A Bayesian Monte Carlo Markov Chain Method for Parameter Estimation of Fractional Differenced Gaussian Processes

We present a Bayesian Monte Carlo Markov Chain method to simultaneously estimate the spectral index and power amplitude of a fractional differenced Gaussian process at low frequency, in presence of white noise, and a linear trend and periodic signals. This method provides a sample of the likelihood function and thereby, using Monte Carlo integration, all parameters and their uncertainties are estimated simultaneously. We test this method with simulated and real Global Positioning System height time series and propose it as an alternative to optimization methods currently in use. Furthermore, without any mathematical proof, the results from the simulations suggest that this method is unaffected by the stationary regime and hence, can be used to check whether or not a time series is stationary.

[1]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[2]  Peter Guttorp,et al.  Space‐Time Modelling of Trends in Temperature Series , 2011 .

[3]  D. Agnew,et al.  The time-domain behavior of power-law noises. [of many geophysical phenomena] , 1992 .

[4]  Judith Rousseau,et al.  Bayesian nonparametric estimation of the spectral density of a long or intermediate memory Gaussian process , 2010, 1007.3823.

[5]  Brian D. Rigling.  Maximum Likelihood Estimation of Band-Limited Power Law Spectrums , 2012, IEEE Signal Processing Letters.

[6]  Yehuda Bock,et al.  Error analysis of continuous GPS position time series , 2004 .

[7]  V. Yohai,et al.  Robust Statistics: Theory and Methods , 2006 .

[8]  Geoffrey Blewitt,et al.  Crustal displacements due to continental water loading , 2001 .

[9]  Manfredo Massironi,et al.  Average Strain Rate In The Italian Crust Inferred From A Permanent GPS Network , 2001 .

[10]  P. Lafrance,et al.  Digital filters , 1974, Proceedings of the IEEE.

[11]  Fallaw Sowell.  Maximum likelihood estimation of stationary univariate fractionally integrated time series models , 1992 .

[12]  U. Hugentobler,et al.  Estimation of velocity uncertainties from GPS time series: Examples from the analysis of the South African TrigNet network , 2011 .

[13]  Juliette Marais,et al.  Dirichlet Process Mixtures for Density Estimation in Dynamic Nonlinear Modeling: Application to GPS Positioning in Urban Canyons , 2012, IEEE Transactions on Signal Processing.

[14]  J. R. Wallis,et al.  Computer Experiments with Fractional Gaussian Noises: Part 2, Rescaled Ranges and Spectra , 1969 .

[15]  W. Michael Conklin,et al.  Monte Carlo Methods in Bayesian Computation , 2001, Technometrics.

[16]  Peter Green,et al.  Markov chain Monte Carlo in Practice , 1996 .

[17]  Carlos Velasco Gómez.  Gaussian semiparametric estimation of non-stationary time series , 1998 .

[18]  Fast and reliable MCMC for cosmological parameter estimation , 2004, astro-ph/0405462.

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

[20]  D. McLaughlin,et al.  Crustal motions in Great Britain: evidence from continuous GPS, absolute gravity and Holocene sea level data , 2009 .

[21]  Bradley P. Carlin,et al.  Markov Chain Monte Carlo conver-gence diagnostics: a comparative review , 1996 .

[22]  Tianshuang Qiu,et al.  On the robustness of Hurst estimators , 2011 .

[23]  J. Rousseau,et al.  ASYMPTOTIC THEORY FOR MAXIMUM LIKELIHOOD ESTIMATION OF THE MEMORY PARAMETER IN STATIONARY GAUSSIAN PROCESSES , 2011, Econometric Theory.

[24]  Geoffrey Blewitt,et al.  Effect of annual signals on geodetic velocity , 2002 .

[25]  Yehuda Bock,et al.  Southern California permanent GPS geodetic array: Error analysis of daily position estimates and site velocities , 1997 .

[26]  J. R. Wallis,et al.  Computer Experiments With Fractional Gaussian Noises: Part 1, Averages and Variances , 1969 .

[27]  Rui Fernandes,et al.  The influence of seasonal signals on the estimation of the tectonic motion in short continuous GPS time-series , 2010 .

[28]  Simon D. P. Williams,et al.  CATS: GPS coordinate time series analysis software , 2008 .

[29]  Simon D. P. Williams,et al.  Fast error analysis of continuous GPS observations , 2008 .

[30]  A. Amiri-Simkooei,et al.  Noise in multivariate GPS position time-series , 2009 .

[31]  John Langbein,et al.  Correlated errors in geodetic time series: Implications for time‐dependent deformation , 1997 .

[32]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[33]  Brunero Liseo,et al.  Bayesian semiparametric inference on long-range dependence , 2001 .

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

[35]  C. Velasco,et al.  Non-stationary log-periodogram regression , 1999 .

[36]  C. Skordis,et al.  Fast and reliable Markov chain Monte Carlo technique for cosmological parameter estimation , 2005 .

[37]  Gary Koop,et al.  Bayesian Analysis of Long Memory and Persistence using ARFIMA Models , 1995 .

[38]  Jan Beran,et al.  On a class of M-estimators for Gaussian long-memory models , 1994 .

[39]  R. Dahlhaus.  Efficient parameter estimation for self-similar processes , 1989, math/0607078.

[40]  T. Dixon,et al.  Noise in GPS coordinate time series , 1999 .

[41]  Jan Beran,et al.  Statistics for long-memory processes , 1994 .