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.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/manifold/plot_compare_methods.py"
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.. only:: html
.. note::
:class: sphx-glr-download-link-note
:ref:`Go to the end <sphx_glr_download_auto_examples_manifold_plot_compare_methods.py>`
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.. rst-class:: sphx-glr-example-title
.. _sphx_glr_auto_examples_manifold_plot_compare_methods.py:
=========================================
Comparison of Manifold Learning methods
=========================================
An illustration of dimensionality reduction on the S-curve dataset
with various manifold learning methods.
For a discussion and comparison of these algorithms, see the
:ref:`manifold module page <manifold>`
For a similar example, where the methods are applied to a
sphere dataset, see :ref:`sphx_glr_auto_examples_manifold_plot_manifold_sphere.py`
Note that the purpose of the 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.
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.. code-block:: Python
# Author: Jake Vanderplas -- <vanderplas@astro.washington.edu>
.. GENERATED FROM PYTHON SOURCE LINES 26-30
Dataset preparation
-------------------
We start by generating the S-curve dataset.
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.. code-block:: Python
import matplotlib.pyplot as plt
# unused but required import for doing 3d projections with matplotlib < 3.2
import mpl_toolkits.mplot3d # noqa: F401
from matplotlib import ticker
from sklearn import datasets, manifold
n_samples = 1500
S_points, S_color = datasets.make_s_curve(n_samples, random_state=0)
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Let's look at the original data. Also define some helping
functions, which we will use further on.
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.. code-block:: Python
def plot_3d(points, points_color, title):
x, y, z = points.T
fig, ax = plt.subplots(
figsize=(6, 6),
facecolor="white",
tight_layout=True,
subplot_kw={"projection": "3d"},
)
fig.suptitle(title, size=16)
col = ax.scatter(x, y, z, c=points_color, s=50, alpha=0.8)
ax.view_init(azim=-60, elev=9)
ax.xaxis.set_major_locator(ticker.MultipleLocator(1))
ax.yaxis.set_major_locator(ticker.MultipleLocator(1))
ax.zaxis.set_major_locator(ticker.MultipleLocator(1))
fig.colorbar(col, ax=ax, orientation="horizontal", shrink=0.6, aspect=60, pad=0.01)
plt.show()
def plot_2d(points, points_color, title):
fig, ax = plt.subplots(figsize=(3, 3), facecolor="white", constrained_layout=True)
fig.suptitle(title, size=16)
add_2d_scatter(ax, points, points_color)
plt.show()
def add_2d_scatter(ax, points, points_color, title=None):
x, y = points.T
ax.scatter(x, y, c=points_color, s=50, alpha=0.8)
ax.set_title(title)
ax.xaxis.set_major_formatter(ticker.NullFormatter())
ax.yaxis.set_major_formatter(ticker.NullFormatter())
plot_3d(S_points, S_color, "Original S-curve samples")
.. image-sg:: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_001.png
:alt: Original S-curve samples
:srcset: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_001.png
:class: sphx-glr-single-img
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Define algorithms for the manifold learning
-------------------------------------------
Manifold learning is an approach to non-linear dimensionality reduction.
Algorithms for this task are based on the idea that the dimensionality of
many data sets is only artificially high.
Read more in the :ref:`User Guide <manifold>`.
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.. code-block:: Python
n_neighbors = 12 # neighborhood which is used to recover the locally linear structure
n_components = 2 # number of coordinates for the manifold
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Locally Linear Embeddings
^^^^^^^^^^^^^^^^^^^^^^^^^
Locally linear embedding (LLE) can be thought of as a series of local
Principal Component Analyses which are globally compared to find the
best non-linear embedding.
Read more in the :ref:`User Guide <locally_linear_embedding>`.
