197-222.
Matheron, G. (1963). Principles of Geostatistics, Econom. Geol. 58, 1246-1266.
Nychka, D. and Saltzman, N. (1998). Design of Air Quality Monitoring Networks. Case Studies in Environmental Statistics, 132, 51-76.
Nychka D., Wikle C., and Royle, J. (2002). Multiresolution Models for Nonstationary Spatial Sovariance Functions, Statistical Modelling, 2(4), 315.
Paciorek, C. and Schervish, M. (2006). Spatial modelling using a new class of nonstationary covariance functions. Environmetrics Journal, 17, 483-506.
Parker, R. J., Reich, B. J. and Eidsvik, J. (2016). A Fused Lasso Approach to Nonstationary Spatial Covariance Estimation. Journal of Agricultural, Biological, and Environmental Statistics, DOI: 10.1007/s13253-016-0251-8
Pickle, S. M., Robinson, T. J., Birich, J. B. and Anderson-Cook, C. M. (2008). Semi-parametric Approach to Robust Parameter Design, Journal of Statistical Planning and Inference, 138, 114-131.
Risser, D. M. and Calder, A. C. (2015). Regression Based Covariance Functions for Nonstationary Spatial Modeling. Environmetrics, 26, 284--297.
Sampson, P., Damian, D., Guttorp, P., and Holland, D. M. (2001). Deformation Based Nonstationary Spatial Covariance Modelling and Network Design. In Spatio-temporal Modelling of Environmental processes, Colecion Treballs D Informatica I Technologia, Num. 10, J Mateu,M Fuentes (eds). Universitat Jaume I:Castellon, Spain; 125-132.
Sampson, P. and Guttorp, P. (1992). Nonparametric Estimation of Nonstationary Spatial Covariance Structure. Journal of the American Statistical Association, 87,108-119.
Schmidt, M. A., Guttorp, P. and O Hagan, A. (2011). Considering Covariates in the Covariance Structure of Spatial Processes. Environmetrics Journal, 22, 487-500.
Shand, L. and Li, B. (2017). Modeling Nonstationarity in Space and Time. Biometrics, 1-10.
Shapiro, A. and Botha, J. D. (1991) Variogram fitting with a general class of conditionally nonnegative definite functions. Comput. Statist. Data Anal., 11, 87 -96.