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Pilipauskait˙e V, Surgailis D. Local scaling limits of Lévy driven fractional random fields. BERNOULLI 2022. [DOI: 10.3150/21-bej1439] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- Vytaut˙e Pilipauskait˙e
- University of Luxembourg, Department of Mathematics, 6 Avenue de la Fonte, 4364 Esch-sur-Alzette, Luxembourg
| | - Donatas Surgailis
- Vilnius University, Faculty of Mathematics and Informatics, Naugarduko 24, 03225 Vilnius, Lithuania
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Shen J, Stoev S, Hsing T. Tangent fields, intrinsic stationarity, and self similarity. ELECTRON J PROBAB 2022. [DOI: 10.1214/22-ejp754] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Affiliation(s)
- Jinqi Shen
- Department of Statistics, University of Michigan, Ann Arbor, United States of America
| | - Stilian Stoev
- Department of Statistics, University of Michigan, Ann Arbor, United States of America
| | - Tailen Hsing
- Department of Statistics, University of Michigan, Ann Arbor, United States of America
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Rabiei H, Coulon O, Lefevre J, Richard FJP. Surface Regularity via the Estimation of Fractional Brownian Motion Index. IEEE TRANSACTIONS ON IMAGE PROCESSING : A PUBLICATION OF THE IEEE SIGNAL PROCESSING SOCIETY 2020; 30:1453-1460. [PMID: 33326381 DOI: 10.1109/tip.2020.3043892] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
The recent definition of fractional Brownian motions on surfaces has raised the statistical issue of estimating the Hurst index characterizing these models. To deal with this open issue, we propose a method which is based on a spectral representation of surfaces built upon their Laplace-Beltrami operator. This method includes a first step where the surface supporting the motion is recovered using a mean curvature flow, and a second one where the Hurst index is estimated by linear regression on the motion spectrum. The method is evaluated on synthetic surfaces. The interest of the method is further illustrated on some fetal cortical surfaces extracted from magnetic resonance images as a means to quantify the brain complexity during the gestational age.
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Sönmez E. Sample Path Properties of Generalized Random Sheets with Operator Scaling. J THEOR PROBAB 2020. [DOI: 10.1007/s10959-020-01045-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
Abstract
Abstract
We consider operator scaling $$\alpha $$
α
-stable random sheets, which were introduced in Hoffmann (Operator scaling stable random sheets with application to binary mixtures. Dissertation Universität Siegen, 2011). The idea behind such fields is to combine the properties of operator scaling $$\alpha $$
α
-stable random fields introduced in Biermé et al. (Stoch Proc Appl 117(3):312–332, 2007) and fractional Brownian sheets introduced in Kamont (Probab Math Stat 16:85–98, 1996). We establish a general uniform modulus of continuity of such fields in terms of the polar coordinates introduced in Biermé et al. (2007). Based on this, we determine the box-counting dimension and the Hausdorff dimension of the graph of a trajectory over a non-degenerate cube $$I \subset {\mathbb {R}}^d$$
I
⊂
R
d
.
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Kremer D, Scheffler HP. About atomless random measures on δ-rings. Stat Probab Lett 2020. [DOI: 10.1016/j.spl.2020.108805] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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Shen Y, Wang Y. Operator-scaling Gaussian random fields via aggregation. BERNOULLI 2020. [DOI: 10.3150/19-bej1133] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Affiliation(s)
- Dustin Kremer
- Department Mathematik, Universität Siegen, Siegen, Germany
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Directional Thermodynamic Formalism. Symmetry (Basel) 2019. [DOI: 10.3390/sym11060825] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022] Open
Abstract
The usual thermodynamic formalism is uniform in all directions and, therefore, it is not adapted to study multi-dimensional functions with various directional behaviors. It is based on a scaling function characterized in terms of isotropic Sobolev or Besov-type norms. The purpose of the present paper was twofold. Firstly, we proved wavelet criteria for a natural extended directional scaling function expressed in terms of directional Sobolev or Besov spaces. Secondly, we performed the directional multifractal formalism, i.e., we computed or estimated directional Hölder spectra, either directly or via some Legendre transforms on either directional scaling function or anisotropic scaling functions. We obtained general upper bounds for directional Hölder spectra. We also showed optimal results for two large classes of examples of deterministic and random anisotropic self-similar tools for possible modeling turbulence (or cascades) and textures in images: Sierpinski cascade functions and fractional Brownian sheets.
