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Merge pull request #13570 from CarstenSchelp/confidence_ellipse
Add new example for plotting a confidence_ellipse
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‎examples/statistics/confidence_ellipse.py

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"""
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======================================================
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Plot a confidence ellipse of a two-dimensional dataset
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======================================================
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This example shows how to plot a confidence ellipse of a
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two-dimensional dataset, using its pearson correlation coefficient.
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The approach that is used to obtain the correct geometry is
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explained and proved here:
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https://carstenschelp.github.io/2018/09/14/Plot_Confidence_Ellipse_001.html
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The method avoids the use of an iterative eigen decomposition algorithm
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and makes use of the fact that a normalized covariance matrix (composed of
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pearson correlation coefficients and ones) is particularly easy to handle.
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"""
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#############################################################################
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#
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# The plotting function itself
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# """"""""""""""""""""""""""""
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#
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# This function plots the confidence ellipse of the covariance of the given
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# array-like variables x and y. The ellipse is plotted into the given
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# axes-object ax.
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#
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# The radiuses of the ellipse can be controlled by n_std which is the number
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# of standard deviations. The default value is 3 which makes the ellipse
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# enclose 99.7% of the points (given the data is normally distributed
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# like in these examples).
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import numpy as np
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import matplotlib.pyplot as plt
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from matplotlib.patches import Ellipse
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import matplotlib.transforms as transforms
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def confidence_ellipse(x, y, ax, n_std=3.0, facecolor='none', **kwargs):
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"""
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Create a plot of the covariance confidence ellipse of `x` and `y`
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Parameters
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----------
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x, y : array_like, shape (n, )
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Input data.
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ax : matplotlib.axes.Axes
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The axes object to draw the ellipse into.
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n_std : float
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The number of standard deviations to determine the ellipse's radiuses.
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Returns
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-------
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matplotlib.patches.Ellipse
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Other parameters
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----------------
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kwargs : `~matplotlib.patches.Patch` properties
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"""
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if x.size != y.size:
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raise ValueError("x and y must be the same size")
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cov = np.cov(x, y)
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pearson = cov[0, 1]/np.sqrt(cov[0, 0] * cov[1, 1])
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# Using a special case to obtain the eigenvalues of this
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# two-dimensionl dataset.
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ell_radius_x = np.sqrt(1 + pearson)
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ell_radius_y = np.sqrt(1 - pearson)
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ellipse = Ellipse((0, 0),
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width=ell_radius_x * 2,
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height=ell_radius_y * 2,
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facecolor=facecolor,
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**kwargs)
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# Calculating the stdandard deviation of x from
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# the squareroot of the variance and multiplying
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# with the given number of standard deviations.
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scale_x = np.sqrt(cov[0, 0]) * n_std
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mean_x = np.mean(x)
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# calculating the stdandard deviation of y ...
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scale_y = np.sqrt(cov[1, 1]) * n_std
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mean_y = np.mean(y)
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transf = transforms.Affine2D() \
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.rotate_deg(45) \
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.scale(scale_x, scale_y) \
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.translate(mean_x, mean_y)
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ellipse.set_transform(transf + ax.transData)
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return ax.add_patch(ellipse)
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#############################################################################
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#
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# A helper function to create a correlated dataset
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# """"""""""""""""""""""""""""""""""""""""""""""""
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#
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# Creates a random two-dimesional dataset with the specified
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# two-dimensional mean (mu) and dimensions (scale).
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# The correlation can be controlled by the param 'dependency',
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# a 2x2 matrix.
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def get_correlated_dataset(n, dependency, mu, scale):
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latent = np.random.randn(n, 2)
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dependent = latent.dot(dependency)
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scaled = dependent * scale
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scaled_with_offset = scaled + mu
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# return x and y of the new, correlated dataset
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return scaled_with_offset[:, 0], scaled_with_offset[:, 1]
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#############################################################################
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#
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# Positive, negative and weak correlation
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# """""""""""""""""""""""""""""""""""""""
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#
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# Note that the shape for the weak correlation (right) is an ellipse,
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# not a circle because x and y are differently scaled.
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# However, the fact that x and y are uncorrelated is shown by
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# the axes of the ellipse being aligned with the x- and y-axis
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# of the coordinate system.
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np.random.seed(0)
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PARAMETERS = {
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'Positive correlation': np.array([[0.85, 0.35],
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[0.15, -0.65]]),
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'Negative correlation': np.array([[0.9, -0.4],
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[0.1, -0.6]]),
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'Weak correlation': np.array([[1, 0],
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[0, 1]]),
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}
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mu = 2, 4
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scale = 3, 5
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fig, axs = plt.subplots(1, 3, figsize=(9, 3))
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for ax, (title, dependency) in zip(axs, PARAMETERS.items()):
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x, y = get_correlated_dataset(800, dependency, mu, scale)
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ax.scatter(x, y, s=0.5)
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ax.axvline(c='grey', lw=1)
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ax.axhline(c='grey', lw=1)
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confidence_ellipse(x, y, ax, edgecolor='red')
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ax.scatter(mu[0], mu[1], c='red', s=3)
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ax.set_title(title)
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plt.show()
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#############################################################################
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#
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# Different number of standard deviations
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# """""""""""""""""""""""""""""""""""""""
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#
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# A plot with n_std = 3 (blue), 2 (purple) and 1 (red)
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fig, ax_nstd = plt.subplots(figsize=(6, 6))
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dependency_nstd = np.array([
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[0.8, 0.75],
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[-0.2, 0.35]
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])
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mu = 0, 0
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scale = 8, 5
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ax_nstd.axvline(c='grey', lw=1)
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ax_nstd.axhline(c='grey', lw=1)
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x, y = get_correlated_dataset(500, dependency_nstd, mu, scale)
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ax_nstd.scatter(x, y, s=0.5)
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confidence_ellipse(x, y, ax_nstd, n_std=1,
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label=r'$1\sigma$', edgecolor='firebrick')
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confidence_ellipse(x, y, ax_nstd, n_std=2,
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label=r'$2\sigma$', edgecolor='fuchsia', linestyle='--')
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confidence_ellipse(x, y, ax_nstd, n_std=3,
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label=r'$3\sigma$', edgecolor='blue', linestyle=':')
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ax_nstd.scatter(mu[0], mu[1], c='red', s=3)
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ax_nstd.set_title('Different standard deviations')
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ax_nstd.legend()
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plt.show()
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#############################################################################
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#
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# Using the keyword arguments
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# """""""""""""""""""""""""""
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#
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# Use the kwargs specified for matplotlib.patches.Patch in order
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# to have the ellipse rendered in different ways.
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fig, ax_kwargs = plt.subplots(figsize=(6, 6))
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dependency_kwargs = np.array([
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[-0.8, 0.5],
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[-0.2, 0.5]
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])
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mu = 2, -3
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scale = 6, 5
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ax_kwargs.axvline(c='grey', lw=1)
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ax_kwargs.axhline(c='grey', lw=1)
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x, y = get_correlated_dataset(500, dependency_kwargs, mu, scale)
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# Plot the ellipse with zorder=0 in order to demonstrate
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# its transparency (caused by the use of alpha).
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confidence_ellipse(x, y, ax_kwargs,
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alpha=0.5, facecolor='pink', edgecolor='purple', zorder=0)
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ax_kwargs.scatter(x, y, s=0.5)
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ax_kwargs.scatter(mu[0], mu[1], c='red', s=3)
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ax_kwargs.set_title(f'Using kwargs')
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fig.subplots_adjust(hspace=0.25)
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plt.show()

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