{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:20:40Z","timestamp":1760145640687,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,11]],"date-time":"2024-08-11T00:00:00Z","timestamp":1723334400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico\u2014CNPq"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Many techniques have been proposed to model space-varying observation processes with a nonstationary spatial covariance structure and\/or anisotropy, usually on a geostatistical framework. Nevertheless, there is an increasing interest in point process applications, and methodologies that take nonstationarity into account are welcomed. In this sense, this work proposes an extension of a class of spatial Cox process using spatial deformation. The proposed method enables the deformation behavior to be data-driven, through a multivariate latent Gaussian process. Inference leads to intractable posterior distributions that are approximated via MCMC. The convergence of algorithms based on the Metropolis\u2013Hastings steps proved to be slow, and the computational efficiency of the Bayesian updating scheme was improved by adopting Hamiltonian Monte Carlo (HMC) methods. Our proposal was also compared against an alternative anisotropic formulation. Studies based on synthetic data provided empirical evidence of the benefit brought by the adoption of nonstationarity through our anisotropic structure. A real data application was conducted on the spatial spread of the Spodoptera frugiperda pest in a corn-producing agricultural area in southern Brazil. Once again, the proposed method demonstrated its benefit over alternatives.<\/jats:p>","DOI":"10.3390\/e26080678","type":"journal-article","created":{"date-parts":[[2024,8,12]],"date-time":"2024-08-12T11:23:46Z","timestamp":1723461826000},"page":"678","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Bayesian Modeling for Nonstationary Spatial Point Process via Spatial Deformations"],"prefix":"10.3390","volume":"26","author":[{"given":"Dani","family":"Gamerman","sequence":"first","affiliation":[{"name":"DME-Instituto de Matem\u00e1tica, Universidade Federal do Rio de Janeiro, Rio de Janeiro 21941-909, RJ, Brazil"}]},{"given":"Marcel de Souza Borges","family":"Quintana","sequence":"additional","affiliation":[{"name":"DME-Instituto de Matem\u00e1tica, Universidade Federal do Rio de Janeiro, Rio de Janeiro 21941-909, RJ, Brazil"},{"name":"Instituto Nacional de Infectologia Evandro Chagas-FIOCRUZ, Rio de Janeiro 21040-360, RJ, Brazil"}]},{"given":"Mariane Branco","family":"Alves","sequence":"additional","affiliation":[{"name":"DME-Instituto de Matem\u00e1tica, Universidade Federal do Rio de Janeiro, Rio de Janeiro 21941-909, RJ, Brazil"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Diggle, P. (2013). Statistical Analysis of Spatial and Spatiotemporal Point Patterns, Taylor & Francis Inc.","DOI":"10.1201\/b15326"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Illian, J., Penttinen, A., Stoyan, H., and Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns (Statistics in Practice), Wiley-Interscience.","DOI":"10.1002\/9780470725160"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1016\/j.spasta.2017.05.003","article-title":"A three-dimensional anisotropic point process characterization for pharmaceutical coatings","volume":"22","author":"Rajala","year":"2017","journal-title":"Spat. Stat."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1007\/s11009-013-9358-3","article-title":"Estimating Second-Order Characteristics of Inhomogeneous Spatio-Temporal Point Processes","volume":"16","author":"Gabriel","year":"2014","journal-title":"Methodol. Comput. Appl. Probab."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1111\/sjos.12185","article-title":"Hidden Second-order Stationary Spatial Point Processes","volume":"43","author":"Hahn","year":"2016","journal-title":"Scand. J. Stat."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1111\/jmi.13142","article-title":"Pairwise interaction Markov model for 3D epidermal nerve fibre endings","volume":"288","author":"Konstantinou","year":"2022","journal-title":"J. Microsc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/j.spasta.2015.12.005","article-title":"Estimating geometric anisotropy in spatial point patterns","volume":"15","author":"Rajala","year":"2016","journal-title":"Spat. Stat."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"100456","DOI":"10.1016\/j.spasta.2020.100456","article-title":"Second order analysis of geometric anisotropic point processes revisited","volume":"38","author":"Sormani","year":"2020","journal-title":"Spat. Stat."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1420","DOI":"10.1111\/sjos.12640","article-title":"Multivariate geometric anisotropic Cox processes","volume":"50","author":"Martin","year":"2023","journal-title":"Scand. J. Stat."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"100728","DOI":"10.1016\/j.spasta.2023.100728","article-title":"Flexible spatio-temporal Hawkes process models for earthquake occurrences","volume":"54","author":"Kwon","year":"2023","journal-title":"Spat. Stat."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1093\/jrsssc\/qlad013","article-title":"Non-stationary spatio-temporal point process modeling for high-resolution COVID-19 data","volume":"72","author":"Dong","year":"2023","journal-title":"J. R. Stat. Soc. Ser. C Appl. Stat."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1239\/jap\/1409932670","article-title":"Fractional Poisson Process: Long-Range Dependence and Applications in Ruin Theory","volume":"51","author":"Biard","year":"2014","journal-title":"J. Appl. Probab."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1111\/cgf.13111","article-title":"General Point Sampling with Adaptive Density and Correlations","volume":"36","author":"Roveri","year":"2017","journal-title":"Comput. Graph. Forum"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Higdon, D.M., Swall, J., and Kern, J.C. (1999). Non-stationary spatial modeling. Bayesian Statistics 6, Proceedings of the Sixth Valencia International Meeting, Valencia, Spain, 6\u201310 June 1998, Oxford University Press.","DOI":"10.1093\/oso\/9780198504856.003.0036"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1002\/env.2336","article-title":"Regression-based covariance functions for nonstationary spatial modeling","volume":"26","author":"Risser","year":"2015","journal-title":"Environmetrics"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1002\/env.473","article-title":"A high frequency kriging approach for nonstationary environmental processes","volume":"12","author":"Fuentes","year":"2001","journal-title":"Environmetrics"},{"key":"ref_17","unstructured":"Pintore, A., and Holmes, C.C. (2003). Constructing Localized Non-Stationary Covariance Functions Through the Frequency Domain, Imperial College. Technical Report."