arXiv subjects

Evgeny Spodarev

Publications and source records attributed to Evgeny Spodarev.

At least 19 recordsLinked to original sources

arXiv2026

DFS-based fast crack pre-detection

This paper develops a computationally efficient pre-detection method for cracks in three-dimensional CT images of concrete. Instead of attempting full voxel-wise crack segmentation, the method focuses on locating cubic subregions where crack structures are likely to be present and should be analyzed further. The proposed pipeline combines multiscale Maximal Hessian Entry filtering with graph-based connectivity analysis. After binarization, each subregion is represented by the boundary face with the largest foreground pixels, which transforms the local detection problem from a three-dimensional image task into a two-dimensional graph problem. A sparse lattice graph is constructed on the selected face, and Depth-First Search is applied to detect connected components corresponding to possible crack cross-sections. The choice of mesh size is justified by a probabilistic upper bound on a lattice-miss event. Experiments on semi-synthetic and real CT data show that the method gives fast, interpretable crack pre-localization while avoiding exhaustive analysis of the full image.

eess.IV↗

arXiv2026

Extrapolation of max-stable random fields with Fréchet marginals

We propose a method for the prediction of stationary max--stable random fields with $α$-Fréchet marginal distribution $H_α$. The method is suitable to cope with heavy tails for $α\in(0,2)$ and is (approximately) exact in marginal distributions. It is based on a recent extrapolation approach via level sets which requires no moment assumptions. An explicit connection between the excursion metric and the Davis-Resnick distance is established. The existence of the predictor is proven. The non-uniqueness of the forecast is demonstrated on several examples. The method is tested on multiple simulated time series and random fields as well as applied to real data of annual maximum precipitation.

math.PR↗

arXiv2025

A statistical method for crack pre-detection in 3D concrete images

In practical applications, effectively segmenting cracks in large-scale computed tomography (CT) images holds significant importance for understanding the structural integrity of materials. Classical image-processing techniques and modern deep-learning models both face substantial computational challenges when applied directly to high resolution big data volumes. This paper introduces a statistical framework for crack pre-localization, whose purpose is not to replace or compete with segmentation networks, but to identify, with controlled error rates, the regions of a 3D CT image that are most likely to contain cracks. The method combines a simple Hessian-based filter, geometric descriptors computed on a regular spatial partition, and a spatial multiple testing procedure to detect anomalous regions while relying only on minimal calibration data, rather than large annotated datasets. Experiments on semi-synthetic and real 3D CT scans demonstrate that the proposed approach reliably highlights regions likely to contain cracks while preserving linear computational complexity. By restricting subsequent high resolution segmentation to these localized regions, deep-learning models can be trained and operate more efficiently, reducing both training runtime as well as resource consumption. The framework thus offers a practical and interpretable preprocessing step for large-scale CT inspection pipelines.

cs.CV↗

arXiv2025

Tube formula for spherically contoured random fields with subexponential marginals

It is widely known that the tube method, or equivalently the Euler characteristic heuristic, provides a very accurate approximation for the tail probability that the supremum of a smooth Gaussian random field exceeds a threshold value $c$. The relative approximation error $Δ(c)$ is exponentially small as a function of $c$ when $c$ tends to infinity. On the other hand, little is known about non-Gaussian random fields. In this paper, we obtain the approximation error of the tube method applied to the canonical isotropic random fields on a unit sphere defined by $u\mapsto\langle u,ξ\rangle$, $u\in M\subset\mathbb{S}^{n-1}$, where $ξ$ is a spherically contoured random vector. These random fields have statistical applications in multiple testing and simultaneous regression inference when the unknown variance is estimated. The decay rate of the relative error $Δ(c)$ depends on the tail of the distribution of $\|ξ\|^2$ and the critical radius of the index set $M$. If this distribution is subexponential but not regularly varying, $Δ(c)\to 0$ as $c\to\infty$. However, in the regularly varying case, $Δ(c)$ does not vanish and hence is not negligible. To address this limitation, we provide simple upper and lower bounds for $Δ(c)$ and for the tube formula itself. Numerical studies are conducted to assess the accuracy of the asymptotic approximation.

