IDEAS home Printed from https://ideas.repec.org/a/bla/scjsta/v49y2022i1p211-235.html
   My bibliography  Save this article

Nonstationary space–time covariance functions induced by dynamical systems

Author

Listed:
  • Rachid Senoussi
  • Emilio Porcu

Abstract

This article provides a novel approach to nonstationarity by considering a bridge between differential equations and spatial fields. We consider the dynamical transformation of a given spatial process undergoing the action of a temporal flow of space diffeomorphisms. Such dynamical deformations are shown to be connected to certain classes of ordinary and partial differential equations. The natural question arises of how such dynamical diffeomorphisms convert the original spatial covariance function, specifically if the original covariance is spatially stationary or isotropic. We first challenge this question from a general perspective, and then turn into the special cases of both d‐dimensional Euclidean spaces, and hyperspheres. Several examples of dynamical diffeomorphisms defined in these spaces are given and some emphasis has been put on the stationary reducibility problem. We provide a simple illustration to show the performance of the maximum likelihood estimation of the parameters of a family of dynamically deformed covariance functions.

Suggested Citation

  • Rachid Senoussi & Emilio Porcu, 2022. "Nonstationary space–time covariance functions induced by dynamical systems," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(1), pages 211-235, March.
    • Handle: RePEc:bla:scjsta:v:49:y:2022:i:1:p:211-235
    DOI: 10.1111/sjos.12513
    as

    Download full text from publisher

    File URL: https://doi.org/10.1111/sjos.12513
    Download Restriction: no
    File URL: https://libkey.io/10.1111/sjos.12513?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Perrin, Olivier & Senoussi, Rachid, 2000. "Reducing non-stationary random fields to stationarity and isotropy using a space deformation," Statistics & Probability Letters, Elsevier, vol. 48(1), pages 23-32, May.
    2. Emilio Porcu & Alfredo Alegria & Reinhard Furrer, 2018. "Modeling Temporally Evolving and Spatially Globally Dependent Data," International Statistical Review, International Statistical Institute, vol. 86(2), pages 344-377, August.
    3. Alexandre Rodrigues & Peter J. Diggle, 2010. "A Class of Convolution‐Based Models for Spatio‐Temporal Processes with Non‐Separable Covariance Structure," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(4), pages 553-567, December.
    4. Perrin, Olivier & Senoussi, Rachid, 1999. "Reducing non-stationary stochastic processes to stationarity by a time deformation," Statistics & Probability Letters, Elsevier, vol. 43(4), pages 393-397, July.
    5. Christakos, George, 1987. "The space transformation in the simulation of multidimensional random fields," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 29(3), pages 313-319.
    6. Kleiber, William & Nychka, Douglas, 2012. "Nonstationary modeling for multivariate spatial processes," Journal of Multivariate Analysis, Elsevier, vol. 112(C), pages 76-91.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kleiber, William, 2016. "High resolution simulation of nonstationary Gaussian random fields," Computational Statistics & Data Analysis, Elsevier, vol. 101(C), pages 277-288.
    2. Liu, Jialuo & Chu, Tingjin & Zhu, Jun & Wang, Haonan, 2021. "Semiparametric method and theory for continuously indexed spatio-temporal processes," Journal of Multivariate Analysis, Elsevier, vol. 183(C).
    3. Lim, Chae Young & Wu, Wei-Ying, 2022. "Conditions on which cokriging does not do better than kriging," Journal of Multivariate Analysis, Elsevier, vol. 192(C).
    4. Laura M. Sangalli, 2021. "Spatial Regression With Partial Differential Equation Regularisation," International Statistical Review, International Statistical Institute, vol. 89(3), pages 505-531, December.
    5. Michel Harel & Jean-François Lenain & Joseph Ngatchou-Wandji, 2016. "Asymptotic behaviour of binned kernel density estimators for locally non-stationary random fields," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 28(2), pages 296-321, June.
    6. Cleanthous, Galatia & Georgiadis, Athanasios G. & Lang, Annika & Porcu, Emilio, 2020. "Regularity, continuity and approximation of isotropic Gaussian random fields on compact two-point homogeneous spaces," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4873-4891.
    7. Taylor, Benjamin M. & Davies, Tilman M. & Rowlingson, Barry S. & Diggle, Peter J., 2013. "lgcp: An R Package for Inference with Spatial and Spatio-Temporal Log-Gaussian Cox Processes," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 52(i04).
    8. Monterrubio-Gómez, Karla & Roininen, Lassi & Wade, Sara & Damoulas, Theodoros & Girolami, Mark, 2020. "Posterior inference for sparse hierarchical non-stationary models," Computational Statistics & Data Analysis, Elsevier, vol. 148(C).
    9. Alessia Caponera, 2021. "SPHARMA approximations for stationary functional time series on the sphere," Statistical Inference for Stochastic Processes, Springer, vol. 24(3), pages 609-634, October.
    10. Bingham, N.H. & Symons, Tasmin L., 2019. "Dimension walks on Sd×R," Statistics & Probability Letters, Elsevier, vol. 147(C), pages 12-17.
    11. Takahiro Yoshida & Morito Tsutsumi, 2018. "On the effects of spatial relationships in spatial compositional multivariate models," Letters in Spatial and Resource Sciences, Springer, vol. 11(1), pages 57-70, March.
    12. Pilar García-Soidán & Tomás R. Cotos-Yáñez, 2020. "Use of Correlated Data for Nonparametric Prediction of a Spatial Target Variable," Mathematics, MDPI, vol. 8(11), pages 1-20, November.
    13. Bolin, David & Lindgren, Finn, 2013. "A comparison between Markov approximations and other methods for large spatial data sets," Computational Statistics & Data Analysis, Elsevier, vol. 61(C), pages 7-21.
    14. Philip A. White & Durban G. Keeler & Daniel Sheanshang & Summer Rupper, 2022. "Improving piecewise linear snow density models through hierarchical spatial and orthogonal functional smoothing," Environmetrics, John Wiley & Sons, Ltd., vol. 33(5), August.
    15. Bissiri, Pier Giovanni & Cleanthous, Galatia & Emery, Xavier & Nipoti, Bernardo & Porcu, Emilio, 2022. "Nonparametric Bayesian modelling of longitudinally integrated covariance functions on spheres," Computational Statistics & Data Analysis, Elsevier, vol. 176(C).
    16. Jun, Mikyoung, 2014. "Matérn-based nonstationary cross-covariance models for global processes," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 134-146.
    17. Jordan Richards & Jennifer L. Wadsworth, 2021. "Spatial deformation for nonstationary extremal dependence," Environmetrics, John Wiley & Sons, Ltd., vol. 32(5), August.
    18. Monica Palma & Claudia Cappello & Sandra De Iaco & Daniela Pellegrino, 2019. "The residential real estate market in Italy: a spatio-temporal analysis," Quality & Quantity: International Journal of Methodology, Springer, vol. 53(5), pages 2451-2472, September.
    19. Perrin, Olivier & Senoussi, Rachid, 2000. "Reducing non-stationary random fields to stationarity and isotropy using a space deformation," Statistics & Probability Letters, Elsevier, vol. 48(1), pages 23-32, May.
    20. De Iaco, S., 2023. "Spatio-temporal generalized complex covariance models based on convolution," Computational Statistics & Data Analysis, Elsevier, vol. 183(C).

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:bla:scjsta:v:49:y:2022:i:1:p:211-235. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: http://www.blackwellpublishing.com/journal.asp?ref=0303-6898 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.