.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "user-guide/model-gallery/spatial/plot_gen_gauss.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_user-guide_model-gallery_spatial_plot_gen_gauss.py>`
        to download the full example code. or to run this example in your browser via Binder

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_user-guide_model-gallery_spatial_plot_gen_gauss.py:


.. _generalized-gaussian-spatial-model:

Generalized gaussian spatial model
==================================

This is a spatial model parametrising a generalized Gaussian function.

By default, the Generalized Gaussian is defined as :

.. math::
    \phi(\text{lon}, \text{lat})  = \phi(\text{r}) = N \times \exp \left[ - \left( \frac{r}{r_{\rm eff}} \right)^ \left( 1/\eta \right) \right] \,,

the normalization is expressed as:

.. math::
    N = \frac{1}{ 2 \pi \sqrt(1-e^2) r_{0}^2 \eta \Gamma(2\eta)}\,

where :math:`\Gamma` is the gamma function.
This analytical norm is approximated so it may not integrate to unity in extreme cases
if ellipticity tend to one and radius is large or :math:`\eta` much larger than one (outside the default range).

The effective radius is given by:

.. math::
    r_{rm eff}(\text{lon}, \text{lat}) = \sqrt{
        (r_M \sin(\Delta \phi))^2 +
        (r_m \cos(\Delta \phi))^2
    }.

where :math:`r_M` (:math:`r_m`) is the major (minor) semiaxis, and
:math:`\Delta \phi` is the difference between `phi`, the position angle of the model, and the
position angle of the evaluation point.
If the eccentricity (:math:`e`) is null it reduces to :math:`r_0`.

.. GENERATED FROM PYTHON SOURCE LINES 40-43

Example plot
------------
Here is an example plot of the model for different shape parameter:

.. GENERATED FROM PYTHON SOURCE LINES 43-90

.. code-block:: Python


    from astropy import units as u
    import matplotlib.pyplot as plt
    from gammapy.maps import Map, WcsGeom
    from gammapy.modeling.models import (
        GeneralizedGaussianSpatialModel,
        Models,
        PowerLawSpectralModel,
        SkyModel,
    )

    lon_0 = 20
    lat_0 = 0
    reval = 3
    dr = 0.02
    geom = WcsGeom.create(
        skydir=(lon_0, lat_0),
        binsz=dr,
        width=(2 * reval, 2 * reval),
        frame="galactic",
    )

    tags = [r"Disk, $\eta=0.01$", r"Gaussian, $\eta=0.5$", r"Laplace, $\eta=1$"]
    eta_range = [0.01, 0.5, 1]
    r_0 = 1
    e = 0.5
    phi = 45 * u.deg
    fig, axes = plt.subplots(1, 3, figsize=(9, 6))
    for ax, eta, tag in zip(axes, eta_range, tags):
        model = GeneralizedGaussianSpatialModel(
            lon_0=lon_0 * u.deg,
            lat_0=lat_0 * u.deg,
            eta=eta,
            r_0=r_0 * u.deg,
            e=e,
            phi=phi,
            frame="galactic",
        )
        meval = model.evaluate_geom(geom)
        Map.from_geom(geom=geom, data=meval.value, unit=meval.unit).plot(ax=ax)
        pixreg = model.to_region().to_pixel(geom.wcs)
        pixreg.plot(ax=ax, edgecolor="g", facecolor="none", lw=2)
        ax.set_title(tag)
        ax.set_xticks([])
        ax.set_yticks([])
    plt.tight_layout()




.. image-sg:: /user-guide/model-gallery/spatial/images/sphx_glr_plot_gen_gauss_001.png
   :alt: Disk, $\eta=0.01$, Gaussian, $\eta=0.5$, Laplace, $\eta=1$
   :srcset: /user-guide/model-gallery/spatial/images/sphx_glr_plot_gen_gauss_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 91-94

YAML representation
-------------------
Here is an example YAML file using the model:

.. GENERATED FROM PYTHON SOURCE LINES 94-102

.. code-block:: Python


    pwl = PowerLawSpectralModel()
    gengauss = GeneralizedGaussianSpatialModel()

    model = SkyModel(spectral_model=pwl, spatial_model=gengauss, name="pwl-gengauss-model")
    models = Models([model])

    print(models.to_yaml())




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    components:
    -   name: pwl-gengauss-model
        type: SkyModel
        spectral:
            type: PowerLawSpectralModel
            parameters:
            -   name: index
                value: 2.0
            -   name: amplitude
                value: 1.0e-12
                unit: cm-2 s-1 TeV-1
            -   name: reference
                value: 1.0
                unit: TeV
        spatial:
            type: GeneralizedGaussianSpatialModel
            frame: icrs
            parameters:
            -   name: lon_0
                value: 0.0
                unit: deg
            -   name: lat_0
                value: 0.0
                unit: deg
            -   name: r_0
                value: 1.0
                unit: deg
            -   name: eta
                value: 0.5
            -   name: e
                value: 0.0
            -   name: phi
                value: 0.0
                unit: deg
    metadata:
        creator: Gammapy 1.3
        date: '2024-11-26T10:08:39.908583'
        origin: null






.. _sphx_glr_download_user-guide_model-gallery_spatial_plot_gen_gauss.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: binder-badge

      .. image:: images/binder_badge_logo.svg
        :target: https://mybinder.org/v2/gh/gammapy/gammapy-webpage/v1.3?urlpath=lab/tree/notebooks/1.3/user-guide/model-gallery/spatial/plot_gen_gauss.ipynb
        :alt: Launch binder
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_gen_gauss.ipynb <plot_gen_gauss.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_gen_gauss.py <plot_gen_gauss.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_gen_gauss.zip <plot_gen_gauss.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_