math.PR↗

arXiv2025

Random eigenvalues of graphenes and the triangulation of plane

We analyse the numbers of closed paths of length $k\in\mathbb{N}$ on two important regular lattices: the hexagonal lattice (also called $\textit{graphene}$ in chemistry) and its dual triangular lattice. These numbers form a moment sequence of specific random variables connected to the distance of a position of a planar random flight (in three steps) from the origin. Here, we refer to such a random variable as a $\textit{random eigenvalue}$ of the underlying lattice. Explicit formulas for the probability density and characteristic functions of these random eigenvalues are given for both the hexagonal and the triangular lattice. Furthermore, it is proven that both probability distributions can be approximated by a functional of the random variable uniformly distributed on increasing intervals $[0,b]$ as $b\to\infty$. This yields a straightforward method to simulate these random eigenvalues without generating graphene and triangular lattice graphs. To demonstrate this approximation, we first prove a key integral identity for a specific series containing the third powers of the modified Bessel functions $I_n$ of $n$th order, $n\in\mathbb{Z}$. Such series play a crucial role in various contexts, in particular, in analysis, combinatorics, and theoretical physics.

math.SP↗

arXiv2024

Some open mathematical problems on fullerenes

Fullerenes are hollow carbon molecules where each atom is connected to exactly three other atoms, arranged in pentagonal and hexagonal rings. Mathematically, they can be combinatorially modeled as planar, 3-regular graphs with facets composed only of pentagons and hexagons. In this work, we outline a few of the many open questions about fullerenes, beginning with the problem of generating fullerenes randomly. We then introduce an infinite family of fullerenes on which the generalized Stone-Wales operation is inapplicable. Furthermore, we present numerical insights on a graph invariant, called \textit{character} of a fullerene, derived from its adjacency and degree matrices. This descriptor may lead to a new method for linear enumeration of all fullerenes.

math.CO↗

arXiv2024

Random eigenvalues of nanotubes

The hexagonal lattice and its dual, the triangular lattice, serve as powerful models for comprehending the atomic and ring connectivity, respectively, in \textit{graphene} and \textit{carbon $(p,q)$--nanotubes}. The chemical and physical attributes of these two carbon allotropes are closely linked to the average number of closed paths of different lengths $k\in\mathbb{N}_0$ on their respective graph representations. Considering that a carbon $(p,q)$--nanotube can be thought of as a graphene sheet rolled up in a matter determined by the \textit{chiral vector} $(p,q)$, our findings are based on the study of \textit{random eigenvalues} of both the hexagonal and triangular lattices presented in \cite{bille2023random}. This study reveals that for any given \textit{chiral vector} $(p,q)$, the sequence of counts of closed paths forms a moment sequence derived from a functional of two independent uniform distributions. Explicit formulas for key characteristics of these distributions, including probability density function (PDF) and moment generating function (MGF), are presented for specific choices of the chiral vector. Moreover, we demonstrate that as the \textit{circumference} of a $(p,q)$--nanotube approaches infinity, i.e., $p+q\rightarrow \infty$, the $(p,q)$--nanotube tends to converge to the hexagonal lattice with respect to the number of closed paths for any given length $k$, indicating weak convergence of the underlying distributions.

math.SP↗

arXiv2024

Non-ergodic inference for stationary-increment harmonizable stable processes

We consider the class of stationary-increment harmonizable stable processes with infinite control measure, which most notably includes real harmonizable fractional stable motions. We give conditions for the integrability of the paths of such processes with respect to a finite, absolutely continuous measure and derive the distributional characteristics of the path integral with respect to said measure. The convolution of the path of a stationary-increment harmonizable stable process with a suitable measure yields a real stationary harmonizable stable process with finite control measure. This allows us to construct consistent estimators for the index of stability as well as the kernel function in the integral representation of a stationary increment harmonizable stable process (up to a constant factor). For real harmonizable fractional stable motions consistent estimators for the index of stability and its Hurst parameter are given. These are computed directly from the periodogram frequency estimates of the smoothed process.

