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

Abstract

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.

Markov Chain Monte Carlo and the Application to Geodetic Time Series Analysis

“…The better correlation shown at high values in Fig. 7 suggests that, like the MCMC method, CATS is also a good estimator of the spectral index for non-stationary time series (Olivares and Teferle, 2013).…”

“…The synthetic time series were generated in order to assess the MCMC method through investigating the ability of the algorithm to recover the input values when generating the time series. Aspects of this analysis were published in Olivares and Teferle (2013).…”

“…Here we provide an evaluation of the MCMC method as implemented by Olivares and Teferle (2013) using Hector and the Benchmark Synthetic GNSS (BSG) time series. While providing the results from MCMC by themselves, we also carry out a basic comparison with the results provided and computed in-house with Hector.…”

Coloured noise effects on deformation parameters of permanent GPS networks

“…In addition, several studies have shown the existence of random walk noise in geodetic data (Wyatt 1982(Wyatt , 1989Wyatt et al 1989;Johnson & Agnew 2000;Caporali et al 2003). There are also research ongoing in which Gaussian noise processes have been identified and estimated in geodetic timeseries analysis (Olivares & Teferle 2013). Another active field of research in permanent GPS networks is the strain analysis of deformation parameters (Caporali et al 2003).…”

Complex-valued Bayesian parameter estimation via Markov chain Monte Carlo

“…The so-called 'Markov chain Monte Carlo (MCMC)' sampling, a kind of simulation method, is effective in handling these intractable integrals [13,25,29,32]. It can facilitate the implementation of Bayesian analysis of complex data sets containing missing observations and multidimensional outcomes.…”