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+"""
+Galaxy Ellipticity Distributions
+================================
+
+This example demonstrate how to sample ellipticity distributions
+in SkyPy.
+
+"""
+
+
+# %%
+# 3D Ellipticity Distribution
+# ---------------------------
+#
+# In Ryden 2004 [1]_, the ellipticity is sampled by randomly projecting
+# a 3D ellipsoid with principal axes :math:`A > B > C`.
+#
+# The distribution of the axis ratio :math:`\gamma = C/A` is a truncated
+# normal with mean :math:`\mu_\gamma` and standard deviation
+# :math:`\sigma_\gamma`.
+#
+# The distribution of :math:`\epsilon = \log(1 - B/A)` is truncated normal
+# with mean :math:`\mu` and standard deviation :math:`\sigma`.
+#
+#
+# The func:`skypy.galaxies.morphology.ryden04_ellipticity()` model samples
+# projected 2D axis ratios. Specifically, it samples the axis ratios of the
+# 3D ellipsoid according to the description above and
+# then randomly projects them using
+# :func:`skypy.utils.random.triaxial_axis_ratio()`.
+#
+#
+# Validation plot with SDSS Data
+# ------------------------------
+#
+# Here we validate our function by comparing our ``SkyPy`` simulated galaxy
+# ellipticities against observational data from SDSS DR1 in Figure 1 of Ryden 2004.
+# You can download the data file
+# :download:`SDSS_ellipticity <../../../examples/galaxies/SDSS_ellipticity.npz>`,
+# stored as a 2D array: :math:`e_1`, :math:`e_2`.
+#
+
+import numpy as np
+
+# Load SDSS data from Fig. 1 in Ryden 2004
+data = np.load('SDSS_ellipticity.npz')
+e1, e2 = data['sdss']['e1'], data['sdss']['e2']
+
+Ngal = len(e1)
+e = np.hypot(e1, e2)
+q_amSDSS = np.sqrt((1 - e)/(1 + e))
+
+
+# %%
+#
+# In this example, we generate 100 galaxy ellipticity samples using the
+# ``SkyPy`` function
+# :func:`skypy.galaxies.morphology.ryden04_ellipticity()` and the
+# best fit parameters
+# given in Ryden 2004 [1]:
+# :math:`\mu_\gamma =0.222`, :math:`\sigma_\gamma=0.057`, :math:`\mu =-1.85`
+# and :math:`\sigma=0.89`.
+#
+
+from skypy.galaxies.morphology import ryden04_ellipticity
+
+# Best fit parameters from Fig. 1 in Ryden 2004
+mu_gamma, sigma_gamma, mu, sigma = 0.222, 0.057, -1.85, 0.89
+
+# Binning scheme of Fig. 1
+bins = np.linspace(0, 1, 41)
+mid = 0.5 * (bins[:-1] + bins[1:])
+
+# Mean and variance of sampling
+mean = np.zeros(len(bins)-1)
+var = np.zeros(len(bins)-1)
+
+# %%
+#
+
+# Create 100 SkyPy realisations
+for i in range(100):
+    # sample ellipticity
+    ellipticity = ryden04_ellipticity(mu_gamma, sigma_gamma, mu, sigma, size=Ngal)
+    axis_ratio = (1 - ellipticity) / (1 + ellipticity)
+    n, _ = np.histogram(axis_ratio, bins=bins)
+
+    # update mean and variance
+    x = n - mean
+    mean += x/(i+1)
+    y = n - mean
+    var += x*y
+
+# finalise variance and standard deviation
+var = var/i
+std = np.sqrt(var)
+
+
+# %%
+#
+# We now plot the distribution of axis ratio :math:`𝑞_{am}`
+# using adaptive moments in the i band, for exponential galaxies in the SDSS DR1
+# (solid line). The data points with error bars represent
+# the `SkyPy` simulation:
+#
+
+import matplotlib.pyplot as plt
+
+plt.hist(q_amSDSS, range=[0, 1], bins=40, histtype='step',
+         ec='k', lw=0.5, label='SDSS data')
+plt.errorbar(mid, mean, yerr=std, fmt='.r', ms=4, capsize=3,
+             lw=0.5, mew=0.5, label='SkyPy model')
+
+plt.xlabel(r'Axis ratio, $q_{am}$')
+plt.ylabel(r'N')
+plt.legend(frameon=False)
+plt.show()
+
+# %%
+# References
+# ----------
+#
+#
+# .. [1] Ryden, Barbara S., 2004, `The Astrophysical Journal, Volume 601, Issue 1, pp. 214-220`_
+#
+# .. _The Astrophysical Journal, Volume 601, Issue 1, pp. 214-220: https://arxiv.org/abs/astro-ph/0310097