Gaussian Process regression

Gaussian Process regression (GPR) is a nonparametric, Bayesian approach to regression that has been very successful in the analysis of radial velocity datasets in the presence of stellar activity, e.g., Haywood at al. 2014, Grunblatt et al. 2015. The first stable and tested implementation of GPR in PyORBIT dates back to 2018 and was showcased in Malavolta et al. 2018. Over the years, several kernels and new packages have been implemented.

In the following, I will assume that you are already familiar with the mathematical basis of GP. The Gaussian Process website and the review by Aigrain & Foreman-Mackey 2022 represent good starting points to learn more about GPs.

In this section, we will discuss only models encompassing independent covariance matrix among datasets, with some hyperparameters in common. I will refer to these approaches as classic or standard Gaussian processes regression. Multi-dimensional GP, formerly known as GP framework, will be presented in a dedicated section.

In PyORBIT unlimited number of additional datasets can be included for the simultaneous training of the hyperparameters. Unless differently specified, all the hyperparameters will be shared (if referred to the same model and common model) except the amplitude of covariance matrix, which is dataset-dependent. Each dataset will be characterized by its own covariance matrix.

• Quasi-periodic kernel
• Quasi-periodic with cosine kernel