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 proccess \(G(t)\) and its first derivative \(G^\prime (t)\).

(1)#\[\begin{split}\Delta \mathrm{RV} & = V_c G(t) + V_r G^\prime (t) \\ \mathrm{BIS} & = B_c G(t) + B_r G^\prime (t) \\ \log{R^{\prime}_\mathrm{HK}} & = L_c G(t) \\\end{split}\]

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 defintion of \(\Delta t = (t_i-t_j)\).

Give credits to the authors

If you use the multidimensional GP, remeber to cite Rajpaul et al. 2015 and Barragán et al. 2022. For more details on the PyORBIT implementation of multidimensional GP, please refer to Nardiello et al. 2022, appendix D

Kernels#

The only implemented kernel for the underlying GP is the quasi-periodic one, following the expression given by Grunblatt et al. 2015:

(2)#\[\gamma ^{(G,G)}_{i,j} = \exp{ \left \{-\frac{\sin^2{[\pi(t_i - t_j)/\theta]}}{2 w ^2} - \left (\frac{t_i-t_j}{\lambda} \right )^2 \right \} }\]

where \(\theta\) is equivalent to the rotation period of the star, \(w\) is the coherence scale, and \(\lambda\) is usually associated to the decay time scale fo the active regions

Important

It is common to have a factor 2 in the denominator of the aperiodic variation (i.e., \(2 \lambda\) rather than \(\lambda\)) in (2). In such a case, it is sufficient to multiply the value of \(\lambda\) of PyORBIT by a factor \(\sqrt(2)\) - keep it in mind when assigning priors!

Improvements#

In PyORBIT the framework implementation has been expanded to support an unlimited number of datasets. Additionally, it is not required for the datasets to have the same dimensions, thus allowing the use of several RV datasets even if activity indicators are not available for all of them.

The coefficients \(X_c\) and \(X_c\) are thus renamed in con_amp and rot_amp, with the reference dataset easily identifiable from the terminal output.

Model definition and requirements#

  • model name: gp_multidimensional_quasiperiodic

  • required common objects : activity

Keywords#

Model-wide keywords, with the default value in bold face.

hyperparameters_condition

  • accepted values: True | False

  • activate the conditions \( \lambda ^ 2 > (3/4 \pi) \theta ^2 w ^ 2 \) (adapted from Rajpaiul 2017 and Rajpaul et al. 2021 to take into account the factor 2 in the denominator of the aperiodic variation) to ensure that the QP function has at least one non-trivial turning point.

rotation_decay_condition

  • accepted values: True | False

  • if activated, it ensures that the decay time scale of the activity regions \(\lambda\) is at least twice the rotational period of the star \(\theta\)

derivative

  • accepted values: list of dataset using the model

  • if not provided, default is True for all datasets except H-alpha S_index Ca_HK, and FWHM. If provided, default is False for all the dataset not explicitly mentioned.

  • If True, the first derivative of the ubderlying Gaussian process is included, otherwise rot_ampF is fixed to zero.

Examples#

In the following example, ore RV dataset comes together with two activity indicators, the BIS and the FWHM of the CCF, the latter as replacement of \(\log{R^{\prime}_\mathrm{HK}}\).

Here gp_multidimensional is the label that we assign to the gp_multidimensional_quasiperiodic model. The label is assigned to each dataset, including the RV dataset for which also the RV model is included.

The common:planets section includes three planets, of which one is transiting - with Gaussian priors on the period and time of inferior conjunction - and two non-transiting planets with a prior on the eccentricity. The common:activity section provides the hyperparameters for the GP shared among all the datasets. The example shows how to assign boundaries and priors to the parameters. The model keywords and the boundaries for the dataset-specific parameters are listed in models:gp_multidimensional

