Quasi-periodic plus derivative kernel

Quasi-periodic plus derivative kernel#

The kernel employed in the gp_quasiperiodic_derivative model is a combination of the quasi-periodic kernel and its first derivative, not different from using the multivariate approachon a single dataset. If we consider the correlated noise of a given dataset \(\mathrm{D}\) as the combination of a Gaussian process and its first derivative:

(11)#\[\begin{split}\Delta \mathrm{D} & = H_\mathrm{amp} G(t) + C_\mathrm{amp} G^\prime (t) \\\end{split}\]

Then the final covariance between observations of G and its derivative at times \(t_i\) and \(t_j\) is given by:

(12)#\[\gamma (t_i, t_j)_{\rm full} = H_\mathrm{amp}^2 \gamma_{\rm GP}^{(G,G)} (t_i, t_j) + C_\mathrm{amp}^2 \gamma_{\rm GP}^{(dG,dG)} (t_i, t_j) (t)\]

Where \(\gamma_{\rm GP}^{(G,G)} (t_i, t_j)\) is the quasi-periodic kernel and \( \gamma^{(dG,dG)}(t_i, t_j) = \left.\left.\frac{\partial}{\partial t'} \frac{\partial}{\partial t} \gamma^{(G,G)}(t, t') \right|_{t=t_i} \right|_{t'=t_j} \). Whit a single dataset, the terms \( \gamma^{(G,dG)}(t_i, t_j)\) and \( \gamma^{(dG,G)}(t_i, t_j)\) cancel each other. See Section 3.3 of Rajpaul et al. 2015 for more details.

This model can be useful when a single dataset needs a more flexible stellar-activity covariance than the standard quasi-periodic kernel, without switching to a multidimensional GP where the first derivative is usually employed.

Model definition and requirements#

The fastest implementation relies on tinyGP, but it requires a few extra tricks in the configuration file and execution (see Caveats on the use of tinyGP). If you use this model, cite the Zenodo repository.

An independent implementation that relies only on basic packages is maintained for legacy reasons; however, it is much slower, and I don’t recommend using it.

model name: tinygp_quasiperiodic_derivative

model name: gp_quasiperiodic_derivative

  • required common object: activity

  • direct implementation relying only on numpy and scipy

  • independent covariance matrix for each dataset

Model parameters#

Name

Parameter

Common?

Definition

Notes

Prot

Rotational period of the star

common

activity

Replaced by rotation_period when use_stellar_rotation_period: True

Pdec

Decay timescale of active regions

common

activity

Replaced by activity_decay when use_stellar_activity_decay: True

Oamp

Coherence scale

common

activity

Hamp

Amplitude of the quasi-periodic component

dataset

activity

Camp

Amplitude of the derivative component

dataset

activity

Keywords#

Model-wide keywords, with the default value in boldface.

hyperparameters_condition

  • accepted values: True | False

  • activates the quasi-periodic hyperparameter condition described in the quasi-periodic kernel.

rotation_decay_condition

  • accepted values: True | False

  • if activated, requires Pdec > 2 Prot.

halfrotation_decay_condition

  • accepted values: True | False

  • if activated, requires Pdec > 0.5 Prot.

decay_rotation_factor or rotation_decay_factor

  • accepted values: float | not used

  • if provided, requires Pdec to be larger than the specified factor times Prot.

use_stellar_rotation_period

  • accepted values: True | False

  • replaces Prot with rotation_period from star_parameters.

use_stellar_activity_decay

  • accepted values: True | False

  • replaces Pdec with activity_decay from star_parameters.

Example#

 1inputs:
 2  RVdata:
 3    file: datasets/star_RV_PyORBIT.dat
 4    kind: RV
 5    models:
 6      - radial_velocities
 7      - gp_qp_derivative
 8common:
 9  activity:
10    boundaries:
11      Prot: [10.0, 20.0]
12      Pdec: [20.0, 1000.0]
13      Oamp: [0.001, 1.0]
14models:
15  gp_qp_derivative:
16    model: gp_quasiperiodic_derivative
17    common: activity
18    hyperparameters_condition: True
19    rotation_decay_condition: True
20    boundaries:
21      Hamp: [0.0, 100.0]
22      Camp: [0.0, 100.0]