The Annals of Statistics

Estimation of fractal dimension for a class of non-Gaussian stationary processes and fields

Grace Chan and Andrew T. A. Wood

Full-text: Open access

Abstract

We present the asymptotic distribution theory for a class of increment-based estimators of the fractal dimension of a random field of the form g{X(t)}, where g:RR is an unknown smooth function and X(t) is a real-valued stationary Gaussian field on Rd, d=1 or 2, whose covariance function obeys a power law at the origin. The relevant theoretical framework here is “fixed domain” (or “infill”) asymptotics. Surprisingly, the limit theory in this non-Gaussian case is somewhat richer than in the Gaussian case (the latter is recovered when g is affine), in part because estimators of the type considered may have an asymptotic variance which is random in the limit. Broadly, when g is smooth and nonaffine, three types of limit distributions can arise, types (i), (ii) and (iii), say. Each type can be represented as a random integral. More specifically, type (i) can be represented as the integral of a certain random function with respect to Lebesgue measure; type (ii) can be represented as the integral of a second random function with respect to an independent Gaussian random measure; and type (iii) can be represented as a Wiener–Itô integral of order 2. Which type occurs depends on a combination of the following factors: the roughness of X(t), whether d=1 or d=2 and the order of the increment which is used. Another notable feature of our results is that, even though the estimators we consider are based on a variogram, no moment conditions are required on the observed field g{X(t)} for the limit theory to hold. The results of a numerical study are also presented.

Article information

Source
Ann. Statist. Volume 32, Number 3 (2004), 1222-1260.

Dates
First available in Project Euclid: 24 May 2004

Permanent link to this document
https://projecteuclid.org/euclid.aos/1085408501

Digital Object Identifier
doi:10.1214/009053604000000346

Mathematical Reviews number (MathSciNet)
MR2065204

Zentralblatt MATH identifier
1046.62103

Subjects
Primary: 62M99: None of the above, but in this section
Secondary: 62E20: Asymptotic distribution theory

Keywords
Gaussian random measure Hermite polynomial increments stationary random field variogram Wiener–Itô integral

Citation

Chan, Grace; Wood, Andrew T. A. Estimation of fractal dimension for a class of non-Gaussian stationary processes and fields. Ann. Statist. 32 (2004), no. 3, 1222--1260. doi:10.1214/009053604000000346. https://projecteuclid.org/euclid.aos/1085408501.


Export citation

References

  • Adler, R. J. (1981). The Geometry of Random Fields. Wiley, New York.
  • Arcones, M. A. (1994). Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann. Probab. 22 2242--2274.
  • Chan, G., Hall, P. and Poskitt, D. S. (1995). Periodogram-based estimators of fractal properties. Ann. Statist. 23 1684--1711.
  • Chan, G. and Wood, A. T. A. (1999). Simulation of stationary Gaussian vector fields. Statist. Comput. 9 265--268.
  • Chan, G. and Wood, A. T. A. (2000). Increment-based estimators of fractal dimension for two-dimensional surface data. Statist. Sinica 10 343--376.
  • Constantine, A. G. and Hall, P. (1994). Characterising surface smoothness via estimation of effective fractal dimension. J. Roy. Statist. Soc. Ser. B 56 97--113.
  • Davies, S. and Hall, P. (1999). Fractal analysis of surface roughness by using spatial data (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 61 3--37.
  • Dobrushin, R. L. and Major, P. (1979). Noncentral limit theorems for nonlinear functions of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50 27--52.
  • Dudley, R. M. (1989). Real Analysis and Probability. Wadsworth, Belmont, CA.
  • Feuerverger, A., Hall, P. and Wood, A. T. A. (1994). Estimation of fractal index and fractal dimension of a Gaussian process by counting the number of level crossings. J. Time Ser. Anal. 15 587--606.
  • Hall, P. and Roy, R. (1994). On the relationship between fractal dimension and fractal index for stationary stochastic processes. Ann. Appl. Probab. 4 241--253.
  • Hall, P. and Wood, A. T. A. (1993). On the performance of box counting estimators of fractal dimension. Biometrika 80 246--252.
  • Istas, J. and Lang, G. (1997). Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. H. Poincaré Probab. Statist. 33 407--436.
  • Jakeman, E. and Jordan, D. L. (1990). Statistical accuracy of measurements on Gaussian random fractals. J. Phys. D. 23 397--405.
  • Kent, J. T. and Wood, A. T. A. (1995). Estimating the fractal dimension of a locally self-similar Gaussian process by using increments. Statistics Research Report SRR 034-95, Centre for Mathematics and Its Applications, Australian National Univ., Canberra.
  • Kent, J. T. and Wood, A. T. A. (1997). Estimating the fractal dimension of a locally self-similar Gaussian process by using increments. J. Roy. Statist. Soc. Ser. B 59 679--699.
  • Major, P. (1981). Multiple Wiener--Itô Integrals. Springer, Berlin.
  • Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, New York.
  • Rogers, L. C. G. and Williams, D. (1994). Diffusions, Markov Processes, and Martingales 1 Foundations, 2nd ed. Wiley, New York.
  • Stein, M. L. (1999). Interpolation of Spatial Data. Springer, New York.
  • Taqqu, M. S. (1977). Law of the iterated logarithm for sums of non-linear functions of Gaussian variables that exhibit a long range dependence. Z. Wahrsch. Verw. Gebiete 40 203--238.
  • Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50 53--83.
  • Wood, A. T. A. and Chan, G. (1994). Simulation of stationary Gaussian processes in $[0,1]^d$. J. Comput. Graph. Statist. 3 409--432.