A note on isotropic random flights moving in mixed Poisson environments

2017 ◽  
Vol 129 ◽  
pp. 311-317 ◽  
Author(s):  
Alessandro De Gregorio
Keyword(s):  
Random Flights ◽  
Isotropic Random

scholarly journals Needlet approximation for isotropic random fields on the sphere

2017 ◽  
Vol 216 ◽  
pp. 86-116 ◽  
Author(s):  
Quoc T. Le Gia ◽  
Ian H. Sloan ◽  
Yu Guang Wang ◽  
Robert S. Womersley
Keyword(s):  
Random Fields ◽  
Isotropic Random

scholarly journals Identification and isotropy characterization of deformed random fields through excursion sets

2018 ◽  
Vol 50 (3) ◽  
pp. 706-725
Author(s):  
Julie Fournier
Keyword(s):  
Random Field ◽  
Euler Characteristic ◽  
Random Fields ◽  
Weak Form ◽  
Invariance Property ◽  
Basic Domain ◽  
Excursion Sets ◽  
The Mean ◽  
Isotropic Random

Abstract A deterministic application θ:ℝ2→ℝ2 deforms bijectively and regularly the plane and allows the construction of a deformed random field X∘θ:ℝ2→ℝ from a regular, stationary, and isotropic random field X:ℝ2→ℝ. The deformed field X∘θ is, in general, not isotropic (and not even stationary), however, we provide an explicit characterization of the deformations θ that preserve the isotropy. Further assuming that X is Gaussian, we introduce a weak form of isotropy of the field X∘θ, defined by an invariance property of the mean Euler characteristic of some of its excursion sets. We prove that deformed fields satisfying this property are strictly isotropic. In addition, we are able to identify θ, assuming that the mean Euler characteristic of excursion sets of X∘θ over some basic domain is known.

scholarly journals Wicksell's Problem in Local Stereology

2013 ◽  
Vol 45 (4) ◽  
pp. 925-944
Author(s):  
Ó. Thórisdóttir ◽  
M. Kiderlen
Keyword(s):  
Reference Point ◽  
Simulated Data ◽  
Domain Of Attraction ◽  
Spherical Particles ◽  
Section Profile ◽  
Particle Parameters ◽  
The Individual ◽  
The Relationship ◽  
Theoretical Results ◽  
Isotropic Random

Wicksell's classical corpuscle problem deals with the retrieval of the size distribution of spherical particles from planar sections. We discuss the problem in a local stereology framework. Each particle is assumed to contain a reference point and the individual particle is sampled with an isotropic random plane through this reference point. Both the size of the section profile and the position of the reference point inside the profile are recorded and used to recover the distribution of the corresponding particle parameters. Theoretical results concerning the relationship between the profile and particle parameters are discussed. We also discuss the unfolding of the arising integral equations, uniqueness issues, and the domain of attraction relations. We illustrate the approach by providing reconstructions from simulated data using numerical unfolding algorithms.

Variance prediction for pseudosystematic sampling on the sphere

2002 ◽  
Vol 34 (03) ◽  
pp. 469-483
Author(s):  
Ximo Gual-Arnau ◽  
Luis M. Cruz-Orive
Keyword(s):  
Fixed Point ◽  
Prediction Accuracy ◽  
Single Sample ◽  
Particle Model ◽  
Strict Sense ◽  
Biological Cell ◽  
Systematic Design ◽  
Arbitrary Size ◽  
Equal Area ◽  
Isotropic Random

Geometric sampling, and local stereology in particular, often require observations at isotropic random directions on the sphere, and some sort of systematic design on the sphere becomes necessary on grounds of efficiency and practical applicability. Typically, the relevant probes are of nucleator type, in which several rays may be contained in a sectioning plane through a fixed point (e.g. through a nucleolus within a biological cell). The latter requirement considerably reduces the choice of design in practice; in this paper, we concentrate on a nucleator design based on splitting the sphere into regions of equal area, but not of identical shape; this design is pseudosystematic rather than systematic in a strict sense. Firstly, we obtain useful exact representations of the variance of an estimator under pseudosystematic sampling on the sphere. Then we adopt a suitable covariogram model to obtain a variance predictor from a single sample of arbitrary size, and finally we examine the prediction accuracy by way of simulation on a synthetic particle model.

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