.. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:here  to download the full example code .. rst-class:: sphx-glr-example-title .. _sphx_glr_gallery_statistics_confidence_ellipse.py: ====================================================== Plot a confidence ellipse of a two-dimensional dataset ====================================================== This example shows how to plot a confidence ellipse of a two-dimensional dataset, using its pearson correlation coefficient. The approach that is used to obtain the correct geometry is explained and proved here: https://carstenschelp.github.io/2018/09/14/Plot_Confidence_Ellipse_001.html The method avoids the use of an iterative eigen decomposition algorithm and makes use of the fact that a normalized covariance matrix (composed of pearson correlation coefficients and ones) is particularly easy to handle. .. code-block:: default import numpy as np import matplotlib.pyplot as plt from matplotlib.patches import Ellipse import matplotlib.transforms as transforms The plotting function itself """""""""""""""""""""""""""" This function plots the confidence ellipse of the covariance of the given array-like variables x and y. The ellipse is plotted into the given axes-object ax. The radiuses of the ellipse can be controlled by n_std which is the number of standard deviations. The default value is 3 which makes the ellipse enclose 99.4% of the points if the data is normally distributed like in these examples (3 standard deviations in 1-D contain 99.7% of the data, which is 99.4% of the data in 2-D). .. code-block:: default 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-dimensionl 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 stdandard 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 stdandard 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) A helper function to create a correlated dataset """""""""""""""""""""""""""""""""""""""""""""""" Creates a random two-dimesional dataset with the specified two-dimensional mean (mu) and dimensions (scale). The correlation can be controlled by the param 'dependency', a 2x2 matrix. .. code-block:: default def get_correlated_dataset(n, dependency, mu, scale): 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] Positive, negative and weak correlation """"""""""""""""""""""""""""""""""""""" Note that the shape for the weak correlation (right) is an ellipse, not a circle because x and y are differently scaled. However, the fact that x and y are uncorrelated is shown by the axes of the ellipse being aligned with the x- and y-axis of the coordinate system. .. code-block:: default 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, mu, c='red', s=3) ax.set_title(title) plt.show() .. image:: /gallery/statistics/images/sphx_glr_confidence_ellipse_001.png :alt: Positive correlation, Negative correlation, Weak correlation :class: sphx-glr-single-img Different number of standard deviations """"""""""""""""""""""""""""""""""""""" A plot with n_std = 3 (blue), 2 (purple) and 1 (red) .. code-block:: default 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, mu, c='red', s=3) ax_nstd.set_title('Different standard deviations') ax_nstd.legend() plt.show() .. image:: /gallery/statistics/images/sphx_glr_confidence_ellipse_002.png :alt: Different standard deviations :class: sphx-glr-single-img Using the keyword arguments """"""""""""""""""""""""""" Use the kwargs specified for matplotlib.patches.Patch in order to have the ellipse rendered in different ways. .. code-block:: default 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, mu, c='red', s=3) ax_kwargs.set_title('Using kwargs') fig.subplots_adjust(hspace=0.25) plt.show() .. image:: /gallery/statistics/images/sphx_glr_confidence_ellipse_003.png :alt: Using kwargs :class: sphx-glr-single-img .. admonition:: References The use of the following functions, methods, classes and modules is shown in this example: - matplotlib.transforms.Affine2D - matplotlib.patches.Ellipse .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 1.290 seconds) .. _sphx_glr_download_gallery_statistics_confidence_ellipse.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-python :download:Download Python source code: confidence_ellipse.py  .. container:: sphx-glr-download sphx-glr-download-jupyter :download:Download Jupyter notebook: confidence_ellipse.ipynb  .. only:: html .. rst-class:: sphx-glr-signature Keywords: matplotlib code example, codex, python plot, pyplot Gallery generated by Sphinx-Gallery _