割球面上的流形学习方法¶
不同流形学习技术在球面数据集上的应用。在这里，人们可以看到维数约简的使用，以获得一些关于流形学习方法的直觉。关于数据集，两极被从球体中切下来，以及沿着球体一侧的一片薄片。这使得多种学习技术能够‘spread it open’ ，同时将其投射到两个维度。
对于类似的示例，将这些方法应用于S曲线数据集，参看流形学习方法的比较
注意，MDS的目的是找到数据的低维表示(这里是2D)，其中距离很好地尊重原始高维空间中的距离，与其他流形学习算法不同，它不寻求低维空间中数据的各向同性表示。在这里，流形问题与表示地球平面图的问题相当匹配，就像地图投影一样。
/home/circleci/project/sklearn/utils/validation.py:68: 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: 0.086 sec
/home/circleci/project/sklearn/utils/validation.py:68: 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: 0.15 sec
/home/circleci/project/sklearn/utils/validation.py:68: 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: 0.24 sec
/home/circleci/project/sklearn/utils/validation.py:68: 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: 0.19 sec
/home/circleci/project/sklearn/utils/validation.py:68: 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: 0.27 sec
MDS: 1.2 sec
Spectral Embedding: 0.087 sec
t-SNE: 6.2 sec
# Author: Jaques Grobler <jaques.grobler@inria.fr>
# 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()
脚本的总运行时间：(0分9.266秒)
