Spatio-Temporal Covariance and Cross-Covariance Functions of the Great Circle Distance on a Sphere

Porcu, Emilio; Bevilacqua, Moreno; Genton, Marc G.

doi:10.1080/01621459.2015.1072541

Cited by 150 publications

(188 citation statements)

References 30 publications

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“…Mappings (6) and (7) are known as Exponential and Askey models, respectively. The former decreases exponentially to zero and it takes values less than 0.05 for θ > c, whereas the second is compactly supported, that is, it is identically equal to zero beyond the cut-off distance c. Explicit parametric conditions for the validity of the bivariate Exponential model are provided by Porcu et al (2016). On the other hand, Appendix B illustrates a construction principle that justify the use of bivariate Askey models on spheres.…”

Section: Parameterizationmentioning

confidence: 99%

“…In the univariate case (m = 1), Gneiting (2013) establishes that some classical covariances such as the Cauchy, Matérn, Askey and Spherical models, given in the classical literature in terms of Euclidean metrics, can be coupled with the geodesic distance under specific constraints for the parameters. Furthermore, Porcu et al (2016) propose several covariance models for space-time and multivariate RFs on spherical spatial domains.…”

Section: Skew-gaussian Rfs On the Unit Spherementioning

confidence: 99%

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Estimating covariance functions of multivariate skew-Gaussian random fields on the sphere

et al. 2017

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This paper considers a multivariate spatial random field, with each component having univariate marginal distributions of the skew-Gaussian type. We assume that the field is defined spatially on the unit sphere embedded in R 3 , allowing for modeling data available over large portions of planet Earth. This model admits explicit expressions for the marginal and cross covariances. However, the n-dimensional distributions of the field are difficult to evaluate, because it requires the sum of 2 n terms involving the cumulative and probability density functions of a n-dimensional Gaussian distribution. Since in this case inference based on the full likelihood is computationally unfeasible, we propose a composite likelihood approach based on pairs of spatial observations. This last being possible thanks to the fact that we have a closed form expression for the bivariate distribution. We illustrate the effectiveness of the method through simulation experiments and the analysis of a real data set of minimum and maximum surface air temperatures.

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Section: Parameterizationmentioning

confidence: 99%

Section: Skew-gaussian Rfs On the Unit Spherementioning

confidence: 99%

Estimating covariance functions of multivariate skew-Gaussian random fields on the sphere

et al. 2017

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“…We first define the Gneiting class of functions (Gneiting, ),

{scriptG}_{\dim} : {false[0, \infty false)}^{2} \to {double-struckR}_{+}

, as

{scriptG}_{\dim} false(x_{1}, x_{2} false) : = \frac{1}{ψ {false(x_{2} false)}^{\dim false/ 2}} φ ((), \frac{x_{1}}{ψ false(x_{2} false)}), x_{1}, x_{2} \geq 0,

where dim is the dimension of the space over which x 1 is computed and x 1 and x 2 are placeholders for h 2 , u 2 , or θ . A similar class is proposed by Porcu et al (), which we call the modified Gneiting class,

scriptP : false[0, \infty false) \times false[0, π false] \to double-struckR

, and is given by

scriptP false(x_{1}, x_{2} false) : = \frac{1}{ψ_{false[0, π false]} {false(x_{2} false)}^{1 false/ 2}} φ ((), \frac{x_{1}}{ψ_{false[0, π false]} false(x_{2} false)}), x_{1} \geq 0, x_{2} \in false[0, π false] .

For both classes, the function

φ : false[0, \infty false) \to {double-struckR}_{+}

is completely monotonic; that is, φ is infinitely differentiable on (0, ∞ ), satisfying ( −1) n φ ( n ) ( t ) ≥ 0,

n \in double-struckN

. The function ψ is strictly positive and has a completely monotonic derivative.…”

Section: Covariance Modelsmentioning

confidence: 99%

Nonseparable covariance models on circles cross time: A study of Mexico City ozone

White

Porcu

2019

Environmetrics

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We propose a continuous spatiotemporal model for Mexico City ozone levels that account for distinct daily seasonality, as well as variation across the city and over the peak ozone season (April and May) of 2017. To account for these patterns, we use covariance models over space, circles, and time. We review relevant existing covariance models and develop new classes of nonseparable covariance models appropriate for seasonal data collected at many locations. We compare the predictive performance of a variety of models that utilize various nonseparable covariance functions. We use the best model to predict hourly ozone levels at unmonitored locations in April and May to infer compliance with Mexican air quality standards and to estimate the respiratory health risk associated with ozone exposure. We find that predicted compliance with air quality standards and estimated respiratory health risk vary greatly over space and time. In some regions, we predict exceedance of national standards for more than a third of the hours in April and May, and on many days, we predict that nearly all of Mexico City exceeds nationally legislated ozone thresholds at least once. In southern Mexico City, we estimate the respiratory risk for ozone to be 55% higher, on average, than the annual average risk and as much at 170% higher on some days.

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“…For stochastic processes on a sphere, the reader is referred to Jones (), Marinucci and Peccati () and the thorough reviews in Gneiting () and Jeong et al (). For space–time stochastic processes on the sphere, we refer the reader to the more recent approaches in Porcu et al (), Berg and Porcu () and Jeong and Jun (). Generalisations to multivariate space–time processes have been considered in Alegria et al ().…”

Section: Introductionmentioning

confidence: 99%

Modeling Temporally Evolving and Spatially Globally Dependent Data

Porcu

Alegría

Furrer

2018

Int Statistical Rev

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Summary The last decades have seen an unprecedented increase in the availability of data sets that are inherently global and temporally evolving, from remotely sensed networks to climate model ensembles. This paper provides an overview of statistical modeling techniques for space–time processes, where space is the sphere representing our planet. In particular, we make a distintion between (a) second order‐based approaches and (b) practical approaches to modeling temporally evolving global processes. The former approaches are based on the specification of a class of space–time covariance functions, with space being the two‐dimensional sphere. The latter are based on explicit description of the dynamics of the space–time process, that is, by specifying its evolution as a function of its past history with added spatially dependent noise. We focus primarily on approach (a), for which the literature has been sparse. We provide new models of space–time covariance functions for random fields defined on spheres cross time. Practical approaches (b) are also discussed, with special emphasis on models built directly on the sphere, without projecting spherical coordinates onto the plane. We present a case study focused on the analysis of air pollution from the 2015 wildfires in Equatorial Asia, an event that was classified as the year's worst environmental disaster. The paper finishes with a list of the main theoretical and applied research problems in the area, where we expect the statistical community to engage over the next decade.

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Spatio-Temporal Covariance and Cross-Covariance Functions of the Great Circle Distance on a Sphere

Cited by 150 publications

References 30 publications

Estimating covariance functions of multivariate skew-Gaussian random fields on the sphere

Estimating covariance functions of multivariate skew-Gaussian random fields on the sphere

Nonseparable covariance models on circles cross time: A study of Mexico City ozone

Modeling Temporally Evolving and Spatially Globally Dependent Data

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