On the Problem of Permissible Covariance and Variogram Models

By George Christakos. 1984

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Citations in the corpus

On the Schoenberg Transformations in Data Analysis: Theory and Illustrations (2010, neutral): “Under uniform linear dilatation of the original right-angled triangle by a factor ǫ > 0, (11) readily yields that limǫ→∞ α̃(ǫ) = π/3 whenever φ is bounded, and limǫ→0 α̃(ǫ) = π/2 whenever φ is rectifiable.”
Classes of Kernels for Machine Learning: A Statistics Perspective (2002, background): “Cressie and Huang (1999) and Gneiting (2002b) use (10) to derive nonseparable space-time stationary kernels, see also Christakos (2000) for illustrative examples.”
Efficient methods for large‐scale linear inversion using a geostatistical approach (2012, background): “For a more detailed discussion on covariance kernels that permissible, we refer the reader to the following references [Christakos, 1984; Matheron, 1973].”
Spatio-temporal variograms and covariance models (2005, background): “Email address: cma@math.twsu.edu 706 family whose members are known to be positive-definite functions; see, for example, Christakos (1984), Weber and Talkner (1993), and Gaspari and Cohn (1999).”
On the Use of Non-Euclidean Distance Measures in Geostatistics (2007, background): “Valid variograms are necessarily conditionally negative definite as in (2) and also must grow more slowly than � h� 2 (Matheron, 1973; Christakos, 1984 ).”
Compactly Supported Correlation Functions (2002, methodology): “We refer to Christakos (1984), Weber and Talkner (1993), Gaspari and Cohn (1999),”
Space–time analysis using a general product–sum model (2001, background): “The function Cs,t in (1) must satisfy a positive-definiteness condition in order to be an admissible covariance model.”
Methods for generating non-separable spatiotemporal covariance models with potential environmental applications (2004, background): “In particular, [4,6] introduced space transformation operators, which were later applied to various problems (see [12], for a detailed review of the relevant literature).”
Statistical Inference and Simulation for Spatial Point Processes (2003, unknown): “Hastings algorithm is more sensitive to the choice of initial state Γ0, see Christensen, Roberts & Rosenthal (2003) and Example 10.”
Teacher's Aide Variogram Interpretation and Modeling (2001, background): “For these reasons, geostatisticians have fit variograms with specific known positive definite functions like the spherical, exponential, Gaussian, and hole effect variogram models (Journel and Huijbregts, 1978; Cressie, 1993; Christakos, 1984).”
On the effect of covariance function estimation on the accuracy of kriging predictors (2001, background): “For a detailed account of permissibility issues see Christakos (1984). Hall & Patil (1994) propose permissible kernel-type nonparametric estimators of covariance functions. There is a slightly worrying issue concerning covariance function estimation under infill asymptotics, pointed out by Lahiri (1”
Applications of percolation theory to porous media with distributed local conductances (2001, methodology): “7)) can be analyzed using Bochner's theorem [57], which says that for a stationary conductivity the Fourier transform of the correlation function (proportional to the negative of the second term) must exist and be positive de®nite.”