Add technical analysis document on RV-BIS independence in Bayesian joint modeling

    else:

    else:

        visit_filter = np.ones(len(residuals), dtype=bool)

        visit_filter = np.ones(len(residuals), dtype=bool)

    # Determine y-axis label based on instrument data_type

    # Use the data_type from plModel for the specific instrument

    # plModel.data_types is a dict mapping inst_id -> 'RV' or 'LC'

    dtype = getattr(plModel, 'data_types', {}).get(instrument_to_plot, None)

    y_label='Residual '+dtype

    visit_mask = inst_filter & visit_filter

    visit_mask = inst_filter & visit_filter

    residuals = residuals[visit_mask]

    residuals = residuals[visit_mask]

                    pass

                    pass

        ax.set_xlabel(var_name)

        ax.set_xlabel(var_name)

        ax.set_ylabel('Residual Flux')

        ax.set_ylabel(y_label)

        ax.axhline(0, color='gray', linestyle='--', linewidth=0.5, alpha=0.5)

        ax.axhline(0, color='gray', linestyle='--', linewidth=0.5, alpha=0.5)

        ax.grid(True, alpha=0.3)

        ax.grid(True, alpha=0.3)

- check the checkpoint_interval logic

- check the checkpoint_interval logic

- implement different limb darkening laws

- implement different limb darkening laws

- better arrangement of legends and text in plots

- better arrangement of legends and text in plots

- remove double INFO messages in the log

## Contact

## Contact

CHEOPS sees the sum $K_\text{GP\_time} + K_\text{GP\_roll}$.  TESS sees

CHEOPS sees the sum $K_\text{GP\_time} + K_\text{GP\_roll}$.  TESS sees

only $K_\text{GP\_time}$.

only $K_\text{GP\_time}$.

> **Note:** When several `GP_<name>` blocks list the **same instrument**, their

`x_variable`s are stacked into a multidimensional input and kernels are

summed for that instrument:

$K_{\text{total}} = K_1(x_{:,0}) + K_2(x_{:,1}) + \ldots$\

> When GP sections target **different instruments**, each instrument gets

only the kernel(s) from sections that include it.

#### Scenario 5 — Spectral Mixture GP

#### Scenario 5 — Spectral Mixture GP

The Spectral Mixture kernel can efficiently model complex variability and periodic signals with a flexible mixture of components (see https://tinygp.readthedocs.io/en/latest/tutorials/kernels.html). Unlike other kernels, each parameter $\sigma$, $l_s$, and $f$ in SM can be specified independently for multiple components, or broadcast to all components as needed.

The Spectral Mixture kernel can efficiently model complex variability and periodic signals with a flexible mixture of components (see https://tinygp.readthedocs.io/en/latest/tutorials/kernels.html). Unlike other kernels, each parameter $\sigma$, $l_s$, and $f$ in SM can be specified independently for multiple components, or broadcast to all components as needed.

For detailed benchmarks and numerical analysis, see

For detailed benchmarks and numerical analysis, see

[GP Per-Visit Analysis](gp_per_visit_analysis/GP_PER_VISIT_ANALYSIS.md).

[GP Per-Visit Analysis](gp_per_visit_analysis/GP_PER_VISIT_ANALYSIS.md).

#### Multiple GPs

### About Multi-GPs

When several `GP_<name>` blocks list the **same instrument**, their

`x_variable`s are stacked into a multidimensional input and kernels are

summed for that instrument:

$K_{\text{total}} = K_1(x_{:,0}) + K_2(x_{:,1}) + \ldots$

When GP sections target **different instruments**, each instrument gets

only the kernel(s) from sections that include it.

#### About Multi-GPs

PyAETNA’s GP system is fundamentally an **instrumentwise GP framework**.

PyAETNA’s GP system is fundamentally an **instrumentwise GP framework**.

Each GP section defines a kernel for one or more instruments, and the

Each GP section defines a kernel for one or more instruments, and the

- PyAETNA does not currently do **full joint output covariance** between

- PyAETNA does not currently do **full joint output covariance** between

  RV and BIS.

  RV and BIS.

### Comparison with Rajpaul et al. (2015)

#### Comparison with Rajpaul et al. (2015)

Rajpaul et al. build a model from a single latent stellar activity GP

Rajpaul et al. build a model from a single latent stellar activity GP

$G(t)$ and its derivative $\dot{G}(t)$, then express RV and BIS as linear

$G(t)$ and its derivative $\dot{G}(t)$, then express RV and BIS as linear

This is a stronger coupling than the “same period/lengthscale” fit, and it is

This is a stronger coupling than the “same period/lengthscale” fit, and it is

mathematically different from PyAETNA’s instrumentwise approach.

mathematically different from PyAETNA’s instrumentwise approach.

### Comparison with PyORBIT multiGP

> **Important note:** For a detailed technical analysis of this formalism, advantages and risks, see [RV-BIS Independence Analysis](rajpaul_bis_analysis.md).

#### Comparison with PyORBIT multiGP

PyORBIT does provide a true multidimensional TinyGP implementation. Its

PyORBIT does provide a true multidimensional TinyGP implementation. Its

multidimensional GP classes use a single shared kernel family for all

multidimensional GP classes use a single shared kernel family for all

  kernel forms for different outputs; the kernel family is shared across the

  kernel forms for different outputs; the kernel family is shared across the

  joint dataset.

  joint dataset.

### Pros and cons

#### Pros and cons

**PyAETNA pros**

**PyAETNA pros**

- Very flexible: different instruments may use different kernels.

- Very flexible: different instruments may use different kernels.

