(Local) self-similarity is a seminal concept, especially for Euclidean random fields. We study in this paper the extension of these notions to manifold indexed fields. We give conditions on the (local) self-similarity index that ensure the existence of fractional fields. Moreover, we explain how to identify the self-similar index. We describe a way of simulating Gaussian fractional fields.
Mots-clés : self-similarity, stochastic fields, manifold
@article{PS_2012__16__222_0,
author = {Istas, Jacques},
title = {Manifold indexed fractional fields},
journal = {ESAIM: Probability and Statistics},
pages = {222--276},
publisher = {EDP-Sciences},
volume = {16},
year = {2012},
doi = {10.1051/ps/2011106},
mrnumber = {2956575},
zbl = {1275.60041},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2011106/}
}
Istas, Jacques. Manifold indexed fractional fields. ESAIM: Probability and Statistics, Tome 16 (2012), pp. 222-276. doi : 10.1051/ps/2011106. http://archive.numdam.org/articles/10.1051/ps/2011106/
[1] , and , Wavelets, spectrum analysis and 1 / f processes. Lect. Note Stat. 103 (1995) 15-29. | Zbl
[2] and , The Multifractional Brownian motion. Stat. Inference Stoch. Process. 1 (2000) 7-18. | MR | Zbl
[3] and , On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion. Stoc. Proc. Appl. 111 (2004) 119-156. | MR | Zbl
[4] , and , A central limit theorem for the generalized quadratic variation of the step fractional Brownian. Stat. Inference Stoch. Process. 10 (2007) 1-27. | MR | Zbl
[5] , Testing for the presence of self-similarity of Gaussian time series having stationary increments. J. Time Ser. Anal. 25 (2000) 497-515. | MR | Zbl
[6] and , Identification of the multiscale fractional Brownian motion with biomechanical applications. J. Time Ser. Anal. 28 (2007) 1-52. | MR | Zbl
[7] , and , Kazhdan's Property (T). Cambridge University Press (2008). | MR | Zbl
[8] , and , Gaussian processes and Pseudodifferential Elliptic operators. Revista Mathematica Iberoam. 13 (1997) 19-90. | EuDML | MR | Zbl
[9] , and , Identifying the multifractional function of a Gaussian process. Stat. Probab. Lett. 39 (1998) 337-345. | MR | Zbl
[10] , , and , Identification of filtered white noises. Stoc. Proc. Appl. 75 (1998) 31-49. | MR | Zbl
[11] , , and , Identification of the Hurst index of a step fractional Brownian motion. Stat. Inference Stoch. Process 3 (2000) 101-111. | MR | Zbl
[12] , and , Identification and properties of real harmonizable fractional Lévy motions. Bernoulli 8 (2002) 97-115. | MR | Zbl
[13] , and , On roughness indices for fractional fields. Bernoulli 10 (2004) 357-373. | MR | Zbl
[14] , Quadratic variations along irregular subdivisions for Gaussian processes. Electron. J. Probab. 10 (2005) 691-717. | EuDML | MR | Zbl
[15] , Asymptotic expansion and central limit theorem for quadratic variations of Gaussian processes. Bernoulli 13 (2007) 712-753. | MR | Zbl
[16] , Functional limit theorems for generalized quadratic variations of Gaussian processes. Stoc. Proc. Appl. 117 (2007) 1848-1869. | MR | Zbl
[17] and , Estimating the Hurst parameter. Stat. Inference Stock. Process. 10 (2007) 49-73. | MR | Zbl
[18] and , Anisotropic analysis of Gaussian models. J. Fourier Anal. Appl. 9 (2004) 215-236. | MR | Zbl
[19] , and , On the angular defect of triangulations and the pointwise approximation of curvatures, curves and surfaces'02. Comput. Aid. Geom. Des. 20 319-341. | MR | Zbl
[20] , and , Lois stables et espaces Lp. Ann. Inst. Henri Poincaré 2 (1969) 231-259. | EuDML | Numdam | MR | Zbl
[21] , and , On fractional Gaussian random fields simulation. J. Stat. Soft. 1 (2007) 1-23.
[22] , and , On simulation of fractional Brownian motion indexed by a manifold. J. Stat. Soft. 36 (2010).
[23] , Lévy's Brownian motion of several parameters and generalized white noise. Theory Probab. Appl. 2 (1957) 265-266.
[24] , Simulation and identification of the fractional Brownian motion : a bibliographical and comparative study. J. Stat. Software 5 (2000) 1-53.
