On the non-reducibility of non-stationary correlation functions to stationary ones under a class of mean-operator transformations

Porcu, Emilio; Matkowski, Janusz; Mateu, Jorge

doi:10.1007/s00477-009-0347-6

Cited by 8 publications

(3 citation statements)

References 30 publications

(28 reference statements)

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“…The dimension q in Proposition 1 is important, in that EVecchia approximations become more challenging as the input dimension increases. There have been studies on necessary and sufficient conditions for reducibility and how large q must be (e.g., Perrin and Senoussi, 2000;Curriero, 2006), and sufficient conditions for related concepts have been identified (e.g., Porcu et al, 2010;Perrin and Meiring, 2003;Perrin and Schlather, 2007). In some settings, theoretical guarantees depending on q on the performance of CVecchia for reducible GPs can be provided using recent results for isotropic GPs (Schäfer et al, 2021).…”

Section: Properties Of Cvecchia In the Special Case Of Reducible Gpsmentioning

confidence: 99%

Correlation-based sparse inverse Cholesky factorization for fast Gaussian-process inference

Myeongjong¹,

Katzfuß²

2021

Preprint

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Gaussian processes are widely used as priors for unknown functions in statistics and machine learning. To achieve computationally feasible inference for large datasets, a popular approach is the Vecchia approximation, which is an ordered conditional approximation of the data vector that implies a sparse Cholesky factor of the precision matrix. The ordering and sparsity pattern are typically determined based on Euclidean distance of the inputs or locations corresponding to the data points. Here, we propose instead to use a correlation-based distance metric, which implicitly applies the Vecchia approximation in a suitable transformed input space. The correlation-based algorithm can be carried out in quasilinear time in the size of the dataset, and so it can be applied even for iterative inference on unknown parameters in the correlation structure. The Euclidean-and correlation-based Vecchia approximations are equivalent for strictly decreasing isotropic covariances, but the correlation-based approach has two advantages for more complex settings: It can result in more accurate approximations, and it offers a simple, automatic strategy that can be applied to any covariance, even when Euclidean distance is not applicable. We demonstrate these advantages in several settings, including anisotropic, nonstationary, multivariate, and spatio-temporal processes. We also illustrate our method on multivariate spatio-temporal temperature fields produced by a regional climate model.

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Section: Properties Of Cvecchia In the Special Case Of Reducible Gpsmentioning

confidence: 99%

Correlation-based sparse inverse Cholesky factorization for fast Gaussian-process inference

Myeongjong¹,

Katzfuß²

2021

Preprint

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“…Meiring and Perrin, Perrin and Senoussi, Genton and Perrin, and Porcu et al gave some theoretical properties about uniqueness and richness of this class of nonstationary covariance functions. In particular, this class does not cover all nonstationary smooth covariances, but the subclass of smooth spatial covariances that can be written in this fashion appears relatively large.…”

Section: Covariance Structure For Spatial Processesmentioning

confidence: 99%

Covariance structure of spatial and spatiotemporal processes

Guttorp

Schmidt

2013

WIREs Computational Stats

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An important aspect of statistical modeling of spatial or spatiotemporal data is to determine the covariance function. It is a key part of spatial prediction (kriging). The classical geostatistical approach uses an assumption of isotropy, which yields circular isocorrelation curves. However, this is inappropriate for many applications, and several nonstationary approaches have been developed. Adding the temporal aspect, there is often interaction between time and space, requiring classes of nonseparable covariance structures.where δ is spatio-temporal Gaussian white noise with variance σ 2 . The solution Z(s,t) is a stationary Gaussian process whose spatial margins have Matérn covariance structure, and whose space-time Volume 5,

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“…Maximum likelihood and Bayesian variants of this approach have been developed by Mardia and Goodall [9], Smith [10], Damian et al [11], Schmidt and O'Hagan [12]. Perrin and Meiring [13], Perrin and Senoussi [14], Perrin and Meiring [15], Genton and Perrin [16], Porcu et al [17] established some theoretical properties about uniqueness and richness of this class of non-stationary models. Some adaptations have been proposed recently by Castro Morales et al [18], Bornn et al [19], Schmidt et al [20], Vera et al [21,22].…”

Section: Introductionmentioning

confidence: 99%

Estimation of space deformation model for non-stationary random functions

2015

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Stationary Random Functions have been successfully applied in geostatistical applications for decades. In some instances, the assumption of a homogeneous spatial dependence structure across the entire domain of interest is unrealistic. A practical approach for modelling and estimating non-stationary spatial dependence structure is considered. This consists in transforming a non-stationary Random Function into a stationary and isotropic one via a bijective continuous deformation of the index space. So far, this approach has been successfully applied in the context of data from several independent realizations of a Random Function. In this work, we propose an approach for non-stationary geostatistical modelling using space deformation in the context of a single realization with possibly irregularly spaced data. The estimation method is based on a non-stationary variogram kernel estimator which serves as a dissimilarity measure between two locations in the geographical space. The proposed procedure combines aspects of kernel smoothing, weighted non-metric multi-dimensional scaling and thin-plate spline radial basis functions. On a simulated data, the method is able to retrieve the true deformation. Performances are assessed on both synthetic and real datasets. It is shown in particular that our approach outperforms the stationary approach. Beyond the prediction, the proposed method can also serve as a tool for exploratory analysis of the non-stationarity.

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On the non-reducibility of non-stationary correlation functions to stationary ones under a class of mean-operator transformations

Cited by 8 publications

References 30 publications

Correlation-based sparse inverse Cholesky factorization for fast Gaussian-process inference

Correlation-based sparse inverse Cholesky factorization for fast Gaussian-process inference

Covariance structure of spatial and spatiotemporal processes

Estimation of space deformation model for non-stationary random functions

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