Plotting Confidence Ellipses with Matplotlib

Introduction

This lab will demonstrate how to plot confidence ellipses of a two-dimensional dataset using Python Matplotlib. A confidence ellipse is a graphical representation of the covariance of a dataset, showing the uncertainty of the estimated mean and standard deviation. The ellipses are plotted using the Pearson correlation coefficient.

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Import Required Libraries

The first step is to import the necessary libraries. We will need numpy and matplotlib.pyplot for this lab.

import matplotlib.pyplot as plt
import numpy as np

Define the confidence_ellipse Function

Next, we define the confidence_ellipse function that will take in the x and y coordinates of the dataset, the axes object to draw the ellipse into, and the number of standard deviations. It returns a Matplotlib patch object representing the ellipse.

def confidence_ellipse(x, y, ax, n_std=3.0, facecolor='none', **kwargs):
    """
    Create a plot of the covariance confidence ellipse of *x* and *y*.

    Parameters
    ----------
    x, y : array-like, shape (n, )
        Input data.

    ax : matplotlib.axes.Axes
        The axes object to draw the ellipse into.

    n_std : float
        The number of standard deviations to determine the ellipse's radiuses.

    **kwargs
        Forwarded to `~matplotlib.patches.Ellipse`

    Returns
    -------
    matplotlib.patches.Ellipse
    """
    if x.size != y.size:
        raise ValueError("x and y must be the same size")

    cov = np.cov(x, y)
    pearson = cov[0, 1]/np.sqrt(cov[0, 0] * cov[1, 1])
    ## Using a special case to obtain the eigenvalues of this
    ## two-dimensional dataset.
    ell_radius_x = np.sqrt(1 + pearson)
    ell_radius_y = np.sqrt(1 - pearson)
    ellipse = Ellipse((0, 0), width=ell_radius_x * 2, height=ell_radius_y * 2,
                      facecolor=facecolor, **kwargs)

    ## Calculating the standard deviation of x from
    ## the squareroot of the variance and multiplying
    ## with the given number of standard deviations.
    scale_x = np.sqrt(cov[0, 0]) * n_std
    mean_x = np.mean(x)

    ## calculating the standard deviation of y ...
    scale_y = np.sqrt(cov[1, 1]) * n_std
    mean_y = np.mean(y)

    transf = transforms.Affine2D() \
        .rotate_deg(45) \
        .scale(scale_x, scale_y) \
        .translate(mean_x, mean_y)

    ellipse.set_transform(transf + ax.transData)
    return ax.add_patch(ellipse)

Define the get_correlated_dataset Function

We also need a function to generate a two-dimensional dataset with specified mean, dimensions, and correlation.

def get_correlated_dataset(n, dependency, mu, scale):
    """
    Creates a random two-dimensional dataset with the specified
    two-dimensional mean (mu) and dimensions (scale).
    The correlation can be controlled by the param 'dependency',
    a 2x2 matrix.
    """
    latent = np.random.randn(n, 2)
    dependent = latent.dot(dependency)
    scaled = dependent * scale
    scaled_with_offset = scaled + mu
    ## return x and y of the new, correlated dataset
    return scaled_with_offset[:, 0], scaled_with_offset[:, 1]

Plotting Positive, Negative, and Weak Correlations

Now, we can use these functions to plot the confidence ellipses of datasets with positive, negative, and weak correlations.

np.random.seed(0)

PARAMETERS = {
    'Positive correlation': [[0.85, 0.35],
                             [0.15, -0.65]],
    'Negative correlation': [[0.9, -0.4],
                             [0.1, -0.6]],
    'Weak correlation': [[1, 0],
                         [0, 1]],
}

mu = 2, 4
scale = 3, 5

fig, axs = plt.subplots(1, 3, figsize=(9, 3))
for ax, (title, dependency) in zip(axs, PARAMETERS.items()):
    x, y = get_correlated_dataset(800, dependency, mu, scale)
    ax.scatter(x, y, s=0.5)

    ax.axvline(c='grey', lw=1)
    ax.axhline(c='grey', lw=1)

    confidence_ellipse(x, y, ax, edgecolor='red')

    ax.scatter(mu[0], mu[1], c='red', s=3)
    ax.set_title(title)

plt.show()

Plotting Different Number of Standard Deviations

We can also plot the confidence ellipses with different number of standard deviations.

fig, ax_nstd = plt.subplots(figsize=(6, 6))

dependency_nstd = [[0.8, 0.75],
                   [-0.2, 0.35]]
mu = 0, 0
scale = 8, 5

ax_nstd.axvline(c='grey', lw=1)
ax_nstd.axhline(c='grey', lw=1)

x, y = get_correlated_dataset(500, dependency_nstd, mu, scale)
ax_nstd.scatter(x, y, s=0.5)

confidence_ellipse(x, y, ax_nstd, n_std=1,
                   label=r'$1\sigma$', edgecolor='firebrick')
confidence_ellipse(x, y, ax_nstd, n_std=2,
                   label=r'$2\sigma$', edgecolor='fuchsia', linestyle='--')
confidence_ellipse(x, y, ax_nstd, n_std=3,
                   label=r'$3\sigma$', edgecolor='blue', linestyle=':')

ax_nstd.scatter(mu[0], mu[1], c='red', s=3)
ax_nstd.set_title('Different standard deviations')
ax_nstd.legend()
plt.show()

Using Keyword Arguments

Lastly, we can customize the appearance of the ellipses using keyword arguments.

fig, ax_kwargs = plt.subplots(figsize=(6, 6))
dependency_kwargs = [[-0.8, 0.5],
                     [-0.2, 0.5]]
mu = 2, -3
scale = 6, 5

ax_kwargs.axvline(c='grey', lw=1)
ax_kwargs.axhline(c='grey', lw=1)

x, y = get_correlated_dataset(500, dependency_kwargs, mu, scale)
## Plot the ellipse with zorder=0 in order to demonstrate
## its transparency (caused by the use of alpha).
confidence_ellipse(x, y, ax_kwargs,
                   alpha=0.5, facecolor='pink', edgecolor='purple', zorder=0)

ax_kwargs.scatter(x, y, s=0.5)
ax_kwargs.scatter(mu[0], mu[1], c='red', s=3)
ax_kwargs.set_title('Using keyword arguments')

fig.subplots_adjust(hspace=0.25)
plt.show()

Summary

In this lab, we learned how to plot confidence ellipses of a two-dimensional dataset using Python Matplotlib. We defined the confidence_ellipse and get_correlated_dataset functions, and used them to plot ellipses of datasets with different correlations and number of standard deviations. We also showed how to customize the appearance of the ellipses using keyword arguments.