Analysis of spherical monofractal and multifractal random fields
- PDF / 2,022,671 Bytes
- 21 Pages / 595.276 x 790.866 pts Page_size
- 37 Downloads / 210 Views
ORIGINAL PAPER
Analysis of spherical monofractal and multifractal random fields Nikolai Leonenko1
•
Ravindi Nanayakkara2
•
Andriy Olenko2
Accepted: 16 October 2020 Ó Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The Re´nyi function plays an important role in the analysis of multifractal random fields. For random fields on the sphere, there are three models in the literature where the Re´nyi function is known explicitly. The theoretical part of the article presents multifractal random fields on the sphere and develops specific models where the Re´nyi function can be computed explicitly. For all considered models explicit expressions of their multifractal spectrum are obtained. Properties of the models and dependencies of their characteristics on parameters are investigated. Then these results are applied to the Cosmic Microwave Background Radiation data collected from the Planck mission. The main statistical model used to describe these data in the literature is isotropic Gaussian fields. We present numerical multifractality studies and methodology based on simulating random fields, computing the Re´nyi function and the multifractal spectrum for different scenarios and actual CMB data. The obtained results can also find numerous potential applications for other geoscience, environmental and directional data. Keywords Re´nyi function Random field Multifractality Monofractality Cosmic microwave background radiation
1 Introduction Recent years have witnessed an enormous amount of attention, in the environmental, earth science, biological and astrophysical literature, on investigating spherical random fields. Excellent overviews of some novel geostatistics directions and applications can be found in Christakos (2017), Jeong et al. (2017), Marinucci and Peccati (2011), Porcu et al. (2018) and references therein. From a statistical point of view, random fields on Euclidean spaces is a rather well studied area. However, the majority of available results is not directly translatable to manifolds (where a sphere is an obvious first important
& Andriy Olenko [email protected] Nikolai Leonenko [email protected] Ravindi Nanayakkara [email protected] 1
School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK
2
Department of Mathematics and Statistics, La Trobe University, Melbourne, VIC 3086, Australia
candidate for investigations) and requires new stochastic models and tools, see, for example, Emery and Porcu (2019), Emery et al. (2019), Lang and Schwab (2015), Malyarenko (2012) and Marinucci and Peccati (2011). This research investigates multifractal properties of spherical random fields and provides practical methodology and examples of applications to actual data. The concept of multifractality initially emerged in the context of physics. B. Mandelbrot showed the significance of scaling relations in turbulence modelling. Subsequently this concept developed to mathematical models and examining their fine scale characteristics. A multif
Data Loading...