math.ST↗

arXiv2024

A short memory condition for infinitely divisible random fields

This small note yields a sufficient condition for the short range dependence of measurable stationary infinitely divisible moving average random fields with $d$--dimensional index space. Here, the short/long range dependence concept in borrowed from the paper of Kulik and Spodarev (2021). In the special case of symmetric stable moving averages, our new condition coincides with the one from the paper of Makogin et al. (2021).

math.PR↗

arXiv2024

A Fubini-type limit theorem for the integrated hyperuniform infinitely divisible moving averages

This short note shows a limiting behavior of integrals of some centered antipersistent stationary infinitely divisible moving averages as the compact integration domain in $d\ge 1$ dimensions extends to the whole positive quadrant $\mathbb{R}^d_+$. Namely, the weak limit of their finite dimensional distributions is again a moving average with the same infinitely divisible purely jump integrator measure (i.e., possessing no Gaussian component), but with an integrated kernel function. The results apply equally to time series ($d=1$) as well as to random fields ($d>1$). Apart from the existence of the expectation, no moment assumptions on the moving average are imposed allowing it to have an infinite variance as e.g. in the case of $α$-stable moving averages with $α\in(1,2)$ . If the field is additionally square integrable, its covariance integrates to zero (hyperuniformity).

math.PR↗

arXiv2023

Non-ergodic statistics and spectral density estimation for stationary real harmonizable symmetric $α$-stable processes

We consider non-ergodic class of stationary real harmonizable symmetric $α$-stable processes $X=\left\{X(t):t\in\mathbb{R}\right\}$ with a finite symmetric and absolutely continuous control measure. We refer to its density function as the spectral density of $X$. These processes admit a LePage series representation and are conditionally Gaussian, which allows us to derive the non-ergodic limit of sample functions on $X$. In particular, we give an explicit expression for the non-ergodic limits of the empirical characteristic function of $X$ and the lag process $\left\{X(t+h)-X(t):t\in\mathbb{R}\right\}$ with $h>0$, respectively. The process admits an equivalent representation as a series of sinusoidal waves with random frequencies which are i.i.d. with the (normalized) spectral density of $X$ as their probability density function. Based on strongly consistent frequency estimation using the periodogram we present a strongly consistent estimator of the spectral density. The periodogram's computation is fast and efficient, and our method is not affected by the non-ergodicity of $X$.

math.ST↗

arXiv2023

Detection of Long Range Dependence in the Time Domain for (In)Finite-Variance Time Series

Empirical detection of long range dependence (LRD) of a time series often consists of deciding whether an estimate of the memory parameter $d$ corresponds to LRD. Surprisingly, the literature offers numerous spectral domain estimators for $d$ but there are only a few estimators in the time domain. Moreover, the latter estimators are criticized for relying on visual inspection to determine an observation window $[n_1, n_2]$ for a linear regression to run on. Theoretically motivated choices of $n_1$ and $n_2$ are often missing for many time series models. In this paper, we take the well-known variance plot estimator and provide rigorous asymptotic conditions on $[n_1, n_2]$ to ensure the estimator's consistency under LRD. We establish these conditions for a large class of square-integrable time series models. This large class enables one to use the variance plot estimator to detect LRD for infinite-variance time series (after suitable transformation). Thus, detection of LRD for infinite-variance time series is another novelty of our paper. A simulation study indicates that the variance plot estimator can detect LRD better than the popular spectral domain GPH estimator.

math.ST↗

arXiv2022

Prediction of random variables by excursion metric projections

We use the concept of excursions for the prediction of random variables without any moment existence assumptions. To do so, an excursion metric on the space of random variables is defined which appears to be a kind of a weighted $L^1$-distance. Using equivalent forms of this metric and the specific choice of excursion levels, we formulate the prediction problem as a minimization of a certain target functional which involves the excursion metric. Existence of the solution and weak consistency of the predictor are discussed. An application to the extrapolation of stationary heavy-tailed random functions illustrates the use of the aforementioned theory. Numerical experiments with the prediction of Gaussian, $α$-stable and further heavy--tailed time series round up the paper.

