arXiv subjects

Andriy Olenko

Publications and source records attributed to Andriy Olenko.

At least 37 records · Page 2Linked to original sources

arXiv2021

On Multifractionality of Spherical Random Fields with Cosmological Applications

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.

physics.gen-ph↗

arXiv2021

Asymptotics of running maxima for $φ$-subgaussian random double arrays

The article studies the running maxima $Y_{m,j}=\max_{1 \le k \le m, 1 \le n \le j} X_{k,n} - a_{m,j}$ where $\{X_{k,n}, k \ge 1, n \ge 1\}$ is a double array of $φ$-subgaussian random variables and $\{a_{m,j}, m\ge 1, j\ge 1\}$ is a double array of constants. Asymptotics of the maxima of the double arrays of positive and negative parts of $\{Y_{m,j}, m \ge 1, j \ge 1\}$ are studied, when $\{X_{k,n}, k \ge 1, n \ge 1\}$ have suitable "exponential-type" tail distributions. The main results are specified for various important particular scenarios and classes of $φ$-subgaussian random variables.

math.PR↗

arXiv2020

Asymptotic normality of simultaneous estimators of cyclic long-memory processes

Spectral singularities at non-zero frequencies play an important role in investigating cyclic or seasonal time series. The publication [2] introduced the generalized filtered method-of-moments approach to simultaneously estimate singularity location and long-memory parameters. This paper continues studies of these simultaneous estimators. A wide class of Gegenbauer-type semi-parametric models is considered. Asymptotic normality of several statistics of the cyclic and long-memory parameters is proved. New adjusted estimates are proposed and investigated. The theoretical findings are illustrated by numerical results. The methodology includes wavelet transformations as a particular case.

math.ST↗

arXiv2020

Strong Law of Large Numbers for Functionals of Random Fields With Unboundedly Increasing Covariances

The paper proves the Strong Law of Large Numbers for integral functionals of random fields with unboundedly increasing covariances. The case of functional data and increasing domain asymptotics is studied. Conditions to guarantee that the Strong Law of Large Numbers holds true are provided. The considered scenarios include wide classes of non-stationary random fields. The discussion about application to weak and long-range dependent random fields and numerical examples are given.

math.PR↗

arXiv2020

Analysis of Spherical Monofractal and Multifractal Random Fields

The Rényi 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 Rényi function is known explicitly. The theoretical part of the article presents multifractal random fields on the sphere and develops specific models where the Rényi function can be computed explicitly. 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 Rényi function and the multifractal spectrum for different scenarios and actual CMB data.

math.PR↗

arXiv2020

On stationarity properties of generalized Hermite-type processes

The paper investigates properties of generalized Hermite-type processes that arise in non-central limit theorems for integral functionals of long-range dependent random fields. The case of increasing multidimensional domain asymptotics is studied. Three approaches to investigate properties of these processes are discussed. Contrary to the classical one-dimensional case, it is shown that for any choice of a multidimensional observation window the generalized Hermite-type process has non-stationary increments.

math.PR↗

arXiv2020

Reduction principle for functionals of strong-weak dependent vector random fields

We prove the reduction principle for asymptotics of functionals of vector random fields with weakly and strongly dependent components. These functionals can be used to construct new classes of random fields with skewed and heavy-tailed distributions. Contrary to the case of scalar long-range dependent random fields, it is shown that the asymptotic behaviour of such functionals is not necessarily determined by the terms at their Hermite rank. The results are illustrated by an application to the first Minkowski functional of the Student random fields. Some simulation studies based on the theoretical findings are also presented.

math.PR↗

arXiv2019

Spherically Restricted Random Hyperbolic Diffusion

This paper investigates solutions of hyperbolic diffusion equations in $\mathbb{R}^3$ with random initial conditions. The solutions are given as spatial-temporal random fields. Their restrictions to the unit sphere $S^2$ are studied. All assumptions are formulated in terms of the angular power spectrum or the spectral measure of the random initial conditions. Approximations to the exact solutions are given. Upper bounds for the mean-square convergence rates of the approximation fields are obtained. The smoothness properties of the exact solution and its approximation are also investigated. It is demonstrated that the Hölder-type continuity of the solution depends on the decay of the angular power spectrum. Conditions on the spectral measure of initial conditions that guarantee short or long-range dependence of the solutions are given. Numerical studies are presented to verify the theoretical findings.

math.PR↗

arXiv2019

Random spherical hyperbolic diffusion

The paper starts by giving a motivation for this research and justifying the considered stochastic diffusion models for cosmic microwave background radiation studies. Then it derives the exact solution in terms of a series expansion to a hyperbolic diffusion equation on the unit sphere. The Cauchy problem with random initial conditions is studied. All assumptions are stated in terms of the angular power spectrum of the initial conditions. An approximation to the solution is given and analysed by finitely truncating the series expansion. The upper bounds for the convergence rates of the approximation errors are derived. Smoothness properties of the solution and its approximation are investigated. It is demonstrated that the sample Hölder continuity of these spherical fields is related to the decay of the angular power spectrum. Numerical studies of approximations to the solution and applications to cosmic microwave background data are presented to illustrate the theoretical results.

stat.AP↗

arXiv2019

Modeling of Electrical Resistivity of Soil Based on Geotechnical Properties

Determining the relationship between the electrical resistivity of soil and its geotechnical properties is an important engineering problem. This study aims to develop methodology for finding the best model that can be used to predict the electrical resistivity of soil, based on knowing its geotechnical properties. The research develops several linear models, three non-linear models, and three artificial neural network models (ANN). These models are applied to the experimental data set comprises 864 observations and five variables. The results show that there are significant exponential negative relationships between the electrical resistivity of soil and its geotechnical properties. The most accurate prediction values are obtained using the ANN model. The cross-validation analysis confirms the high precision of the selected predictive model. This research is the first rigorous systematic analysis and comparison of difference methodologies in ground electrical resistivity studies. It provides practical guidelines and examples of design, development and testing non-linear relationships in engineering intelligent systems and applications.

