Radial Positive Definite Functions Generated by Euclid's Hat

Gneiting, Tilmann

doi:10.1006/jmva.1998.1800

Cited by 71 publications

(91 citation statements)

References 22 publications

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“…(4.10)]). Other compactly supported covariance functions can be found in Gneiting [23][24][25], Wendland [53,54], and Wu [56]. These functions are parameterized by their support length and sill (i.e.…”

Section: Pseudo-optimal Tapermentioning

confidence: 98%

Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants

Furrer

Bengtsson²

2007

Journal of Multivariate Analysis

345

332

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This work studies the effects of sampling variability in Monte Carlo-based methods to estimate very highdimensional systems. Recent focus in the geosciences has been on representing the atmospheric state using a probability density function, and, for extremely high-dimensional systems, various sample-based Kalman filter techniques have been developed to address the problem of real-time assimilation of system information and observations. As the employed sample sizes are typically several orders of magnitude smaller than the system dimension, such sampling techniques inevitably induce considerable variability into the state estimate, primarily through prior and posterior sample covariance matrices. In this article, we quantify this variability with mean squared error measures for two Monte Carlo-based Kalman filter variants: the ensemble Kalman filter and the ensemble square-root Kalman filter. Expressions of the error measures are derived under weak assumptions and show that sample sizes need to grow proportionally to the square of the system dimension for bounded error growth. To reduce necessary ensemble size requirements and to address rank-deficient sample covariances, covariance-shrinking (tapering) based on the Schur product of the prior sample covariance and a positive definite function is demonstrated to be a simple, computationally feasible, and very effective technique. Rules for obtaining optimal taper functions for both stationary as well as non-stationary covariances are given, and optimal taper lengths are given in terms of the ensemble size and practical range of the forecast covariance. Results are also presented for optimal covariance inflation. The theory is verified and illustrated with extensive simulations.

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Section: Pseudo-optimal Tapermentioning

confidence: 98%

Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants

Furrer

Bengtsson²

2007

Journal of Multivariate Analysis

345

332

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“…These are generated by convolving the characteristic function of a disc in R 2 , and of a ball in R 3 , with themselves. The Euclid hat functions, see Wu [17] and Gneiting [7], are a continuation of this method of construction beyond R 3 . Such a self convolution will automatically have a nonnegative Fourier transform.…”

Section: Proof Of Theorem 41mentioning

confidence: 99%

Dimension hopping and families of strictly positive definite zonal basis functions on spheres

Beatson

Castell

2017

Journal of Approximation Theory

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Positive definite functions of compact support are widely used for radial basis function approximation as well as for estimation of spatial processes in geostatistics. Several constructions of such functions for R d are based upon recurrence operators. These map functions of such type in a given space dimension onto similar ones in a space of lower or higher dimension. We provide analogs of these dimension hopping operators for positive definite, and strictly positive definite, zonal (radial) functions on the sphere. These operators are then used to provide new families of strictly positive definite functions with local support on the sphere.

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“…It is worth indicating that the multiple monotonicity on IR + is a much more standard concept. Williamson (1956) has investigated in detail the properties of such functions when n ≥ 1 is an integer (as here) or even any real; see also Lévy (1962) and Gneiting (1999). In probability, n-monotonicity of continuous distributions corresponds to the so-called beta(1, n)-unimodality, defined for n real ≥ 0 (Bertin et al (1997), page 72).…”

Section: Introductionmentioning

confidence: 99%

Discrete Schur-constant models

Castañer

Bielsa

Lefèvre

et al. 2015

Journal of Multivariate Analysis

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This paper introduces a class of Schur-constant survival models, of dimension n, for arithmetic non-negative random variables. Such a model is defined through a univariate survival function that is shown to be n-monotone. Two general representations are obtained, by conditioning on the sum of the n variables or through a doubly mixed multinomial distribution. Several other properties including correlation measures are derived. Three processes in insurance theory are discussed for which the claim interarrival periods form a Schur-constant model.

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Radial Positive Definite Functions Generated by Euclid's Hat

Cited by 71 publications

References 22 publications

Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants

Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants

Dimension hopping and families of strictly positive definite zonal basis functions on spheres

Discrete Schur-constant models

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