Gaussian Random Processes

Ибрагимов, И. А.; Rozanov, Yuriĭ A.

doi:10.1007/978-1-4612-6275-6

Cited by 311 publications

(269 citation statements)

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“…A major step towards a theoretically founded answer to the kernel misspecification issue was made in [74]: if K andK are compatible, then the approximation based onK will have the same asymptotic efficiency as the optimal approximation, and the relative deviation of the true expected squared approximation error from the one calculated under the false assumption thatK is correct is asymptotically negligible. A full explanation of the concept of compatibility is beyond the scope of this article, for details consider [76,37,77]. To compare with the statement above, we shall however give a sufficient condition for compatibility in the important special case where…”

Section: Interpolation With Misspecified Kernelsmentioning

confidence: 99%

Interpolation of spatial data – A stochastic or a deterministic problem?

2013

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Interpolation of spatial data is a very general mathematical problem with various applications. In geostatistics, it is assumed that the underlying structure of the data is a stochastic process which leads to an interpolation procedure known as kriging. This method is mathematically equivalent to kernel interpolation, a method used in numerical analysis for the same problem, but derived under completely different modelling assumptions. In this article we present the two approaches and discuss their modelling assumptions, notions of optimality and the different concepts to quantify the interpolation accuracy. Their relation is much closer than has been appreciated so far, and even results on convergence rates of kernel interpolants can be translated to the geostatistical framework. We sketch the different answers obtained in the two fields concerning the issue of kernel misspecification, present some methods for kernel selection, and discuss the scope of these methods with a data example from the computer experiments literature.

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Section: Interpolation With Misspecified Kernelsmentioning

confidence: 99%

Interpolation of spatial data – A stochastic or a deterministic problem?

2013

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“…The answer is affirmative, and one can prove this as a corollary to Theorem 0 by using the results in Chapters 4 and 5 of [3]. It also follows easily from the following theorem.…”

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confidence: 73%

“…In [3,Note 2,p. 190] this is stated for stationary Gaussian sequences, but the extension to general weakly stationary sequences is easy.…”

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confidence: 99%

A Sufficient Condition for Linear Growth of Variances in a Stationary Random Sequence

Bradley¹

1981

Proceedings of the American Mathematical Society

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“…It is known (4.31, Chapter III, [29]) that there is a function φ ∈ L 2 (R) with bounded support such…”

Section: A4 Proofs For Sectionmentioning

confidence: 99%

Optimal change point detection in Gaussian processes

Keshavarz

Scott

Nguyen

2018

Journal of Statistical Planning and Inference

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We study the problem of detecting a change in the mean of one-dimensional Gaussian process data. This problem is investigated in the setting of increasing domain (customarily employed in time series analysis) and in the setting of fixed domain (typically arising in spatial data analysis). We propose a detection method based on the generalized likelihood ratio test (GLRT), and show that our method achieves nearly asymptotically optimal rate in the minimax sense, in both settings. The salient feature of the proposed method is that it exploits in an efficient way the data dependence captured by the Gaussian process covariance structure. When the covariance is not known, we propose the plug-in GLRT method and derive conditions under which the method remains asymptotically near optimal. By contrast, the standard CUSUM method, which does not account for the covariance structure, is shown to be asymptotically optimal only in the increasing domain. Our algorithms and accompanying theory are applicable to a wide variety of covariance structures, including the Matern class, the powered exponential class, and others. The plug-in GLRT method is shown to perform well for maximum likelihood estimators with a dense covariance matrix.

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Gaussian Random Processes

Cited by 311 publications

References 0 publications

Interpolation of spatial data – A stochastic or a deterministic problem?

Interpolation of spatial data – A stochastic or a deterministic problem?

A Sufficient Condition for Linear Growth of Variances in a Stationary Random Sequence

Optimal change point detection in Gaussian processes

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