101 lines
2.6 KiB
Python
101 lines
2.6 KiB
Python
|
"""
|
||
|
|
=========================
|
||
|
|
Multi-dimensional scaling
|
||
|
|
=========================
|
||
|
|
|
||
|
|
An illustration of the metric and non-metric MDS on generated noisy data.
|
||
|
|
|
||
|
|
The reconstructed points using the metric MDS and non metric MDS are slightly
|
||
|
|
shifted to avoid overlapping.
|
||
|
|
|
||
|
|
"""
|
||
|
|
|
||
|
|
# Author: Nelle Varoquaux <nelle.varoquaux@gmail.com>
|
||
|
|
# License: BSD
|
||
|
|
|
||
|
|
import numpy as np
|
||
|
|
from matplotlib import pyplot as plt
|
||
|
|
from matplotlib.collections import LineCollection
|
||
|
|
|
||
|
|
from sklearn import manifold
|
||
|
|
from sklearn.decomposition import PCA
|
||
|
|
from sklearn.metrics import euclidean_distances
|
||
|
|
|
||
|
|
EPSILON = np.finfo(np.float32).eps
|
||
|
|
n_samples = 20
|
||
|
|
seed = np.random.RandomState(seed=3)
|
||
|
|
X_true = seed.randint(0, 20, 2 * n_samples).astype(float)
|
||
|
|
X_true = X_true.reshape((n_samples, 2))
|
||
|
|
# Center the data
|
||
|
|
X_true -= X_true.mean()
|
||
|
|
|
||
|
|
similarities = euclidean_distances(X_true)
|
||
|
|
|
||
|
|
# Add noise to the similarities
|
||
|
|
noise = np.random.rand(n_samples, n_samples)
|
||
|
|
noise = noise + noise.T
|
||
|
|
noise[np.arange(noise.shape[0]), np.arange(noise.shape[0])] = 0
|
||
|
|
similarities += noise
|
||
|
|
|
||
|
|
mds = manifold.MDS(
|
||
|
|
n_components=2,
|
||
|
|
max_iter=3000,
|
||
|
|
eps=1e-9,
|
||
|
|
random_state=seed,
|
||
|
|
dissimilarity="precomputed",
|
||
|
|
n_jobs=1,
|
||
|
|
)
|
||
|
|
pos = mds.fit(similarities).embedding_
|
||
|
|
|
||
|
|
nmds = manifold.MDS(
|
||
|
|
n_components=2,
|
||
|
|
metric=False,
|
||
|
|
max_iter=3000,
|
||
|
|
eps=1e-12,
|
||
|
|
dissimilarity="precomputed",
|
||
|
|
random_state=seed,
|
||
|
|
n_jobs=1,
|
||
|
|
n_init=1,
|
||
|
|
)
|
||
|
|
npos = nmds.fit_transform(similarities, init=pos)
|
||
|
|
|
||
|
|
# Rescale the data
|
||
|
|
pos *= np.sqrt((X_true**2).sum()) / np.sqrt((pos**2).sum())
|
||
|
|
npos *= np.sqrt((X_true**2).sum()) / np.sqrt((npos**2).sum())
|
||
|
|
|
||
|
|
# Rotate the data
|
||
|
|
clf = PCA(n_components=2)
|
||
|
|
X_true = clf.fit_transform(X_true)
|
||
|
|
|
||
|
|
pos = clf.fit_transform(pos)
|
||
|
|
|
||
|
|
npos = clf.fit_transform(npos)
|
||
|
|
|
||
|
|
fig = plt.figure(1)
|
||
|
|
ax = plt.axes([0.0, 0.0, 1.0, 1.0])
|
||
|
|
|
||
|
|
s = 100
|
||
|
|
plt.scatter(X_true[:, 0], X_true[:, 1], color="navy", s=s, lw=0, label="True Position")
|
||
|
|
plt.scatter(pos[:, 0], pos[:, 1], color="turquoise", s=s, lw=0, label="MDS")
|
||
|
|
plt.scatter(npos[:, 0], npos[:, 1], color="darkorange", s=s, lw=0, label="NMDS")
|
||
|
|
plt.legend(scatterpoints=1, loc="best", shadow=False)
|
||
|
|
|
||
|
|
similarities = similarities.max() / (similarities + EPSILON) * 100
|
||
|
|
np.fill_diagonal(similarities, 0)
|
||
|
|
# Plot the edges
|
||
|
|
start_idx, end_idx = np.where(pos)
|
||
|
|
# a sequence of (*line0*, *line1*, *line2*), where::
|
||
|
|
# linen = (x0, y0), (x1, y1), ... (xm, ym)
|
||
|
|
segments = [
|
||
|
|
[X_true[i, :], X_true[j, :]] for i in range(len(pos)) for j in range(len(pos))
|
||
|
|
]
|
||
|
|
values = np.abs(similarities)
|
||
|
|
lc = LineCollection(
|
||
|
|
segments, zorder=0, cmap=plt.cm.Blues, norm=plt.Normalize(0, values.max())
|
||
|
|
)
|
||
|
|
lc.set_array(similarities.flatten())
|
||
|
|
lc.set_linewidths(np.full(len(segments), 0.5))
|
||
|
|
ax.add_collection(lc)
|
||
|
|
|
||
|
|
plt.show()
|