(quasiperiodic_derivative_kernel)=

# 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

```{math}
\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](quasiperiodic_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](quasiperiodic_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

```yaml
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` | |