Non-stationary Gaussian models with physical barriers

Bakka, Haakon; Vanhatalo, Jarno; Illian, Janine B.; Simpson, Daniel; Rue, Haåvard

doi:10.1016/j.spasta.2019.01.002

Cited by 101 publications

(91 citation statements)

References 27 publications

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“…Further, stopping the mesh at the coastline imposes the Neumann boundary conditions, also leading to unrealistic models. Bakka, Vanhatalo, Illian, Simpson, and Rue (2018) develop the Barrier model, defining the operator…”

Section: The Barrier Modelmentioning

confidence: 99%

“…This conversion can be automated Simulation of anisotropic field from the model by . In the west part of the plot there is a strong horisontal dependence, while in the east part, there is a strong vertical dependence 4 Example correlation surface for the barrier model by Bakka et al (2018). The gray region acts as a physical barrier to spatial correlation, forcing the model to smooth around this barrier by using the coordinate reference system (CRS) specification that usually accompany large-scale spatial data, that specify which planar projection was used for the planar coordinates, such as UTM or longitude and latitude.…”

Section: Spdes On Manifoldsmentioning

confidence: 99%

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Spatial modeling with R‐INLA: A review

Bakka

Rue

Fuglstad

et al. 2018

WIREs Computational Stats

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278

246

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Coming up with Bayesian models for spatial data is easy, but performing inference with them can be challenging. Writing fast inference code for a complex spatial model with realistically‐sized datasets from scratch is time‐consuming, and if changes are made to the model, there is little guarantee that the code performs well. The key advantages of R‐INLA are the ease with which complex models can be created and modified, without the need to write complex code, and the speed at which inference can be done even for spatial problems with hundreds of thousands of observations. R‐INLA handles latent Gaussian models, where fixed effects, structured and unstructured Gaussian random effects are combined linearly in a linear predictor, and the elements of the linear predictor are observed through one or more likelihoods. The structured random effects can be both standard areal model such as the Besag and the BYM models, and geostatistical models from a subset of the Matérn Gaussian random fields. In this review, we discuss the large success of spatial modeling with R‐INLA and the types of spatial models that can be fitted, we give an overview of recent developments for areal models, and we give an overview of the stochastic partial differential equation (SPDE) approach and some of the ways it can be extended beyond the assumptions of isotropy and separability. In particular, we describe how slight changes to the SPDE approach leads to straight‐forward approaches for nonstationary spatial models and nonseparable space–time models. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory Statistical Models > Bayesian Models Data: Types and Structure > Massive Data

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Section: The Barrier Modelmentioning

confidence: 99%

Section: Spdes On Manifoldsmentioning

confidence: 99%

Spatial modeling with R‐INLA: A review

Bakka

Rue

Fuglstad

et al. 2018

WIREs Computational Stats

Self Cite

278

246

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“…Should one need to account for the effect of a "rigid" barrier (e.g., a river, a major divide, a main lithologic or tectonic discontinuity) on landslide intensity or susceptibility, two solutions are possible. Bakka et al (2019) have developed a model for incorporating physical barriers, which can be fitted using R-INLA, although their method relies on a different type of spatial effect as the one exploited in this work. Alternatively, one can remove manually the links between adjacent SUs in the adjacency matrix ( Figure 2).…”

