Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T02:10:10.706Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalised spectral analysis of fractional random fields

Part of: Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  09 April 2009

V. V. Anh  and
K. E. Lunney
V. V. Anh
Affiliation:
School of Mathematics Queensland University of TechnologyGPO Box 2434 Brisbane, QLD 4001, Australia
K. E. Lunney
Affiliation:
School of Australian Environmental Studies Griffith UniversityNathan Brisbane, QLD 4111, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper considers a large class of non-stationary random fields which have fractal characteristics and may exhibit long-range dependence. Its motivation comes from a Lipschitz-Holder-type condition in the spectral domain.

The paper develops a spectral theory for the random fields, including a spectral decomposition, a covariance representation and a fractal index. From the covariance representation, the covariance function and spectral density of these fields are defined. These concepts are useful in multiscaling analysis of random fields with long-range dependence.

MSC classification

Secondary: 60G60: Random fields 62M15: Spectral analysis
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 62 , Issue 1 , February 1997 , pp. 64 - 83
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Adler, R. J., The geometry of random fields (Wiley, New York, 1981).Google Scholar
[2]Anh, V. V. and Lunney, K. E., ‘Covariance function and ergodicity of asymptotically stationary random fields’, Bull. Austral. Math. Soc. 44 (1991), 49–62.CrossRefGoogle Scholar
[3]Anh, V. V. and Lunney, K. E., ‘Generalised harmonic analysis of a class of nonstationary random fields’, J. Austral. Math. Soc. (Series A) 53 (1992), 385–401.CrossRefGoogle Scholar
[4]Anh, V. V. and Lunney, K. E., ‘Parameter estimation of random fields with long-range dependence’, Math. Comput. Modelling 21 (1995), 67–77.CrossRefGoogle Scholar
[5]Anh, V. V. and Lunney, K. E., ‘Multiscaling analysis of time series with long-range dependence’, (submitted), 1995.Google Scholar
[6]Anh, V. V., Gras, F. and Tsui, H.T., ‘A class of fractional random fields for image texture modelling’, in: Proc. 1994 International Symposium on Speech, Image Processing and Neural Networks(The IEEE Hong Kong Chapter of Signal Processing). (1994) pp. 300–303.Google Scholar
[7]Beran, J., ‘Statistical methods for data with long-range dependence’, Statist. Sci. 7 (1992), 404–427.Google Scholar
[8]Beurling, A., ‘Construction and analysis of some convolution algebras’, Ann. Inst. Fourier (Grenoble) 14 (1994), 1–32.CrossRefGoogle Scholar
[9]Flandrin, P., ‘On the spectrum of fractional Brownian motion’, Bull. Inst. Internat. Statist. 52 (1989), 197–199.Google Scholar
[10]Hampel, F. R., ‘Data analysis and self-similar processes’, Bull. Inst. Internat. Statist. 52 (1987), 235–254.Google Scholar
[11]Haslett, J. and Raftery, A. E., ‘Space-time modelling with long-memory dependence: Assessing Ireland's wind power resource’, Appl. Statist. 38 (1989), 1–50.CrossRefGoogle Scholar
[12]Henninger, J., ‘Functions of bounded mean square and generalized Fourier-Stieltjes transforms’, Canad. J. Math. 22 (1970), 1016–1034.CrossRefGoogle Scholar
[13]Heyde, C. C. and Gay, R., ‘Smoothed periodogram asymptotics and estimation for processes and fields with possible long-range dependence’, Stochastic Process. Appl. 45 (1993), 169–182.CrossRefGoogle Scholar
[14]Lau, K. S., ‘Fractal measures and mean p-variations’, J. Funct. Anal. 108 (1992), 427–457.CrossRefGoogle Scholar
[15]Lau, K. S. and Wang, J., ‘Mean quadratic variations and Fourier asymptotics of self-similar measures’, Monatsh. Math. 115 (1993), 99–132.CrossRefGoogle Scholar
[16]Mandelbrot, B. and van Ness, J. W., ‘Fractional Brownian motion, fractional noises and applications’, SIAM Rev. 10 (1968), 422–437.CrossRefGoogle Scholar
[17]Rudin, W., Functional analysis (McGraw-Hill, New York, 1973).Google Scholar
[18]Solo, V., ‘Intrinsic random functions and the paradox of 1/f noise’, SIAM J. Appl. Math. 52 (1992), 270–291.CrossRefGoogle Scholar
[19]Strichartz, R., ‘Fourier asymptotics of fractal measures’, J. Funct. Anal. 89 (1990), 154–187.CrossRefGoogle Scholar
[20]Wiener, N., The Fourier integral and certain of its applications (Dover, New York, 1958).Google Scholar
[21]Wornell, G. W., ‘Wavelet-based representations for the 1/f family of fractal processes’, Proc. IEEE 81 (1993), 1428–1450.CrossRefGoogle Scholar