  1inputs:
  2  RVdata:
  3    file: RVS_PyORBIT.dat
  4    kind: RV
  5    models:
  6      - radial_velocities
  7      - gp_multidimensional
  8  BISdata:
  9    file: BIS_PyORBIT.dat
 10    kind: BIS
 11    models:
 12      - gp_multidimensional
 13  FWHMdata:
 14    file: FWHM_PyORBIT.dat
 15    kind: FWHM
 16    models:
 17      - gp_multidimensional
 18common:
 19  planets:
 20    b:
 21      orbit: circular
 22      use_time_inferior_conjunction: True
 23      boundaries:
 24        P: [2.21000, 2.240000]
 25        K: [0.001, 20.0]
 26        Tc: [59144.60, 59144.63]
 27      priors:
 28        P: ['Gaussian', 2.2241951, 0.00000030]
 29        Tc: ['Gaussian', 59144.616171, 0.000284]
 30    c:
 31      orbit: keplerian
 32      boundaries:
 33        P: [2.0, 100.0]
 34        K: [0.001, 30.0]
 35        e: [0.00, 0.70]
 36      priors:
 37        e: ['Gaussian', 0.00, 0.098]
 38    d:
 39      orbit: keplerian
 40      boundaries:
 41        P: [2.0, 100.0]
 42        K: [0.001, 30.0]
 43        e: [0.00, 0.70]
 44      priors:
 45        e: ['Gaussian', 0.00, 0.098]
 46  activity:
 47    boundaries:
 48      Prot: [20.0, 30.0]
 49      Pdec: [30.0, 1000.0]
 50      Oamp: [0.01, 1.0]
 51    priors:
 52      Prot: ['Gaussian', 14.00, 0.50]
 53      Oamp: ['Gaussian', 0.35, 0.035]
 54  star:
 55    star_parameters:
 56      priors:
 57        mass: ['Gaussian', 0.65, 0.05]
 58        radius: ['Gaussian', 0.624, 0.005]
 59        density: ['Gaussian', 2.65, 0.08]
 60models:
 61  radial_velocities:
 62    planets:
 63      - b
 64      - c
 65      - d
 66  gp_multidimensional:
 67    model: gp_multidimensional_quasiperiodic
 68    common: activity
 69    hyperparameters_condition: True
 70    rotation_decay_condition: True
 71    RVdata:
 72      boundaries:
 73        rot_amp: [0.0, 20.0] #at least one must be positive definite
 74        con_amp: [-20.0, 20.0]
 75      derivative: True
 76    BISdata:
 77      boundaries:
 78        rot_amp: [-20.0, 20.0]
 79        con_amp: [-20.0, 20.0]
 80      derivative: True
 81    FWHMdata:
 82      boundaries:
 83        con_amp: [-50., 50.]
 84      derivative: False
 85parameters:
 86  Tref: 59200.00
 87solver:
 88  pyde:
 89    ngen: 50000
 90    npop_mult: 4
 91  emcee:
 92    npop_mult: 4
 93    nsteps: 100000
 94    nburn: 25000
 95    nsave: 25000
 96    thin: 100
 97    #use_threading_pool: False
 98  nested_sampling:
 99    nlive: 1000
100    sampling_efficiency: 0.30
101  recenter_bounds: True

Tip

Since the coefficients are always copuled, there is a degenracy between a given set and the opposite one (i.e., all the coefficient with opposite sign). This degeneracy can be solved by force one coefficient to be positive, e.g., rot_amp for the RV_data in the example.

A simpler way to prepare the configuration file if you are not specifying the boundaries is to list the datasets with derivatives under the same keyword:

 1...
 2  gp_multidimensional:
 3    model: gp_multidimensional_quasiperiodic
 4    common: activity
 5    hyperparameters_condition: True
 6    rotation_decay_condition: True
 7    derivative:
 8      RVdata: True
 9      BISdata: True
10      FWHMdata: False
11parameters:
12...

Model parameters#

The following parameters will be inherited from the common model (column Common?: common) or a different value will be assigned for each dataset (column Common?: dataset)

Name

Parameter

Common?

Definition

Notes

Prot

rotational period of the star \(\theta\)

common

activity

Pdec

Decay time scale of active regions \(\lambda\)

common

activity

Oamp

Coherence scale \(w\)

common

activity

rot_amp

coefficient of \(G(t)\) component

dataset

activity

con_amp

coefficient of \(G^\prime (t)\) component

dataset

activity