.. only:: html
.. note::
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Click :ref:`here ` to download the full example code or to run this example in your browser via Binder
.. 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
:alt: Manifold Learning with 1000 points, 10 neighbors, LLE (0.092 sec), LTSA (0.087 sec), Hessian LLE (0.17 sec), Modified LLE (0.14 sec), Isomap (0.22 sec), MDS (0.97 sec), Spectral Embedding (0.048 sec), t-SNE (3.9 sec)
:class: sphx-glr-single-img
.. rst-class:: sphx-glr-script-out
Out:
.. code-block:: none
/home/circleci/project/sklearn/utils/validation.py:67: FutureWarning: Pass n_neighbors=10, n_components=2 as keyword args. From version 0.25 passing these as positional arguments will result in an error
warnings.warn("Pass {} as keyword args. From version 0.25 "
standard: 0.092 sec
/home/circleci/project/sklearn/utils/validation.py:67: FutureWarning: Pass n_neighbors=10, n_components=2 as keyword args. From version 0.25 passing these as positional arguments will result in an error
warnings.warn("Pass {} as keyword args. From version 0.25 "
ltsa: 0.087 sec
/home/circleci/project/sklearn/utils/validation.py:67: FutureWarning: Pass n_neighbors=10, n_components=2 as keyword args. From version 0.25 passing these as positional arguments will result in an error
warnings.warn("Pass {} as keyword args. From version 0.25 "
hessian: 0.17 sec
/home/circleci/project/sklearn/utils/validation.py:67: FutureWarning: Pass n_neighbors=10, n_components=2 as keyword args. From version 0.25 passing these as positional arguments will result in an error
warnings.warn("Pass {} as keyword args. From version 0.25 "
modified: 0.14 sec
/home/circleci/project/sklearn/utils/validation.py:67: FutureWarning: Pass n_neighbors=10 as keyword args. From version 0.25 passing these as positional arguments will result in an error
warnings.warn("Pass {} as keyword args. From version 0.25 "
ISO: 0.22 sec
MDS: 0.97 sec
Spectral Embedding: 0.048 sec
t-SNE: 3.9 sec
|
.. code-block:: default
# 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()
.. rst-class:: sphx-glr-timing
**Total running time of the script:** ( 0 minutes 6.131 seconds)
.. _sphx_glr_download_auto_examples_manifold_plot_manifold_sphere.py:
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.. container:: sphx-glr-download sphx-glr-download-python
:download:`Download Python source code: plot_manifold_sphere.py `
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