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 \(G^\prime (t)\).
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 \(\Delta t = (t_i-t_j)\).
Kernels#
The only implemented kernel for the underlying GP is the quasi-periodic one, following the expression given by Grunblatt et al. 2015:
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 exceptH-alpha
S_index
Ca_HK
, andFWHM
. If provided, default isFalse
for all the dataset not explicitly mentioned.If
True
, the first derivative of the ubderlying Gaussian process is included, otherwiserot_amp
F 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 |
|
|
Pdec |
Decay time scale of active regions \(\lambda\) |
common |
|
|
Oamp |
Coherence scale \(w\) |
common |
|
|
rot_amp |
coefficient of \(G(t)\) component |
dataset |
|
|
con_amp |
coefficient of \(G^\prime (t)\) component |
dataset |
|