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.. code-block:: Python
params = {
"n_neighbors": n_neighbors,
"n_components": n_components,
"eigen_solver": "auto",
"random_state": 0,
}
lle_standard = manifold.LocallyLinearEmbedding(method="standard", **params)
S_standard = lle_standard.fit_transform(S_points)
lle_ltsa = manifold.LocallyLinearEmbedding(method="ltsa", **params)
S_ltsa = lle_ltsa.fit_transform(S_points)
lle_hessian = manifold.LocallyLinearEmbedding(method="hessian", **params)
S_hessian = lle_hessian.fit_transform(S_points)
lle_mod = manifold.LocallyLinearEmbedding(method="modified", **params)
S_mod = lle_mod.fit_transform(S_points)
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.. code-block:: Python
fig, axs = plt.subplots(
nrows=2, ncols=2, figsize=(7, 7), facecolor="white", constrained_layout=True
)
fig.suptitle("Locally Linear Embeddings", size=16)
lle_methods = [
("Standard locally linear embedding", S_standard),
("Local tangent space alignment", S_ltsa),
("Hessian eigenmap", S_hessian),
("Modified locally linear embedding", S_mod),
]
for ax, method in zip(axs.flat, lle_methods):
name, points = method
add_2d_scatter(ax, points, S_color, name)
plt.show()
.. image-sg:: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_002.png
:alt: Locally Linear Embeddings, Standard locally linear embedding, Local tangent space alignment, Hessian eigenmap, Modified locally linear embedding
:srcset: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_002.png
:class: sphx-glr-single-img
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Isomap Embedding
^^^^^^^^^^^^^^^^
Non-linear dimensionality reduction through Isometric Mapping.
Isomap seeks a lower-dimensional embedding which maintains geodesic
distances between all points. Read more in the :ref:`User Guide <isomap>`.
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.. code-block:: Python
isomap = manifold.Isomap(n_neighbors=n_neighbors, n_components=n_components, p=1)
S_isomap = isomap.fit_transform(S_points)
plot_2d(S_isomap, S_color, "Isomap Embedding")
.. image-sg:: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_003.png
:alt: Isomap Embedding
:srcset: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_003.png
:class: sphx-glr-single-img
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Multidimensional scaling
^^^^^^^^^^^^^^^^^^^^^^^^
Multidimensional scaling (MDS) seeks a low-dimensional representation
of the data in which the distances respect well the distances in the
original high-dimensional space.
Read more in the :ref:`User Guide <multidimensional_scaling>`.
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.. code-block:: Python
md_scaling = manifold.MDS(
n_components=n_components,
max_iter=50,
n_init=4,
random_state=0,
normalized_stress=False,
)
S_scaling = md_scaling.fit_transform(S_points)
plot_2d(S_scaling, S_color, "Multidimensional scaling")
.. image-sg:: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_004.png
:alt: Multidimensional scaling
:srcset: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_004.png
:class: sphx-glr-single-img
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Spectral embedding for non-linear dimensionality reduction
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
This implementation uses Laplacian Eigenmaps, which finds a low dimensional
representation of the data using a spectral decomposition of the graph Laplacian.
Read more in the :ref:`User Guide <spectral_embedding>`.
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.. code-block:: Python
spectral = manifold.SpectralEmbedding(
n_components=n_components, n_neighbors=n_neighbors, random_state=42
)
S_spectral = spectral.fit_transform(S_points)
plot_2d(S_spectral, S_color, "Spectral Embedding")
.. image-sg:: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_005.png
:alt: Spectral Embedding
:srcset: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_005.png
:class: sphx-glr-single-img
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T-distributed Stochastic Neighbor Embedding
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
It converts similarities between data points to joint probabilities and
tries to minimize the Kullback-Leibler divergence between the joint probabilities
of the low-dimensional embedding and the high-dimensional data. t-SNE has a cost
function that is not convex, i.e. with different initializations we can get
different results. Read more in the :ref:`User Guide <t_sne>`.
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.. code-block:: Python
t_sne = manifold.TSNE(
n_components=n_components,
perplexity=30,
init="random",
n_iter=250,
random_state=0,
)
S_t_sne = t_sne.fit_transform(S_points)
plot_2d(S_t_sne, S_color, "T-distributed Stochastic \n Neighbor Embedding")
.. image-sg:: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_006.png
:alt: T-distributed Stochastic Neighbor Embedding
:srcset: /auto_examples/manifold/images/sphx_glr_plot_compare_methods_006.png
:class: sphx-glr-single-img
.. rst-class:: sphx-glr-timing
**Total running time of the script:** (0 minutes 11.760 seconds)
.. _sphx_glr_download_auto_examples_manifold_plot_compare_methods.py:
.. only:: html
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:alt: Launch binder
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:alt: Launch JupyterLite
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.. container:: sphx-glr-download sphx-glr-download-jupyter
:download:`Download Jupyter notebook: plot_compare_methods.ipynb <plot_compare_methods.ipynb>`
.. container:: sphx-glr-download sphx-glr-download-python
:download:`Download Python source code: plot_compare_methods.py <plot_compare_methods.py>`
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