# Principal components analysis (PCA)¶

These figures aid in illustrating how a point cloud can be very flat in one direction–which is where PCA comes in to choose a direction that is not flat.

# Authors: Gael Varoquaux
#          Jaques Grobler
#          Kevin Hughes


## Create the data¶

import numpy as np
from scipy import stats

e = np.exp(1)
np.random.seed(4)

def pdf(x):
return 0.5 * (stats.norm(scale=0.25 / e).pdf(x) + stats.norm(scale=4 / e).pdf(x))

y = np.random.normal(scale=0.5, size=(30000))
x = np.random.normal(scale=0.5, size=(30000))
z = np.random.normal(scale=0.1, size=len(x))

density = pdf(x) * pdf(y)
pdf_z = pdf(5 * z)

density *= pdf_z

a = x + y
b = 2 * y
c = a - b + z

norm = np.sqrt(a.var() + b.var())
a /= norm
b /= norm


## Plot the figures¶

import matplotlib.pyplot as plt

# unused but required import for doing 3d projections with matplotlib < 3.2
import mpl_toolkits.mplot3d  # noqa: F401

from sklearn.decomposition import PCA

def plot_figs(fig_num, elev, azim):
fig = plt.figure(fig_num, figsize=(4, 3))
plt.clf()
ax = fig.add_subplot(111, projection="3d", elev=elev, azim=azim)
ax.set_position([0, 0, 0.95, 1])

ax.scatter(a[::10], b[::10], c[::10], c=density[::10], marker="+", alpha=0.4)
Y = np.c_[a, b, c]

# Using SciPy's SVD, this would be:
# _, pca_score, Vt = scipy.linalg.svd(Y, full_matrices=False)

pca = PCA(n_components=3)
pca.fit(Y)
V = pca.components_.T

x_pca_axis, y_pca_axis, z_pca_axis = 3 * V
x_pca_plane = np.r_[x_pca_axis[:2], -x_pca_axis[1::-1]]
y_pca_plane = np.r_[y_pca_axis[:2], -y_pca_axis[1::-1]]
z_pca_plane = np.r_[z_pca_axis[:2], -z_pca_axis[1::-1]]
x_pca_plane.shape = (2, 2)
y_pca_plane.shape = (2, 2)
z_pca_plane.shape = (2, 2)
ax.plot_surface(x_pca_plane, y_pca_plane, z_pca_plane)
ax.xaxis.set_ticklabels([])
ax.yaxis.set_ticklabels([])
ax.zaxis.set_ticklabels([])

elev = -40
azim = -80
plot_figs(1, elev, azim)

elev = 30
azim = 20
plot_figs(2, elev, azim)

plt.show()


Total running time of the script: (0 minutes 0.162 seconds)

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