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Makogin V, Mishura Y. Gaussian multi-self-similar random fields with distinct stationary properties of their rectangular increments. STOCH MODELS 2019. [DOI: 10.1080/15326349.2019.1610664] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
Affiliation(s)
| | - Yuliya Mishura
- Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
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Davydov Y, Paulauskas V. Lamperti-Type Theorems for Random Fields. THEORY OF PROBABILITY AND ITS APPLICATIONS 2019. [DOI: 10.1137/s0040585x97t989155] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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Abry P, Didier G. Wavelet eigenvalue regression for n-variate operator fractional Brownian motion. J MULTIVARIATE ANAL 2018. [DOI: 10.1016/j.jmva.2018.06.007] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
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Two-step wavelet-based estimation for Gaussian mixed fractional processes. STATISTICAL INFERENCE FOR STOCHASTIC PROCESSES 2018. [DOI: 10.1007/s11203-018-9190-z] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
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Scaling Limits of Solutions of Linear Stochastic Differential Equations Driven by Lévy White Noises. J THEOR PROBAB 2018. [DOI: 10.1007/s10959-018-0809-1] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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Didier G, Meerschaert MM, Pipiras V. Domain and range symmetries of operator fractional Brownian fields. Stoch Process Their Appl 2018. [DOI: 10.1016/j.spa.2017.04.003] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
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Pilipauskaitė V, Surgailis D. Scaling transition for nonlinear random fields with long-range dependence. Stoch Process Their Appl 2017. [DOI: 10.1016/j.spa.2016.12.011] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Biermé H, Durieu O, Wang Y. Invariance principles for operator-scaling Gaussian random fields. ANN APPL PROBAB 2017. [DOI: 10.1214/16-aap1229] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Abstract
AbstractThe main purpose of this paper is to define and characterize random fields of bounded variation, that is, random fields with sample paths in the space of functions of bounded variation, and to study their mean total variation. Simple formulas are obtained for the mean total directional variation of random fields, based on known formulas for the directional variation of deterministic functions. It is also shown that the mean variation of random fields with stationary increments is proportional to the Lebesgue measure, and an expression of the constant of proportionality, called thevariation intensity, is established. This expression shows, in particular, that the variation intensity depends only on the family of two-dimensional distributions of the stationary increment random field. When restricting to random sets, the obtained results give generalizations of well-known formulas from stochastic geometry and mathematical morphology. The interest of these general results is illustrated by computing the variation intensities of several classical stationary random field and random set models, namely Gaussian random fields and excursion sets, Poisson shot noises, Boolean models, dead leaves models, and random tessellations.
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Puplinskaitė D, Surgailis D. Aggregation of autoregressive random fields and anisotropic long-range dependence. BERNOULLI 2016. [DOI: 10.3150/15-bej733] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Abstract
AbstractWe obtain a complete description of anisotropic scaling limits of the random grain model on the plane with heavy-tailed grain area distribution. The scaling limits have either independent or completely dependent increments along one or both coordinate axes and include stable, Gaussian, and ‘intermediate’ infinitely divisible random fields. The asymptotic form of the covariance function of the random grain model is obtained. Application to superimposed network traffic is included.
