Multitaper spectral estimation of power law processes

In many branches of science, particularly astronomy and geophysics, power spectra of the form f/sup -/spl beta//, where /spl beta/ is a positive power-law exponent, are common. This form of spectrum is characterized by a sharp increase in the spectral density as the frequency f decreases toward zero. A power spectrum analysis method that has proven very powerful wherever the spectrum of interest is detailed and/or varies rapidly with a large dynamic range is the multitaper method. With multitaper spectral estimation, a set of orthogonal tapers are applied to the time series, and the resulting direct spectral estimators ("eigenspectra") are averaged, thus, reducing the variance. One class of processes with spectra of the power-law type are fractionally differenced Gaussian processes that are stationary and can model certain types of long-range persistence. Spectral decay f/sup -/spl beta// can be modeled for 0</spl beta/<1. Estimation of the spectral slope parameter by regression on multitaper spectral ordinates is examined for this class of processes. It is shown that multitapering, or using sine or Slepian tapers, produces much better results than using the periodogram and is attractive compared with other competing methods. The technique is applied to a geophysical estimation problem.

[1]  Andrew T. Walden,et al.  An investigation of the spectral properties of primary reflection coefficients , 1985 .

[2]  Clifford M. Hurvich,et al.  AUTOMATIC SEMIPARAMETRIC ESTIMATION OF THE MEMORY PARAMETER OF A LONG‐MEMORY TIME SERIES , 1994 .

[3]  Donald B. Percival,et al.  Spectrum estimation by wavelet thresholding of multitaper estimators , 1998, IEEE Trans. Signal Process..

[4]  Donald B. Percival,et al.  The variance of multitaper spectrum estimates for real Gaussian processes , 1994, IEEE Trans. Signal Process..

[5]  I. McHardy,et al.  Fractal X-ray time variability and spectral invariance of the Seyfert galaxy NGC5506 , 1987, Nature.

[6]  Donald B. Percival,et al.  Spectral Analysis for Physical Applications , 1993 .

[7]  D. B. Percival,et al.  Characterization of frequency stability: frequency-domain estimation of stability measures , 1991, Proc. IEEE.

[8]  Paul F. Fougere,et al.  On the accuracy of spectrum analysis of red noise processes using maximum entropy and periodogram methods: Simulation studies and application to geophysical data , 1985 .

[9]  A. Raftery,et al.  Space-time modeling with long-memory dependence: assessing Ireland's wind-power resource. Technical report , 1987 .

[10]  J. Beran.  Fitting long-memory models by generalized linear regression , 1993 .

[11]  Kurt S. Riedel,et al.  Minimum bias multiple taper spectral estimation , 2018, IEEE Trans. Signal Process..

[12]  D. Thomson,et al.  Spectrum estimation and harmonic analysis , 1982, Proceedings of the IEEE.

[13]  J. Geweke,et al.  THE ESTIMATION AND APPLICATION OF LONG MEMORY TIME SERIES MODELS , 1983 .

[14]  Yoshihiro Yajima,et al.  A CENTRAL LIMIT THEOREM OF FOURIER TRANSFORMS OF STRONGLY DEPENDENT STATIONARY PROCESSES , 1989 .

[15]  R. Kashyap,et al.  ESTIMATION IN LONG‐MEMORY TIME SERIES MODEL , 1988 .

[16]  Peter Harris,et al.  Scattering attenuation in sediments modeled by ARMA processes-Validation of simple Q models , 1994 .

[17]  Kurt S. Riedel.  Adaptive smoothing of the log-spectrum with multiple tapering , 1996, IEEE Trans. Signal Process..

[18]  Georges Oppenheim,et al.  LONG-RANGE DEPENDENCE AND MIXING FOR DISCRETE TIME FRACTIONAL PROCESSES , 1995 .

[19]  S. K. Mayo,et al.  The influence of reflection coefficient statistics on the seismic method: scattering and the minimum detectable reflection coefficient , 1993, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[20]  P. Robinson.  Log-Periodogram Regression of Time Series with Long Range Dependence , 1995 .

[21]  J. Hosking.  Modeling persistence in hydrological time series using fractional differencing , 1984 .

[22]  Bovas Abraham,et al.  LAG WINDOW ESTIMATION OF THE DEGREE OF DIFFERENCING IN FRACTIONALLY INTEGRATED TIME SERIES MODELS , 1994 .

[23]  John B. Carlin,et al.  On models and methods for Bayesian time series analysis , 1985 .

[24]  D. B. Preston.  Spectral Analysis and Time Series , 1983 .

[25]  Oliver G. Jensen,et al.  Joseph geology and seismic deconvolution , 1988 .

[26]  Mohamed A. Deriche,et al.  Maximum likelihood estimation of the parameters of discrete fractionally differenced Gaussian noise process , 1993, IEEE Trans. Signal Process..

[27]  Richard A. Davis,et al.  Time Series: Theory and Methods (2nd ed.). , 1992 .

[28]  Walter Munk,et al.  The rotation of the earth , 1960 .

[29]  Donald B. Percival,et al.  The effective bandwidth of a multitaper spectral estimator , 1995 .

[30]  A. Holden.  Models of the stochastic activity of neurones , 1976 .

[31]  V. Reisen,et al.  ESTIMATION OF THE FRACTIONAL DIFFERENCE PARAMETER IN THE ARIMA(p, d, q) MODEL USING THE SMOOTHED PERIODOGRAM , 1994 .

[32]  D. Thomson.  Quadratic-inverse spectrum estimates: applications to palaeoclimatology , 1990, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[33]  A. Walden,et al.  Calculating Thomson's Spectral Multitapers by Inverse Iteration , 1993 .

[34]  C. Granger,et al.  Spectral Analysis for Economic Time Series , 1964 .

[35]  Andrew T. Walden,et al.  The influence of reflection coefficient statistics on the seismic method: scattering attenuation and transmission wavelets , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[36]  C. Hurvich,et al.  ASYMPTOTICS FOR THE LOW‐FREQUENCY ORDINATES OF THE PERIODOGRAM OF A LONG‐MEMORY TIME SERIES , 1993 .

[37]  D. Kendall,et al.  The Statistical Analysis of Variance‐Heterogeneity and the Logarithmic Transformation , 1946 .

[38]  P. Newbold,et al.  BIAS IN AN ESTIMATOR OF THE FRACTIONAL DIFFERENCE PARAMETER , 1993 .