[25] , Estimating the parameters of a fractional Brownian Motion by discrete variations of its sample paths. Stat. Inference Stoch. Process. 4 (2001) 199-227. | MR | Zbl
[26] , Identification of multifractional Brownian motion. Bernoulli 11 (2005) 987-1008. | MR | Zbl
[27] , Hurst exponent estimation of locally self-similar Gaussian processes using sample quantiles. Ann. Statist. 36 (2008) 1404-1434. | MR | Zbl
[28] and , Cramer-Rao bounds for fractional Brownian motions. Stat. Probab. Lett. 53 (2001) 435-447. | MR | Zbl
[29] , From self-similarity to local self-similarity : the estimation problem. Fractal in Engineering, edited by J. Lévy-Vehel and C. Tricot. Springer Verlag, Delft (1999). | MR | Zbl
[30] and , An universal estimator of local self-similarity. Preprint (2006).
[31] and , Fractional fields : Modelling and statistical applications (Submitted).
[32] and , Stationary Gaussian random fields on hyperbolic spaces and Euclidean spheres. To appear in ESAIM : PS. | EuDML | Numdam | MR | Zbl
[33] , , and , Singularity functions for fractional processes : application to the fractional brownian sheet. Ann. Inst. Henri Poincaré 42 (2006) 187-205. | EuDML | Numdam | MR | Zbl
[34] and , Probabilités et Statistiques tome 2. Masson, Paris (1983). | MR | Zbl
[35] , Efficient parameter estimation for self-similar processes. Ann. Statist. 17 (1989) 1749-1766. | MR | Zbl
[36] , Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41 (1988) 909-996. | MR | Zbl
[37] and , Partial autocorrelation function of a nonstationary time series. J. Multiv. Anal. (2003) 46-59. | MR | Zbl
[38] , Automodel generalized random fields and their renorm group, in Multicomponent Random Systems, edited by R.L. Dobrushin and Ya. G. Sinai. Dekker, New York (1980) 153-198. | MR | Zbl
[39] , and , T-theory : An overview, Eur. J. Comb. 17 (1996) 161-175. | MR | Zbl
[40] , , and , Higher transcendental functions (Bateman manuscript project). McGraw-Hill 2 (1953) | MR | Zbl
[41] , Tangent fields and the local structure of random fields. J. Theor. Probab. 15 (2002) 731-750. | MR | Zbl
[42] , The local structure of random processes. J. Lond. Math. Soc. 67 (2003) 657-672. | MR | Zbl
[43] , Fonction brownienne sur une variété riemannienne. Séminaire de probabilités de Strasbourg 7 (1973) 61-76. | EuDML | Numdam | MR | Zbl
[44] and , Distances hilbertiennes invariantes sur un espace homogène. Ann. Inst. Fourier 24 (1974) 171-217. | EuDML | Numdam | MR | Zbl
[45] , and , Riemannian Geometry, 2nd edition. Springer-Verlag (1993). | Zbl
[46] , Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy's Brownian motion of several parameters. Ann. Inst. Henri Poincaré 3 (1967) 121-226. | EuDML | Numdam | MR | Zbl
[47] and , Convergence en loi des H-variations d'un processus gaussien stationnaire. Ann. Inst. Henri Poincaré 25 (1989) 265-282. | EuDML | Numdam | MR | Zbl
[48] , Differential Geometry and Symmetric spaces. Academic Press (1962). | MR | Zbl
[49] and , A set-indexed fractional Brownian motion. J. Theor. Probab. 19 (2006) 337-364. | MR | Zbl
[50] and , Stationarity and self-similarity characterization of the set-indexed fractional Brownian motion. J. Theor. Probab. 22 (2009) 1010-1029. | MR | Zbl
[51] , Spherical and hyperbolic fractional Brownian motion. Electron. Comm. Probab. 10 (2005) 254-262. | EuDML | MR | Zbl
[52] , On fractional stable fields indexed by metric spaces. Electron. Comm. Probab. 11 (2006) 242-251. | EuDML | MR | Zbl
[53] , Karhunen-Loève expansion of spherical fractional Brownian motions. Stat. Probab. Lett. 76 (2006) 1578-1583. | MR | Zbl
[54] , Quadratic variations of spherical fractional Brownian motions, Stoc. Proc. Appl. 117 (2007) 476-486. | MR | Zbl
[55] , Identifying the anisotropical function of a d-dimensional Gaussian self-similar process with stationary increments. Stat. Inf. Stoc. Proc. 10-1 (2007) 97-106. | MR | Zbl
