Random gaussian fields and systems of stochastic partial differential equations

Sílvia Isabel Belo  Guerra; Fernanda Cipriano; Isabel Natário

doi:10.1007/978-3-031-69622-0_1

Random gaussian fields and systems of stochastic partial differential equations

Sílvia Isabel Belo Guerra, Fernanda Cipriano, Isabel Natário

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

Abstract

In this work we consider certain systems of Stochastic Partial Differential Equations, that allow us to generate multivariate Gaussian random fields (GF),
. We consider a theoretical case were we have observations of a vector field that displays spatial dependency given by a GRF which is approximated by a Gaussian Markov random field (GMRF), applying the Finite Element Method. Considering a hierarchical model for the observations with a latent Gaussian model for x, under a Bayesian framework, we can obtain the posterior distribution of the GMRF. The main goal of this work is to explicitly present the calculations needed to obtain the posterior distributions for the multivariate case, as they are not gathered all together, neither fully detailed, in a single source in the literature. This can prove to be very useful for new future applications of this methodology.

Original language	English
Title of host publication	Statistical modeling and applications
Subtitle of host publication	Multivariate, heavy-tailed, skewed distributions and mixture modeling
Publisher	Springer
Pages	3-24
Volume	2
Edition	1
ISBN (Electronic)	978-3-031-69622-0
ISBN (Print)	978-3-031-69621-3, 978-3-031-69624-4
DOIs	https://doi.org/10.1007/978-3-031-69622-0_1
Publication status	Published - Dec 2024

Publication series

Name	Emerging topics in statistics and biostatistics
ISSN (Print)	2524-7735
ISSN (Electronic)	2524-7743

Access to Document

10.1007/978-3-031-69622-0_1Licence: Unspecified

Cite this

Guerra, S. I. B., Cipriano, F., & Natário, I. (2024). Random gaussian fields and systems of stochastic partial differential equations. In Statistical modeling and applications: Multivariate, heavy-tailed, skewed distributions and mixture modeling (1 ed., Vol. 2, pp. 3-24). (Emerging topics in statistics and biostatistics). Springer. https://doi.org/10.1007/978-3-031-69622-0_1

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title = "Random gaussian fields and systems of stochastic partial differential equations",

abstract = "In this work we consider certain systems of Stochastic Partial Differential Equations, that allow us to generate multivariate Gaussian random fields (GF), . We consider a theoretical case were we have observations of a vector field that displays spatial dependency given by a GRF which is approximated by a Gaussian Markov random field (GMRF), applying the Finite Element Method. Considering a hierarchical model for the observations with a latent Gaussian model for x, under a Bayesian framework, we can obtain the posterior distribution of the GMRF. The main goal of this work is to explicitly present the calculations needed to obtain the posterior distributions for the multivariate case, as they are not gathered all together, neither fully detailed, in a single source in the literature. This can prove to be very useful for new future applications of this methodology.",

author = "Guerra, {S{\'i}lvia Isabel Belo} and Fernanda Cipriano and Isabel Nat{\'a}rio",

note = "{\textcopyright} 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG This work is funded by national funds through the FCT—Funda{\c c}{\~a}o para a Ci{\^e}ncia e a Tecnologia, I.P., under the scope of the projects: UIDB/00297/2020 (https://doi.org/10.54499/UIDB/00297/2020) and UIDP/00297/2020 (https://doi.org/10.54499/UIDP/00297/2020) (Center for Mathematics and Applications). We would like to thank Doctor Ricardo Tom{\'e}, from Instituto Dom Luiz, for the wind data set: the Hersbach et al. (2023).",

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Random gaussian fields and systems of stochastic partial differential equations. / Guerra, Sílvia Isabel Belo ; Cipriano, Fernanda ; Natário, Isabel.
Statistical modeling and applications: Multivariate, heavy-tailed, skewed distributions and mixture modeling. Vol. 2 1. ed. Springer, 2024. p. 3-24 (Emerging topics in statistics and biostatistics).

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

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N2 - In this work we consider certain systems of Stochastic Partial Differential Equations, that allow us to generate multivariate Gaussian random fields (GF), . We consider a theoretical case were we have observations of a vector field that displays spatial dependency given by a GRF which is approximated by a Gaussian Markov random field (GMRF), applying the Finite Element Method. Considering a hierarchical model for the observations with a latent Gaussian model for x, under a Bayesian framework, we can obtain the posterior distribution of the GMRF. The main goal of this work is to explicitly present the calculations needed to obtain the posterior distributions for the multivariate case, as they are not gathered all together, neither fully detailed, in a single source in the literature. This can prove to be very useful for new future applications of this methodology.

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