stat.AP↗

arXiv2019

Fast Tensor Needlet Transforms for Tangent Vector Fields on the Sphere

This paper constructs a semi-discrete tight frame of tensor needlets associated with a quadrature rule for tangent vector fields on the unit sphere $\mathbb{S}^2$ of $\mathbb{R}^3$ --- tensor needlets. The proposed tight tensor needlets provide a multiscale representation of any square integrable tangent vector field on $\mathbb{S}^2$, which leads to a multiresolution analysis (MRA) for the field. From the MRA, we develop fast algorithms for tensor needlet transforms, including the decomposition and reconstruction of the needlet coefficients between levels, via a set of filter banks and scalar FFTs. The fast tensor needlet transforms have near linear computational cost proportional to $N\log \sqrt{N}$ for $N$ evaluation points or coefficients. Numerical examples for the simulated and real data demonstrate the efficiency of the proposed algorithm.

math.NA↗

arXiv2019

Spherical data handling and analysis with R package rcosmo

The R package rcosmo was developed for handling and analysing Hierarchical Equal Area isoLatitude Pixelation(HEALPix) and Cosmic Microwave Background(CMB) radiation data. It has more than 100 functions. rcosmo was initially developed for CMB, but also can be used for other spherical data. This paper discusses transformations into rcosmo formats and handling of three types of non-CMB data: continuous geographic, point pattern and star-shaped. For each type of data we provide a brief description of the corresponding statistical model, data example and ready-to-use R code. Some statistical functionality of rcosmo is demonstrated for the example data converted into the HEALPix format. The paper can serve as the first practical guideline to transforming data into the HEALPix format and statistical analysis with rcosmo for geo-statisticians, GIS and R users and researches dealing with spherical data in non-HEALPix formats.

stat.AP↗

arXiv2019

rcosmo: R Package for Analysis of Spherical, HEALPix and Cosmological Data

The analysis of spatial observations on a sphere is important in areas such as geosciences, physics and embryo research, just to name a few. The purpose of the package rcosmo is to conduct efficient information processing, visualisation, manipulation and spatial statistical analysis of Cosmic Microwave Background (CMB) radiation and other spherical data. The package was developed for spherical data stored in the Hierarchical Equal Area isoLatitude Pixelation (Healpix) representation. rcosmo has more than 100 different functions. Most of them initially were developed for CMB, but also can be used for other spherical data as rcosmo contains tools for transforming spherical data in cartesian and geographic coordinates into the HEALPix representation. We give a general description of the package and illustrate some important functionalities and benchmarks.

stat.CO↗

arXiv2019

Asymptotic Behaviour of Discretised Functionals of Long-Range Dependent Functional Data

The paper studies the asymptotic behaviour of weighted functionals of long-range dependent data over increasing observation windows. Various important statistics, including sample means, high order moments, occupation measures can be given by these functionals. It is shown that in the discrete sampling case additive functionals have the same asymptotic distribution as the corresponding integral functionals for the continuous functional data case. These results are applied to obtain non-central limit theorems for weighted additive functionals of random fields. As the majority of known results concern the discrete sampling case the developed methodology helps in translating these results to functional data without deriving them again. Numerical studies suggest that the theoretical findings are valid for wider classes of long-range dependent data.

math.ST↗

arXiv2019

On LSE in regression model for long-range dependent random fields on spheres

We study the asymptotic behaviour of least squares estimators in regression models for long-range dependent random fields observed on spheres. The least squares estimator can be given as a weighted functional of long-range dependent random fields. It is known that in this scenario the limits can be non-Gaussian. We derive the limit distribution and the corresponding rate of convergence for the estimators. The results were obtained under rather general assumptions on the random fields. Simulation studies were conducted to support theoretical findings.

math.ST↗

arXiv2018

Limit theorems for filtered long-range dependent random fields

This article investigates general scaling settings and limit distributions of functionals of filtered random fields. The filters are defined by the convolution of non-random kernels with functions of Gaussian random fields. The case of long-range dependent fields and increasing observation windows is studied. The obtained limit random processes are non-Gaussian. Most known results on this topic give asymptotic processes that always exhibit non-negative auto-correlation structures and have the self-similar parameter $H\in(\frac{1}{2},1)$. In this work we also obtain convergence for the case $H\in(0,\frac{1}{2})$ and show how the Hurst parameter $H$ can depend on the shape of the observation windows. Various examples are presented.

math.PR↗

arXiv2018

Non-central limit theorems for functionals of random fields on hypersurfaces

This paper derives non-central asymptotic results for non-linear integral functionals of homogeneous isotropic Gaussian random fields defined on hypersurfaces in $\mathbb{R}^d$. We obtain the rate of convergence for these functionals. The results extend recent findings for solid figures. We apply the obtained results to the case of sojourn measures and demonstrate different limit situations.

math.PR↗

arXiv2018

Estimation of seasonal long-memory parameters

This paper studies seasonal long-memory processes with Gegenbauer-type spectral densities. Estimates for singularity location and long-memory parameters based on general filter transforms are proposed. It is proved that the estimates are almost surely convergent to the true values of parameters. Solutions of the estimation equations are studied and adjusted statistics are proposed. Numerical results are presented to confirm the theoretical findings.

math.ST↗