流形学习方法应用于被切开的球体#

在球形数据集上应用不同的流形学习技术。在这里，我们可以看到降维的应用，以便对流形学习方法获得一些直观的理解。关于数据集，球体的两极被切除，以及沿着侧面的一薄片。这使得流形学习技术能够将其“展开”并投影到二维空间中。

对于类似示例，其中方法应用于 S 曲线数据集，请参阅流形学习方法的比较。

请注意，MDS的目的是找到数据的低维表示（此处为 2D），其中距离与原始高维空间中的距离保持良好一致，这与其他流形学习算法不同，它不寻求数据在低维空间中的各向同性表示。这里的流形问题与表示地球平面图的问题非常吻合，就像地图投影一样。

Manifold Learning with 1000 points, 10 neighbors, LLE (0.06 sec), LTSA (0.8 sec), Hessian LLE (0.69 sec), Modified LLE (1.2 sec), Isomap (0.1 sec), MDS (0.89 sec), Non-metric MDS (11 sec), Classical MDS (0.039 sec), Spectral Embedding (0.041 sec), t-SNE (3.7 sec)

standard: 0.06 sec
ltsa: 0.8 sec
hessian: 0.69 sec
modified: 1.2 sec
ISO: 0.1 sec
MDS: 0.89 sec
Non-metric MDS: 11 sec
Classical MDS: 0.039 sec
Spectral Embedding: 0.041 sec
t-SNE: 3.7 sec

# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause

from time import time

import matplotlib.pyplot as plt

# Unused but required import for doing 3d projections with matplotlib < 3.2
import mpl_toolkits.mplot3d  # noqa: F401
import numpy as np
from matplotlib.ticker import NullFormatter

from sklearn import manifold
from sklearn.utils import check_random_state

# 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, 12))
plt.suptitle(
    "Manifold Learning with %i points, %i neighbors" % (1000, n_neighbors), fontsize=14
)

ax = fig.add_subplot(351, 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=n_neighbors, n_components=2, method=method, random_state=42
        )
        .fit_transform(sphere_data)
        .T
    )
    t1 = time()
    print("%s: %.2g sec" % (methods[i], t1 - t0))

    ax = fig.add_subplot(352 + 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_neighbors, n_components=2)
    .fit_transform(sphere_data)
    .T
)
t1 = time()
print("%s: %.2g sec" % ("ISO", t1 - t0))

ax = fig.add_subplot(357)
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, n_init=1, random_state=42, init="classical_mds")
trans_data = mds.fit_transform(sphere_data).T
t1 = time()
print("MDS: %.2g sec" % (t1 - t0))

ax = fig.add_subplot(358)
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")

t0 = time()
mds = manifold.MDS(2, n_init=1, random_state=42, metric_mds=False, init="classical_mds")
trans_data = mds.fit_transform(sphere_data).T
t1 = time()
print("Non-metric MDS: %.2g sec" % (t1 - t0))

ax = fig.add_subplot(359)
plt.scatter(trans_data[0], trans_data[1], c=colors, cmap=plt.cm.rainbow)
plt.title("Non-metric MDS (%.2g sec)" % (t1 - t0))
ax.xaxis.set_major_formatter(NullFormatter())
ax.yaxis.set_major_formatter(NullFormatter())
plt.axis("tight")

t0 = time()
mds = manifold.ClassicalMDS(2)
trans_data = mds.fit_transform(sphere_data).T
t1 = time()
print("Classical MDS: %.2g sec" % (t1 - t0))

ax = fig.add_subplot(3, 5, 10)
plt.scatter(trans_data[0], trans_data[1], c=colors, cmap=plt.cm.rainbow)
plt.title("Classical 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, random_state=42
)
trans_data = se.fit_transform(sphere_data).T
t1 = time()
print("Spectral Embedding: %.2g sec" % (t1 - t0))

ax = fig.add_subplot(3, 5, 12)
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, random_state=0)
trans_data = tsne.fit_transform(sphere_data).T
t1 = time()
print("t-SNE: %.2g sec" % (t1 - t0))

ax = fig.add_subplot(3, 5, 13)
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()

脚本总运行时间： (0 分钟 19.112 秒)

流形学习方法应用于被切开的球体#

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