Quasi-periodic plus derivative kernel

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The gp_quasiperiodic_derivative model is a single-output GP kernel built from the quasi-periodic covariance and the covariance of its first derivative. It 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.

The covariance has the form

\[\gamma(t_i,t_j) = H_\mathrm{amp}^2 G(t_i,t_j) + C_\mathrm{amp}^2 \frac{\partial^2 G(t_i,t_j)}{\partial t_i \partial t_j}\]

where G is the quasi-periodic kernel described in the quasi-periodic kernel page.

Model definition and requirements

model name: gp_quasiperiodic_derivative

• required common object: activity

• direct implementation using numpy and scipy

• independent covariance matrix for each dataset

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

inputs:
  RVdata:
    file: datasets/star_RV_PyORBIT.dat
    kind: RV
    models:
      - radial_velocities
      - gp_qp_derivative
common:
  activity:
    boundaries:
      Prot: [10.0, 20.0]
      Pdec: [20.0, 1000.0]
      Oamp: [0.001, 1.0]
models:
  gp_qp_derivative:
    model: gp_quasiperiodic_derivative
    common: activity
    hyperparameters_condition: True
    rotation_decay_condition: True
    boundaries:
      Hamp: [0.0, 100.0]
      Camp: [0.0, 100.0]

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