Analysis of spherical monofractal and multifractal random fields

Leonenko, Nikolai; Nanayakkara, Ravindi; Olenko, Andriy

doi:10.1007/s00477-020-01911-z

Cited by 9 publications

(15 citation statements)

References 33 publications

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“…Multifractal temporal and spatial data have been observed in many applications, for example, environmental processes (precipitations fields), engineering (teletraffic), and cosmology [18,21,24]. Jaffard [9] showed that the class of multifractal random processes is wide and all Lévy processes except Brownian motion and Poisson processes are multifractal.…”

Section: Introductionmentioning

confidence: 99%

“…Sufficient conditions for the pointwise convergence of cumulative fields in the space L 2 were obtained. In [18], Leonenko et al, motivated by the analysis of cosmic microwave background radiation data, studied multifractal random fields and measures defined on spheres. They developed specific models of multifractal fields where the Rényi function can be computed explicitly.…”

Section: Introductionmentioning

confidence: 99%

“…-rates of convergence in the spaces L q , q > 0, and their analysis, -unified assumptions that are stated in the same terms for all results, -examples for the case of ϕ-sub-Gaussian random fields. To our best knowledge, only the geometric Gaussian and related scenarios have been studied in the existing literature [18]. The class of ϕ-sub-Gaussian distributions includes compactly supported distributions, centred Weibull distributions, Gaussian, centred Poisson distributions, etc.…”

Section: Introductionmentioning

confidence: 99%

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Limit theorems for multifractal products of random fields

Donhauzer¹,

Olenko²

2022

Preprint

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This paper investigates asymptotic properties of multifractal products of random fields. The obtained limit theorems provide sufficient conditions for the convergence of cumulative fields in the spaces Lq. New results on the rate of convergence of cumulative fields are presented. Simple unified conditions for the limit theorems and the calculation of the Rényi function are given. They are less restrictive than those in the known one-dimensional results. The developed methodology is also applied to multidimensional multifractal measures. Finally, a new class of examples of geometric ϕ-sub-Gaussian random fields is presented. In this case, the general assumptions have a simple form and can be expressed in terms of covariance functions only.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Limit theorems for multifractal products of random fields

Donhauzer¹,

Olenko²

2022

Preprint

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“…It has been in debate for several years by using the CMB data. Numerous research suggested that the CMB data are either non-Gaussian or cannot be accurately described by statistical or mathematical models with few constant parameters, see [16,17,18,19,20,21]. The classical book by Weinberg [22] explained that this anisotropy in the plasma universe was significant enough to produce anisotropy in current galaxy distributions.…”

Section: Introductionmentioning

confidence: 99%

“…For some recent results and discussion of fundamental cosmological models of the universe, see [23]. To detect departures from the isotropic model in actual CMB data several statistical approaches can be employed, see, for example, [24,18]. Different approaches can result in different results and suggest to cosmologists sky regions for further investigations.…”

Section: Introductionmentioning

confidence: 99%

On Multifractionality of Spherical Random Fields with Cosmological Applications

Broadbridge¹,

Nanayakkara²,

Olenko³

2021

Preprint

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This paper studies random fields on the unit sphere. Traditionally, isotropic Gaussian random fields are considered as the underlying statistical model of the cosmic microwave background (CMB) data. This paper discusses the generalized multifractional Brownian motion and its pointwise Hölder exponent on the sphere.The multifractional approach is used to investigate the CMB data from the Planck mission. These data consist of CMB radiation measurements at narrow angles of the sky sphere. The obtained results suggest that the estimated Hölder exponents for different CMB regions do change from location to location. Therefore, CMB data are multifractional. Then the developed methodology is used to suggest two approaches for detecting regions with anomalies in cleaned CMB maps.

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Point pattern analysis and classification on compact two-point homogeneous spaces evolving time

Frías

Torres

Ruiz-Medina

2023

Stoch Environ Res Risk Assess

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This paper introduces a new modeling framework for the statistical analysis of point patterns on a manifold defined by a connected and compact two-point homogeneous space, including the special case of the sphere. The presented approach is based on temporal Cox processes driven by a -valued log-intensity. Different aggregation schemes on the manifold of the spatiotemporal point-referenced data are implemented in terms of the time-varying discrete Jacobi polynomial transform of the log-risk process. The n -dimensional microscale point pattern evolution in time at different manifold spatial scales is then characterized from such a transform. The simulation study undertaken illustrates the construction of spherical point process models displaying aggregation at low Legendre polynomial transform frequencies (large scale), while regularity is observed at high frequencies (small scale). K -function analysis supports these results under temporal short, intermediate and long range dependence of the log-risk process.

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Analysis of spherical monofractal and multifractal random fields

Abstract: Le o n e n k o, N ikol ai, N a n a y a k k a r a , R avi n di a n d Ol e n k o, And ryi 2 0 2 1. An alysis of s p h e ri c al m o n of r a c t al a n d m ul tifr a c t al r a n d o m fiel d s.

Cited by 9 publications

References 33 publications

Limit theorems for multifractal products of random fields

Limit theorems for multifractal products of random fields

On Multifractionality of Spherical Random Fields with Cosmological Applications

Point pattern analysis and classification on compact two-point homogeneous spaces evolving time

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