SHO and Matern-3/2 kernels

The Stochastically-driven Harmonic Oscillator (SHO) and Matern-3/2 kernels are generic GP kernels commonly used to model correlated noise regardless of the specific origin. By default, their parameters are dataset-specific, but it is possible to change this behaviour using the available keywords.

Tip

For their general purpose, these kernels will not inherit the hyperparameters from the activity common model. However, it is possible to relate the hyperparameters to the star’s rotation period and activity-decay timescale.

The kernels have been implemented using celerite2, tinygp, and S+LEAF.

• If you use tinygp, cite the Zenodo repository and check the Caveats on the use of tinyGP

• If you use celerite2, please cite Foreman-Mackey et al. 2017 and Foreman-Mackey 2018

• If you use S+LEAF kernels, please cite Delisle et al. 2020 and Delisle et al. 2022.

SHO Kernel

This kernel has been introduced by Foreman-Mackey et al. 2017 with the following power spectral density:

(15)\[ S(\omega) = \sqrt{\frac{2}{\pi}} \frac{S_0\,\omega_0^4} {(\omega^2-{\omega_0}^2)^2 + {\omega_0}^2\,\omega^2/Q^2}\]

Where the hyperparameters represent:

• \(\omega_0\): The undamped angular frequency

• \(Q\): the quality factor

• \(S0\): scale factor so that the power of the PSD is \(S(0) =S_0\sqrt{2/\pi}\)

In PyORBIT, the alternative parametrisation implemented in celerite2 has been extended to all the SHO models:

• \(\rho\): the undamped period of the oscillator, defined as \(\rho = 2\,\pi / \omega_0\),

• \(\tau\); the damping timescale of the process, defined as \(\tau = 2\,Q / \omega_0\)

• \(sigma\)_ the standard deviation of the process, defined as \(\sigma = \sqrt{S_0\,\omega_0\,Q} \)

If you use the SHO kernel, please cite Foreman-Mackey et al. 2017

Model definition and requirements

model name: tinygp_sho

• required common object: activity

• implemented with tinygp

model name: celerite2_sho

• required common object: activity

• implemented with celerite2

model name: spleaf_sho

• required common object: activity

• implemented with S+LEAF

model name: tinygp_matern32

• required common object: activity

• implemented with tinygp

model name: celerite2_matern32

• required common object: activity

• implemented with celerite2

Model parameters

SHO models

Name	Parameter	Common?	Definition	Notes
sho_scale	SHO undamped period, also known as rho	dataset	activity	Common when use_shared_scale: True or use_shared_hyperparameters: True
sho_decay	SHO damping timescale, also known as tau	dataset	activity	Common when use_shared_decay: True or use_shared_hyperparameters: True
sho_sigma	Standard deviation of the process	dataset	activity	Common when use_shared_hyperparameters: True

Matern-3/2 models

Name	Parameter	Common?	Definition	Notes
matern32_scale	Matern-3/2 scale, also known as rho	dataset	activity	Common when use_shared_scale: True or use_shared_hyperparameters: True
matern32_sigma	Standard deviation of the process	dataset	activity	Common when use_shared_hyperparameters: True

Tip

Some older result files may contain matern32_rho. The current models map it internally to matern32_scale.

Keywords

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

use_shared_hyperparameters

• accepted values: True | False

• moves all kernel parameters from dataset-specific to common parameters. For SHO models this includes sho_scale, sho_decay, and sho_sigma; for Matern-3/2 models this includes matern32_scale and matern32_sigma.

use_shared_scale

• accepted values: True | False

• makes only the kernel timescale common. For SHO models this is sho_scale; for Matern-3/2 models this is matern32_scale.

use_shared_decay

• accepted values: True | False

• makes only the SHO damping timescale sho_decay common.

use_activity_Prot

• accepted values: True | False

• replaces the kernel scale with Prot from the activity common object.

use_activity_Pdec

• accepted values: True | False

• replaces the kernel decay or scale with Pdec from the activity common object.

use_stellar_rotation_period

• accepted values: True | False

• replaces the kernel scale with rotation_period from star_parameters.

use_stellar_activity_decay

• accepted values: True | False

• replaces the kernel decay or scale with activity_decay from star_parameters.

For Matern-3/2 models, the replacement flags are mutually exclusive because there is only one timescale.

The SHO models also support some of the hyperparameter-condition keywords involving Prot and Pdec described in the quasi-periodic kernel. In that case, the SHO scale is interpreted as Prot, and the SHO decay as Pdec.

Examples

Dataset-specific SHO kernel:

models:
  celerite2_sho:
    model: celerite2_sho
    boundaries:
      sho_scale: [0.1, 100.0]
      sho_decay: [0.1, 1000.0]
      sho_sigma: [0.0, 1.0]

Matern-3/2 kernel with a shared timescale:

common:
  activity:
    boundaries:
      matern32_scale: [0.1, 100.0]
models:
  tinygp_matern32:
    model: tinygp_matern32
    common: activity
    use_shared_scale: True
    boundaries:
      matern32_sigma: [0.0, 1.0]

SHO kernel using stellar rotation and activity decay from star_parameters:

common:
  activity:
    model: activity
  star:
    star_parameters:
      boundaries:
        rotation_period: [10.0, 20.0]
        activity_decay: [20.0, 1000.0]
models:
  spleaf_sho:
    model: spleaf_sho
    common:
      - activity
      - star_parameters
    use_stellar_rotation_period: True
    use_stellar_activity_decay: True
    boundaries:
      sho_sigma: [0.0, 1.0]