[56] and , On locally self-similar fractional random fields indexed by a manifold. preprint. | MR | Zbl
[57] and , Variations quadratiques et estimation de l'exposant de Hölder local d'un processus gaussien. C. R. Acad. Sci. Sér. I Paris 319 (1994) 201-206. | MR | Zbl
[58] and , Quadratic variations and estimation of the Hölder index of a Gaussian process. Ann. Inst. Henri Poincaré 33 (1997) 407-436. | EuDML | Numdam | MR | Zbl
[59] and , Estimating the fractal dimension of a locally self-similar Gaussian process using increments. J. Roy. Statist. Soc. B 59 (1997) 679-700. | MR | Zbl
[60] , Schoenberg's problem on positive definite functions. Algebra Anal. 3 (1991) 78-85. | MR | Zbl
[61] and , A short proof of Schoenberg's conjecture on positive definite functions. Bull. Lond. Math. Soc. (1999) 693-699. | MR | Zbl
[62] , Wienersche Spiralen und einige andere interessante Kurven im Hilbertsche Raum (German). C. R. (Dokl.) Acad. Sci. URSS 26 (1940) 115-118. | MR | Zbl
[63] , Real harmonizable multifractional Lévy motions. Ann. Inst. Henri Poincaré 40 (2004) 259-277. | EuDML | Numdam | MR | Zbl
[64] and , Semi-parametric estimation of the Hölder exponent of a stationary Gaussian process with minimax rates. Stat. Inf. Stoc. Proc. 4-3 (2001) 283-306. | MR | Zbl
[65] , Processus stochastiques et mouvement Brownien. Gauthier-Vilars (1965). | Zbl
[66] , Fractional Brownian fields as integrals of white noise. Bull. Lond. Math. Soc. 25 (1993) 83-88. | MR | Zbl
[67] , A remark on self-similar processes with stationary increments. Can. J. Stat. 14 (1986) 81-82. | MR | Zbl
[68] and , Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10 (1968) 422-437. | MR | Zbl
[69] and , Multifractional Brownian motion : definition and preliminary results. Rapport de recherche de l'INRIA 2645 (1996).
[70] , Riemannian Geometry. Graduate Texts in Mathematics, Springer (1998). | MR | Zbl
[71] , Testing (non-)existence of input-output relationships by estimating fractal dimensions. IEEE Trans. Signal Process. 52 (2004) 3151-3159. | MR
[72] , Crofton formulae and geodesic distance in hyperbolic spaces. J. Lie Theory 8 (1998) 163-172. | EuDML | MR | Zbl
[73] and , Negative definite kernels and a dynamical characterization of property T for countable groups. Ergod. Theory Dyn. Syst. 18 (1998) 247-253. | MR | Zbl
[74] , Fourier analysis on groups. Wiley (1962). | MR | Zbl
[75] , Long memory and self-similar processes. Annales de la Faculté des Sciences Toulouse 15 (2006) 107-123. | EuDML | Numdam | MR | Zbl
[76] and , Stable non-Gaussian random processes : stochastic models with infinite variance. Chapman & Hall, New York (1994). | MR | Zbl
[77] , Metric spaces and positive definite functions. Ann. Math. 39 (1938) 811-841. | JFM | Zbl
[78] , Spherical harmonics. Am. Math. Mon. 73 (1966) 115-121. | MR | Zbl
[79] and , Stochastic properties of the linear multifractional stable motion. Adv. Appl. Prob. 36 (2004) 1085-1115. | MR | Zbl
[80] , Orthogonal Polynomials, 4th edition, in Amer. Math. Soc. Providence, RI (1975). | MR
[81] , Integral-geometric construction of self-similar stable processes. Nagoya Math. J. 123 (1991) 1-12. | MR | Zbl
[82] , and , Brownian motion parametrized with metric space of constant curvature. Nagoya Math. J. 82 (1981) 131-140. | MR | Zbl
[83] , Les représentations uniformément bornées associées à un arbre réel. Bull. Soc. Math. Belgique 42 (1990) 747-760. | MR | Zbl
[84] , Two-point homogeneous spaces. Ann. Math. 2 (1952) 177-191. | MR | Zbl
[85] , Some classes of random fields in n-dimensional space, related to stationary random processes. Theory Probab. Appl. 2 (1957) 273-320.
[86] , Linear Anal. North Holland Publishing Co (1960).
Cité par Sources :
