Principal Component Analysis

These figures aid in illustrating how a the point cloud can be very flad in one direction - which is where PCA would come in to choose a direction that is not flat.

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Python source code: plot_pca_3d.py

print __doc__


# Code source: Gael Varoqueux
# Modified for Documentation merge by Jaques Grobler
# License: BSD


import pylab as pl
import numpy as np
from scipy import stats
from mpl_toolkits.mplot3d import Axes3D

e = np.exp(1)

np.random.seed(4)


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)), )
def pdf(x):
    return 0.5*(  stats.norm(scale=0.25/e).pdf(x)
                + stats.norm(scale=4/e).pdf(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
def plot_figs(fig_num, elev, azim):
    fig = pl.figure(fig_num, figsize=(4, 3))
    pl.clf()
    ax = Axes3D(fig, rect=[0, 0, .95, 1], elev=elev, azim=azim)

    pl.set_cmap(pl.cm.hot_r)

    pts = ax.scatter(a[::10], b[::10], c[::10], c=density,
                     marker='+', alpha=.4)

    Y = np.c_[a, b, c]
    U, pca_score, V = np.linalg.svd(Y, full_matrices=False)
    x_pca_axis, y_pca_axis, z_pca_axis = V.T*pca_score/pca_score.min()

#ax.quiver(0.1*x_pca_axis, 0.1*y_pca_axis, 0.1*z_pca_axis,
#                x_pca_axis, y_pca_axis, z_pca_axis,
#                color=(0.6, 0, 0))

    x_pca_axis, y_pca_axis, z_pca_axis = 3*V.T
    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.w_xaxis.set_ticklabels([])
    ax.w_yaxis.set_ticklabels([])
    ax.w_zaxis.set_ticklabels([])


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

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

pl.show()