Simulation of Stationary Gaussian Processes in [0, 1]<sup><i>d</i></sup>

Wood, Andrew; Grace, Clare

doi:10.1080/10618600.1994.10474655

Cited by 223 publications

(222 citation statements)

References 29 publications

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“…Thus, we have included outliers which are not far away from the rest of the curves in terms of any distance but differ in shape. We use a family of models to generate shape outliers which was introduced in López-Pintado and Romo (2009) based on the covariance kernels presented in Wood and Chan (1994). More concretely, the curves are generated from a Gaussian process with covariance kernel γ (s, t) = k exp{−c|t − s| µ }, with s, t ∈ [0, 1], and k, c, µ > 0.…”

Section: Simulation Resultsmentioning

confidence: 99%

A half-region depth for functional data

López‐Pintado

Romo

2011

Computational Statistics & Data Analysis

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a b s t r a c tA new definition of depth for functional observations is introduced based on the notion of ''half-region'' determined by a curve. The half-region depth provides a simple and natural criterion to measure the centrality of a function within a sample of curves. It has computational advantages relative to other concepts of depth previously proposed in the literature which makes it applicable to the analysis of high-dimensional data. Based on this depth a sample of curves can be ordered from the center-outward and order statistics can be defined. The properties of the half-region depth, such as consistency and uniform convergence, are established. A simulation study shows the robustness of this new definition of depth when the curves are contaminated. Finally, real data examples are analyzed.

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Section: Simulation Resultsmentioning

confidence: 99%

A half-region depth for functional data

López‐Pintado

Romo

2011

Computational Statistics & Data Analysis

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“…The sample autocorrelation sequence shown by the circles in both plots of Figure 4 is the average of all 10,000 individual sequences. While there is a small systematic difference between the average sample autocorrelation sequences generated by the two models individually, the variation in the individual sequences is quite small (generally less than the diameter of the circles), particularly when compared to the uncertainty due to the finite sample size N. To assess this latter source of uncertainty, we generated 25,000 realizations of length N from both an autoregressive process and a fractionally differenced process (with values of f and d again dictated by estimates from the SCICEX 97 profile) using an appropriate exact simulation procedure [Kay, 1981;Davies and Harte, 1987;Wood and Chan, 1994;Dietrich and Newsam, 1997;Gneiting, 2000;Craigmile, 2003]. We computedr d for each realization and then formed the sample average of ther d 's to estimate E{r d }, which are displayed as thick curves in Figure 4.…”

Section: Appendix A: Three Models For Scicex Draft Profilesmentioning

confidence: 99%

The variance of mean sea‐ice thickness: Effect of long‐range dependence

Percival¹,

Rothrock²,

Thorndike³

et al. 2008

J. Geophys. Res.

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[1] Measured sea-ice draft exhibits variations on all scales. We regard draft profiles up to several hundred kilometers in length as being drawn from a stationary stochastic process. We focus on the estimation of the mean draft H of the process. This elementary statistic is typically computed from a profile segment of length L and has some uncertainty, or sampling error, that is quantified by its variance s L 2 . How efficiently can the variance of H be reduced by the use of more data, that is, by increasing L? Three properties of the data indicate the need for a non-standard statistical model: the variance s 2 L of H falls off more slowly than L À1 ; the autocorrelation sequence does not fall rapidly to zero; and the spectrum does not flatten off with decreasing wave number. These indicate that ice draft exhibits, as a fundamental geometric property, 'long-range dependence.' One good model for this dependence is a fractionally differenced process, whose variance s L 2 is proportional to L À1+2d . From submarine ice draft data in the Arctic Ocean, we find d = 0.27. Mean draft estimated from a 50-km sample has a sample standard deviation of 0.29 m; for 200 km, it is 0.21 m. Tabulated values provide the sample standard deviation s L for various values of L for samples both in a straight line and in a rosette or spoke pattern, allowing for the efficient design of observational programs to measure draft to a desired accuracy.

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“…Under this form, the problem of characterising numerically stationary covariances is the same without stationarity. Nevertheless it might be possible to take advantage of the special Toeplitz form of stationary field covariances to make efficient computations, for instance by imbedding it in a circulant matrix, as it has been made in [21,2]. For the same reasons, the numerical complexity of the problem reduces since the dimension of the space where live stationary covariances is smaller, but the combinatorial symmetry is lost, thus there is no gain on the theoretical point of view.…”

Section: Theorem 28 a Function α On H Is The Reduced Covariance Of mentioning

confidence: 99%

Realisability conditions for second-order marginals of biphased media

Lachièze-Rey

2014

Random Struct. Alg.

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This paper concerns the second order marginals of biphased random media. We give discriminating necessary conditions for a bivariate function to be such a valid marginal, and illustrate our study with two practical applications: (1) the spherical variograms are valid indicator variograms if and only if they are multiplied by a sufficiently small constant, which upper bound is estimated, and (2) not every covariance/indicator variogram can be obtained with a Gaussian level set. The theoretical results backing this study are contained in a companion paper.

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Simulation of Stationary Gaussian Processes in [0, 1]^d

Cited by 223 publications

References 29 publications

A half-region depth for functional data

A half-region depth for functional data

The variance of mean sea‐ice thickness: Effect of long‐range dependence

Realisability conditions for second-order marginals of biphased media

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Simulation of Stationary Gaussian Processes in [0, 1]d

Cited by 223 publications

References 29 publications

A half-region depth for functional data

A half-region depth for functional data

The variance of mean sea‐ice thickness: Effect of long‐range dependence

Realisability conditions for second-order marginals of biphased media

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Simulation of Stationary Gaussian Processes in [0, 1]^d