:: Volume 17, Issue 1 (8-2020) ::
JSRI 2020, 17(1): 235-251 Back to browse issues page
A New Class of Spatial Covariance Functions Generated by Higher-order Kernels
Nafiseh Vafaei 1, Jorge Mateu2 , Masoud Ganji3 , Mohammad Ghorbani4
1- University of Mohaghegh Ardabili , n.vafaei@uma.ac.ir
2- Jaume I University
3- University of Mohaghegh Ardabili
4- Lulera University of Technology
Abstract:   (417 Views)
Covariance functions and variograms play a fundamental role in exploratory analysis and statistical modelling of spatial and spatio-temporal datasets. In this paper, we construct a new class of spatial covariance functions using the Fourier transform of some higher-order kernels. Moreover, we extend this class of spatial covariance functions to the spatio-temporal setting using the idea used in Ma (2003).
Keywords: Bochner's theorem, characteristic function, covariance model, higher-order Kernels, spatial data.
Full-Text [PDF 792 kb]   (409 Downloads)    
Type of Study: Applicable | Subject: General
Received: 2022/11/20 | Accepted: 2023/02/26 | Published: 2020/08/22
References
1. Brown, P. E., Karesen, K. F., Roberts, G. O. and Tonellato, S. (2000). Blur-generated nonseparable space-time models, Journal of the Royal Statistical Society. Series B (Statistical [DOI:10.1111/1467-9868.00269]
2. Methodology) 62, 847-860.
3. Cherian, A., Koniusz, P. and Gould, S. (2017). Higher-order pooling of cnn features via kernel linearization for action recognition, IEEE Winter Conference on Applications of Computer Vision p. 130-138. [DOI:10.1109/WACV.2017.22]
4. Christakos, G. (2000). Modern spatiotemporal geostatistics, Vol. 6, Oxford university press.
5. Cressie, N.A.C. (1993). Statistics for Spatial Data, Second edn, Wiley, New York. [DOI:10.1002/9781119115151]
6. Cressie, N. and Huang, H. C. (1999). Classes of nanseparable, spatio-temporal stationary covariance function, Journal of the American Statistical Association 94, 1330-1340. [DOI:10.1080/01621459.1999.10473885]
7. Das, D., Nayak, M. and Pani, S.K. (2019). Missing value imputation-a review, Int J Comput Sci Eng 7, 548-558. [DOI:10.26438/ijcse/v7i4.548558]
8. Dubois, G. (1998). Spatial interpolation comparison 97: foreword and introduction, Journal of Geographic Information and Decision Analysis, 2, 1-10.
9. Fasshauer, G.E. (2007). Meshfree Approximation Methods with MATLAB, World Scientific Publishing Company. [DOI:10.1142/6437]
10. Finkenstadt, B., Held, L. and Isham, V. (2007). Statistical methods for spatio-temporal system, Taylor & Francis Group, LLC. [DOI:10.1201/9781420011050]
11. Fuentes, M. and Smith, R. (2001). A new class of nonstationary models, Technical report, North Carolina State University .
12. Granovsky, B.L. and M¨uller, H.G. (1991). Optimizing kernel methods: A unifying variational principle, International Statistical Review, 59, 373-388. [DOI:10.2307/1403693]
13. Hansen, B. E. (2005). Exact mean integrated squared error of higher order kernel estimators, Econometric Theory 21, 1031-1057. [DOI:10.1017/S0266466605050528]
14. Hurst, S. (1995). The characteristic function of the student t-distribution. Financial Mathematics Research Report 006-95, Australian National University, Canberra ACT 0200, Australia.
15. Jones, R.H. and Zhang, Y. (1997). Models for continuous stationary space-time processes, Modelling longitudinal and spatially correlated data, Springer, pp. 289-298. [DOI:10.1007/978-1-4612-0699-6_25]
16. Lindgren, G. (2012). Stationary Stochastic Processes: Theory and Applications, Chapman & [DOI:10.1201/b12171]
17. Hall/CRC Press.
18. Ma, C. (2003). Families of spatio-temporal stationary covariance models, Journal of Statistical [DOI:10.1016/S0378-3758(02)00353-1]
19. Planning and Inference, 116, 489-501.
20. Marron, J. (1994). Visual understanding of higher-order kernels, Journal of Computational and Graphical Statistics, 3, 447-458. [DOI:10.1080/10618600.1994.10474657]
21. Matern, B. (1960). Spatial variation: stochastic models and their applications to problems in forest surveys and other sampling investigation, Meddelanden fran Statens Skogforsknings institute 49.
22. Matheron, G. (1962). Traite de geostatistique, Mem. BRGM (14).
23. Moller, J. and Waagepetersen, R.P. (2004). Statistical Inference and Simulation for Spatial Point Processes, Chapman and Hall/CRC, Boca Raton. [DOI:10.1201/9780203496930]
24. Muller, H.G. (1984). Smooth optimum kernel estimators of densities, regression curves and modes, The Annals of Statistics 12, 766-774. [DOI:10.1214/aos/1176346523]
25. Powell, M.J.D. (2009). The bobyqa algorithm for bound constrained optimization without derivatives. Research report NA2009/06, Department of Applied Mathematics and Theoretical Physics, Cambridge, England.
26. Ribeiro Jr,P.J. and Diggle, P.J. (2001). geoR: a package for geostatistical analysis, R-NEWS, 1, 15-18. URL: http://cran.R-project.org/doc/Rnews
27. Ripley, B.D. (1981). Spatial Statistics, Wiley, New York. [DOI:10.1002/0471725218]
28. Salvi, C., Lemercier, M., Liu, C., Horvath, B., Damoulas, T. and Lyons, T. (2021). Higher order kernel mean embeddings to capture filtrations of stochastic processes, Advances in Neural Information Processing Systems, 34, 16635-16647.
29. Sampson, P.D. and Guttorp, P. (1992). Nonparametric estimation of nonstationary spatial covariance structure, Journal of the American Statistical Association, 87, 108-119. [DOI:10.1080/01621459.1992.10475181]
30. Schucany, W. and Sommers, J.P. (1977). Improvement of kernel type density estimators, Journal of the American Statistical Association, 72, 420-423. [DOI:10.1080/01621459.1977.10481012]
31. Tsuruta, Y. and Sagae, M. (2017). Higher order kernel density estimation on the circle, Statistics Probability Letters, 131, 46-50. [DOI:10.1016/j.spl.2017.08.003]
32. Wackernagel, H. (2003). Multivariate Geostatistics, Springer-Verlag, Berlin Heidelberg. [DOI:10.1007/978-3-662-05294-5]
33. Wand, M.P. and Schucany, W.R. (1990). Gaussian-based kernels, The Canadian Journal of Statistics, 18, 197-204. [DOI:10.2307/3315450]
34. Wikle, C.K., Z.M.A. and Cressie, N. (2019). Spatio-Temporal Statistics with R, Boca Raton, FL: Chapman Hall/CRC. [DOI:10.1201/9781351769723]
35. Yaglom, A.M. (1987). Correlation Theory of Stationary and Related Random Functions. Volume I: Basic Results, Springer-Verlag, New York Inc. [DOI:10.1007/978-1-4612-4620-6]
36. Ypma, J. (2017). nloptr: R interface to nlopt. R package version 1.0.4.

XML     Print

Rights and permissions
Creative Commons License This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Volume 17, Issue 1 (8-2020) Back to browse issues page