.. note:: :class: sphx-glr-download-link-note Click :ref:`here ` to download the full example code .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_manifold_plot_manifold_sphere.py: ============================================= Manifold Learning methods on a severed sphere ============================================= An application of the different :ref:`manifold` techniques on a spherical data-set. Here one can see the use of dimensionality reduction in order to gain some intuition regarding the manifold learning methods. Regarding the dataset, the poles are cut from the sphere, as well as a thin slice down its side. This enables the manifold learning techniques to 'spread it open' whilst projecting it onto two dimensions. For a similar example, where the methods are applied to the S-curve dataset, see :ref:`sphx_glr_auto_examples_manifold_plot_compare_methods.py` Note that the purpose of the :ref:`MDS ` is to find a low-dimensional representation of the data (here 2D) in which the distances respect well the distances in the original high-dimensional space, unlike other manifold-learning algorithms, it does not seeks an isotropic representation of the data in the low-dimensional space. Here the manifold problem matches fairly that of representing a flat map of the Earth, as with `map projection `_ .. image:: /auto_examples/manifold/images/sphx_glr_plot_manifold_sphere_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out Out: .. code-block:: none standard: 0.069 sec ltsa: 0.12 sec hessian: 0.29 sec modified: 0.16 sec ISO: 0.27 sec MDS: 1.2 sec Spectral Embedding: 0.15 sec t-SNE: 3.5 sec | .. code-block:: python # Author: Jaques Grobler # License: BSD 3 clause print(__doc__) from time import time import numpy as np import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D from matplotlib.ticker import NullFormatter from sklearn import manifold from sklearn.utils import check_random_state # Next line to silence pyflakes. Axes3D # Variables for manifold learning. n_neighbors = 10 n_samples = 1000 # Create our sphere. random_state = check_random_state(0) p = random_state.rand(n_samples) * (2 * np.pi - 0.55) t = random_state.rand(n_samples) * np.pi # Sever the poles from the sphere. indices = ((t < (np.pi - (np.pi / 8))) & (t > ((np.pi / 8)))) colors = p[indices] x, y, z = np.sin(t[indices]) * np.cos(p[indices]), \ np.sin(t[indices]) * np.sin(p[indices]), \ np.cos(t[indices]) # Plot our dataset. fig = plt.figure(figsize=(15, 8)) plt.suptitle("Manifold Learning with %i points, %i neighbors" % (1000, n_neighbors), fontsize=14) ax = fig.add_subplot(251, projection='3d') ax.scatter(x, y, z, c=p[indices], cmap=plt.cm.rainbow) ax.view_init(40, -10) sphere_data = np.array([x, y, z]).T # Perform Locally Linear Embedding Manifold learning methods = ['standard', 'ltsa', 'hessian', 'modified'] labels = ['LLE', 'LTSA', 'Hessian LLE', 'Modified LLE'] for i, method in enumerate(methods): t0 = time() trans_data = manifold\ .LocallyLinearEmbedding(n_neighbors, 2, method=method).fit_transform(sphere_data).T t1 = time() print("%s: %.2g sec" % (methods[i], t1 - t0)) ax = fig.add_subplot(252 + i) plt.scatter(trans_data[0], trans_data[1], c=colors, cmap=plt.cm.rainbow) plt.title("%s (%.2g sec)" % (labels[i], t1 - t0)) ax.xaxis.set_major_formatter(NullFormatter()) ax.yaxis.set_major_formatter(NullFormatter()) plt.axis('tight') # Perform Isomap Manifold learning. t0 = time() trans_data = manifold.Isomap(n_neighbors, n_components=2)\ .fit_transform(sphere_data).T t1 = time() print("%s: %.2g sec" % ('ISO', t1 - t0)) ax = fig.add_subplot(257) plt.scatter(trans_data[0], trans_data[1], c=colors, cmap=plt.cm.rainbow) plt.title("%s (%.2g sec)" % ('Isomap', t1 - t0)) ax.xaxis.set_major_formatter(NullFormatter()) ax.yaxis.set_major_formatter(NullFormatter()) plt.axis('tight') # Perform Multi-dimensional scaling. t0 = time() mds = manifold.MDS(2, max_iter=100, n_init=1) trans_data = mds.fit_transform(sphere_data).T t1 = time() print("MDS: %.2g sec" % (t1 - t0)) ax = fig.add_subplot(258) plt.scatter(trans_data[0], trans_data[1], c=colors, cmap=plt.cm.rainbow) plt.title("MDS (%.2g sec)" % (t1 - t0)) ax.xaxis.set_major_formatter(NullFormatter()) ax.yaxis.set_major_formatter(NullFormatter()) plt.axis('tight') # Perform Spectral Embedding. t0 = time() se = manifold.SpectralEmbedding(n_components=2, n_neighbors=n_neighbors) trans_data = se.fit_transform(sphere_data).T t1 = time() print("Spectral Embedding: %.2g sec" % (t1 - t0)) ax = fig.add_subplot(259) plt.scatter(trans_data[0], trans_data[1], c=colors, cmap=plt.cm.rainbow) plt.title("Spectral Embedding (%.2g sec)" % (t1 - t0)) ax.xaxis.set_major_formatter(NullFormatter()) ax.yaxis.set_major_formatter(NullFormatter()) plt.axis('tight') # Perform t-distributed stochastic neighbor embedding. t0 = time() tsne = manifold.TSNE(n_components=2, init='pca', random_state=0) trans_data = tsne.fit_transform(sphere_data).T t1 = time() print("t-SNE: %.2g sec" % (t1 - t0)) ax = fig.add_subplot(2, 5, 10) plt.scatter(trans_data[0], trans_data[1], c=colors, cmap=plt.cm.rainbow) plt.title("t-SNE (%.2g sec)" % (t1 - t0)) ax.xaxis.set_major_formatter(NullFormatter()) ax.yaxis.set_major_formatter(NullFormatter()) plt.axis('tight') plt.show() **Total running time of the script:** ( 0 minutes 5.954 seconds) .. _sphx_glr_download_auto_examples_manifold_plot_manifold_sphere.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: sphx-glr-download :download:`Download Python source code: plot_manifold_sphere.py ` .. container:: sphx-glr-download :download:`Download Jupyter notebook: plot_manifold_sphere.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_