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Power-law correlations, related models for long-range dependence and their simulation

Part of: Inference from stochastic processes Stochastic processes Harmonic analysis in one variable

Published online by Cambridge University Press:  14 July 2016

Tilmann Gneiting
Tilmann Gneiting*
Affiliation:
University of Washington
*
∗ Postal address: Department of Statistics, Box 354322, University of Washington, Seattle, Washington 98195-4322, USA.
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Abstract

Martin and Walker ((1997) J. Appl. Prob.34, 657–670) proposed the power-law ρ(v) = c|v|-β, |v| ≥ 1, as a correlation model for stationary time series with long-memory dependence. A straightforward proof of their conjecture on the permissible range of c is given, and various other models for long-range dependence are discussed. In particular, the Cauchy family ρ(v) = (1 + |v/c|α)-β/α allows for the simultaneous fitting of both the long-term and short-term correlation structure within a simple analytical model. The note closes with hints at the fast and exact simulation of fractional Gaussian noise and related processes.

Keywords

Cauchy familycorrelation functionfractional noiselong-memorypower-lawsimulationtime series

MSC classification

Primary: 62M10: Time series, auto-correlation, regression, etc.
Secondary: 42A32: Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.) 60G10: Stationary processes 60G18: Self-similar processes
Type
Short Communications
Information
Journal of Applied Probability , Volume 37 , Issue 4 , December 2000 , pp. 1104 - 1109
Copyright
Copyright © by the Applied Probability Trust 2000 

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References

Beran, J. (1992). Statistical methods for data with long-range dependence (with discussion). Statist. Sci. 4, 404–427.Google Scholar
Beran, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall, London.Google Scholar
Bras, R. L. and Rodríguez-Iturbe, I. (1984). Random Functions and Hydrology. Addison-Wesley, Reading.Google Scholar
Caccia, D. C., Percival, D., Cannon, M. J., Raymond, G., and Bassingthwaighte, J. B. (1997). Analyzing exact fractal time series: evaluating dispersional analysis and rescaled range methods. Physica A 246, 609–632.CrossRefGoogle ScholarPubMed
Chan, G., and Wood, A. T. A. (1997). An algorithm for simulating stationary Gaussian random fields. Appl. Statist. 46, 171–181.Google Scholar
Chilès, J.-P., and Delfiner, P. (1999). Geostatistics. Modeling Spatial Uncertainty. John Wiley, New York.CrossRefGoogle Scholar
Davies, R. B., and Harte, D. S. (1987). Tests for Hurst effect. Biometrika 74, 95–101.CrossRefGoogle Scholar
Dietrich, C. R., and Newsam, G. N. (1997). Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix. SIAM J. Sci. Comput. 18, 1088–1107.CrossRefGoogle Scholar
Gneiting, T. (1997). Normal scale mixtures and dual probability densities. J. Statist. Comput. Simulation 59, 375–384.CrossRefGoogle Scholar
Granger, C. W. J., and Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. J. Time Ser. Anal. 1, 15–29.CrossRefGoogle Scholar
Haslett, J., and Raftery, A. E. (1989). Space–time modelling with long-memory dependence (with discussion). J. R. Statist. Soc. Ser. C Appl. Statist. 38, 1–50.Google Scholar
Hosking, J. R. M. (1981). Fractional differencing. Biometrika 68, 165–172.CrossRefGoogle Scholar
Isaacson, E., and Keller, H. B. (1966). Analysis of Numerical Methods. John Wiley, New York.Google Scholar
Mandelbrot, B. B. (1971). A fast fractional Gaussian noise generator. Water Resources Res. 7, 543–553.CrossRefGoogle Scholar
Martin, R. J., and Walker, A. M. (1997). A power-law model and other models for long-range dependence. J. Appl. Prob. 34, 657–670.CrossRefGoogle Scholar
Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I. (1986). Integrals and Series, Vol. 1. Gordon and Breach, New York.Google Scholar
Wackernagel, H. (1998). Multivariate Geostatistics, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
Wintner, A. (1947). The Bernoullian Fourier diagrams. Phil. Mag. (7) 38, 495–504.CrossRefGoogle Scholar
Wintner, A. (1957). On the reduction (mod 1) of completely monotone functions (0,∞). Ann. Mat. Pura Appl. (4) 43, 299–312.CrossRefGoogle Scholar