scholarly journals Smoothness and monotonicity of the excursion set density of planar Gaussian fields

2020 ◽  
Vol 25 (0) ◽  
Author(s):  
Dmitry Beliaev ◽  
Michael McAuley ◽  
Stephen Muirhead
Keyword(s):  
Gaussian Fields ◽  
Excursion Set

The size of the connected components of excursion sets of χ2, t and F fields

1999 ◽  
Vol 31 (03) ◽  
pp. 579-595 ◽  
Author(s):  
J. Cao
Keyword(s):  
Positron Emission Tomography ◽  
Two Dimensions ◽  
Emission Tomography ◽  
Connected Components ◽  
Gaussian Fields ◽  
Connected Component ◽  
Test Statistic ◽  
Excursion Sets ◽  
Positron Emission ◽  
Excursion Set

The distribution of the size of one connected component and the largest connected component of the excursion set is derived for stationary χ2, t and F fields, in the limit of high or low thresholds. This extends previous results for stationary Gaussian fields (Nosko 1969, Adler 1981) and for χ2 fields in one and two dimensions (Aronowich and Adler 1986, 1988). An application of this is to detect regional changes in positron emission tomography (PET) images of blood flow in human brain, using the size of the largest connected component of the excursion set as a test statistic.

The size of the connected components of excursion sets of χ2, t and F fields

1999 ◽  
Vol 31 (3) ◽  
pp. 579-595 ◽  
Author(s):  
J. Cao
Keyword(s):  
Positron Emission Tomography ◽  
Two Dimensions ◽  
Emission Tomography ◽  
Connected Components ◽  
Gaussian Fields ◽  
Connected Component ◽  
Test Statistic ◽  
Excursion Sets ◽  
Positron Emission ◽  
Excursion Set

The distribution of the size of one connected component and the largest connected component of the excursion set is derived for stationary χ2, t and F fields, in the limit of high or low thresholds. This extends previous results for stationary Gaussian fields (Nosko 1969, Adler 1981) and for χ2 fields in one and two dimensions (Aronowich and Adler 1986, 1988). An application of this is to detect regional changes in positron emission tomography (PET) images of blood flow in human brain, using the size of the largest connected component of the excursion set as a test statistic.

scholarly journals Sufficiency of a Gaussian power spectrum likelihood for accurate cosmology from upcoming weak lensing surveys

2021 ◽  
Author(s):  
Robin E Upham ◽  
Michael L Brown ◽  
Lee Whittaker
Keyword(s):  
Power Spectrum ◽  
Random Noise ◽  
Power Spectra ◽  
Stage Iv ◽  
Wishart Distribution ◽  
Weak Lensing ◽  
Dependence Structure ◽  
Gaussian Fields ◽  
Non Gaussian ◽  
Parameter Constraints

Abstract We investigate whether a Gaussian likelihood is sufficient to obtain accurate parameter constraints from a Euclid-like combined tomographic power spectrum analysis of weak lensing, galaxy clustering and their cross-correlation. Testing its performance on the full sky against the Wishart distribution, which is the exact likelihood under the assumption of Gaussian fields, we find that the Gaussian likelihood returns accurate parameter constraints. This accuracy is robust to the choices made in the likelihood analysis, including the choice of fiducial cosmology, the range of scales included, and the random noise level. We extend our results to the cut sky by evaluating the additional non-Gaussianity of the joint cut-sky likelihood in both its marginal distributions and dependence structure. We find that the cut-sky likelihood is more non-Gaussian than the full-sky likelihood, but at a level insufficient to introduce significant inaccuracy into parameter constraints obtained using the Gaussian likelihood. Our results should not be affected by the assumption of Gaussian fields, as this approximation only becomes inaccurate on small scales, which in turn corresponds to the limit in which any non-Gaussianity of the likelihood becomes negligible. We nevertheless compare against N-body weak lensing simulations and find no evidence of significant additional non-Gaussianity in the likelihood. Our results indicate that a Gaussian likelihood will be sufficient for robust parameter constraints with power spectra from Stage IV weak lensing surveys.

scholarly journals Derivatives of local times for some Gaussian fields II

2021 ◽  
Vol 172 ◽  
pp. 109063
Author(s):  
Minhao Hong ◽  
Fangjun Xu
Keyword(s):  
Local Times ◽  
Gaussian Fields ◽  
Derivatives Of

scholarly journals Excursion sets of infinitely divisible random fields with convolution equivalent Lévy measure

2017 ◽  
Vol 54 (3) ◽  
pp. 833-851 ◽  
Author(s):  
Anders Rønn-Nielsen ◽  
Eva B. Vedel Jensen
Keyword(s):  
Random Field ◽  
Large Class ◽  
Random Fields ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Excursion Sets ◽  
Excursion Set ◽  
Fixed Radius ◽  
Asymptotic Probability ◽  
The Right

Abstract We consider a continuous, infinitely divisible random field in ℝd, d = 1, 2, 3, given as an integral of a kernel function with respect to a Lévy basis with convolution equivalent Lévy measure. For a large class of such random fields, we compute the asymptotic probability that the excursion set at level x contains some rotation of an object with fixed radius as x → ∞. Our main result is that the asymptotic probability is equivalent to the right tail of the underlying Lévy measure.

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