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Biermé H, Lacaux C. Modulus of continuity of some conditionally sub-Gaussian fields, application to stable random fields. BERNOULLI 2015. [DOI: 10.3150/14-bej619] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Puplinskaitė D, Surgailis D. Scaling transition for long-range dependent Gaussian random fields. Stoch Process Their Appl 2015. [DOI: 10.1016/j.spa.2014.12.011] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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Li Y, Wang W, Xiao Y. Exact moduli of continuity for operator-scaling Gaussian random fields. BERNOULLI 2015. [DOI: 10.3150/13-bej593] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Roux SG, Clausel M, Vedel B, Jaffard S, Abry P. Self-similar anisotropic texture analysis: the hyperbolic wavelet transform contribution. IEEE TRANSACTIONS ON IMAGE PROCESSING : A PUBLICATION OF THE IEEE SIGNAL PROCESSING SOCIETY 2013; 22:4353-4363. [PMID: 23864204 DOI: 10.1109/tip.2013.2272515] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
Textures in images can often be well modeled using self-similar processes while they may simultaneously display anisotropy. The present contribution thus aims at studying jointly selfsimilarity and anisotropy by focusing on a specific classical class of Gaussian anisotropic selfsimilar processes. It will be first shown that accurate joint estimates of the anisotropy and selfsimilarity parameters are performed by replacing the standard 2D-discrete wavelet transform with the hyperbolic wavelet transform, which permits the use of different dilation factors along the horizontal and vertical axes. Defining anisotropy requires a reference direction that needs not a priori match the horizontal and vertical axes according to which the images are digitized; this discrepancy defines a rotation angle. Second, we show that this rotation angle can be jointly estimated. Third, a nonparametric bootstrap based procedure is described, which provides confidence intervals in addition to the estimates themselves and enables us to construct an isotropy test procedure, which can be applied to a single texture image. Fourth, the robustness and versatility of the proposed analysis are illustrated by being applied to a large variety of different isotropic and anisotropic self-similar fields. As an illustration, we show that a true anisotropy built-in self-similarity can be disentangled from an isotropic self-similarity to which an anisotropic trend has been superimposed.
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Benson DA, Meerschaert MM, Revielle J. Fractional calculus in hydrologic modeling: A numerical perspective. ADVANCES IN WATER RESOURCES 2013; 51:479-497. [PMID: 23524449 PMCID: PMC3603590 DOI: 10.1016/j.advwatres.2012.04.005] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
Fractional derivatives can be viewed either as handy extensions of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Lévy motion, and that fractional integration is the corresponding inverse operator. Fractional integration, and its multi-dimensional extensions derived in this way, are intimately tied to fractional Brownian (and Lévy) motions and noises. By following these general principles, we discuss the Eulerian and Lagrangian numerical solutions to fractional partial differential equations, and Eulerian methods for stochastic integrals. These numerical approximations illuminate the essential nature of the fractional calculus.
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Affiliation(s)
- David A. Benson
- Hydrological Science and Engineering, Colorado School of Mines, Golden, CO 80401, USA
| | - Mark M. Meerschaert
- Department of Statistics and Probability, Michigan State University, East Lansing, MI, USA
| | - Jordan Revielle
- Hydrological Science and Engineering, Colorado School of Mines, Golden, CO 80401, USA
- Ward Petroleum Corporation, Fort Collins, CO 80521, USA
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Meerschaert MM, Wang W, Xiao Y. Fernique-type inequalities and moduli of continuity for anisotropic Gaussian random fields. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 2012; 2013:1081-1107. [PMID: 24825922 PMCID: PMC4016971 DOI: 10.1090/s0002-9947-2012-05678-9] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
This paper is concerned with sample path properties of anisotropic Gaussian random fields. We establish Fernique-type inequalities and utilize them to study the global and local moduli of continuity for anisotropic Gaussian random fields. Applications to fractional Brownian sheets and to the solutions of stochastic partial differential equations are investigated.
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Didier G, Pipiras V. Integral representations and properties of operator fractional Brownian motions. BERNOULLI 2011. [DOI: 10.3150/10-bej259] [Citation(s) in RCA: 56] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Luan N, Xiao Y. Chung’s law of the iterated logarithm for anisotropic Gaussian random fields. Stat Probab Lett 2010. [DOI: 10.1016/j.spl.2010.08.016] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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Ayache A, Roueff F, Xiao Y. Linear fractional stable sheets: Wavelet expansion and sample path properties. Stoch Process Their Appl 2009. [DOI: 10.1016/j.spa.2008.06.004] [Citation(s) in RCA: 23] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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