A non-homogeneous Poisson process geostatistical model with spatial deformation
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A non-homogeneous Poisson process geostatistical model with spatial deformation Fidel Ernesto Castro Morales1
· Lorena Vicini2
Received: 20 November 2019 / Accepted: 28 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we propose a geostatistical model for the counting process using a nonhomogeneous Poisson model. This work aims to model the intensity function as the sum of two components: spatial and temporal. The spatial component is modeled using a Gaussian process in which the covariance structure is assumed to be anisotropic. Anisotropy is incorporated by applying a spatial deformation approach. The temporal component is modeled in such a way that its behavior concerning time has the structure of a Goel process. The inferences for the proposed model are obtained from a Bayesian perspective. The parameter estimation is obtained using Markov Chain Monte Carlo methods. The proposed model is adjusted to a set of real data, referring to the rain precipitation in 29 monitoring stations, distributed in the states of Maranhão and Piauí, in the northeast region of Brazil, in 31 years, from 01/01/1980 to 12/31/2010. The objective is to estimate the frequency of rain that exceeded a certain threshold. Keywords Non-homogeneous Poisson processes · Geostatistical data · Spatial deformation · Bayesian inference · Markov Chain Monte Carlo
1 Introduction Several counting processes have been highlighted in the literature, including counting processes with different formats for spatial data, some of which are modeled with a non-homogeneous Poisson process, see, for example, Lawson (2009), Cox (1955), and Moller et al. (1998). For counting processes with spatial components in the geostatistical context, we have the model proposed by Morales et al. (2017). In this work, the authors proposed a non-homogeneous Poisson geostatistical model, where the spatial component is modeled by a Gaussian process, where the spatial correlation structure
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Fidel Ernesto Castro Morales [email protected]
1
Universidade Federal do Rio Grande do Norte, Natal, Brazil
2
Universidade Federal de Santa Maria, Santa Maria, Brazil
123
F. E. C. Morales, L. Vicini
depends on a set of parameters and ensures that the resulting correlation matrix is positive definite. Besides, they established that the correlation between two locations in space depends only on the Euclidean distance between them; this type of process is called isotropic. But, the assumption of isotropy is often not satisfied with real data. For example, in some applications with environmental data, the spatial correlation structure can be affected by topographic characteristics, wind direction, and other climatic phenomena (Rodrigues et al. 2014; Ingebrigtsen et al. 2014). In this case, it is more appropriate to assume that the spatial correlation between two locations does not depend on the Euclidean distance, and this type of process is called anisotropic. Spatial deformation is a good alternative for modeling anisotropic processes. Thi
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