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Register a new userTowards a Complete Picture of Stationary Covariance Functions on Spheres Cross Time
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Added by
Philip White
Publication date
2018
fields
Mathematical Statistics
and research's language is
English
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With the advent of wide-spread global and continental-scale spatiotemporal datasets, increased attention has been given to covariance functions on spheres over time. This paper provides results for stationary covariance functions of random fields defined over $d$-dimensional spheres cross time. Specifically, we provide a bridge between the characterization in cite{berg-porcu} for covariance functions on spheres cross time and Gneitings lemma citep{gneiting2002} that deals with planar surfaces. We then prove that there is a valid class of covariance functions similar in form to the Gneiting class of space-time covariance functions citep{gneiting2002} that replaces the squared Euclidean distance with the great circle distance. Notably, the provided class is shown to be positive definite on every $d$-dimensional sphere cross time, while the Gneiting class is positive definite over $R^d times R$ for fixed $d$ only. In this context, we illustrate the value of our adapted Gneiting class by comparing examples from this class to currently established nonseparable covariance classes using out-of-sample predictive criteria. These comparisons are carried out on two climate reanalysis datasets from the National Centers for Environmental Prediction and National Center for Atmospheric Research. For these datasets, we show that examples from our covariance class have better predictive performance than competing models.
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Multivariate space-time data are increasingly available in various scientific disciplines. When analyzing these data, one of the key issues is to describe the multivariate space-time dependencies. Under the Gaussian framework, one needs to propose relevant models for multivariate space-time covariance functions, i.e. matrix-valued mappings with the additional requirement of non-negative definiteness. We propose a flexible parametric class of cross-covariance functions for multivariate space-time Gaussian random fields. Space-time components belong to the (univariate) Gneiting class of space-time covariance functions, with Matern or Cauchy covariance functions in the spatial margins. The smoothness and scale parameters can be different for each variable. We provide sufficient conditions for positive definiteness. A simulation study shows that the parameters of this model can be efficiently estimated using weighted pairwise likelihood, which belongs to the class of composite likelihood methods. We then illustrate the model on a French dataset of weather variables.
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The Mat{e}rn family of isotropic covariance functions has been central to the theoretical development and application of statistical models for geospatial data. For global data defined over the whole sphere representing planet Earth, the natural distance between any two locations is the great circle distance. In this setting, the Mat{e}rn family of covariance functions has a restriction on the smoothness parameter, making it an unappealing choice to model smooth data. Finding a suitable analogue for modelling data on the sphere is still an open problem. This paper proposes a new family of isotropic covariance functions for random fields defined over the sphere. The proposed family has a parameter that indexes the mean square differentiability of the corresponding Gaussian field, and allows for any admissible range of fractal dimension. Our simulation study mimics the fixed domain asymptotic setting, which is the most natural regime for sampling on a closed and bounded set. As expected, our results support the analogous results (under the same asymptotic scheme) for planar processes that not all parameters can be estimated consistently. We apply the proposed model to a dataset of precipitable water content over a large portion of the Earth, and show that the model gives more precise predictions of the underlying process at unsampled locations than does the Mat{e}rn model using chordal distances.
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