Multidimensional GPs#
In the original Gaussian process framework (Rajpaul et al. 2015, Rajpaul et al. 2021) the radial velocity datasets (\(\Delta \mathrm{RV}\), after removing the deterministic part) and two activity indicators (in this example, \(\mathrm{BIS}\) and \(\log{R^{\prime}_\mathrm{HK}}\)) are modeled as a liner combination of an underlying Gaussian process \(G(t)\) and its first derivative \(G^\prime (t)\).
PyORBIT implementation produces results that are perfectly consistent with the GP framework by Rajpaul et al. 2015 and the multidimensional GP by Barragán et al. 2022, as shown in Nardiello et al. 2022, appendix D. Note that the signs of the coefficients may vary according to the employed definition of \(\Delta t = (t_i-t_j)\).
Warning
In the previous version of the documentation and in several papers, it was reported that PyORBIT was relying on the quasi-periodic kernel definiton by Grunblatt et al. 2015. I came to realize only recently that the factor \(2\) accompanying the decay time-scale of the activity regions \(P_\mathrm{dec}\) was implicitely included in the exponenitial-squared kernel as defined in george and tinygp, making the kernel definiton identical to the one reported in Rajpaul et al. 2015. Please keep this problem in mind when comparing your results with other analysis.