math.ST↗

arXiv2022

Intrinsic volumes of ellipsoids

We deduce explicit formulae for the intrinsic volumes of an ellipsoid in $\mathbb R^d$, $d\ge 2$, in terms of elliptic integrals. Namely, for an ellipsoid ${\mathcal E}\subset \mathbb R^d$ with semiaxes $a_1,\ldots, a_d$ we show that \begin{align*} V_k({\mathcal E})=κ_k\sum_{i=1}^da_i^2s_{k-1}(a_1^2,\dots,a_{i-1}^2,a_{i+1}^2,\dots,a_d^2)\int_0^{\infty}{t^{k-1}\over(a_i^2t^2+1)\prod_{j=1}^d\sqrt{a_j^2t^2+1}}\,\rm{d}t \end{align*} for all $k=1,\ldots,d$, where $s_{k-1}$ is the $(k-1)$-th elementary symmetric polynomial and $κ_k$ is the volume of the $k$-dimensional unit ball. Some examples of the intrinsic volumes $V_k$ with low and high $k$ are given where our formulae look particularly simple. As an application we derive new formulae for the expected $k$-dimensional volume of random $k$-simplex in an ellipsoid and random Gaussian $k$-simplex.

math.MG↗

arXiv2021

Extrapolation of Stationary Random Fields Via Level Sets

In this paper, we use the concept of excursion sets for the extrapolation of stationary random fields. Doing so, we define excursion sets for the field and its linear predictor, and then minimize the expected volume of the symmetric difference of these sets under the condition that the univariate distributions of the predictor and of the field itself coincide. We illustrate the new approach on Gaussian random fields.

math.PR↗

arXiv2021

Inversion of $α$-sine and $α$-cosine transforms on $\mathbb{R}$

We consider the $α$-sine transform of the form $T_αf(y)=\int_0^\infty\vert\sin(xy)\vert^αf(x)dx$ for $α>-1$, where $f$ is an integrable function on $\mathbb{R}_+$. First, the inversion of this transform for $α>1$ is discussed in the context of a more general family of integral transforms on the space of weighted, square-integrable functions on the positive real line. In an alternative approach, we show that the $α$-sine transform of a function $f$ admits a series representation for all $α>-1$, which involves the Fourier transform of $f$ and coefficients which can all be explicitly computed with the Gauss hypergeometric theorem. Based on this series representation we construct a system of linear equations whose solution is an approximation of the Fourier transform of $f$ at equidistant points. Sampling theory and Fourier inversion allow us to compute an estimate of $f$ from its $α$-sine transform. The same approach can be extended to a similar $α$-cosine transform on $\mathbb{R}_+$ for $α>-1$, and the two-dimensional spherical $α$-sine and cosine transforms for $α>-1$, $α\neq 0,2,4,\dots$. In an extensive numerical analysis, we consider a number of examples, and compare the inversion results of both methods presented.

math.FA↗

arXiv2021

Limit theorems for excursion sets of subordinated Gaussian random fields with long-range dependence

This paper considers the asymptotic behaviour of volumes of excursion sets of subordinated Gaussian random fields with (possibly) infinite variance. Actually, we consider integral functionals of such fields and obtain their limiting distribution using the Hermite expansion of the integrand. We consider the general non-stationary Gaussian random fields, including stationary and anisotropic special cases. The limiting random variables in our limit theorems have the form of multiple Wiener-Itô integrals. We illustrate most results with corresponding examples.

math.PR↗

arXiv2020

Long range dependence of heavy tailed random functions

We introduce a definition of long range dependence of random processes and fields on an (unbounded) index space $T\subseteq \R^d$ in terms of integrability of the covariance of indicators that a random function exceeds any given level. This definition is particularly designed to cover the case of random functions with infinite variance. We show the value of this new definition and its connection to limit theorems on some examples including subordinated Gaussian as well as random volatility fields and time series.

math.PR↗