Section: A New Landslide Predictive Modelling Frameworkmentioning

confidence: 99%

Space-Time Landslide Predictive Modelling

Lombardo¹,

Opitz²,

Ardizzone³

et al. 2020

Preprint

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Landslides are nearly ubiquitous phenomena and pose severe threats to people, properties, and the environment in many areas. Investigators have for long attempted to estimate landslide hazard in an effort to determine where, when (or how frequently), and how large (or how destructive) landslides are expected to be in an area. This information may prove useful to design landslide mitigation strategies, and to reduce landslide risk and societal and economic losses. In the geomorphology literature, most of the attempts at predicting the occurrence of populations of landslides by adopting statistical approaches are based on the empirical observation that landslides occur as a result of multiple, interacting, conditioning and triggering factors. Based on this observation, and under the assumption that at the spatial and temporal scales of our investigation individual landslides are discrete "point" events in the landscape, we propose a novel Bayesian modelling framework for the prediction of the spatio-temporal occurrence of landslides of the slide type caused by weather triggers. We build our modelling effort on a Log-Gaussian Cox Process (LGCP) by assuming that individual landslides in an area are the result of a point process described by an unknown intensity function. The modelling framework has two stochastic components: (i) a Poisson component, which models the observed (random) landslide count in each terrain subdivision for a given landslide "intensity", i.e., the expected number of landslides per terrain subdivision (which may be transformed into a corresponding landslide "susceptibility"); and (ii) a Gaussian component, used to account for the spatial distribution of the local environmental conditions that influence landslide occurrence, and for the spatio-temporal distribution of "unobserved" latent environmental controls on landslide occurrence. We tested our prediction framework in the Collazzone area, Umbria, Central Italy, for which a detailed multi-temporal landslide inventory covering the period from before 1941 to 2014 is available together with lithological and bedding data. We subdivided the 79 km 2 area into 889 slope units (SUs). In each SU, we computed the percentage of 16 morphometric covariates derived from a 10 m × 10 m digital elevation model, and 13 lithological and bedding attitude covariates obtained from a 1:10,000 scale geological map. We further counted how many of the 3,379 landslides in the multi-temporal inventory affect each SU and grouped them into six periods. We used this complex space-time information to prepare five models of increasing complexity. Our "baseline" model (Mod1) carries the spatial information only through the covariates mentioned above. It does not include any additional information about the spatial and temporal structure of the data, and it is therefore equivalent to a "traditional" landslide susceptibility model. The second model (Mod2) is analogous, but it allows for time-interval-specific regression constants. Our next two models are more complex. In partic...

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“…on the differential operator L are satisfied, e.g., by the Matérn operator L = κ 2 − ∆, in which case the condition β > d/4 on the fractional exponent in (1.2) corresponds to a positive smoothness parameter ν, i.e., to a nondegenerate field. Moreover, the equation (1.2) as considered in our work includes several previously proposed non-fractional non-stationary models as special cases, such as the non-stationary Matérn models by Lindgren et al (2011), the models with locally varying anisotropy by Fuglstad et al (2015), and the barrier models by Bakka et al (2019). Thus, Proposition 3.1 shows existence and uniqueness of the fractional versions of all these models, which can be treated in practice by using the results of the following sections.…”

Section: )mentioning

confidence: 99%

The Rational SPDE Approach for Gaussian Random Fields With General Smoothness

Bolin

Kirchner

2019

Journal of Computational and Graphical Statistics

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A popular approach for modeling and inference in spatial statistics is to represent Gaussian random fields as solutions to stochastic partial differential equations (SPDEs) of the form L β u = W, where W is Gaussian white noise, L is a second-order differential operator, and β > 0 is a parameter that determines the smoothness of u. However, this approach has been limited to the case 2β ∈ N, which excludes several important models and makes it necessary to keep β fixed during inference.We propose a new method, the rational SPDE approach, which in spatial dimension d ∈ N is applicable for any β > d/4, and thus remedies the mentioned limitation. The presented scheme combines a finite element discretization with a rational approximation of the function x −β to approximate u. For the resulting approximation, an explicit rate of convergence to u in mean-square sense is derived. Furthermore, we show that our method has the same computational benefits as in the restricted case 2β ∈ N. Several numerical experiments and a statistical application are used to illustrate the accuracy of the method, and to show that it facilitates likelihood-based inference for all model parameters including β.

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Non-stationary Gaussian models with physical barriers

Cited by 101 publications

References 27 publications

Spatial modeling with R‐INLA: A review

Spatial modeling with R‐INLA: A review

Space-Time Landslide Predictive Modelling

The Rational SPDE Approach for Gaussian Random Fields With General Smoothness

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