- The multiGP block must use a single kernel family for all datasets it

- The multiGP block must use a single kernel family for all datasets it

  couples.

  couples.

### Practical guidance

#### Practical guidance

- If you need speed and flexibility, and you only want to share physical

- If you need speed and flexibility, and you only want to share physical

  timescales like rotation period or decay lengthscale, PyAETNA’s existing

  timescales like rotation period or decay lengthscale, PyAETNA’s existing

> `INSTRUMENT` ID/name or placing them in a separate dataset entry, rather

> `INSTRUMENT` ID/name or placing them in a separate dataset entry, rather

> than keeping them as an ancillary column attached to the RV rows.

> than keeping them as an ancillary column attached to the RV rows.

> **Important note:** When analyzing multiple indicators extracted from the same CCF profile (e.g., RV and BIS), be aware of the risk of information redundancy. For a detailed technical analysis of when this is a concern and how it relates to the model structure, see [RV-BIS Independence Analysis](rajpaul_bis_analysis.md).

---

---

## Jitter

## Jitter

"""

Convert *.md files in the current directory to PDF.

Pipeline:

  1. pandoc  : Markdown -> self-contained HTML (with embedded CSS for readability)

  2. google-chrome --headless : HTML -> PDF

Requirements (must be in PATH):

  - pandoc  (https://pandoc.org)

  - google-chrome  (any Chrome/Chromium binary)

"""

import os

import subprocess

import sys

import tempfile

from pathlib import Path

# Get the directory where this script is located

SCRIPT_DIR = Path(__file__).parent

CHROME_CANDIDATES = ["google-chrome", "chromium-browser", "chromium", "google-chrome-stable"]

CSS = """

body {

    font-family: Georgia, 'Times New Roman', serif;

    font-size: 11pt;

    line-height: 1.6;

    max-width: 900px;

    margin: 40px auto;

    padding: 0 20px;

    color: #111;

}

h1 { font-size: 1.8em; border-bottom: 2px solid #333; padding-bottom: 6px; }

h2 { font-size: 1.3em; border-bottom: 1px solid #aaa; padding-bottom: 3px; margin-top: 1.6em; }

h3 { font-size: 1.1em; margin-top: 1.2em; }

img { max-width: 100%; height: auto; display: block; margin: 1em auto; }

table { border-collapse: collapse; width: 100%; margin: 1em 0; font-size: 10pt; }

th, td { border: 1px solid #bbb; padding: 6px 10px; text-align: left; }

th { background: #eee; font-weight: bold; }

tr:nth-child(even) td { background: #f9f9f9; }

code { background: #f0f0f0; padding: 1px 4px; border-radius: 3px; font-size: 0.92em; }

pre { background: #f0f0f0; padding: 12px; border-radius: 4px; overflow-x: auto; }

blockquote { border-left: 4px solid #aaa; margin-left: 0; padding-left: 16px; color: #555; }

a { color: #1a5a99; }

"""

def find_chrome():

    for name in CHROME_CANDIDATES:

        result = subprocess.run(["which", name], capture_output=True, text=True)

        if result.returncode == 0:

            return result.stdout.strip()

    sys.exit(

        "ERROR: No Chrome/Chromium binary found. "

        "Tried: " + ", ".join(CHROME_CANDIDATES)

    )

def md_to_html(md_path, html_path, css_path):

    cmd = [

        "pandoc",

        str(md_path),

        "--from=markdown+tex_math_single_backslash",

        "--to=html5",

        "--standalone",

        "--css=" + str(css_path),

        "--metadata", "title=" + md_path.stem,

        "--output", str(html_path),

        "--mathml",  # Use MathML for native browser rendering (better for headless PDF)

        "--embed-resources",  # Embed images and CSS directly into the HTML

        "--resource-path=" + str(md_path.parent)

    ]

    subprocess.run(cmd, check=True)

def html_to_pdf(chrome, html_path, pdf_path):

    cmd = [

        chrome,

        "--headless",

        "--disable-gpu",

        "--no-sandbox",

        "--disable-dev-shm-usage",

        "--allow-file-access-from-files",

        "--disable-web-security",

        "--run-all-compositor-stages-before-draw",

        "--virtual-time-budget=10000",

        "--print-to-pdf=" + str(pdf_path),

        "--print-to-pdf-no-header",

        str(html_path),

    ]

    subprocess.run(cmd, check=True, capture_output=True)

def main():

    chrome = find_chrome()

    print("Using Chrome:", chrome)

    # Find all markdown files in the current directory

    md_files = sorted(

        p for p in SCRIPT_DIR.iterdir()

        if p.suffix == ".md"

    )

    if not md_files:

        print("No .md files found.")

        return

    with tempfile.TemporaryDirectory() as tmpdir:

        css_path = Path(tmpdir) / "style.css"

        css_path.write_text(CSS)

        for md_path in md_files:

            if not 'rajpaul' in md_path.name:

                continue

            pdf_path = md_path.with_suffix(".pdf")

            html_path = Path(tmpdir) / (md_path.stem + ".html")

            print("  " + md_path.name + "  ->  " + pdf_path.name + " ... ", end="", flush=True)

            try:

                md_to_html(md_path, html_path, css_path)

                html_to_pdf(chrome, html_path, pdf_path)

                print("OK")

            except subprocess.CalledProcessError as exc:

                print("FAILED (return code " + str(exc.returncode) + ")")

                print("    Command: " + " ".join(str(a) for a in exc.cmd))

                if exc.stderr:

                    print("    stderr: " + exc.stderr[:400])

    print("\nDone.")

if __name__ == "__main__":

    main()