Article

Bayesian exoplanet tests of a new method for MCMC sampling in highly correlated model parameter spaces

Physics and Astronomy Department, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada
Monthly Notices of the Royal Astronomical Society (Impact Factor: 5.52). 12/2010; 410(1):94 - 110. DOI: 10.1111/j.1365-2966.2010.17428.x

ABSTRACT The Markov chain Monte Carlo (MCMC) method is a powerful technique for facilitating Bayesian non-linear model fitting. In many cases, the MCMC exploration of the parameter space is very inefficient, because the model parameters are highly correlated. Differential evolution MCMC is one technique that addresses this problem by employing multiple parallel chains. We present a new method that automatically achieves efficient MCMC sampling in highly correlated parameter spaces, which does not require additional chains to accomplish this. It was designed to work with an existing hybrid MCMC (HMCMC) algorithm, which incorporates parallel tempering, simulated annealing and genetic cross-over operations. These features, together with the new correlated parameter sampler, greatly facilitate the detection of a global minimum in χ2. The new HMCMC algorithm is very general in scope. Two tests of the algorithm are described employing (a) exoplanet precision radial velocity (RV) data and (b) simulated space astrometry data. The latter test explores the accuracy of parameter estimates obtained with the Bayesian HMCMC algorithm on the assumed astrometric noise.

0 Bookmarks
 · 
106 Views
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: A re-analysis of Gliese 667C HARPS precision radial velocity data was carried out with a Bayesian multi-planet Kepler periodogram (from 0 to 7 planets) based on a fusion Markov chain Monte Carlo algorithm. The most probable number of signals detected is 6 with a Bayesian false alarm probability of 0.012. The residuals are shown to be consistent with white noise. The 6 signals detected include two previously reported with periods of 7.198 (b) and 28.14 (c) days, plus additional periods of 30.82 (d), 38.82 (e), 53.22, and 91.3 (f) days. The 53 day signal is probably the second harmonic of the stellar rotation period and is likely the result of surface activity. The existence of the additonal Keplerian signals suggest the possibilty of further planets, two of which (d and e) could join Gl 667Cc in the central region of the habitable zone. N-body simulations are required to determine which of these signals are consistent with a stable planetary system. $M \sin i$ values corresponding to signals b, c, d, e, and f are $\sim$ 5.4, 4.8, 3.1, 2.4, and 5.4 M$_{\earth}$, respectively.
    12/2012;
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: A re-analysis of Gliese 581 HARPS and HIRES precision radial velocity data was carried out with a Bayesian multi-planet Kepler periodogram (from 1 to 6 planets) based on a fusion Markov chain Monte Carlo algorithm. In all cases the analysis included an unknown parameterized stellar jitter noise term. For the HARPS data set the most probable number of planetary signals detected is 5 with a Bayesian false alarm probability of 0.01. These include the $3.1498\pm0.0005$, $5.3687\pm0.0002$, $12.927_{-0.004}^{+0.006}$, and $66.9\pm0.2$d periods reported previously plus a $399_{-16}^{+14}$d period. The orbital eccentricities are $0.0_{-0.0}^{+0.2}$, $0.00_{-0.00}^{+0.02}$, $0.10_{-0.10}^{+0.06}$, $0.33_{-0.10}^{+0.09}$, and $0.02_{-0.02}^{+0.30}$, respectively. The semi-major axis and $M sin i$ of the 5 planets are ($0.0285\pm0.0006$ au, $1.9\pm0.3$M$_{\earth}$), ($0.0406\pm0.0009$ au, $15.7\pm0.7$M$_{\earth}$), ($0.073\pm0.002$ au, $5.3\pm0.4$M$_{\earth}$), ($0.218\pm0.005$ au, $6.7\pm0.8$M$_{\earth}$), and ($0.7\pm0.2$ au, $6.6_{-2.7}^{+2.0}$M$_{\earth}$), respectively. The analysis of the HIRES data set yielded a reliable detection of only the strongest 5.37 and 12.9 day periods. The analysis of the combined HIRES/HARPS data again only reliably detected the 5.37 and 12.9d periods. Detection of 4 planetary signals with periods of 3.15, 5.37, 12.9, and 66.9d was only achieved by including an additional unknown but parameterized Gaussian error term added in quadrature to the HIRES quoted errors. The marginal distribution for the sigma of this additional error term has a well defined peak at $1.8\pm0.4$m s$^{-1}$. It is possible that this additional error arises from unidentified systematic effects. We did not find clear evidence for a fifth planetary signal in the combined HIRES/HARPS data set.
    Monthly Notices of the Royal Astronomical Society 01/2011; · 5.52 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We describe work in progress by a collaboration of astronomers and statisticians developing a suite of Bayesian data analysis tools for extrasolar planet (exoplanet) detection, planetary orbit estimation, and adaptive scheduling of observations. Our work addresses analysis of stellar reflex motion data, where a planet is detected by observing the "wobble" of its host star as it responds to the gravitational tug of the orbiting planet. Newtonian mechanics specifies an analytical model for the resulting time series, but it is strongly nonlinear, yielding complex, multimodal likelihood functions; it is even more complex when multiple planets are present. The parameter spaces range in size from few-dimensional to dozens of dimensions, depending on the number of planets in the system, and the type of motion measured (line-of-sight velocity, or position on the sky). Since orbits are periodic, Bayesian generalizations of periodogram methods facilitate the analysis. This relies on the model being linearly separable, enabling partial analytical marginalization, reducing the dimension of the parameter space. Subsequent analysis uses adaptive Markov chain Monte Carlo methods and adaptive importance sampling to perform the integrals required for both inference (planet detection and orbit measurement), and information-maximizing sequential design (for adaptive scheduling of observations). We present an overview of our current techniques and highlight directions being explored by ongoing research.
    Statistical Methodology 07/2011; 9(1):101-114.

Full-text

View
4 Downloads
Available from