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Leuangthong, O., and Deutsch, C.V. (2005). A Statistical Technique for Modelling Non-stationary Spatial Processes. Geostatistics Banff 2004, Springer.","DOI":"10.1007\/978-1-4020-3610-1"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1080\/01621459.1992.10475181","article-title":"Nonparametric Estimation of Nonstationary Spatial Covariance Structure","volume":"87","author":"Sampson","year":"1992","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1111\/1467-9868.00413","article-title":"Bayesian inference for non-stationary spatial covariance structure via spatial deformations","volume":"65","author":"Schmidt","year":"2003","journal-title":"J. R. Stat. Society. Ser. B (Stat. Methodol.)"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1111\/1467-9469.00115","article-title":"Log Gaussian Cox Processes","volume":"25","author":"Moller","year":"1998","journal-title":"Scand. J. Stat."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"823","DOI":"10.1111\/1467-9868.00315","article-title":"Spatiotemporal prediction for log-Gaussian Cox processes","volume":"63","author":"Brix","year":"2001","journal-title":"J. R. Stat. Society. Ser. B (Stat. Methodol.)"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"981","DOI":"10.1002\/env.976","article-title":"Cox processes for estimating temporal variation in disease risk","volume":"20","author":"Paez","year":"2009","journal-title":"Environmetrics"},{"key":"ref_24","first-page":"9","article-title":"Tractable nonparametric Bayesian inference in Poisson processes with Gaussian process intensities","volume":"Volume 382","author":"Danyluk","year":"2009","journal-title":"Proceedings of the 26th Annual International Conference on Machine Learning, ICML 2009"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1111\/rssb.12237","article-title":"Exact Bayesian inference in spatio-temporal Cox processes driven by multivariate Gaussian processes","volume":"80","author":"Gamerman","year":"2018","journal-title":"J. R. Stat. Soc. Ser. B (Stat. Methodol.)"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/s10651-012-0215-2","article-title":"State space models with spatial deformation","volume":"20","author":"Morales","year":"2012","journal-title":"Environ. Ecol. Stat."},{"key":"ref_27","unstructured":"Daley, D.J., and Vere-Jones, D. (2006). An Introduction to the Theory of Point Processes, Springer."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Gupta, A.K., and Nagar, D.K. (2018). Matrix Variate Distributions, Taylor & Francis Group.","DOI":"10.1201\/9780203749289"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1002\/1099-095X(200103)12:2<161::AID-ENV452>3.0.CO;2-G","article-title":"Bayesian estimation of semi-parametric non-stationary spatial covariance structures","volume":"12","author":"Damian","year":"2001","journal-title":"Environmetrics"},{"key":"ref_30","unstructured":"Sampson, P.D., and Meiring, W. (2014). Nonstationary Spatial Covariance Modeling Through Spatial Deformation, Pan-American Advanced Study Institute on Spatio-Temporal Statistics."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1214\/088342307000000014","article-title":"A General Framework for the Parametrization of Hierarchical Models","volume":"22","author":"Papaspiliopoulos","year":"2007","journal-title":"Stat. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v076.i01","article-title":"Stan: A Probabilistic Programming Language","volume":"76","author":"Carpenter","year":"2017","journal-title":"J. Stat. Softw."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1111\/j.2517-6161.1990.tb01780.x","article-title":"The Conditional Predictive Ordinate for the Normal Distribution","volume":"52","author":"Pettit","year":"1990","journal-title":"J. R. Stat. Society. Ser. B (Methodol.)"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1111\/1467-9868.00353","article-title":"Bayesian measures of model complexity and fit","volume":"64","author":"Spiegelhalter","year":"2002","journal-title":"J. R. Stat. Soc. Ser. B (Stat. Methodol.)"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1080\/01621459.1972.10481224","article-title":"A Decision-Theoretic Approach to Interval Estimation","volume":"67","author":"Winkler","year":"1972","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_36","unstructured":"Quintana, M.S.B. (2022). Bayesian Modeling for Spatial Point Process with Nonstationary Covariance Structure via Spatial Deformations. [Unpublished D.Sc. Thesis, Graduate Program of Statistics, Universidade Federal do Rio de Janeiro]."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"e1003","DOI":"10.5424\/sjar\/2018161-11916","article-title":"Statistical methods for identifying anisotropy in the Spodoptera frugiperda spatial distribution","volume":"16","author":"Nava","year":"2018","journal-title":"Span. J. Agric. Res."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1111\/j.1541-0420.2005.00436.x","article-title":"Assessing Isotropy for Spatial Point Processes","volume":"62","author":"Guan","year":"2005","journal-title":"Biometrics"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1080\/00949655.2013.838565","article-title":"Multiresolution analysis of linearly oriented spatial point patterns","volume":"85","author":"Mateu","year":"2013","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Banerjee, S., Carlin, B.P., and Gelfand, A.E. (2014). Hierarchical Modeling and Analysis for Spatial Data, Chapman and Hall\/CRC.","DOI":"10.1201\/b17115"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/8\/678\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:34:52Z","timestamp":1760110492000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/8\/678"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,11]]},"references-count":40,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2024,8]]}},"alternative-id":["e26080678"],"URL":"https:\/\/doi.org\/10.3390\/e26080678","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2024,8,11]]}}}